Acme Mathematics 8

Acme Mathematics 8 516. There are 34201 books in library A and 10011 books in library B.(a) Which library used number of books of only binary digits?(b) Which library used number of books of only quinary digits?(c) Convert 342015 into decimal number system.7. (a) Convert 15 into binary number system.(b) Convert 11012 into decimal number system.(c) Write in expanded form of 346.(d) Convert 1112 into quinary number system.8. The distance between Tandi and Parsa is 5000 m.(a) Is this number rational? Why?(b) Give an example of irrational number.(c) Write this distance in scientific method.(d) Write 2.34 × 106 into simple decimal number.9. (a) Are the digits 0 and 1 used in binary number system?(b) Convert 45 into binary number system.(c) 12345 into binary number system.10. (a) In the fraction 34 , are 3 and 4 both integers? What type of number 34 is?(b) Convert 34 into decimal.(c) What is the nature of that decimal?(d) Write the conclusion from the above result of 'a' and 'c'.11. (a) Write the fraction 7510000 in scientific notation(b) What is the product of 7510000 and 0.005? Write it in scientific notation.(c) Using the results of 'b' simplify: 7510000 × 0.0052.5 × 10–612. Mr. Yadav is a student of class 8. He writes his monthly fee in the expanded form as:3 ×53 + 2 × 52 + 4 × 51 + 1 × 50(a) Write the short form of this quinary number.(b) Convert his quinary number into decimal number.(c) Convert this decimal number into binary number.

52 Acme Mathematics 813. 25and 24are two real numbers.(a) Which one is a rational number? Give reason.(b) Which one is an irrational number? Give reason.(c) Compare these two real numbers.14. (a) Show 335 in a number line.(b) Insert four rational numbers between 134 and 133 .(c) 3 and 12are irrational numbers. Then verify their product is an rational number.15. (a) Is –72 a rational or irrational number ? Give reason.(b) Convert 0.15 in to fraction.(c) Locate 5in the number line16. (a) the scientific notation of 0.000034 is ...............(i) 3.4 ×10–6 (ii) 3.4 × 105 (iii) 3.4 × 10–5 (iv) 3.4 × 106(b) Write in scientific notation of 85000.(c) Simplify and write in scientific notation.2.1 × 103 + 4.3 × 1031.6 × 10– 517. (a) Is 20 rational number ? Why?(b) Insert any three rational number between 5 and 7.18. Two numbers 5 and 43 are given,(a) Which one among them is a rational number? (b) Locate the rational number in a number line. (c) Prove that the product of 5and 6is an irrational number. 19. The weight of boy is 45 kg.(a) Convert his weight into milligram. (b) Express in scientific notation. (c) If he gains 5 kg weight in a year then express his new weight in scientific notation.

Acme Mathematics 8 5320. In quinary system, the digits used are 0, 1, 2, 3, & 4.(a) What is binary system? (b) Differentiate between binary and quinary system in terms of digits used. (c) Perform the addition of quinary number 32415 and 14235 by converting into decimal number and find the result in binary. 21. (a) Define rational number.(b) What is the relation between rational and irrational number? (c) Put any two rational numbers between 5 and 6. 22. (a) If 5and 7are an irrational numbers, show their product is also an irrational number.(b) Which types of decimals are irrational number? (c) Write in general from: 4.7 × 10–4 (d) Simplify: 2.4 × 105 + 1.5 × 1053.0 × 10223. (a) The value of 205 in denary number system is(i) 10 (ii) 20(iii) 30 (iv) 40(b) For what value of A, the binary number 1A012 represent 13? (c) Put any two rational numbers between 4 and 6.24. 9and 10are two real numbers.(a) Which one is a rational number? Give reason.(b) Which one is an irrational number? Give reason. (c) Compare these two real numbers. 25. From the given diagram, answer the following questions.(a) Convert in decimal. (b) Convert into binary number.(4321)526. The monthly salary of Manisha is Rs. 43689.(a) Write the amount in Scientific notation.(b) Express the number in quinary number system.(c) Convert 3.34 into fraction.

