The Leading Maths - 9

Arithmetic Arithmetic (ii) An agriculture firm earned Rs. 35,50,000 in the year 2019 AD and distributed Rs. 17,750 as dividend to its each 30 labours. (a) What is the rate of dividend when the dividend amount and total profit are given ? (b) What was the rate of payment of the bonus? (c) If the rate is increased by 3%, what amount will each labour get as bonus? 10. (i) The profit of a bank increased from 35% to 48% and made a profit of Rs. 6,50,00,000. The management decides to distribute 6% bonus increased to 15% equally among 150 staffs. (a) Why is bonus different from dividend of a company? Give reason. (b) Find the amount of bonus received by each staff. (c) If the staffs demanded Rs. 200 more, what percentage increase will be there in the given rate of bonus ? (ii) The CG's profit increased from 45% to 65% and made a profit of Rs. 76,55,00,000. The management decides to distribute 4% bonus increased to 20% equally among 1531 employees. (a) Why is bonus different from dividend of a company? Give reason. (b) Find the amount of bonus received by each employees. (c) If the employees demanded Rs. 5000 more, what percent increase will be there in the given rate of the bonus? 2. (a) Rs. 132250 (b) Rs. 7565000 (c) Rs. 299490000 (d) Rs. 94550000 3. (i) (b) Rs. 6395400 (c) Rs. 51163.20 (ii) (b) Rs. 8520000 (c) Rs. 4260 4. (i) (b) 23% (c) Rs. 119641.07 5. (i) (b) Rs. 1426500, Rs. 951000, Rs. 891562.50 (c) Rs. 951000 (ii) (b) Rs. 1051500 (c) Rs. 525750 6. (i) (b) Rs. 5875000 (c) 47 (ii) (b) Rs. 1125000 (c) 25 7. (i) (b) Rs. 5280000 (c) Rs. 24000000 (ii) (b) Rs. 10610 (c) Rs. 2122000 8. (i) (b) Rs. 20000 (c) 40% (ii) (b) Rs. 60000000 (c) 100% 9. (i) (b) Rs. 13125000 (c) 14.4% (ii) (b) Rs. 532500 (c) Rs 21800 10. (i) (b) Rs. 65000 (c) 0.046% (ii) (b) Rs. 100000 (c) 1% Answers 100 Allied The Leading Mathematics-9 Commission and Dividend 101

Arithmetic Arithmetic Uses of Home Arithmetic in Our Daily Life Activities We carry different activities within our home too such as payment of various bills and buying goods. For instances, if 12 pencils cost Rs. 60, what is the price of 1 pencil ? What will be the total payable amount if 10% fine is added to a charge of Rs. 100? What will be the amount after allowing 10 % discount in Rs. 500 ? These questions are related to unitary method and percentage, in which mathematics rules and laws are followed. In our daily activities, we generally talk about arithmetic and not about mathematics. General rules and laws such as four simple rules of arithmetic may also be violated or may both be applicable in our day-to-day dealings. Study the following trisections: SN Unit Item Rate More Items Amount Remarks 1. 1 Egg Rs. 15 30 Eggs Rs. 430, pay How ? (Rs. 15 × 30 = Rs. 450, not) Nobody feels it wrong. They are not unitary methods. 2. 1 Copy Rs. 40 1 Dozen Rs. 450, pay (Rs. 40 × 12 = Rs. 480) How ? 3. 20 units Electricity Rs. 80 25 units Rs. 80 + Rs. 7.30×5 = Rs.116.50, pay Why ? (Rs. 80/20 × 25 = Rs. 100, not) 4. 175 Telephone calls Rs. 200 200 Telephone calls Rs. 200 + Re. 1 × 25 = Rs. 225, pay Why ? (Rs. 200/175 × 200 = Rs. 228.57, not) In the same activities, the simple rules of mathematics may be partly applicable and partly not applicable. So, arithmetic at our home or household arithmetic or domestic arithmetic is not an exact science but it is a social science. Household arithmetic plays some meaningful role only in those Nepalese towns and cities where facilities for water, electricity, telephone, taxi, IT, etc. and good markets are available. The cost or price is also recorded against each such item. The total of all costs are calculated and mentioned in a slip. Such a slip of paper is called a bill. An important aspect of household arithmetic is reading of a bill. In our day-to-day life, we come across various types of bills by social requirement. In all cities and some villages of Nepal the government has provided the facilities mentioned below: � Electricity � Water � Telephone � Taxi Why, why ? CHAPTER 4 HOME ARITHMETIC 102 Allied The Leading Mathematics-9 Home Arithmetic 103

Arithmetic Arithmetic 4.1 Household Expenses for Use of Electricity At the end of this topic, the students will be able to: ¾ solve problems on the consumed electricity in household. Learning Objectives Uses of Electricity in Our Daily Activities Playing Game in Computer Electric Rice Cooker Electric Meter Normal Meter Digital Smart Meter If you played the game in the computer for 5 hours everyday in a month, how many charge should pay for it? Do you know? How can we find out? When you cooked the rice by using electric rice cooker for morning and night everyday in a month, how many electric units charge for it ? Do you know ? How can we find out ? We can read consumed electricity after using the computer,rice cooker, etc. Can you find the consumed electricity in the electric meter? How do you calculate the charge of consumed electricity? Introduction to Charge of Consumed Electricity The consumption of the electricity by a household is measured in the number of units by the electric meter. The number of units of electricity consumed is calculated as, Units of consumed electricity = Reading of the recent month – Reading of the previous month For the services of electricity, we have to pay charges to the Nepal Electricity Authority (NEA). How do you pay the electricity charge? The meter reader from the electricity authority reads the electricity meter kept at our home and leaves the bill. We have to go to these authorities monthly and pay the bill. The electricity bill is prepared as the NEA tariff rates given below: Service and Energy Charge : Single Phase (Effective from Paush 1, 2078) KW/hr (Unit) 5 Ampere 15 Ampere 30 Ampere 60 Ampere Service Charge Energy Charge Service Charge Energy Charge Service Charge Energy Charge Service Charge Energy Charge 0 - 20 30 0.00/3.00 50 4.00 75 5.00 125 6.00 21 - 30 50 6.50 75 6.50 100 6.50 125 6.50 31 - 50 50 8.00 75 8.00 100 8.00 125 8.00 Bottom 102 Allied The Leading Mathematics-9 Home Arithmetic 103

Arithmetic Arithmetic 51 - 100 75 9.50 100 9.50 125 9.50 150 9.50 101 - 250 100 9.50 125 9.50 150 9.50 200 9.50 above 250 150 11.00 175 11.00 200 11.00 250 11.00 Billing System (for 5 Ampere) Energy Consumption Block Billing Method Units Charge up to 20 units Service Charge (SC) 30 and Rs. 0/ unit from 0-20 units = Rs. 30 0-20 units charge Rs. 30 @ Rs. 0 21-30 units SC Rs. 50 + Rs. 60 @ Rs. 3 for 0-20 units + Rs. 6.50 / unit from 21-30 units = Rs. 175 0-30 units charge Rs. 125 @ Rs. 4.17 31-50 units SC Rs. 50 + Rs. 125 + Rs. 8.00 / unit from 31-50 units = Rs. 335 0-50 units charge Rs. 285 @ Rs. 5.70 51-100 units SC Rs. 75 + Rs. 285 + Rs. 9.50 / unit from 51-150 units = Rs. 835 0-150 units charge Rs. 760 @ Rs. 7.60 101-250 units SC Rs. 100 + Rs. 760 + Rs. 9.50/ unit from 151-250 units = Rs. 2285 0-250 units charge Rs. 2185 @ Rs. 8.74 above 250 units SC Rs. 150 + Rs. 2185 + Rs. 11/ unit from 251-400 units = Rs. 2335 + 11x; x = Units more than 250 ...... Calculation of Electricity charge for 425 units on 5 A meter Bill amount = Service charge+Unit charge up 400 units+Rate for above 400 units = Rs. 150 + Rs. 2185 + Rs. 11 × (425 – 250) = Rs 150 + Rs. 2185 + Rs. 1925 = Rs. 150 + Rs. 4110 = Rs. 4260 Rebate and fine system for Payment Schedule ™ Within 7 days of meter reading 2% rebate ™ From 8th to 22nd day according to the bill ™ From 23rd to 30th day 5% late fee ™ From 31st to 40th day 10% late fee ™ From 41st to 60th day 25% late fee ™ If the bill is not paid up to 60th day, the electric line will be disconnected. Important Formulae on Use of Electricity Consumed electricity = Reading of the recent month – Reading of the previous month Bill amount of electricity = Minimum charge + Additional Units × Rate + Service charge Amount of Bill payment = Bill Amount +/- Rebate/0/Fine percent of Bill amount NEPAL ELECTRICITY AUTHORITY Distribution & Consumer Services ELECTRICITY BILL BRANCH: BILL MTH & YEAR: BL NO: SC NO: CONSUMER ID: CATG: NAME ADDRESS: METER NO: MF: 1.00 MTR STS: NOR APPROVED LOAD: 5.00 PRESENT RDG: 52415 PREVIOUS RDG: 51990 UNITS: 425 RECORDED DMD : 0.00 BILLABLE DMD : 5.00 ENERGY CHARGES: 4110.00 DEMAND CHARGES : 0.00 SUBSIDY CHARGES: 0.00 SERVICE CHARGES : 150.00 OTHER CHARGES: 0.00 MTR RENT AMOUNT: 0.00 CURRENT AMOUNT: 4260.00 ARREARS AMOUNT: ± 00.00 BILL AMOUNT: 4260.00 INSTALLMENT: 0.00 SBM ID: ATIL7640VER:1.23 READER: RAMAN Mr. JIB LAL DANGOL SAIBU 00095778 BAD-2079 104061127305000176 061.12.038 Bhaisepati 104005568 Domestic (5 A) CFL lrdsf] k|of]u u/L laB'tsf] dxz'n 36fcf}+ Rebate rate is 2% effected from Shrawan, 2074. 104 Allied The Leading Mathematics-9 Home Arithmetic 105

Arithmetic Arithmetic Points to be Remembered Electric Meter: A device that shows the consumed electricity in units or figures when used the electric appliances, is called electric meter. One Unit Electricity: When an electric appliance of 1000 watts is used for 1 hour, it consumes one unit of electricity. Example-1 The meter readings of the consumption of electricity of a house are 3615 units on the 20th Jestha and 3725 on the 20th Ashadh of the same year 2067. If the electricity authority charges are Minimum charge (up to 20 units) = Rs. 80. Cost per unit (for each unit above 20 units) = Rs. 7.30. (a) How can you find the consumed electricity? (b) Find the consumed electricity units in Ashadh. (c) Find the total cost. Solution: (a) We find the consumed electricity by subtracting the reading of the previous month from the reading of the electric meter of the recent month. (b) Given, Meter reading on the 20th Ashadh = 3725 units. Meter reading on 20th Jestha = 3615 units. ∴ Electric energy consumed = 3725 – 3615 = 110 units. (c) Since the rates are different for 110 units, so 110 = 20 units + 90 units. ∴ Charge for first 20 units = Rs. 80 Charge for 90 units at Rs. 7.30/unit = 90 × Rs. 7.30 = Rs. 657. So, the cost of total consumed electricity = Rs. 80 + Rs. 657 = Rs. 737. Example-2 According to the rate of electricity charge, the minimum cost up to 20 units is Rs. 80 and cost per unit from 21 units to 250 units is Rs. 8.60. (a) Find the charge for extra units. (b) How many units can be consumed by Rs. 1757 ? (c) What is the consumed electricity when 15 bulbs of 100 watts are used for 3 hours? Find it. Solution: (a) Given, Total amount = Rs. 1757, Minimum charge up to 20 units = Rs. 80 ∴ Charge for extra units = Rs. 1757 – Rs. 80 = Rs. 1677 (b) Rate of extra units 21 – 250 units = Rs. 8.60 ∴ Extra consumed electricity = Rs. 1677 Rs. 8.60 = 195 units. ∴ Total consumed electricity = 20 + 195 = 215 units Thus, 215 units electricity can be consumed by Rs. 1757. (c) Given, Resistance of 1 bulb for 1 hour = 100 watts 104 Allied The Leading Mathematics-9 Home Arithmetic 105

Arithmetic Arithmetic ∴ Resistance of 15 bulbs for 3 hours = 100 × 15 × 3 = 4500 watts We have, used of 1000 watts = 1 unit ∴ Use of 4500 watts = 4500 1000 = 4.5 units Example-3 The table below shows the reading of an electricity meter of 5 ampere capacity. 20th Baisakh 20th Jestha 20th Ashadh 20th Shrawan 1026 1055 1285 1585 The rules for charging electricity bill are as along side: (a) Calculate the electricity bills for the months of Jestha. (b) Calculate the electricity bills for the months of Ashadh. (c) Calculate the electricity bills for the months of Shrawan. (d) In which months is electricity consumption maximum and minimum ? Solution: (a) Given, Meter reading of 20th Jestha = 1055 Meter reading of 20th Baishakh = 1026 ∴ Consumed electricity in Jestha = 1055 – 1026 = 29 units Extra units = 29 – 20 = 9 units Electric bill of Jestha = Service Charge + Minimum units × Rate + Extra units × Rate of energy charge = Rs. 50 + 20 × Rs. 3 + 9 × Rs. 6.50 = Rs. 50 + Rs. 60 + Rs. 58.50 = Rs. 168.50 (b) Given, Meter reading of 20th Ashadh = 1285 Meter reading of 20th Jestha = 1055 ∴ Consumed electricity in Ashadh = 1285 – 1055 = 230 units Consumed electricity in Ashadh divides according to charge indicator as, 230 units = (20 + 10 + 20 + 50 + 130 ) units Electric bill of Ashadh = Service Charge + Minimum units × Rate + Extra units × Rates of energy charge = Rs. 100 + 20 × Rs. 3 + 10 × Rs. 6.50 + 20 × Rs. 8.00 + 130 × Rs. 9.50 = Rs. 100 + Rs. 60 + Rs. 65 + Rs. 160 + Rs. 1235 = Rs. 1620 (c) Given, Meter reading of 20th Shrawan = 1585 Meter reading of 20th Ashadh = 1285 ∴ Consumed electricity in Shrawan = 1585 – 1285 = 300 units KW/hr (Unit) 5 Ampere Service Charge Energy Charge 0 - 20 30 0/3.00 21 - 30 50 6.5 31 - 50 50 8.00 51 - 100 75 9.50 101 - 250 100 9.50 Above 250 125 11.00 106 Allied The Leading Mathematics-9 Home Arithmetic 107