54 Acme Mathematics 8EvaluationTime: 84 minutes Full Marks: 351. (a) Write down the numbers used in quianary number System. [1](b) Express 43 in the power of 2. [1](c) Convert in decimal number system : 12345. [2](d) Show in place value : 1012 [1]2. The number 110102 only use two numerals 0 and 1. (a) What type of Number system is this? [1](b) Convert it into decimal number system. [2](c) Convert 145 into quinary number system. [2]3. A shopkeeper has kept the price of two notebooks A and B in Rs. as 1111112 and 1345 respectively. (a) What is the price of notebook A in decimal number? [2] (b) What is the price of notebook B in decimal number? [2] (c) Compare the price of notebook A and B. [1]4. The monthly salary of Arun Karki is 45000.(a) Write the amount in scientific notation. [1] (b) Express the number in quinary number system. [2] 5. The distance between Tandi and Parsa is 5000m. (a) Is this number rational ? Why? [1] (b) Give an eample of irrational number. [1] (c) Write this distance in scientific method. [1] (d) Write 2.34×106 into simple decimal number. [2] 6. In the followinig table, some numbers are given.2 5 25 0.25 0. 75(a) Write a rational number from the above table. [1] (b) Convert 0. 75 into a fraction. [2] (c) Show 2 in a number line and determine which whole number is close to itsvalue. [2]7. According to the National Census 2078, the population of Kathmandu district was 2041587. (a) Identify and write whether the number 2041587 is a rational or irrational number. [1](b) Write 2041587 in scientific notation. [1](c) Prove that: 2078 = 313032 [2]

Acme Mathematics 8 552.3 Ratio and ProportionA. RatioACTIVITYObserve the following table and discus the question given below.Goods CostPen Rs. 25A book Rs. 200A copy Rs. 100(a) How many times the cost of a copy is more than cost of pen?(b) Compare the cost of a book and a copy.Comparison of two quantities of the same kind is called 'ratio'.The ratio of two quantities a and b of the same kind (units) is written as a : b or ab and read as a is to b. It is unit less quantity and comparison between two quantities. For example, Binaya obtained 90 marks and Bidur obtained 45 marks in the same examination, we say that Binaya obtained marks just double of Bidur and Bidur obtained marks just half of Binaya. Therefore, the ratio of marks obtained by Binaya to that of Bidur is 9045 or 21 and we read it as 2 is to 1 and write as 2 :1. Similarly, the ratio of marks obtained by Bidur to that of Binaya is 4590 or 12 and we read it as 1 is to 2 and write as 1:2. In the ratio 2:1, 2 is called an antecedent and 1 is called the consequent.Solved ExamplesExample 1 : Find the ratio of 3 days to 3 weeks.Solution : 3 days to 3 weeks= 3 days 3 weeks = 3 days 3 × 7 days = 17 = 1 : 7∴ The required ratio is 1 : 7

56 Acme Mathematics 8Example 2 : Divide 30 mangoes among two boys Ramji and Shyamji in the ratio 2 : 3. Solution : Let Ramji get 2x and Shyamji get 3x mangoes. Then by the question, 2x + 3x = 30 or, 5x = 30 or, x = 305 or, x = 6 Hence, Ramji gets 2x = 2 × 6 = 12 mangoes. Shyamji gets 3x = 3 × 6 = 18 mangoes.Example 3 : Four people bought a printing press with sharing the investments in the ratio 2 : 3 : 4 : 5. If the press costs Rs. 1680000, what is the share of each person? Solution : Since given ratio is 2 : 3 : 4 : 5, let their shares be 2x, 3x, 4x and 5x. Then, by the question, 2x + 3x + 4x + 5x = 1680000or, 14x = 1680000or, x = 168000014or, x = 120000 Now, 2x = 2 × 120000 = 2,40,0003x = 3 × 120000 = 3,60,000 4x = 4 × 120000 = 4,80,000 5x = 5 × 120000 = 6,00,000∴ Required shares are Rs 2,40,000, Rs 3,60,000, Rs 4,80,000 and Rs 6,00,000. Example 4 : Two numbers are in the ratio 2:3. If 6 is subtracted to each number the new ratio becomes 17:27. Find the numbers. Solution : Let two numbers are 2x and 3x. Then according to question, 2x – 6 3x – 6 = 17 27or, 54x – 162 = 51x – 102or, 54x – 51x = 162 – 102 or, 3x = 60 or, x = 20

Acme Mathematics 8 57Now, 2x = 2 × 20 = 40, 3x = 3 × 20 = 60Hence, required two numbers are 40 and 60.Classwork1. Tick () the correct answer.(a) The ratio of 8 books to 20 books is .......(i) 2 : 5 (ii) 5 : 2 (iii) 4 : 5 (iv) 5 : 4(b) The ratio of the number of sides of a square to the number of edges of a cube is(i) 1 : 2 (ii) 3 : 2 (iii) 4 : 1 (iv) 1 : 3(c) A picture is 60 cm wide and 1.8 m long. The ratio of its width to the perimeter in lowest form is .......(i) 1 : 2 (ii) 1 : 3 (iii) 1 : 4 (iv) 1 : 8(d) Niranjan's annual income is Rs. 288000. His annual saving amount is Rs. 36000. The ratio of his savings to his expenditure is .......(i) 1 : 8 (ii) 1 : 7 (iii) 1 : 6 (iv) 1 : 5(e) Mathematics textbook for class VI has 320 pages. the chapter 'Symmetry\" runs from page 261 to page 272. The ratio of the numbers of pages of this chapter to the total number of pages of the book is .......(i) 11 : 320 (ii) 3 : 40 (iii) 3 : 80 (iv) 272 : 320(f) In a box, the ratio of red marbles to blue marbles is 7 : 4. Which of the following could be the total number of marbles in the box?(i) 18 (ii) 19 (iii) 21 (iv) 22(g) On a shelf, books with green cover and that with brown cover are in the ratio 2 : 3. If there are 18 books with green cover, then the number of books with brown cover is .......(i) 12 (ii) 24 (iii) 27 (iv) 36(h) The greatest ratio among the ratios 2 : 3, 5 : 8, 75 : 121 and 40 : 25 is .......(i) 2 : 3 (ii) 5 : 8 (iii) 75 : 121 (iv) 40 : 25(i) There are 'b' boys and 'g' girls in a class. the ratio of the numbers of boys to the total numbers of students in the class is .......(i) bb + g (ii) gb + g (iii) bg (iv) b + gb

58 Acme Mathematics 8(j) If a bus travels 160 km in 4 hours and a microbus travels 320 km in 5 hours at uniform speeds, then the ratio of the distance travelled by them in one hour is .......(i) 1 : 2 (ii) 4 : 5 (iii) 5 : 8 (iv) 8 : 5Exercise 2.71. Simplify the following ratios. (a) 3 : 6 (b) 5 : 20 (c) 8 : 16 (d) 4 : 12 (e) 4 : 10 (f) 7 : 21 (g) 3 : 9 (h) 3 : 15 (i) 42 : 49 (j) 25 : 30 2. Simplify the following ratios: (a) 300 cm : 100 cm (b) 500 cm : 1m (c) 40 g : 1 kg (d) Rs 1.50 : 50 p (e) 6 kg : 60 g (f) 2 km : 500 m(g) 4 km : 800 m (h) 100 g : 0.5 kg (i) 30 p : 90 p(j) 75 p : 25 p 3. Simplify these ratios:(a) 34 : 13 (b) 15 : 110 (c) 12 : 38 (d) 16 : 112 (e) 2 214 : 14(f) 145 : 310 (g) 112 : 13 (h) 112 : 12 (i) 312 : 112 (j) 56 : 1134. Find the ratio of: (a) 40 and 35 (b) 24 and 40 (c) 70 and 300 (d) 300 and 80 (e) 25 and 55 (f) Rs. 4 and 350 Paisa(g) 240 cm and 4 m (h) 75 paisa and Rs. 3 (i) 3 kg and 800 g (j) Rs. 30 and 900 paisa (k) 400 m and 14 km (l) 3000 cm and 15 m. 5. Given that x : y = 4 : 5 and y : z = 6 : 7(a) Find x in terms of y. (b) Find y in terms of x. (c) Express y in terms of z (d) Find x : z.