Arithmetic Arithmetic Consumed electricity in Shrawan divides according to charge indicator as, 230 units = (20 + 10 + 20 + 50 + 150 + 50) units Electric bill of Shrawan = Service Charge + Minimum units × Rate + Extra units × Rates of energy charge = Rs. 100 + 20 × Rs. 3 + 10 × Rs. 6.50 + 20 × Rs. 8 + 150 × Rs. 9.50 + 50 × Rs. 11 = Rs. 100 + Rs. 60 + Rs. 65 + Rs. 160 + Rs. 1425 + Rs. 550 = Rs. 2360 (d) The electricity consumption is maximum and minimum in Shrawan and Jestha respectively. Example-4 The electricity meter at a home showed the following units in the meter. 01527 At the end of Baisakh 2080 01790 At the end of Jestha 2080 The electricity authority has charged the following rate and scheme of payment. Up to 20 units a minimum of Rs. 80. From 21 units to 250 units @ Rs. 7.30 per unit. Above 250 units @ Rs. 9.90 per unit. Mode of payments and the rebates: l Within 7 days of meter reading 2% rebate. l From 8th to 22nd days of meter reading according to the bill. l From 23rd to 30th days additional 5% of the bill as late fee. l From 31st day to 40th day, additional 10% of the bill as late fee. l From 41st day to 60th day, additional 25% of the bill as late fee. According to the above units used in a family, (a) Find total number of units used throughout the month. (b) Calculate the amount to be paid if the bill is paid within 7 days. (c) Calculate the amount to be paid if the bill is paid on 36th day. Solution: (a) Here, the number of units used may be worked out as; present reading = 01790 and previous reading = 01527 ∴ Number of consumed units = 01790 – 01527 = 263 units (b) Since the rates are different 263 units are divided into three parts: 20 units, 230 units and 13 units. Cost of 20 units = Rs. 80 Cost of 230 units = Rs. 7.30 x 230 = Rs. 11679 Cost of 13 units = Rs. 9.90 x 13 = Rs. 128.70 Total cost = Rs. 80 + Rs. 1679 + Rs. 128.70 = Rs. 1887.70 106 Allied The Leading Mathematics-9 Home Arithmetic 107

Arithmetic Arithmetic Paying bill within 7 days: If the amount of bill is paid within 7 days, 3% rebate in the billing amount. Rebate amount = 2% of Amount of Bill = 2 100 × Rs. 1887.7 = Rs. 37.754 Final amount to be paid after 2% rebate = Amount of Bill – Rebate amount = Rs. 1887.70 – Rs. 37.754 = Rs. 1849.946 (c) Paying bill on 36th day: If the amount of bill is paid on 36th days, 10% late fine includes in the billing amount. 10% late fine amount = 10% of Amount of Bill = 10 100 × Rs. 1887.7 = Rs. 188.77 Final amount with late fee = Rs. 1887.7 + Rs. 188.77 = Rs. 2076.47 Example-5 The following table shows the average daily use of electric energy in a certain household having an electricity meter of 60 ampere capacity. Equipment Number of Items Power in watts Hours used per day Bulbs 10 100 6 Water pump 1 750 3 Washing machine 1 1000 3 Geyser 1 2000 1 Television 1 100 8 Rice cooker 1 1500 4 (a) Find out the total electric energy used in one month, assuming that 1 month = 30 days. (b) Calculate the electricity bill for the given month for the cost of electric energy indicated below. Minimum charge (up to 80 units) = Rs. 360 Cost per unit = Rs. 5 (in the range 81–170 units) Cost per unit = Rs. 7.75 (above 170 units) Note : 1 unit = 1000 watt hr. Solution: (a) Given, Electric energy consumed in 1 day = 10 × 100 × 6 1000 + 1 × 750 × 3 1000 + 1 × 1000 × 3 1000 + 1 × 2000 × 1 1000 + 1 × 100 × 8 1000 + 1 × 1500 × 4 1000 = 6 + 2.25 + 3 + 2 + 0 + 0.8 + 6 = 20.5 units Electric energy consumed in 1 month = 20.05 x 30 units = 601.5 units. 108 Allied The Leading Mathematics-9 Home Arithmetic 109

Arithmetic Arithmetic (b) Since, the rates are different 601.5 units are divided into three parts: 80 units, 90 units and 431.5 units. Cost of 80 units = Rs. 360 Cost of 170 units = 90 x 5 = Rs. 450 Cost of 351.5 units = 7.75 x 431.5 = Rs. 3344.125 Hence, total cost = Rs 360 + Rs. 450 + Rs. 3344.125 = Rs. 4154.13 (approx.) PRACTICE 4.1 Read Think Understand Do Keeping Skill Sharp 1. (a) What is an electricity meter? (b) What is one unit electricity ? Define it. (c) What formula do you use to calculate the bill amount of the consumed electricity when the minimum charge rate, additional units with its rate and service charge are given? 2. (a) In an electricity meter reading, it is found at the previous reading is 12345 units and the consumed units is 445 units. What is the present reading? (b) Whatisthe consumed electricity charge of 53 unitsin the sub-meter of a roomatthe rate of Rs. 13 per unit? (c) What is the electricity consumed when the electric appliances of 3000 watts are used for 2 hours? Check Your Performance 3. (i) The reading of a 15A meter of a certain household was 1978 on 9th Ashwin and it is 2050 on 9th Kartik. [Use the Service and Energy Charge of NEA from the page no. 103] (a) What is the Service Charge for 96 units of electricity consumption? (b) Find the total consumed electric units of the household. (c) Calculate the amount of money to be paid if the rules for charging electricity bill are as above. (ii) Initial reading for the consumption of electricity (5A) of a certain household was 3085 on the 8th Shrawan and current reading is 3225 on the 8th of Bhadra. [Use the Service and Energy Charge of NEA from the page no. 103] (a) What is the Energy Charge for 67 units of electricity consumption. (b) Find the total electric consumption units of the household. (c) Calculate the amount of money to be paid if the rules for charging electricity bill are as above. 108 Allied The Leading Mathematics-9 Home Arithmetic 109

Arithmetic Arithmetic (iii) According to the rate of electricity charge, the minimum charge up to 20 units is Rs. 3 per unit, the charge per unit from 21 units to 30 units is Rs. 7 and from 31 units to 50 units is Rs. 8.50 for a domestic meter of 5 A. (a) What is the rate of charge up to 30 units of consumption? (b) Calculate the bill for the consumption of 45 units according to the rate, if the service charge up to 50 units is Rs. 75. (c) If the house maintained the consumption up to 35 units what percent of money did it save? 4. (i) According to the rate of electricity charge, the minimum cost up to 50 units is Rs. 365 and the cost per extra unit from 51 units to 250 units is Rs. 9.50 for a room rental. (a) Define the "1 unit" electric consumption. (b) How many extra units did the tenant consume if he paid Rs. 621.50 for a month? (c) Calculate the bill of next month, if the tenant consumed 105 units of electricity. (d) Compare the bills of two successive months' electricity consumption. (ii) According to the rate of electricity charge, the cost up to 20 units is Rs. 3 per unit, 21 – 30 units is Rs. 7 per unit, 31 – 50 units is Rs. Rs. 8.50 per unit, 51 – 150 units is Rs. 10 per unit, 151 – 250 units is Rs. 11 per unit. The service charge up to 250 units is Rs. 125. (a) What ampere of current is distributed for domestic use by NEA? (b) If the monthly charge of consumed electricity is Rs. 1225 with Rs. 125 service charge, how much units was consumed? (c) If service charge was not imposed what is the percent of charge to reduce? (iii) The present reading in the electricity sub-meter is 999 and the charge rates are as; Minimum charge (up to 20 units) = Rs. 80, Charge of extra units (21 – 50 units) = Rs. 7.30 (a) What charge should be paid for 15 units? (b) If the bill amount is Rs. 116.50 for a room rental, find the former reading of the sub-meter. (c) What percent of charge will be increased if Rs. 50 was paid as service charge? 5. (i) A hostel has 5 TV sets of 150 watts each. Each TV is used for 4 hours daily. (a) How many 1 watts per hour are there in one unit? (b) Find the daily consumed electricity by 5 TVs. (c) Calculate the monthly charge to be paid by using TVs if the per unit charge is Rs. 13. (ii) A hotel uses 2 rice cookers of 1000 watts each and 3 refrigerators of 800 watts each. (a) For how many hours can we use 1000 watts of electric appliance with 1 unit of electricity? (b) Find the consumed electricity by rice cookers and refrigerators separately, if they are used for 5 hours. (c) Find the total monthly charge by those appliances if the rate is Rs. 17 per unit. [2] (d) Which of the appliance charges more? Compare them. 110 Allied The Leading Mathematics-9 Home Arithmetic 111

Arithmetic Arithmetic (iii) Study the table: SN Electric Appliance Number of sets Capacity (watts) Used Time (hours) Consumed units 1. Television 2 100 5 2. Iron 3 2000 0.5 3. Bulb 2 15 8 4. Rice Cooker 1 1000 1 5. Total (a) How much units will be consumed if an appliance of 1000 watts is used for 1 hour? (b) Complete the table. (c) How much amount does a room rental pay at Rs. 15 per units ? Find it. 6. (i) The table below shows the reading of an electricity meter of 15 ampere capacity. [Use the Service and Energy Charge of NEA from the page no. 103] 1nd Asoj 1nd Kartik 1nd Mansir 1nd Paush 2145 2352 2560 2845 The rules for charging electricity bill are as given. (a) How much service charge should be paid for 325 units? (b) Calculate the electricity Consumption for the months of Asoj, Kartik and Mansir separately. (c) Calculate the bill of month of Kartik. (d) Find the percentage of difference of maximum and minimum consumption in those three months. (ii) The table below shows the reading of an electricity meter of 5 ampere capacity. [Use the Service and Energy Charge of NEA from the page no. 103 and 104] 1nd Magh 1nd Falgun 3450 3678 (a) How much fine should be paid if the payment is not done before 60 days? (b) Estimate the bill of Magh. (c) Calculate the amount if the payment was made on the 5th Chaitra. 7. Read the information and answer the questions given below: [Use the Service and Energy Charge of NEA from the page no. 103 and 104] (i) The electricity consumption of a family for a month is 20 units. (a) What is the minimum charge the family needs to pay? (b) Find the total amount paid if the bill is paid within 7 days. (c) How much more money should be paid if the bill was paid on 10th day? 110 Allied The Leading Mathematics-9 Home Arithmetic 111

Arithmetic Arithmetic (ii) The electricity consumption for a month is 200 units. (a) When should the bill pay if the family wants rebate? (b) Find a total amount paid by the family if the bill is paid on 25th day. (c) How much less money should be paid if the bill was paid on 15th day? (iii) The electricity consumption for a month is 350 units. (a) What percent of late fee would be charged if the bill was paid on 42nd day? (b) Find a total amount paid by the family if the bill is paid on 42nd day. (c) How much money would be saved if the bill was paid on 20th day? 8. Read the information and answer the questions given below: [Use the Service and Energy Charge of NEA from the page no. 103 and 104] Find a total amount paid by a family if the electricity consumption for a month is 20 units and the bill is paid (I) within 7 days (II) on 25th day (III) on 42nd day. (i) The electricity meter readings in a house for the two consecutive months are : End of Magh End of Falgun 07582 08325 (a) What is the consumption units of the family? (b) Find a total amount paid if the bill is paid within 7 days. (c) How much more money should be paid if the bill was paid on 10th day? (ii) The electricity meter readings in a house for the two consecutive months are : Start of Falgun End of Falgun 07585 08328 (a) When should the bill pay if the family wants to pay a bill amount? (b) Find a total amount paid by the family if the bill is paid on 15th day. (c) How much less money should be paid if the bill was paid on 5th day? (iii) The electricity meter readings in a house for the two consecutive months are : 1st of Chaitra 2079 1st of Baishakh 2080 083231 08540 (a) What percent of late fee would be charged if the bill was paid on 32nd day? (b) Find a total amount paid by the family if the bill is paid on 32nd day. (c) How much money would be saved if the bill was paid on 20th day? 112 Allied The Leading Mathematics-9 Home Arithmetic 113