Acme Mathematics 8 596. If a : b = 2 : 5 and b : c = 3 : 4, find the a : c.7. Solve: (a) x3 = 56 (b) 4x = 87 (c) 721 = x3 (d) 100 200 = 300x8. Divide the following in the given ratio(a) Rs. 15 in the ratio of 2 : 1. (b) Rs. 16 in the ratio of 3 : 1.(c) Rs. 125 in the ratio of 1 : 4. (d) 300 g in the ratio of 3 : 2. (e) 3000 g in the ratio of 1 : 2. (f) 45 minutes in the ratio of 4 : 5. (g) 6 km in the ratio o 5 : 1. 9. (a) A man divides Rs. 500 between his two children in the ratio of 2 : 3. How much does each child get ?(b) 700 gram of alloy is made up of copper, and nickel in the ratio of 2 : 3. How many grams of each metal are there ?(c) If 75 cm is divided in the ratio of 4 : 1, how long is each part ? (d) Two numbers are in the ratio of 3 : 4. If their sum is 133, find the numbers. (e) Two numbers are in the ratio 3 : 2. If their difference is 52. Find the numbers. 10. (a) In Ram's family the ratio of income and expenditure is 7 : 5. If the income is Rs. 7000 what is the expenditure ?(b) A sum of money was divided between Bipana and Karishma in the ratio of 3 : 4. If Karishma gets Rs. 152, how much will Bipana get ? (c) The ratio between two quantities is 5 : 8. If the first quantity is 215, find the second quantity. (d) The ratio of two numbers is 5 : 4. If the smaller is 44, find the larger. 11. (a) Divide Rs. 600 among Monika, Astha and Markos in the ratio of 3 : 2 : 1. (b) The ratios of men, women and children in a family are 2 : 2 : 5. If the total population is 11106, find the number of men, women and children.(c) Divide Rs 1000 among Ritesh, Kiran and Subrat so that Ritesh's share is twice the share of Kiran and Kiran's share is three times the share of Subrat. 12. (a) Two numbers are in the ratio 4 : 5. If 6 is added to each, the new ratio becomes 14 : 17, find the numbers.

60 Acme Mathematics 8(b) Two numbers are in the ratio 3 : 2. If 5 is subtracted to each, the new ratio becomes 2: 3, find the numbers. (c) Two numbers are in the ratio of 5 : 8. If 9 is added to each they are in the ratio of 8: 11. Find the numbers. (d) Two numbers are in the ratio of 1 : 2. If 10 is subtracted from each, then the new ratio will be 3 : 7, find the numbers. 13. In a school 128 students sat for the exam; 120 of them stood in the grade A and rest in the grade B, find. (a) the ratio of the grade A to the grade B. (b) the ratio of the grade A to the total students. (c) the ratio of the grade B to the total students. 14. (a) The ages of mother and her daughter are in the ratio 5: 3. Ten years ago the ratio of the age of mother to that of daughter was 2: 1. Find the present ages of the mother and the daughter. (b) The ages of Kundan and Kushal are in the ratio 7 : 1. Nine years later the ratio of the age of Kundan and Kushal will be 8 : 5. Find the present age of Kundan and Kushal.(c) The present ages of two brothers are in the ratio of 2 : 3. After 5 years it will be 3 : 4. Find their present ages.(d) The present ages of two sisters are in the ratio of 3 : 4. After 5 years it will be 4 : 5. Find their present ages. 15. The scale of a map is 1cm : 2500 m. The distance between two places on the map is 3.5 cm.(a) Calculate the actual distance between two places.(b) If the place is 5 km from the first place, how far is the position of the place on the map ?16. A model car is 50 times smaller than a real car. (a) 300 cm on the real car would be represented by how much on the model ? (b) What length on the real car does 16 cm on the model represent ? 17. A map has a scale 1 cm 1 km. What distance does 2 cm on the map represent ? What distance would 5 cm represent ?

Acme Mathematics 8 6118. On a map 5 km distance is represented by 1 cm. What distance is represented by the following ?(a) 20 km (b) 30 km (c) 2.5 km 19. (a) Kasi, Raj Kumar and Pramod bought computer parts sharing in the ratio 4 : 1 : 3. If the total expenditure is Rs. 3,60,000, what is the share of each person ? (b) If the income of A is double of B and three times of C and their total income is Rs. 11000. Find the income of each. 20. The students appeared the test examination is 1726. Among them the ratio of the students who passed with grade A+ and in the grade A were 4 : 57, 246 students were placed in the grade B and 50 grade D, if 92 students passed in grade A+.(a) How many students passed in the grade A ? (b) How many students passed in the grade C ? (c) Find the ratio of the students who passed in the grade B and in the grade C.21. The ratio of the present age of a father and his son is 3:1. If the age of father is 36 years then,(a) Find the age of son.(b) What is the ratio of the father and son after 5 years.(c) After how many years, the ratio of the their ages will be 2:1?22. The ratio of the monthly income and expenditure of a family is 7:4. If the income of the family in a month is Rs. 21,000, find:(a) The monthly expenditure of the family. (Rs 12000)(b) What amount does the family save in a year? (Rs 108000)Project WorkCollect the number of boys and girls of your school and calculate the following:(a) The ratio between total number of boys to girls.(b) The ratio between number of boys and girls of class 8.(c) The ratio between total number of students and number of students of class 8.

62 Acme Mathematics 8B. Proportion ACTIVITYTake two pieces of rectangular shaped paper equal in size. Divide first into 4 equal parts and second into 12 equal parts.Fig (1) Fig (2)Ratio between shaded part and unshaded part = 13 in the fig (1).Ratio between shaded part and unshaded part = 39 in the fig (2). = 39 = 3 × 13 × 3 = 13Hence, 13 and 39 are equal. The equal ratios are called proportion.Ratios are used to compare quantities of the same kind or the same unit. For example, Rajiv bought 30 kg of sugar at Rs. 1050 and Sushil bought 3 kg of sugar at Rs. 105. The ratio between the quantities is given below. 30 kg of sugar to 3 kg of sugar = 30 : 3 = 303 = 10 1Cost of 30 kg of sugar to cost of 3 kg of sugar = 1050:105 = 1050 105= 101

Acme Mathematics 8 63Here, both ratios 30 : 3 and 1050 : 105 are equal. We write it as 30 : 3 = 1050:105 or 30 : 3 :: 1050 : 105 and say that four quantities 30 kg, 3 kg, Rs. 1050 and Rs. 105 are in proportion.Since 30 : 3 = 1050 : 105 are in proportion, we haveor,303 = 1050105or, 30 × 105 = 1050 × 3 or, 3150 = 3150 Similarly, if a : b = c : d, we must have, ad = bc. Thus the equality of two ratios is called proportion. If a, b, c and d are four quantities and ab = cd then a, b, c and d are called proportional. The first and the last terms are called the extremes and the second and the third terms are called the means. Here a and d are extremes and b and c are means. The product of extremes and the product of means are always equal.Continued Proportion Susma has Rs 60, Neetu has Rs 120 and Pratistha has Rs 240 for their tiffin.Ratio of Susma to Neetu = 60 : 120 = 1 : 2Ratio of Neetu to Pratistha = 120 : 240 = 1 : 2 Here, 60 : 120 = 120 : 240 or,60 120 = 120 240 In such a case we say that Rs. 60, Rs 120 and Rs. 240 are in continued proportion. Here, Rs. 120 is called the mean proportion and Rs. 240 is called the third proportion. Thus three quantities are said to be in continued proportion if the ratio of the first two quantities is equal to the ratio of the second and the third. If a, b, c are in continued proportion, we must have, a : b = b : c or,ab = bc. 'b' is called the mean proportional and 'c' is called the third proportional.