Arithmetic Arithmetic 9. (i) The following table shows the average daily use of electricity in a certain household having an electricity meter of 60 amperes capacity. KW/hr (unit) 60 Ampere Service Charge (Rs.) Energy Charge (Rs.) 0 - 20 125 6 21 - 30 150 7 31 - 50 175 8.50 51 - 150 200 10 151 - 250 225 11 251 - 400 250 12 above 400 275 13 (a) Which formula can calculate the electric consumption? a. N × P × T b. N × P T c. P × T N d. T × N P (b) What is the total unit of electricity energy used by the house in a month assuming 1 month = 30 days? (c) Calculate the electricity bill for the given month, if the cost of electricity is as indicated in the above table. (ii) In a family of 5A metre, there are 5 electric LED bulbs each of 3 wattsrunning on continuously for 8 hours 30 minutes, 1 TV of 150 W running on for 7 hours daily 1 laptop of 250 W running continuously for 10 hours daily. Workout the total units of electricity used in the family for a month (a) Which of the following is true? (i) 1watt = 1000 units/hour (ii) 1 unit = 1000 hours/watt (iii) 1 unit = 1000 watts/hour (iv) 1 watt = 1000 hour/unit (b) What is the total bill for the family with a rebate of 2% (c) What is the total bill for the family with a fine of 10%. (d) Compare the difference in percentage. 3.(i) (b) 72 units (c) Rs. 665 (ii) (b) 140 units (c) Rs. 1300 (iii) (b) Rs. 332.50 (c) Rs. 25.56 4.(i) (c) 27 units (d) Rs. 982.50 (ii) (b) 80 units (c) 10.20% (iii) (b) 974 (c) 42.92% 5.(i) (b) 3 units (c) Rs. 1170 (ii) (b) 12 units (c) Rs. 11220 (iii) (c) Rs. 2358 6.(i) (b) 207, 208, 285 (c) 285 units (d) Rs. 1931 (ii) (b) Rs. 2076 (c) Rs. 2283.60 7.(i) (b) Rs. 78.40 (c) Rs. 1.60 (ii) (b) Rs. 1463.70 (c) Rs. 69.70 (iii) (b) Rs. 3436.25 (c) Rs. 687.25 8.(i) (a) 743 units (b) Rs. 6440.51 (c) Rs. 199.19 (ii) (b) Rs. 6639.70 (c) Rs. 199.19 (iii) (b) Rs. 1675.91 (c) Rs. 151.81 9.(i) (b) 600 units (c) Rs. 6860 (ii) (b) Rs. 1136.80 (c) Rs. 1276 (d) 12.24% Answers Equipment Number of items Power in watt Hours used per day Electric bulb 10 100 6 Electric bulb 15 60 8 Washing machine 1 1000 1 Water pump 1 250 2 Rice cooker 1 1500 2 Television 1 100 8 Geyser 1 2000 1 Refrigerator 1 150 24 112 Allied The Leading Mathematics-9 Home Arithmetic 113

Arithmetic Arithmetic 4.2 Household Expenses for Use of Water At the end of this topic, the students will be able to: ¾ solve problems on the consumed drinking water in household. Learning Objectives Uses of Water in Our Daily Activities Water for drinking Water for Sashing Water Meter Water is essential to survive the living things. How much water do we need everyday and one month? How much do we pay for the water to water supplier company? We use water for washing clothes. How much water used for water for that work in a month ? Do you measure it ? How can we find out? We can easily read consumed quantity of water for drinking, washing, bathing, swimming, etc. in the electric meter. How do you calculate the charge of consumed quantity of water ? Introduction to Charge of Consumed Water The quantity of consumed water by a household is measured in the number of units by the water meter. The number of units of used water is calculated as, Units of consumed water = Reading of the recent month – Reading of the previous month For the services of drinking water, we have to pay charges to the government (Kathmandu Upatyaka Khanepani Limited (KUKL), Nepal Khanepani Sasthan (NKS), etc.). How do you pay the charge for drinking water? Likewise, in the electricity charge, the meter reader from water supply authority reads the water meter joined in the water pipe and leaves the bill. We have to go to these authorities monthly and pay the bill. The water bill is prepared as the KUKL/NKS tariff rates as given below: Kathmandu Upatyaka Khanepani Limited (KUKL) (The rate of water charge) Pipe size (in inch) Minimum consumption water (in litre) Minimum charge (Rs) Extra consumption rate per 1000 litre (Rs.) Rate of water without meter as per pipe size 1/2" 10000 100 32 785 3/4" 27000 1910 71 4595 1" 56000 3960 71 9540 114 Allied The Leading Mathematics-9 Home Arithmetic 115

Arithmetic Arithmetic 1.5" 155000 10950 71 26280 2" 320000 22600 71 54255 3" 881000 62240 71 149415 4" 1810000 127865 71 306880 Note : (a) 1 unit = 1000 litre of water. (b) Minimumconsumption = 10000 litres(10000 litresis denoted by 10 units)ruled by NWSC. Kathmandu Upatyaka Khanepani Limited (KUKL) Branch office, Lalitpur Name: Nabin Lama Consumer No. 02 - 24 - 18 Address: Vanasthali, Kathmandu Area - 2 – 3 – 305 Month: Asoj Present Reading Former Reading Unit Consumed Minimum Cost Extra Cost Total 1250 1225 25 units Rs. 100 Rs. 480 Rs. 580 Khil Kumar Meter reading of last day of Asoj Meter reading of last day of Bhadra ( 1250 – 1225) = 25 units Mini. charge i.e. of 10 units 25 – 10 = 15 15 × Rs 32 Mini. cost + Extra unit cost i.e. Rs 100 + Rs 480 Signature Rules of payment and rebate: Within 1st and 2nd month of the billing 3% rebate Within 3rd month according to the bill. Within 4th month 10% late fee. Within 5th month 20% late fee. After 5th month 50% late fee. Important Formulae on Use of Electricity Consumed Quantity of water = Reading of the recent month – Reading of the previous month Bill amount of water = Minimum charge + Additional Units × Its rate + Drain charge Amount of Bill payment = Bill Amount +/- Rebate/0/Fine percent of Bill amount Sewage/Drain charge = 50% of Bill Amount Let's Summarise Drinking Water Meter: A device that shows the quantity of consumed drinking water in units or figures is called drinking water meter. One Unit of Drinking Water: When the quantity of drinking water of 1000 liters is used, it is consumption of one unit of drinking water. 114 Allied The Leading Mathematics-9 Home Arithmetic 115

Arithmetic Arithmetic Example-1 The adjoining water meters shows the consumption of water of a family. The rates of water charge are given below; Minimum charge up to 10 units = Rs. 100 Charge more than 10 units = Rs. 32 per unit (a) Which formula do you use to calculate the amount to be paid for the consumed drinking water when the bill amount and late fee of 10% are given ? (b) Calculate the monthly charge on the rates given below: (c) If the bill is to be paid within second month of billing date, how much amount to be paid? Solution: (a) The amount to be paid for the consumed drinking water = Bill amount + Late fee 10% of Bill amount (b) Here, Present reading = 849 units, Former reading = 789 units ∴ Consumed water = 849 – 789 = 60 units Minimum charge for 10 units = Rs. 100 Extra consumed water = 60 – 10 = 50 units Rates of extra consumed water = Rs. 32 per unit ∴ Charge for extra consumed water = Rs. 32 × 50 = Rs. 1600 ∴ Total charge = Rs. 100 + Rs. 1600 = Rs. 1700 Thus, the monthly charge is Rs. 1700. (c) If the bill is to be paid within second month of billing date, 3% gets rebate. Then the amount to be paid = Bill amount – 3% of bill amount. = Rs. 1700 – 3 100 × Rs. 1700 = Rs. 1700 – Rs. 51 = Rs. 1649. Example-2 According to the rate of Drinking Water Supply Corporation in Biratnagar charge of 3 4 inch pipe, the minimum cost up to 27 units is Rs. 1910 and the charge for each extra unit is Rs. 71. The monthly charge of water supply is Rs. 17175. How many units of water were consumed? Solution: Here, Monthly charge of water supply = Rs. 17175, Minimum cost up to 27 units = Rs. 1910 Charge for each extra unit = Rs. 71 ∴ Amount for extra units = Rs. 17175 – Rs. 1910 = Rs. 15265 ∴ No. of extra units = Amount for extra units Charge for each extra unit = Rs. 15265 Rs. 71 = 215 units ∴ Total no. of units consumed water in the month = 27 units + 215 units = 242 units 1st Mansir 00789 2 1st Paush 00849 3 116 Allied The Leading Mathematics-9 Home Arithmetic 117

Arithmetic Arithmetic Example-3 The water meter indicates the following units of water used in a family: 03598 units (At the end of July 2023), 03669 units (At the end of August 2023) The water supply authority has declared the following rates of payments: Minimum charge up to 10 units = Rs. 100 Above 11 units = Rs. 32 per unit Mode of payment and rebates: Within 1st and 2nd month of the billing 3% rebate Within 3rd month according to the bill. Within 4th month 10% late fee. Within 5th month 20% late fee. After 5th month 50% late fee. (a) Find the bill to be paid by the family. (b) If the bill is filled within 15 days of the billing date, find the amount to be paid. (c) If the bill is filled on 129th day after the billing date, find the amount to be paid. Solution: (a) Here, No. of units used can be worked out as; Present reading = 03669 (–) Previous reading = 03598 Total units used = 71 units Unit charges: First 10 units = Rs. 100 Next (71 – 10) = 61 units @ Rs. 32 per units = Rs. 32 × 61 = Rs. 1952 ∴ Total bill amount = Rs. 100 + Rs. 1952 = Rs. 2052 (b) (–) Rebate 3% = 3% of Rs. 2052 = 3 100 × Rs. 2052 = Rs. 61.56 ∴ Final amount after the deduction rebate = Rs. (2052 – 61.56) = Rs. 1990.44 (c) If the bill is paid on 129th day after billing date, 20% fine is paid. Total amount = Rs. 2052 (+) Late fee at 20% = 20% of Rs. 2052 = 20 100 × Rs. 2052 = Rs. 410.40 ∴ Final payment with late fee = Rs. 2052 + Rs. 410.40 = Rs. 2462.40 Example-4 An office uses the water of the volume of 3 tanks of 1000 liters and 5 drums of 200 liters daily. (Minimum charge up to 10000 liters = Rs. 100. Rate of each extra 1000 liters = Rs. 32, 1 month = 30 days) (a) How many liters are there in 1 unit of water ? (b) Find the quantity of water to be consumed by the office. (c) Find the charge of consumed water in a month. Solution: (a) There are 10000 liters in 1 unit of water. 116 Allied The Leading Mathematics-9 Home Arithmetic 117

Arithmetic Arithmetic (b) Given, Volume of 3 tanks of 1000 liters = 3 × 1000 liters = 3000 liters Volume of 5 drums of 200 liters = 5 × 200 liters = 1000 liters ∴ Total Volume of daily used water = 3000 liters + 1000 liters= 4000 liters ∴ Total Volume of monthly used water by the office = 30 × 4000 liters = 120000 liters (c) Minimum charge up to 10000 liters (10 units) = Rs. 100 Extra volume of water = 120000 liters – 10000 liters = 110000 liters Rate of each extra 1000 liters = Rs. 32 ∴ Amount for extra volume of water = Rs. 32 × 110 = Rs. 3520 ∴ The charge of consumed water in a month = Rs. 100 + Rs. 3520 = Rs. 3620 Example-5 A factory uses 1.5 inches water pipe. The meter reading for the consumption of water was 02345 units on 1st Asoj and 02635 on 1st Kartik. The charges are indicated as below: Minimum charge up to 155 units = Rs. 10950, Charge above 71 units = Rs. 71 per unit (a) How many liters of water are consumed by the factory? Find it. (b) Calculate the bill amount. (c) Calculate the charge to be paid including 50% drain charge in 4th month after the bill issued and 10% fine. Solution: (a) Here, Consumption of water on 1st Asoj = 02345 units Consumption of water on 1st Kartik = 02635 units ∴ Consumed unit of water = 02635 – 02345 = 290 units ∴ Consumed quantity of water = 290 × 1000 = 290000 liters. (b) Minimum charge up to 155 units = Rs. 10950 Extra consumed water = 290 – 155 = 135 units Rate of extra consumed water = Rs. 71 ∴ Charge for extra consumed water = Rs. 71 × 135 = Rs. 9585 (c) Total bill amount for consumed water = Rs. 10950 + Rs. 9585 = Rs. 20535 Charge of drain = 50% of total charge of water = 50 100 × 20535 = Rs. 10267.50 ∴ Total charge with drain = Rs. 20535 + Rs. 10267.50 = Rs. 30802.50 If the payment of the bill is made in 4th month, 10% fine should be paid. ∴ Fine charge = 10% of Rs. 30802.50 = Rs. 3080.25 ∴ Total payment = Rs. 30802.50 + Rs. 3080.25 = Rs. 33882.75 118 Allied The Leading Mathematics-9 Home Arithmetic 119

Arithmetic Arithmetic PRACTICE 4.2 Read Think Understand Do Keeping Skill Sharp 1. (a) Define one unit water. (b) How can you find the consumed water? (c) Which formula do you use to calculate the bill amount of the consumed water when the minimum charge rate, additional charge and its rate, and drain charge are given? (d) Which formula do you use to calculate the bill amount to be paid for the consumed water when the bill amount and rebate rate of 3% are given? 2. (a) In a water meter reading, the present reading 0125 units and the previous reading 0108 units are shown. What is the consumed water units? (b) In a water meter reading, it is found the at the previous reading is 0543 units and the consumed units is 65 units. What is the present reading? (c) In a water meter reading, it is found the at the present reading is 0653 units and the consumed units is 37 units. What is the previous reading? (d) If a rectangular tank contains 7000 litres of water, how many units of water are there? (e) What is the charge of water of 34 units at the rate of Rs. 40 per unit? Check Your Performance 3. (i) A house has water pipe line of 1 2 ''. The minimum charge up to 10000 ℓ water is Rs. 100. (a) Define one unit water consumption. (b) If the house consumed 15 units of water, find the bill amount where rate of extra units is Rs. 32 per unit. (c) If the bill of 15 units is paid at the fourth month of billing date, how much amount is paid more? (d) The bill amount of the consumed drinking water is Rs. 292. If the bill is to be paid within first month of billing date, how much amount is to be paid? (ii) A house has pipeline of 1 2''. The minimum charge up to 10000 ℓ water is Rs. 100. The bill amount where rate of extra units is Rs. 32 per unit. The bill amount of the consumed drinking water is Rs. 772. (a) If the bill isto be paid at the third month of billing date, how much amount to be paid? (b) Find the amount of water in litre consumed if the pipe line is of 1 2''. (c) If the bill was paid within the first month of billing, how much money would be save? (d) When should we pay the bill? Give reason. (iii) Given the information: Size of pipeline = 1 2''. Minimum Consumption = 10000 ℓ (10 units) 118 Allied The Leading Mathematics-9 Home Arithmetic 119

Arithmetic Arithmetic Minimum Charge = Rs. 100, Rate of extra units = Rs. 32 per unit. Drain Charge = 50% of bill amount (a) What amount would be charged as drain charge? (b) If the bill of Rs. 324 is to be paid at the first week of billing date, how much amount to be paid with drain charge? (c) If the payment was done at the fourth month of billing date, how much amount is to be paid with drain charge? (d) Find the percentage of difference of drain charges in case (b) and (c). (iv) Find the amount to be paid within 3rd month if the house consumed 13 units of water. Given the information as shown in the above question (iii). Bill Amount = Rs. 420 (a) How many units does the 10000 ℓ of water consumption denote? (b) Find the consumption of water in litre. (c) If the bill is to be paid at the fifth month of billing date, how much amount needs to be paid with drain charge? (d) If the consumption is fixed, how can we save money in payments? 4. (i) The number shown by a meter on a 2 inch pipe of water of a company is 36125 units and the current reading is 37212. If the charge up to 320 units = Rs. 22600 and Rate of extra unit = Rs. 71, find the payment amount. (a) How much units of water was consumed? (b) Find out the bill amount including 50% drain charge.? (c) Find the payment amount if 10% late fee is fined asthe payment was done on 4th month. (d) Find the saving amount if the payment was done in 1st month with 3% rebate. (ii) Reading of water- meter of a certain household is as alongside: The charges according to Water Supply Corporation are as; Cost for water consumed (1 - 10) units = Rs. 100 Rate of extra consumption = Rs. 32 per unit Sewerage charge = 50% of the charge on water consumed, Meter rental charge = Rs. 5. [Note: Discount offers: Connect IPS Rs. 5 cash back and eSewa 2% cash back] (a) Find the units of water consumption. (b) Find the consumed water in litre. (c) Find the total bill amount of the house. (d) Which digital payment method is better to have more discount? (iii) The minimum charge up to 10 units of water through the ½ inch pipe per month is Rs. 100 and Rs. 32 per unit for extra uses. (a) Define one unit water consumption. (b) How much water charge should be paid for a month after using 23 units of water? (c) If the house wantsto save Rs. 160, by how much the consumption should be reduced? 1st Baishakh 01000 2 1st Jestha 01009 3 120 Allied The Leading Mathematics-9 Home Arithmetic 121

Arithmetic Arithmetic (iv) Afamily consumes 356 units of water through a 3 4 inch pipe in a month. The consumtion up to 27000 litres is Rs. 1910 minimum charge and above 27000 litres @ the rate of Rs. 71 per 1000 litres. (a) How much water can be consumed with 1 unit? (b) Find the total bill paid by the family. (c) If the pipe line was of 1 inch, what would be the bill amount? Given the consumption upto 56000 litresis Rs. 3960 and above 56000 litres @ the rate of Rs. 71 per 1000 litres. (d) Find the difference percentage. 5. (i) According to the rate of KUKL charge of 1 2 inch pipe, the minimum cost up to 10 units is Rs. 100 and the extra charge for each extra unit is Rs. 32. (a) How many units are there in 1000 litre? (b) If the monthly charge of water supply is Rs. 1924, find the consumed water. (c) If 5000 ℓ water is consumed less, how many units would be extra consumptions? (ii) A hotel has pipeline of drinking water of 1 inch. According to the rate of Drinking Water Corporation The minimum cost up to 56000 litres is Rs. 3960 and the extra charge for each extra 1000 litres is Rs. 71. (a) How many units can be consumed with Rs. 3960? (b) The monthly charge of the hotel for water supply is Rs. 67079. How many units of water were consumed in the month? (c) Find the difference in price if the hotel consumed 789 units. 6. (i) The water meter indicates the following units of water used by 1/2 inch pipe in a family: 00975 units (At the end of Asar 2076), 01542 units (At the end of Shrawan 2076) [Use the water charge and rules of payment from the page no. 114 and 115] (a) For how many days of billing do we get rebate? (b) Find the bill amount paid by the family within 35th day of the billing. (c) If the another family also had same unit of consumption but paid the bill on 155th day after bill is received, find the paying amount by the second family. (d) Which family would you regard as intelligent family? Give reason. (ii) The water meter of 3 4 inch pipe indicates the following units of water used in a family: 00025 units (At the end of Asar 2076), 00322 units (At the end of Shrawan 2076) [Use the water charge and rules of payment from the page no. 114 and 115] (a) What is the percentage of late fee if the bill is not paid till 180 days? (b) Find the payable amount of the bill is filled within 26th day of the billing. (c) If the family also had same unit of consumption but paid the bill on 6th month after bill is received, find the payment amount paid by the second family. (d) Which family would you regard as intelligent family? Give reason. 120 Allied The Leading Mathematics-9 Home Arithmetic 121

Arithmetic Arithmetic 7. (i) A company uses 5 tanks of water of the volume of 1200 litres and 6 drums of 400 litres daily. (Minimum charge up to 27000 litres = Rs. 1910. Rate of each extra 1000 litres = Rs. 71, 1 month = 30 days) (a) What is the rate of an extra consumption unit? (b) Find the daily water consumption. (c) Find the charge of consumed water in a month. (d) By what percent the units of tanks is higher than that of drums in the same month? (ii) A mineral water company uses 5 green water tanks of volume of 15000 litres and 4 black water tanks of 2500 litre daily. (Minimum charge up to 881 units = Rs. 62240, extra consumption rate per unit = Rs. 71) (a) What is the minimum consumption of water in litres? (b) Find the volume of water used each day. (c) Find the bill amount of consumed water in 30 days. (d) Compare the units raised by green water tank and black water tank. 8. (i) A community supplied 567 units of drinking water through 11 2 inch pipe of KUKL. (Minimum charge up to 155000 = Rs. 10950. Rate for extra per 1000 = Rs. 71. Fine when the bill is paid in 5th month = 20%) (a) What percent of bill amount is charged as drain charge by KUKL? (b) If the community consumed 567 units of water, find the bill amount including drain charge and a miscellaneous charge of Rs. 650. (c) The payment of the bill is made within the 5th month after the bill issued, find the bill paid. (d) KUKL distributes the water without meter through 11 2 inch pipe line for monthly charge of Rs. 26280. Decide which would be better with meter or without meter? Give reason. (ii) The table shows the readings of a water meter of a certain household. Month Last of Baisakh Last of Jestha Last of Ashad Units 6072 6099 6194 The cost for first 10 units of water is Rs. 100, each extra unit costs Rs. 32 and the drain charge is 50% of the bill amount. (a) Write down the foumula to calculate the charge when minimum charge, extra units and rate of extra units are given. (b) Calculate the water bill for the months of Jestha. (c) Calculate the water bill for the months of Ashad if Water Supply Corporation allows a rebate of 3% while the payment is made within the first two months. (d) Which month is economical? Give reason. (iii) The following table shows the readings of water meter of a certain household. Months Baisakh Jestha Ashadh Shrawan Units 1000 1013 1037 1074 122 Allied The Leading Mathematics-9 Home Arithmetic 123

Arithmetic Arithmetic Cost for water consumption (1–10) units = Rs. 100 More than 10 units = Rs. 32 per unit. Drain charge (imposed on water consumption charges) = 50% Water meter rental charge per month = Rs. 15. (a) How can we find the consumed water units from two successive meter reading? (b) Find the water consumed units in month of Jestha. (c) Find the water bill of Ashadh with drain charge and meter rental charge. (d) Which month is most economical? Give reason. 9. (i) A family used 358 units in a month and the payment was made. [Use the water charge and rules of payment from the page no. 114 and 115] (a) How much money should the family pay if payment was made within 62nd day? (b) How much more money should be paid if the payment was made on 2nd month? (c) Compare the difference in percentage when payment is made on 62nd and on 108th day. (ii) The water meter in a house showed the following units used within the fourth month. 02578 units (At the end of Shrawan), 02986 units (At the end of Bhadra) (a) How can we find the consumed water units of Bhadra? Find it. (b) If the pipe line is of 3 4 inch whose minimum charge for 27 units is Rs. 1910 and charge of extra units = 71 per unit. Find the total bill paid by the family if the bill is paid on the 9th day of billing. (c) Compare the difference in percentage when payment is made on 65th and on 128th day. 3. (i) (b) Rs. 260 (c) Rs. 26 (d) 16 units (ii) (a) Rs. 772 (b) 31000 l (c) Rs. 23.16 (iii) (b) Rs. 162 (c) 55% (d) 5% (iv) (b) 20000 l (c) Rs. 945 4. (i) (a) 1087 units (b) Rs. 115585.50 (c) Rs. 127144.05 (d) Rs. 3467.57 (ii) (a) 9 units (b) 9000 l (c) Rs. 150 (d) IPS (iii) (b) Rs. 516 (iv) (b) Rs. 25269 (c) Rs. 25260 (d) 0.036% 5. (i) (b) 67000 l (c) 52 units (ii) (a) 56 units (b) 945 units (c) Rs. 11076 6. (i) (a) within 2 months (b) Rs. 17386.28 (c) Rs. 26886 (d) 1st family (ii) (b) Rs. 21080 (c) Rs. 31620 (d) 1st family 7. (i) (b) 8400 l (c) Rs. 17885 (d) 150% (ii) (b) 85000 l (c) Rs. 180739 (d) 1950 units more 8. (i) (b) Rs. 60953 (c) Rs. 73143.60 (d) without meter (ii) (b) Rs. 644 (c) Rs. 624.68 (d) Jestha (iii) (b) Rs. 196 (c) Rs. 837 (d) Jestha 9. (i) (a) Rs. 11236 (b) Rs. 10898.92 (c) 10% more (ii) (a) 408 units (b) Rs. 28092.17 (c) 20% more Answers 122 Allied The Leading Mathematics-9 Home Arithmetic 123

Arithmetic Arithmetic 4.3 Household Expenses for Use of Telephone At the end of this topic, the students will be able to: ¾ solve problems on tariff of telephone. Learning Objectives Uses of Telephone in Our Daily Activities Telecommunication device What are the devices that can talk between two persons? How much amount charges when used local telephone/ cordless, or post-paid mobile or pre-paid mobile or CDMA mobile? How many calls can two persons talk by local telephone for 6 minutes? Do you find it? Introduction to Charge of Telephone Calls The telephone calls of a household is recorded in Nepal Telecom. The telephone calls are calculated as, Calls of Telephone = Reading of the recent month – Reading of the previous month For the services of communication, we have to pay charges to the Nepal Telecom. How do you pay the charge for telephone calls? On the basis of the time periods of talking by local telephone in each time, the total calls is calculated by subtracting the present calls from the previous calls of preceding month and record the telephone calls and bill from the related office according as the rules of related office including service charge, tax and value added tax on it and the bill of telephone will pay in the office or indicated banks or SMS or eSewa or mobile banking. The rate of telephone calls for Nepal Telecom (NT) Telephone Calls Block Billing Method up to 175 calls Rs. 200 above 175 calls (Additional calls) Re. 1 per call 124 Allied The Leading Mathematics-9 Home Arithmetic 125

Arithmetic Arithmetic Telephone Tariff of National Call (NTC) Time Band Call Period (Time) Rate/call in NRs. Business hour (06:00 hrs. -18:00 hrs.) 2 minutes 1.00 Normal hour (18:00 hrs. -22:00 hrs.) 4 minutes 1.00 Off hour (22:00 hrs. - 06:00 hrs.) 8 minutes 1.00 Bill of Landline/PSTN Telephone 7746732 2079-07-04 Statement for the Month : Kartik 54653 55837 JIWAN RAM AKRKI 01-4333642 CHARGE SUMMARY Rs. 1184 1009 1209 55837 – 54653 = 1184 Local : 1184 STD : ...... ISD : ...... NAT.OA : ...... INT OA : ...... Adjustment : ...... Sub Total 1209 TSC 10% 120.90 Total : 1329.90 VAT 13% : 172.89 Grand Total : 1502.79 1009 calls at Re. 1/call Rules of payment: Within 7th day according to the bill. After 7th day 10% late fee. Important Formulae on Use of Local Telephone Total Telephone Calls = Reading of the recent month – Reading of the previous month Bill Amount of Telephone Calls (Sub-Total) = Minimum charge + Extra Calls × Its rate Bill Amount of with Service Charge (Total) = Sub-total + TSC% of Sub-total = Sub-total (1 + TSC%) = Sub-total (100 + TSC) 100 Bill Amount of with VAT (Grand Total) = Total + VAT% of Total = Total (1 + VAT%) = Sub-total (1 + TSC%)(1 + VAT%) = Sub-total (1 + TSC)(1 + VAT) 10000 Amount of Bill payment = Grand Total/Bill Amount + 0/Fine percent of Bill amount 124 Allied The Leading Mathematics-9 Home Arithmetic 125