64 Acme Mathematics 8Solved ExamplesExample 1 : If the first three quantities of the proportion are 6, 7 and 12, find the fourth quantity. Solution : Let, x be the fourth quantity. Then, 6, 7, 12, x are in proportion, Now, we have, 67 = 12 x or, 6x = 12 × 7or, x = 12 × 7 6or, x = 14∴ The fourth quantity is 14. Example 2 : Find the third proportional of 12 and 18 if they are in continued proportion.Solution : Let x be the third proportional, Then, 12, 18 and x are in continued proportionNow, we have, 12 18 = 18 x or, x = 18 × 18 12or, x = 27∴ The third proportional is 27. Example 3 : The height of Arun is 160 cm and his shadow is 180 cm long in the evening. If the height of Paras is 150 cm, find the length of the shadow of Paras in the same evening. Solution : Let the length of shadow of Paras be x cm Here, four quantities 160 cm, 180 cm, 150cm and x cm are proportional. So, we have, 160 180 = 150 x or, x = 150 × 180 160or, x = 168.75 ∴ The length of shadow of Paras is 168.75 cm.

Acme Mathematics 8 65Classwork1. Tick () the correct answer.(a) 3 : 4 is the same as(i) 4 : 3 (ii) 6 : 8(iii) 5 : 6 (iv) 15 : 12(b) The lowest term of 40 : 50 is(i) 30 : 40 (ii) 20 : 25(iii) 4 : 5 (iv) 8 : 10(c) 1 2 : 6 is equal to ..... (i) 1 : 6 (ii) 1 : 12(iii) 6 : 2 (iv) 2 : 3(d) Ratio is about(i) sharing (ii) adding(iii) multiplying (iv) comparing(e) Ratio should be expressed in ....(i) highest term (ii) lowest term(iii) middle term (iv) last term2. Fill in the blanks.(a) 813 = 39 (b) 5 : 7 = 20 : (c) 2 : 3 = : 18(d) : 7 = 15 : (e) 40 paisa : Re 1 = 2 : (f) 8 hours : 2 days = : 6(g) The first and fourth terms of a proportion are called ................(h) Ratio is a pure number with ................. unit.

66 Acme Mathematics 8Exercise 2.81. Are the following terms in proportion ? (a) 1, 2, 4, 8 (b) 3kg, 4kg, 8kg, 20kg (c) 5m, 20 m, 30 m, 120 m (d) Rs 100, Rs 150, Rs 20, Rs 30. (e) 12 ml, 15 ml, 16 ml, 20 ml (f) 1 second, 2 seconds, 3 seconds, 6 seconds 2. The first three terms of a proportion are given below. Find the fourth term:(a) 40, 30, 8 (b) 3, 4, 6 (c) 13 kg, 52 kg, 1 kg (d) 14 m, 15 m, 42 m (e) 8 l, 10 l, 12 l (f) 100, 200, 300 3. Find the value of x in the following: if the following terms are in proportion: (a) x, 14, 16, 32 (b) 115, x, 230, 112 (c) 140, 150, x, 450 (d) 500, 400, 2500, x (e) 99 kg, 45 kg, 44 kg, x (f) 50, 200, 300, x (g) x, 2, 4, 8 (h) 9, x, x, 4 (i) 2x, 8, 4, x 4. Find the mean proportion between: (a) 27 and 3 (b) 15 and 60 (c) 5 and 125 (d) 3 and 147 (e) 1 and 625 (f) 31 and 1984 5. Following numbers are in continued proportion. Find the value of x.(a) 10, 50, x (b) 2, x, 8 (c) 3, x, 27, (d) x, 4, 8 (e) 6,12, x (f) x, 16, 4