Arithmetic Arithmetic Points to be Remembered Communication: The medium of talking between the persons of two places is called communication. Telephone: A system for transmitting voices over a distance using wire or radio, by converting acoustic vibrations to electrical signals, is called telephone. Phone: The talking device for the communication is called phone. There are different types of phones such as local telephone, cordless, wireless, mobiles, etc. Example-1 In a telephone bill of a house, 21012 calls are shown on the last of Kartik and 21337 calls are shown on the last of Mangsir. The minimum charge for the first 175 telephone calls is Rs. 200 and the charge for each additional call is Re. 1. (a) Write the formula to calculate the grand total amount of telephone calls. (b) How many calls of telephone are made in Mangsir ? (c) How much will be the charge for Mangsir ? (d) If the bill amount is in Magh, how many calls were made in the Magh ? Solution: (a) Grand total amount of telephone calls = Sub-total (1 + TSC%)(1 + VAT%) = Sub-total (1 + TSC)(1 + VAT) 10000 (b) Here, the minimum charge for first 175 calls = Rs. 200 Additional calls = 325 – 175 = 150 calls (c) Rate of each additional call = Re. 1 ∴ Amount of additional calls = Rs. 1 × 150 = Rs. 150 i.e. Sub-total of 325 calls = Rs. 200 + Rs. 150 = Rs. 350 Hence, the charge for 325 calls will be Rs. 350. (d) Here, The charge of telephone calls = Rs. 875 The minimum charge of 175 calls = Rs. 200 ∴ The charge for extra calls = Rs. 875 – Rs. 200 = Rs. 675 Rate of extra call = Re. 1 ∴ Total telephone calls = 175 + 675 = 850 Thus, 850 calls were made by Rs. 875. 126 Allied The Leading Mathematics-9 Home Arithmetic 127

Arithmetic Arithmetic Example-2 The local call details of a certain telephone in the month of Jestha shows the current reading of 1590 units and the previous call reading of 1280 units. The rate of telephone calls for Nepal Telecom (NT) Minimum Rental charge (with 175 free calls) = Rs. 200, Each extra call = Re. 1 Telephone service charge = 10% VAT = 13% of the total of (telephone call charge + service charge) (a) Find the total number of calls within the month. (b) Calculate the telephone bill amount. (c) If you pay the bill on 15th Asar, you pay fine at 5% on billing amount. Now, how much do you pay ? Solution: (a) Here, Current reading = 1590 units, Initial reading = 1280 units ∴ Total number of calls = 1590 – 1280 = 310 (b) Now, we have Number of extra calls = 310 – 175 = 135 Cost of the first 175 calls = Rs. 200 Cost of extra 135 calls at Re. 1 per call = Rs. 135 Total telephone call charge = Rs. 200 + Rs. 135 = Rs. 335 Service charge = 10% of Rs. 335 = 10 100 × Rs. 335 = Rs. 33.50 So, the total of telephone call charge and service charge = Rs. 335 + Rs. 33.50 = Rs. 368.50 VAT = 13% of Rs. 368.50 = 13 100 × Rs. 368.50 = Rs. 47.905. Hence, the final amount to be paid = Rs. 368.50 + Rs. 47.905 = Rs. 416.405. (c) If the bill is paid on 15th Asar, 5% fine is charged on billing amount. So, the payable amount = Rs. 416.405 + 5% of Rs. 416.405 = Rs. 437.23. Example-3 The cost of making a telephone call from Nepal to USA is Rs. 9 for the first one minute and Rs. 3 is the charge for each additional call of 15 seconds. If a person pays Rs. 129 for making a telephone call there, how long did he make the telephone call? Solution: Here,The charge for first 1 minute = Rs. 9 The charge for additional call of 15 seconds = Rs. 3 Total charge = Rs. 129 ∴ Charge for additional calls = Rs. 129 – Rs. 9 = Rs. 120. ∴ No. of additional call extra 15 seconds at Rs. 3 per call = Rs. 120 Rs. 3 = 40 ∴ Duration for additional calls = 40 × 15 seconds = 600 seconds = 600 60 minutes = 10 minutes ∴ Total time for telephone calls = 1 min. + 10 min. = 11 min Thus, he made the telephone calls for 11 minutes. 126 Allied The Leading Mathematics-9 Home Arithmetic 127

Arithmetic Arithmetic Example-4 The minimum charge for the first 175 telephone calls is Rs. 200 and then the additional charge per call is Re. 1. 10% TSC and 13% VAT is included in the bill. How many units can be consumed by Rs. 497.20? Solution: Here, the minimum charge for first 175 telephone calls = Rs. 200 Suppose additional calls = x calls. Then additional charge at Re. 1 per call = Rs. x ∴ Subtotal amount of (175 + x) calls = Rs. (200 + x) ∴ Total amount with 10% TSC = (200 + x) + 10% of (200 + x) = 200 + x + 10 100 × (200 + x) = 2000 + 10x + 200 + x 10 = 11x + 2200 10 ∴ Grand total amount with 10% TSC and 13% VAT = 11x + 2200 10 + 13 100 × 11x + 2200 10 = 1100x + 220000 + 143x + 28600 1000 = 1243x + 248600 1000 By the question, total payment amount = Rs. 497.20 ∴ 497.20 = 1243x + 248600 1000 or, 497200 = 1243x + 248600 or, 1243x = 497200 – 248600 or, x = 248600 1243 = 200 ∴ Total number of calls = 200 + 175 = 375. Thus, 375 calls can be consumed by Rs. 497.20. "Alternatively" The minimum charge for first 175 telephone calls = Rs. 200 Rate of extra calls = Re. 1, TSC = 10%, VAT = 13% Grand total amount = Rs. 497.20 Now, we know that Grand total amount = Subtotal amount × (100 + TSC%) × (100 + VAT%) 100 × 100 or, Rs. 497.20 = Subtotal amount × (100 + 10) × (100 + 13) 10000 or, Rs. 4972000 = Subtotal amount × 110 × 113 or, Subtotal amount = Rs. 4972000 110 × 113 = Rs. 400 Charge of extra calls = Rs. 400 – Rs. 200 = Rs. 200 ∴ No. of extra calls = Rs. 200 Re. 1 = 200 calls ∴ Total no. of calls = 200 + 175 = 375 calls Thus, 375 calls can be made by Rs. 497.20. 128 Allied The Leading Mathematics-9 Home Arithmetic 129

Arithmetic Arithmetic PRACTICE 4.3 Read Think Understand Do Keeping Skill Sharp 1. (a) Define telephone. (b) Define unit call in landline phone. (c) How can you find the telephone calls? (d) Which formula do you use to calculate the bill amount of the telephone calls when the minimum charge rate and rate of extra call are given? (e) Write the formula to calculate the subtotal amount of telephone calls. (f) Write the formula to calculate the grand total amount of telephone calls. 2. (a) In a telephone bill of a house, the previous reading 0952 and the present reading 1095 are shown. How many calls are made there? (b) In a telephone bill of an office, the previous reading is 12345 calls and the total calls is 543 calls. What is the present reading? (c) In a telephone bill of a house, the present reading is 98765 calls and the total calls is 1234 calls. What is the previous reading ? (d) In a telephone bill of a house, 7080 calls is shown on the last of Asar, 2076. If the total calling during the month of Asar is 524 calls, what was the appearing reading on the first of Asar, 2076? Check Your Performance 3. (i) The minimum charge for first 175 telephone calls is Rs. 200. If the charge for each additional call is Re. 1. (a) What percent of Telephone Service Charge (TSC) is imposed by NTC? (b) How much will be the charge for 792 calls? (c) Find the total payment charge including TSC and 13% VAT. (d) Find out the percentage of difference of answers from ii. and iii. (ii) The current and previous calls of the local telephone of a household are 12645 and 11265 respectively. (Minimum charge for first 175 calls = Rs. 200, Rate of additional call = Re. 1) (a) How can we calculate the telephone call? (b) What is the total number of calls done by the household? (c) Find out the total bill with 10% TSC and 13% VAT. (d) By how much the VAT amount is more than TSC amount? 4. (i) The minimum charge for the first 175 telephone calls is Rs. 200 and then additional charge per call is Re. 1. Moreover, 10% TSC and VAT are included in the bill. 128 Allied The Leading Mathematics-9 Home Arithmetic 129

Arithmetic Arithmetic (a) What percent of VAT is levied on telephone bill with service charge? (b) How much should pay for 2368 calls without TSC and VAT? (c) Find the total bill amount to be paid with TSC and VAT. (d) By how much percent the final bill increased from the bill without TSC and VAT? (ii) The previous calls and current calls of the local telephone of a company in a month are 12689 calls and 13965 calls respectively (Minimum charge for first 175 calls = Rs. 200, Rate of additional call = Re. 1, TSC = 10% and VAT = 13%) (a) Write down the formula for calculating total bill amount if subtotal, TSC% and VAT% are given. (b) How many calls did the family make in the given month? (c) Find the grand total amount with 10% TSC and 13% VAT by the household. (d) Compare the TSC amount and VAT amount. 5. (i) A telephone booth charges Rs. 15 for the first minute and Rs. 1.5 for every additional 10 seconds for ISD call from Kathmandu to Canada. (a) What does ISD call stands for? (i) Call in Specific Domain (ii) Call in Specific District (iii) Call in own country (iv) Call out of own country (b) If a girl made the telephone call of 4 minutes and 40 seconds from telephone booth, find the amount to be paid by the girl. (c) If the actual charge of ISD call from Kathmandu to Canada is Rs. 25 for first 2 minutes and Rs. 2 for every additional 20 seconds, find the actual telephone bill of booth owner. (d) Which was better for booth owner, the modified charging system or the actual charging system? (ii) The charge of ISD Telephone Booth, Jomsom are given below: ISD Telephone Booth, Jomshom Call London Call USA Minimum Charge Rs. 45 for a minute Minimum Charge Rs. 25 for a minute Additional Charge Rs. 4.25 per 15 seconds Additional Charge Rs. 2.5 per 10 seconds (a) If a person needs to call Paris from Jomshon which call he need to make? (i) Local (ii) STD (iii) ISD (iv) none of them (b) If a man made the telephone call for 8 minutes 45 seconds, then he made another call for 10 minutes and 30 second to Washington DC, how much amount did he pay altogether? (c) Which call was cheaper for him by how much? Compare them. 130 Allied The Leading Mathematics-9 Home Arithmetic 131

Arithmetic Arithmetic (iii) The charge of ISD call for USA from Pokhara is Rs. 12 for the first minute and Rs. 2 for every additional 15 seconds. A boy made the telephone call and paid Rs. 24 for the bill. (a) Which of them is ISD call? (i) Pokhara−Pokhara (ii) Pokhara−Kathmandu (iii) Pokhara−New York (iv) Pokhara−Dhunche (b) How long did the boy made the call? (c) If the rate was modified as Rs. 24 for first two minutes and Rs. 1.5 for every additional 15 second, then find the difference. (d) Which was best for the boy? Give reason. 6. (i) The minimum charge for 175 local telephone calls is Rs. 200 and Re. 1 is charged for each extra call. (a) How many calls do we need to maintain if we want to pay only Rs. 200 each month? (b) If a person makes 738 calls in a month of Shrawan, then find out the sub-total charge of his local telephone call. (c) If the sub-total charge of Bhadra is Rs. 763, how many calls were made? (d) Compare the calls made by him in two different months mentioned above. (ii) The minimum charge of telephone call up to 175 calls is Rs. 200 and the charge of each additional call is Rs. 1.5. If a costumer paid Rs. 747.50 in a month, find the number of additional calls in the month. (a) Write down the minimum charge of telephone call up to 100 calls. (b) If a costumer paidRs. 747.50 in a month, find the number of additional callsin the month. (c) Later, the owner realized that the actual rate was Rs. 1 for each additional call, then find the amount that the costumer get back. (d) What is the percentage of amount that he get back to amount he paid at first? (iii) The minimum charge for the first 175 telephone calls is Rs. 200 and then the additional charge per call is Re. 1. 10% TSC and 13% VAT are included in the bill. (a) Write down the formula to calculate the total charge of local telephone call if subtotal, TSC% and VAT% are. (b) Using the formula find how many calls can be used by Rs. 1602.227. (c) Find out the TSC and VAT amount separately. (d) Compare TSC and VAT in percentage. (iv) Aman pays Rs. 423.50 in a month for local telephone calls. The charges are indicated below: Minimum rental charge (with 100 free calls) = Rs. 150, Each extra call = Re. 1, Telephone service charge = 10%, (imposed on the telephone call charges ) and VAT = 13% (imposed on the telephone call charges with telephone service charges). (a) If the total calls are limited to 86 calls only, how much charge the man had to pay? (b) Find the total number of local calls made by that man. 130 Allied The Leading Mathematics-9 Home Arithmetic 131

Arithmetic Arithmetic (c) Estimate the number of calls he needs to reduce if he wants to pay only minimum rental charge? 7. (i) Theminimumcharge forthefirst 175 telephone callsRs. 200 and then additional charge per call is Re. 1. 10% TSC and 13% VAT are included in the bill. (Rules of payment: Within 7th day according to the bill. After 7th day 10% late fee. (a) What percent of late fee is charged by NTC if the payment was done after 7th day of billing? (b) How much should pay for 1368 calls with TSC and VAT? (c) If the bill is to be paid within 15th of the billing, find the amount of late fee according to rule of payment. (d) If the bill was paid on 5th day, what amount would be saved by the man? (ii) The current calls and previous calls of the local telephone of a company in a month are 926 calls and 682 calls respectively. (Minimum charge for first 175 calls = Rs. 200, Rate of additional call = Re. 1) (a) On which amount will be the TSC imposed? (b) Find the payment amount with 10% TSC and 13% VAT. (c) Find the total bill of the household if 10% fine is added when the bill is to be paid at the end of second month after the billing month. (d) Compare TSC amount, VAT amount and late fee amount. 3. (i) (b) Rs. 817 (c) Rs. 1015.53 (d) 43% (ii) (b) 1380 units (c) Rs. 1746.42 (d) 43% 4. (i) (b) Rs. 2393 (c) Rs. 2974.50 (d) 43% (ii) (b) 1276 calls (c) Rs. 1617.14 (d) Rs. 55.94 more VAT 5. (i) (b) Rs. 48 (c) Rs. 41 (d) modified changing system (ii) (b) Rs. 310 (c) Rs. 25 (iii) (b) 2 mins 30 secs (c) Rs. 3 (d) 1st 6. (i) (a) 175 calls (b) Rs. 763 (c) 738 calls (d) No (ii) (b) 540 calls (c) Rs. 182.50 (d) 24.41% (iii) (b) 1264 calls (c) Rs. 128.90, Rs. 184.33 (d) 43% more (iv) (b) 290 calls (c) 190 calls 7. (i) (b) Rs. 1731.50 (c) Rs. 1904.65 (d) Rs. 173.15 (ii) (a) sub-total (b) Rs. 334.37 (c) Rs. 367.80 (d) Rs. 26.90, Rs. 38.47, Rs. 33.44 Answers Project Work Study the local telephone bill of your house or neighboursfrom the last 6 months. What difference do you find in these telephone bills? Compare their calls, grand total and payable amount. 132 Allied The Leading Mathematics-9 Home Arithmetic 133

Arithmetic Arithmetic 4.4 Calculation of Taxi Fare At the end of this topic, the students will be able to: ¾ calculate the taxi fare and solve the related problems. Learning Objectives Uses of Taxi in Our Daily Activities Which type of public vehicle do we use in the city while we have emergency to go anywhere if there is no different types of public buses or trains? How do we pay the taxi fare when we take public taxi ? Are the rates of taxi fares the same or different at the day or night ? If we make wait the taxi few minutes on the road for taking something on the footpath, how does its charge pay? Introduction to Taxi Fare Taxi is a type of vehicle for hire with a driver, used by a single passenger or small group of passengers, often for a non-shared ride. A taxi conveys passengers between locations of their choice. This differs from other modes of public transport where the pick-up and drop-off locations are decided by the service provider, not by the passenger. The taxi has a meter which shows and records of the distance travelled and payable amount. Nepal Bureau of Standards and metrology (NBSM) has the responsibility to implement the rules, regulations and the rates of fare and monitoring of meter taxi as follows: Time Minimum fare Fare of 200 meters Waiting charge Day time (6 am to 9 pm) Rs. 60 Rs. 9.80 Rs. 9.80 per 2 minutes Night time (9 pm ot 6 am) Rs. 90 Rs. 14.70 Rs. 14.70 per 2 minutes The information about the taxi fare is given on the hoarding board of passenger's junctions like airpot, buspark, main chowks, etc. In the day time, when the passengersits on the taxi and the taxi driverswitches on the taxi meter then the monitor of the taxi meter shows immediately Rs. 14 and then the fare increases at the rate of Rs. 7.40 per 200 meters. The taxi fare can be checked by the android mobile sets. 132 Allied The Leading Mathematics-9 Home Arithmetic 133

Arithmetic Arithmetic Distance→ 200 m 200 m 200 m Rs. 60 Rs. 60 + Rs. 9.80 = Rs. 69.80 Rs. 69.80 + Rs. 9.80 = Rs. 79.60 Rs. 79.60 + Rs. 9.80 = Rs. 89.40 Important Formulae on Taxi Fare Taxi fare = Minimum fare + Travelled distance × Rate of fare per 200 m Taxi fare = Minimum fare + Travelled distance × Rate of fare per km + Waiting time × Rate of waiting charge per 2 minutes Points to be Remembered Public Transportation: A system of transport for passengers by group travel by the general public typically managed on a schedule, operated on established routes, and that charge a posted fee for each trip is called public transportation. Taxi: A type of public vehicle including meter for hire with a driver, used by a single passenger or small group of passengers, often for a non-shared ride. Taxi Meter: A device used in taxis that automatically records the distance travelled and the fare payable, is called taxi meter. Fare of Taxi: The money paid by a passenger to the taxi driver for a journey in a metered taxi, is called taxi fare. It is a contribution to the operational costs of the transport system. Public transportation fares are organized under various kinds of fare structures which price the service based on criteria such as distance traveled, demand for the service, and time of day. Example-1 A passenger hired a metered taxi and travelled 9.75 km from Balaju to Jawalakhel. The minimum fare is Rs. 60 and the fare goes on at the rate of Rs. 9.80 per 200 meters. (a) What is the taxi fare when the minimum fare, rate of fare per km and travelled distance are given ? (b) Jawalakhel is 4.8 km far from Purano Bus Park. How much taxi fare should pay at the rate of Rs. 100 per km? (c) How much amount does the meter show for the above situation? Solution: (a) When the minimum fare, rate of fare per km and travelled distance are given, the taxi fare is equal to (Minimum fare + Travelled distance × Rate of fare per km) Here, Minimum fare = Rs. 60 134 Allied The Leading Mathematics-9 Home Arithmetic 135

Arithmetic Arithmetic (b) The required taxi fare for travelling from Jawalakhel to Purano Bus Park of the distance 4.8 km ar Rs. 100 per km is Rs. 100 × 4.8 = Rs. 480. (c) Total travelled distance from Balaju to Jawalakhel = 9.75 km = 9750 m Rate per 200 meters = Rs. 9.80 ∴ Taxi fare for 12800 m = Rs. 7.40 200 × 9750 m = Rs. 360.75 ∴ Total fare for taxi = Rs. 14 + Rs. 360.75 = Rs. 374.75 Thus, the meter of taxi shows the amount Rs. 374.75. Example-2 In the meter of a taxi, the initial amount is Rs. 14 and then shows Rs. 7.40 per 200 meters. How long can a passenger travel with Rs. 282.62 from the taxi? Solution: Here, The initial amount shown in the meter of the taxi = Rs. 14 The total amount shown in the meter of the taxi = Rs. 282.62 ∴ The amount only for travelling distance = Rs. 282.62 – Rs. 14 = Rs. 268.62 Again, the taxi fare of 200 m = Rs. 7.40 i.e, Rs. 7.40 is the taxi fare of 200 m or, Re. 1 is the taxi fare of = 200 7.40 m ∴ Rs. 268.62 is the taxi fare of = 200 × 268.62 7.40 m = 7260 m = 7260 1000 km = 7.260 km Thus, the passenger can travel 7.260 km with Rs. 282.62 from the taxi. Example-3 Example-3 A girl hired a taxi and travelled a certain distance at 11:20 pm. He paid the total fare of Rs. 814.65 including waiting charge of 18 minutes. Find the distance travelled by the girl. (Minimum charge = Rs. 21, Rate of fare of 200 meter = Rs. 11.10, Waiting charge for 2 minutes = Rs. 11.10) Solution: Here, Total fare of taxi = Rs. 814.65 Minimum charge at night = Rs. 21 Waiting charge of 2 minutes = Rs. 11.10 ∴ Waiting charge of 18 minutes = Rs. 11.10 2 × 18 = Rs. 99.90 ∴ Fare for only travelled distance = Rs. 814.65 – Rs. 21 – Rs. 99.90 = Rs. 693.75 Rate of fare of 200 meters = Rs. 11.10 i.e, Rs. 11.10 is the fare of 200 m or, Re. 1 is the fare of = 200 11.10 m ∴ Rs. 693.75 is the fare of = 200 × 693.75 11.10 m = 12500 m = 12500 1000 km = 12.5 km Thus, the distance travelled by the girl is 12.5 km. 134 Allied The Leading Mathematics-9 Home Arithmetic 135

Arithmetic Arithmetic Example-4 A man hired a metered taxi and travelled from Lagankhel to Maharajgunj. If the taxi driver received Rs. 1219 waiting with 24 minutes and 15% tips on the total fare, find the distance between these two places. (Minimum taxi fare = Rs. 200, fare per 200 m = Rs. 15, waiting charge per hr = Rs. 350) Solution: Suppose, the distance from Lagankhel to Maharajgunj = x km Since, Minimum taxi fare = Rs. 200, Fare per 200 m = Rs. 15 ∴ Fare per 200 × 5 m (km) = Rs. 15 × 5 = Rs. 75 Waiting charge per hr (60 minutes) = Rs. 350 ∴ Waiting charge per minutes = Rs. 350 60 = Rs. 35 6 Waiting time = 24 minutes So, Taxi fare without tips = Minimum fare + Travelled distance × Rate of fare per km + Waiting time × Rate of waiting charge per minute = Rs. 200 + x × Rs. 75 + 24 × Rs. 35 6 = Rs. 200 + Rs. 75x + Rs. 140 = Rs. 340 + Rs. 75x ∴ Taxi fare with tips = (Taxi fare without tips) + Tips % of (Taxi fare without tips) = (Rs. 340 + Rs. 75x) + 15% × (Rs. 340 + Rs. 75x) = (Rs. 340 + Rs. 75x) + 15 100 × (Rs. 340 + Rs. 75x) = Rs. 34000 + Rs. 7500x + Rs. 5100 + Rs. 1125x 100 = Rs. 39100 + Rs. 8625x 100 By given, Taxi fare with tips = Rs. 1219 ∴ Rs. 39100 + Rs. 8625x 100 = Rs. 1219 or, Rs. 39100 + Rs. 8625x = Rs. 121900 or, Rs. 8625x = Rs. 121900 - Rs. 39100 or, x = Rs. 82800 Rs. 8625 = 9.6 km Hence, the distance between Lagankhel and Maharajgunj is 9.6 km. PRACTICE 4.4 Read Think Understand Do Keeping Skill Sharp 1. (a) What type of device is called taxi meter? (b) How can you find the taxi fare when the minimum fare, rate of fare per 200m an travelled distance are given ? (c) What is the formula to find the rate of taxi fare? Write it. 136 Allied The Leading Mathematics-9 Home Arithmetic 137

Arithmetic Arithmetic 2. (a) What is the taxi fare to travel 8.4 km at the rate of Rs. 125 per km? (b) Kirtipur is 6.2 km far from the Bhaisepati. How much the taxi fare should be paid at the rate of Rs. 25 per 200 m? (c) Deira City Centre, Dubai is 140.25 km far from Grand Sheikh Mosque, Abu Dhabi. How much taxi fare is paid at the rate of Rs. 244 per km? Check Your Performance 3. (i) A meter taxi shows Rs. 60 as the initial fare and Rs. 49 in each km. (a) What is the formula to calculate the taxi fare if initial fare, travelled distance and the rate of fare per km are given? (b) How much amount is paid by a passenger for travelling in the taxi for 20 km? (c) How much more money did he need to travel 4 additional km? (ii) Mr. Gyan hired a metered taxi and travelled 8.6 km. If the minimum fare is Rs. 200 and the fare goes on at the rate of Rs. 15 per 200 m, how much taxi fare is required to him? (a) What is the formula to calculate the taxi fare if initial fare, travelled distance in km and the rate of fare per 200 m are given? (b) How much amount is paid by Mr. Gyan as a taxi fare? (c) How much more distance can he travel with extra Rs. 75? (iii) Mr. Ram Ghimire travelled Purano Bus Park, Ratnapark to Naya Bus Park, Gongabu by a metered taxi at 11 pm. Find the total taxi fare paid by Mr. Ram Ghimire on the basis of the following information: Distance between Purano Bus Park, Ratnapark and Naya Bus Park, Gongabu = 6.2 km, minimum fare (from 9pm to 6am) = Rs. 90, fare for each 200 m = Rs. 14.70. (a) What is the minimum fare of taxi from 9pm to 6am? (b) How much money did Mr Ram Ghimire pay as taxi fare? (c) He had returned from Gongabu to Ratnapark at 8am. Then did he paid the same fare? Give reason? 4. (i) Daya travelled from Balaju to Kakani at 8 am by a metered taxi. The metershowed the fare of Rs. 432.40. (Minimum taxi fare (from 6am to 9pm)= Rs. 16, fare per 200 m = Rs. 9.80) (a) What is the minimum fare of taxi from 6am to 9pm? (b) Find the distance of Balaju to Kakani? (c) He had returned from Kakanni to Balaju at 8pm. Then did he paid the same fare? Give reason? (ii) Miss Sun Keshari paid Rs. 839.70 to the metered taxi driver by travelling some distance at 9:10 pm. (Minimum taxi fare = Rs. 90, fare per 200 m = Rs. 14.70) (a) What is the per 200m fare at night time? (b) Find the travelled distance by her. (c) If the taxi got punctured before 2km from her destination and she left the taxi, how much money would be less? 136 Allied The Leading Mathematics-9 Home Arithmetic 137

Arithmetic Arithmetic 5. (i) The taxi meter shows Rs. 60 at first and then charges Rs. 9.80 every 200 meters and waiting charge for 2 minutes is Rs. 9.80. (a) What is the rate of waiting charge of taxi? (b) A girl travelled 15.4 km, calculate the fare. (c) If the girl had waited for 16 minutes on the travelling and gave one thousand rupee note, how much should she get back? (ii) A tourist travelled 28.6 km distance using metered taxi with the rate of Rs. 11.10 per 200 m plus minimum charge at the start Rs. 21 and Rs. 11.10 per 2 minutes waiting where total waiting time is 32 minutes. (a) Write down the formula to calculate the taxi fare when minimum charge, rate of travel, travelled distance, waiting time and rate of waiting charge are given? (b) What fare should the tourist had to pay? (c) If taxi had not stopped anywhere, how much money would be saved? (d) If he has given one thousand note, how much should he add to? 6. (i) Mr. Kiran hired a metered taxi and travelled some distance at 8:00 pm. (Minimum taxi fare = Rs. 90, fare per 200 m = Rs. 15.20, waiting charge for each 2 minutes = Rs. 15.20) (a) Write down the formula to calculate the taxi fare when minimum charge, rate of travel and travelled distance are given. (b) If he paid the total fare of Rs. 804.40 including waiting charge of 8 minutes, calculate the travelled distance by him. (c) If taxi had not stopped anywhere, how far could he travel with the same money? (ii) Mrs. Shila travelled some distance by a metered taxi at 12:15 pm. She spent 28 minutes at shopping mall during her travel. If the taximeter showed the fare of Rs. 443.20, find the distance travelled by her. (Minimum taxi fare = Rs. 100, fare per 200 m = Rs. 16.50, waiting charge for each 2 min. = Rs. 16.50) (a) Which algorithm does a taxi meter use to calculate the taxi fare when minimum charge, rate of travel, travelled distance, waiting time and rate of waiting charge are given? (b) If the taxi meter shows the total fare of Rs. 727 including waiting charge of 28 minutes, calculate the travelled distance by her. (c) If Mrs Shila ignored shopping, how far could she travel with the same money? 7. (i) The taximeter shows Rs. 14 at first and then charges Rs. 7.40 every 200 meters and waiting charge for 2 minutes is Rs. 7.40. A tourist waited for 28 minutes on the travelling. (a) Define taxi fare. (b) Find the fare if the tourist travelled for 26.2 km. (c) If he adds 10% tips of the fare, what amount will the driver get in total? 138 Allied The Leading Mathematics-9 Home Arithmetic 139

Arithmetic Arithmetic (ii) An American tourist travelled from Tribhuwan International Airport to Thamel by the metered taxi with the rate of Rs. 10 per 200 metres plus minimum charge at the start Rs. 50 and Rs. 10 per 2 minutes waiting where total waiting time is 12 minutes. (a) Write down the rate of waiting charge of taxi. (b) Find the fare with 15% tips if the taxi travelled through the blue line route. (c) The taxi took long route (grey line). Tourist realized that he is being cheated by driver, so he refused to give tips and paid as per meter. How much did he save or loss? 8. (i) A man hired a metered taxi and travelled from Prithvi Chowk to Hemja of Pokhara. (Minimum taxi fare = Rs. 50, fare per 200 m = Rs. 10, waiting charge for each 2 minutes = Rs. 10) (a) Write down the rate of distance travel by taxi. (b) If the fare with 30 minutes waiting charge and 20% tipsis Rs. 924, find the distance between Prithvi Chowk to Hemja. (c) If the driver had not get any tips, how much less amount will he get? (ii) A tourist travelled from Pokhara to Lumbini at 7:15 am by the metered taxi with the rate of Rs. 20 per 200 metres plus minimum charge at the start Rs. 100 and Rs. 20 per 2 minutes waiting where the taxi had waited 1 hour at hotel and 20 minutes at cafeteria. He gave Rs. 34892 to the taxi driver after arrived Lumbini by adding 30% tips. (a) Write down the initial charge of taxi fare. (b) How far is Lumbini from Pokhara ? (c) If the taxi ran non-stop, how much less amount will he get? 3. (i) (b) Rs. 1040 (c) Rs. 196 (ii) (b) Rs. 845 (c) 5 km (iii) (b) Rs. 545.70 (c) No 4. (i) (b) 7.6 km (c) No (ii) (b) 10.2 km (c) Rs. 147 5. (i) (b) Rs. 814.60 (c) Rs. 107 (ii) (b) Rs. 1785.90 (c) Rs. 177.60 (d) Rs. 785.90 6. (i) (b) 8.6 km (c) 800 m (ii) (b) 4.8 km (c) 2.8 km 7. (i) (b) Rs. 1087 (c) Rs. 1195.70 (ii) (b) Rs. 540.50 (c) Loss Rs. 70.50 8. (i) (b) 11.4 km (c) Rs. 154 (ii) (b) 259.4 km (c) Rs. 800 Answers Project Work Collect the electricity bill, drinking water bill, local telephone bill of your home or your neighbourhood. If you pay these bills in related office just after 3 months by taking meter taxi travelling 10.6 km distance in total and 32 minutes waiting it, calculate the total expense altogether according as the present rules of the related office. 138 Allied The Leading Mathematics-9 Home Arithmetic 139

Arithmetic Arithmetic CONFIDENCE LEVEL TEST - II Unit II : Arithmetic Class: 9, The Leading Maths Time: 45 mins. FM: 23 Attempt all questions. 1. The monthly salary of a salesman of a Food Store is Rs. 18,000. If he/she sold the foods more than Rs. 5,00,000 in a month, he/she gets 2% commission. He sold the food of Rs. 8,23,000 in the month of Baisakh. (a) Find the amount of commissions in the month of Baisakh. [2] (b) Find his/her monthly income in the month of Baisakh. [1] (c) What amount of foods he/she sells per month to make a monthly income of Rs.32000 including commission in the month? Calculate it. [2] 2. After allowing 5% discount on the marked price of a radio, it became Rs. 1672. (a) Find the marked price. [1] (b) How much discount was given? [1] (c) If the price of the radio with VAT is Rs. 1889.36, find the VAT amount. [2] (d) Also, find the VAT percent leived on it. [1] 3. A group of young people who have returned from the foreign employment have opened an investment company with the idea to improve the economy of all rural people. The company has sold 11,00,000 total shares of Rs 100 per share for local residents by issuing ordinary share and started working by collecting capital. It has made a profit of Rs. 1,00,00,000 in the first year of operation. From the meeting of board of directors of the company has decided to distribute Rs. 60,00,000 as cash dividend to the shareholders. (a) From what amount is the cash dividend distributed? [1] (b) What percentage of cash dividend was distributed? [2] (c) If the company made a profit of Rs. 1,50,00,000, how much amount will Sanjibani get if she bought 15,000 shares at the same rate? [2] 4. An water refining industry has 3" water pipe line. The owner set the telephone for business purpose. He uses taxi for coming to office and returning home at the distance of 12 km. Answer the following question on the basis of the given rules: Taxi Fare Rule 3" pipe line monthly charge Telephone minimum charge Rs. 60 and Rs. 49 per km Minimum usage (in liter) 1 unit = 1000 liter Minimum Charge per unit cost for access usage Minimum charge Rs. 175 for 200 calls and Rs. 1 per 8,81,000 liter Rs. 62,240 Rs.71 call for access call (a) How much taxi fare should he pay only for going to office from home? [1] (b) If 15,00,000 liter water is used in a month then how much should the industry pay as the drinking water charge? [2] (c) What is the maximum call that can be perform if the bill amount should limit at most Rs. 1000? [2] Best of Luck 140 Allied The Leading Mathematics-9 Home Arithmetic 141

Arithmetic Arithmetic Additional Practice – II 1. Ramesh is an unmarried secondary level teacher. His monthly salary is Rs. 43689. He gets Rs. 2000 per month dearness allowance and once a year one-month salary for festival expenses. A total of the 10% salary of each month is deposited in the employee's provident fund and the same amount is deposited by the government in the same fund. Rate of tax applicable to unmarried persons for the fiscal year 2078/79. Income Rate of tax Income Rate of tax Up to 5 lakhs 1 percentage 10- 20 lakhs up to 30 percentage 5 - 7 lakhs up to 10 percentage more than 20 lakhs 36 percentage 7 - 10 lakhs up to 20 percentage (a) What do you mean by income tax? write it. (b) Find the income of Ramesh in 13 months including festival expenses and EPF. (c) Find out how much income tax he has to pay annually if he also deposits Rs 2000 per month in citizen investment fund from his income. 2. The commission rate given by a company to an agent is mentioned below. Selling Goods Commission Rate Selling Goods Commission Rate Up to 15 lakhs 0.5% 25 lakhs - 54 lakhs 1.5% 15 lakhs - 25 lakhs 1% Above 54 lakhs 2% (a) Write the formula for finding rate of commission. (b) If an agent is selling goods worth Rs.28 lakhs, what amount does he get as commission? (c) If the amount of commission is Rs 25000, calculate the minimum value of goods to be sold. (d) The monthly salary of an agent is Rs. 15000 and he gets commission more than 5 lakhs book sold at 2%, what amounts of goods to be sold for being monthly income Rs. 19000? 3. A man paid Rs. 15,750 for a palmtop with 12.5% VAT. (a) Find the listed price of the palmtop. (b) If 15% discount is given, find the price with 13% VAT. 4. A cost of an article with 15% discount and 10% VAT is Rs. 5610. (a) Find the marked price. (b) Find the difference between discount and VAT. (c) If the marked price is 20% above the cost price, find CP. 140 Allied The Leading Mathematics-9 Home Arithmetic 141

Arithmetic Arithmetic 5. Mr. Lichwa bought a cycle of Rs. 35,000 with 12% discount and some percent of VAT at Rs. 34,804. (a) Find VAT percentage. (b) Find the actual discount percentage in the transaction for the customer. (c) If the discount is double percent of VAT, find the price of the cycle with the same VAT rate. 6. The marked price of a jacket is Rs. 5000. If the discount and VAT rate are the same and is sold at Rs. 4950. Find (a) Rate of VAT. (b) Price of the jacket without bill. 7. A businessman imports EV car from Indra at Rs. 8,00,000 without 13% VAT. He pays 150% tax at Bhairahawa custom office and Rs. 15,000 for transportution to Kathmandu. The businessman sells the EV car to Laxmi Motors at a profit of Rs. 50,000. And Laxmi Motor spend Rs. 15000 for showroom rent, 20,000 for Metropolitan city and made 30,000 profit on it. If a customer buys the EV car, (a) Find the cost of the EV car. (b) What is the total VAT amount? (c) Find the VAT paid by Laxmi motors. 8. Shrestha Carpet Udhog sold a carpet of amount Rs. 75,000 to a dealer with 20% profit and adding 13% VAT. The dealer paid 1% local tax, Rs. 2,000 for transportation and 10% profit is added on it. If the rate of VAT is 13%, find (a) VAT paid by the customer. (b) Cost of the carpet. (c) How much percent is more than actual price for customer? 9. Swikriti sold the photocopy machine to Hari for Rs. 92095 with 13 % VAT. Hari paid Rs. 2000 as transportation cost, Rs. 500 as local tax and Rs 5000 as a profit and delivered it to Shyam's house. (a) Write the formula for finding Value Added Tax (VAT). (b) How much did Hari pay for that machine without VAT? Find it. (c) How much did the machine cost when it reached Shyam's house? Find it. 10. There are 22,500 shares at the rate of Rs. 100 in Sanjeewani Co-operative. The co-operative earned Rs. 15,00,000 in a year and decided to distribute 15% cash dividend of total profit to the shareholders. Sunil is a member and he has 840 shares. (a) Find the cash dividend amount. (b) How much cash dividend amount will Sunil get? (c) If half of the profit amount is distributed as cash dividend to the shareholders, how much additional cash dividend amount would Sunil get? 142 Allied The Leading Mathematics-9 Home Arithmetic 143

Arithmetic Arithmetic 11. In a water processing industry, water comes from a 1 2 inch pipe. To do water business, a person has also connected a telephone set for business communication. He uses a taxi to come and go to the water industry which is 20 km away from his house. Answer the following questions based on the rules given below Calculation of taxi meter Rate of per month for 1 2 '' water pipe Telephone Starting taxi fare is Rs. 60 after that additional charge per km. is Rs, 49 minimum use in liter minimum maintained price The additional charge of per 1 unit water when its use is more than the minimum maintained The minimum charge of first 175 calls is Rs. 200 and then additional charge Rs. 1 per call 10,000 liter (1 unit = 1000 liter) Rs. 100 Rs. 32 (a) How many rupees does the businessman have to pay for using a taxi every day just to go to the water processing industry from home? (b) How much rupees should the company pay for the water tariff, which uses 85,000 liters of water in a month? (c) How many maximum call can he do in one month without increasing the telephone expenses from Rs. 1000? Mention with calculation. 12. Ramesh has been travelling in a taxi from Kalanki to Koteshwor at 9:30 am. The distance between Kalanki and Koteshwor is 10.7 km. The taxi charges are as follows: 6 AM to 9 PM 9 PM to 6 AM Starting rate = Rs. 50 Starting rate = Rs. 75 The rate per 200 meters = Rs. 10 The rate per 200 meters = Rs. 15 (a) How much amount should be paid for travelling only1 kilometer? (b) How much total cost does Ramesh pay for travelling 10.7 km in a taxi at night time? Calculate and mention. (c) Ramesh gave Rs. 1000 to the driver, how much amount does driver return back to Ramesh? Calculate and mention. 13. The Electricity Consumer Tariff rate for 5 amperes on date 2080-07-25 by Nepal Electricity Authority is as follows. Consumed Unit 5 Ampere Service Charge (Rs.) Energy Charge (Rs.) 0 - 20 30 3.00 21 - 30 50 6.50 31 - 50 50 8.00 51 - 100 75 9.50 101-250 100 9.50 142 Allied The Leading Mathematics-9 Home Arithmetic 143

Arithmetic Arithmetic A 5 ampere meter has been installed in Ramesh's house. The electricity bill of the house for the month of Kartik shown as follows. Previous RDG 4283 Present RDG 4523 (a) How much electricity consumption in the month of Kartik? (b) How much rupees should he pay in a month of Kartik? (c) What is the maximum number of units that he can consume to reduce the tariff amount to Rs, 1501 for the month of mangsir? Calculate and mention with reason. 14. A pen drive was sold at a discount of the 10%. If 13% VAT was added to the price and the customer got Rs. 600 as discount. (a) How much was paid on VAT? (b) If VAT amount is the same as discount find the VAT%. 15. A tourist paid Rs. 5610 for decorative window. If the shopkeeper has given 15% discount and 10% VAT was levied on it. (a) Find the marked price of the window. (b) Find the amount when he goes bact his to his country. (c) Also find the discount amount. 16. A telephone set has marked Rs. 2100. Including 20% discount and certain percent VAT added the price reached to Rs. 1848. Find, (a) VAT amount. (b) VAT percentage. 17. The discount rate is twice of VAT rate. If a watch is bought at Rs. 1672.40 of Rs. 2000, find (a) VAT rate. (b) Discount rate. (c) Actual discount amount and rate. 1. (b) Rs. 635426.80 (c) Rs. 5657.32 2. (b) Rs. 22000 (c) Rs. 30 lakhs (d) 7 lakhs 3. (a) Rs. 14,000 (b) 13447 4.(a) Rs. 6000 (b) Rs. 237 (c) Rs. 5000 5. (a) 13% (b) 0.56% (c) Rs. 29267 6. (a) 10% (b) Rs. 4500 7. (a) Rs. 24,06,900 (b) Rs. 2,76,900 (c) Rs. 2,68,450 8. (a) Rs. 13,453.53 (b) Rs. 1,16,942.23 (c) 56% 9. (b) Rs. 81500 (c) Rs. 100570 10. (a) Rs. 225000 (b) Rs. 8400 (c) Rs. 19600 11. (a) Rs. 1030 (b) Rs. 2500 (c) 975 calls 12. (a) Rs. 100 (b) Rs. 885 (c) Rs. 115 13.(a) 240 units (b) Rs. 2190 (c) 178 units 14. (a) Rs. 702 (b) 11% 15. (a) Rs. 5840.71 (b) Rs. 645.34 (c) Rs. 876.11 16. (a) Rs. 168 (b) 10% 17. (a) 13% (b) 26% (c) Rs. 327.60, 16% Answers 144 Allied The Leading Mathematics-9 Home Arithmetic PB

Mensuration Mensuration PB Allied The Leading Mathematics-9 Area 145 analysis and use of the knowledge, skill and concept related to surface area and volume of solid objects on the solution of behaviour problems. Estimated Working Hours : Competency Learning Outcomes At the end of this unit, the students will be able to: find the area of scalene triangle. find the area of four walls, floor and ceiling of the cubic and rectangular rooms. find the surface area and volume of prism. solve the problems related to the surface area and volume of cylindrical and spherical objects. solve the behaviour problems on finding cost. Lessons for Unit 5. Area 5.1 Area of Scalene Triangle 5.2 Surface Area of Room and Cost Estimation 6. Prism and Cylinder 6.1 Surface Area and Volume of Prism 6.2 Surface Area and Volume of Cylinder 7. Sphere 7.1 Surface Area and Volume of Sphere Chapters / Lessons MENSURATION UNIT 28 (Th. + Pr.) HSS Way: Mathematics begins at Home, grows in the Surroundings and takes shape in School. III Specification Grid Unit Areas Total working hour Knowledge Understanding Application Higher ability Total number of items Total number of questions Total Marks No. of items Marks No. of items Marks No. of item Marks No. of item Marks III Mensuration 28 2 2 2 3 2 5 2 3 8 3 13

Mensuration Mensuration CHAPTER 5 AREA Uses of Surface Area in Our Daily Activities A point and a line are two basic geometrical concepts. No measurement is made in the case of a point. A line is measured in terms of what is known as length. We shall be concerned with two kinds of measurements in the case of a plane geometrical figure: Area and Perimeter. What are perimeter and area of the plane geometric shaped figures? Discuss them. In the classroom, how can you find the area of your desk, board, door, window, walls, floor, etc ? How much price for making the board ? What are the shapes of the door and window? How can you find the area of the door and window? Discuss. What is the picture given above ? What are the shapes of land ? How can you find the area of land ? Discuss. What is the picture given above ? How can you find the cost to maintain the paths around the ground ? Discuss. Now, we give the definitions of some of the standard plane figures. The formulae for the perimeter and area should be well known. Remember the formulae of plane surfaces that are studied in the class 8, below: SN Name Definition Properties Figure Perimeter Area 1. Rectangle Equiangular quadrilateral Opposite sides and diagonals are equal, diagonals bisect and every angle is 90o . b l P = 2(l + b) A = l × b 2. Square Rectangle with equal adjacent sides All sides & diagonals are equal, diagonals bisect perpendicularly and every angle is 90o . l P = 4l A = l2 3. Parallelogram Quadrilateral parallel opposite sides Opposite sides and opposite angles are equal and diagonals bisect. a a b b h P = 2(a + b) A = b × h 146 Allied The Leading Mathematics-9 Area 147

Mensuration Mensuration 4. Rhombus Equisided quadrilateral All sides and opposite angles are equal, diagonals bisect perpendicularly. a a a a d1 d2 P = 4l A = 1 2 d1d2 5. Quadrlateral Closed plane figure of four segments There are four sides, vertices and angles and diagonals interest, sum of interior and exterior angles are 360o respectively. a b h1 c h2 e d A = a + b + c + d A = 1 2 d (h1 + h2) 6. Trapezium Quadrilateral with parallel one pair of opposite sides Bases are parallel, No sides, angles and diagonals are equal. But, in isosceles trapezium, legs and base angles are equal, median is half of the sum of bases. a b1 b2 c h P = a + b1 + b2 + c A = 1 2 h (b1 + b2) 7. Kite Convex quadrilateral with equal opposite adjacent sides (Consecutive sides) Vertical diagonal bisects the angles, horizontal opposite angles are equal, vertical diagonal perpendicularly bisects the horizontal diagonal. a a b b d2 d1 P = 2(a + b) A = 1 2 d1d2 8. Arror-Head Concave quadrilateral with equal opposite adjacent sides Vertical diagonal bisects the angles, horizontal opposite angles are equal, vertical diagonal when produced perpendicularly bisects the horizontal diagonal. P = 2(a + b) A = 1 2 d1d2 9. Shaded Region A part of region that is filled with colour Obtained by subtracting the nonshaded region from a whole region r A = A1 – A2, A1 = Area of whole region A2 = Area of non-shaded region 146 Allied The Leading Mathematics-9 Area 147

Mensuration Mensuration 10. Right Triangle Triangle with right angle Sum of two acute angles is 180o , sum of squares of legs is equal to square of hypotenuse (Pythagoras' Theorem p2 + b2 = h2 ). b p h P = p + b + h = p + b + p2 + b2 A = 1 2 pb, h = p2 + b2 11. Isosceles Right Triangle Right triangle with equal legs Two legs are equal, acute/base angles are equal to 45o . a a h P = 2a + h A = 1 2 a2 12. Triangle Closed plane figure of three line segments Sum of length of any two sides is longer than the third side, sum of angles is 180o . b a h c P = a + b + c A = 1 2 bh 14. Isosceles Triangle Triangle with equal any two sides Base angles are equal, median perpendicularly bisects the base and vertical angle, perpendicular of the base passes through the vertex & perpendicular to base. b a a P = 2a + b A = b 4 4a2 – b2 15. Equilateral Triangle Triangle with equal sides All side angles are equal and every angle is equal to 45o . a a a P = 3a A = 3 4 a2 16. Circle Curved line equidistant from a fixed point (centre) Radii are equal, the length of diameter is twice of radius, the ratio of circumference and diameter is π. r P = 2r(π + 1) C = 2πr, d = 2r A = πr2 17. SemiCircle Half of circle divided by diagonal Each diameter bisects the circle, the ratio of half circumference and diameter is π 2. r P = r(π + 2) A = 1 2 πr 2 = πd2 4 = C2 4π 18. Sector A part of circle divided by any two radii Area of sector depends up on the arc length, the central angle is positive. r q P = 2πr × q 360o A = πr 2 × q 360o 148 Allied The Leading Mathematics-9 Area 149

Mensuration Mensuration 5.1 Area of Scalene Triangle At the end of this topic, the students will be able to: ¾ find the area of scalene triangle. ¾ find the area of plane surface or triangular and quadrilateral land by using the relation of square centimeter and square meter and local unts of area (bigaaha, kattha, dhur, ropani, aana, paisa and daam). Learning Objectives Uses of Surface Area in Our Daily Activities Let's discuss the pictures below; What is the name of the picture given above? How can you find the cost of planting grass on it? For this, we can find the area of the triangle. Discuss. What is the area of the triangular traffic plate? How much cost does it need to paint? Discuss. Calculate the area to put jam on the bread? How can we find it ? What is the name of the given solid ? How many triangles are there? If its triangular faces are to be painted, how much does it cost ? Discuss. Let's derive the formula to find the area of scalene triangle with different sides a, b and c as below. This formula is called Heron’s formula. It is, Area of triangle = s(s – a) (s – b) (s – c), where s = a + b + c 2 and a, b, c are the three given sides of a triangle. How to derive the above formula (Heron’s Formula) Let ABC be a triangle with known sides a, b and c. Now, draw a perpendicular AD on BC. The length AD = h is the height of the triangle. From this construction, the given triangle is divided into two right-angled triangles ∆ADB and ∆ADC. Suppose DC = x and thus BD = a–x. Now, from ∆ADB and ∆ADC, we get h2 = c2 – (a – x)2 …… (i) and h2 = b2 – x2 …… (ii) From the eqns (i) into (ii), b2 – x2 = c2 – (a – x)2 A B D C x b h a – x a 148 Allied The Leading Mathematics-9 Area 149

Mensuration Mensuration or, c2 = b2 – x2 + (a – x)2 = b2 + (a – x)2 – x2 = b2 + (a – x + x) (a – x – x) = b2 + a(a – 2x) or, c2 = b2 + a2 – 2ax ∴ x = a2 + b2 – c2 2a ………. (iii) Substituting value of x from the eqn (iii) in the eqn (ii), we get h2 = b2 – a2 + b2 – c2 2a 2 = 22 a2 b2 – (a2 + b2 – c2 ) 2 4a2 = {2ab + (a2 + b2 – c2 )} {2ab – (a2 + b2 – c2 )} 4a2 = {(a + b)2 – c2 )} {c2 – (a – b)2 } 4a2 = (a + b + c) (a + b – c) (c + a – b) (c – a + b) 4a2 …………….(iv) Again, we have a + b +c = 2s ………................ (v) a + b – c = 2s – 2c = 2(s – c) ……. (vi) a + c – b = 2s – 2b = 2(s – b) ……. (vii) b + c – a = 2s – 2a = 2 (s – a) .…. (viii) Substituting the relations of the eqns (v – viii) into the eqn (iv) h2 = 2s . 2(s – c) . 2(s – b) . 2(s – a) 4a2 = 16 s(s – c)(s – b)(s – a) 4a2 ∴ h = 2 s(s – a) (s – b) (s – c) a Hence, the area of the triangle ABC is, ΔABC = 1 2 base × height = 1 2 a . 2 s(s – a) (s – b) (s – c) a ∴ Δ = s(s – a) (s – b) (s – c) This formula is known as Hero's formula for the area of a triangle. Remarks: (i) We put b = a and c = b in the Hero’s formula to compute the area of an isosceles triangle also. Then we obtain the area of an isosceles triangle with equal sides a and base b, ∆ = b 4 4a2 – b2 . (ii) We put b = c = a in the Hero’s formula to compute the area of an equilateral triangle also. Then we obtain the area of an equilateral triangle with sides a, ∆ = 3 4 a2 . Generally, the area of plane surfaces measuresin square meter, centimeter, millimeter or kilometer For isosceles triangle, replace b by a and c by b. For equilateral triangle, replace b & c by a. We obtain the relatrions: 150 Allied The Leading Mathematics-9 Area 151