Acme Mathematics 8 676. (a) A milk man mixed the water and milk in the ratio 2: 7. If the quantity of water is 17 litre, find the quantity of milk.(b) The consumption of 'Samosa' and 'Jerry' for 'Khaja' in a canteen is in the ratio 7:3. If 210 Samosa were used, find the number of Jerry used for Khaja. (c) The cost of 7 kg of potatoes is Rs 84, find the cost of 19 kg of potatoes.(d) The cost of 102 pens is Rs 4080. How many pens can be bought for Rs 8520 ? (e) A bus travels 613 km and consumes 89 litters of diesel. How long will it go on 107 litters of diesel ? 7. (a) What number must be added to each of the numbers 42, 90, 6 and 18 to be in proportion ? (b) What number must be subtracted to each of the numbers 22, 36, 58 and 108 to be in proportion ?8. (a) If the four numbers 3, x, 8 and 18 are in proportion, find x.(b) Find the mean proportional (i.e., x) of given triples 3, x, 48.(c) A store has food stuff for 5 men for 15 days. How long the same food stuff will last for 3 men? 9. A, B, and C invest a sum of Rs. 270,000 on a business in the ratio of 2:3:3.(a) Find the investment of each of them.(b) At the end of the year, if the business gained Rs. 90,000, divide the profit amount them in the ratio of their shares.Project WorkTake A4 paper and measure its length and breadth. Calculate the ratio between its length to its breadth. Take another sheet of paper and make the ratio of length and breadth of it, equal to the ratio of a4 size paper. Paste both paper on 'Display board'.

68 Acme Mathematics 82.4 Profit and Loss, DiscountA. Profit and LossBusinessmen purchase and sell goods in their business. The price at which they purchase goods is called the cost price (C.P.). The price at which they sell goods is called the selling price (S.P.). If the selling price (S.P.) is more than its cost price (C.P.), we say that there is a profit (P) or gain (G). Similarly, if the selling price (S.P.) is less than its cost price (C.P.) we say that there is loss (L). In brief: (i) Cost price (C.P.) = Price at which goods are purchased (ii) Selling price (S.P.) = Price at which goods are sold. (iii) Profit (P) = Selling Price – Cost Price = S.P. – C.P. (iv) Loss (L) = Cost Price – Selling Price = C.P. – S.P. In buying or selling goods, the profit or loss is expressed as percentage of cost price. (v) Percentage profit = actual profitcost price × 100% = S.P.– C.P.C.P. × 100%(vi) Percentage loss = actual losscost price × 100% = C.P.– S.P.C.P. × 100%If Profit (gain) is x% then, (vii) S.P. = 100 + x 100 × C.P. (viii) C.P. = 100 100 + x × S.P. If loss is x% then(ix) S.P. = 100 – x 100 × C.P. (x) C.P. = 100 100 – x × S.P. Note : Expenses which are given for transportation, cartage, labour charges, taxes etc. are overhead expenses. These expenses are also included in the cost price.

Acme Mathematics 8 69Solved ExamplesExample 1 : Rajkumar purchased a radio for Rs 840 and sold to his friend Kashi for Rs 882. Find his profit and profit percent.Solution : C.P. of radio = Rs 840 S.P. of radio = Rs 882 Now, Profit = S.P. – C.P.= Rs (882 – 840)= Rs 42Profit percent = actual profit C.P. × 100% = Rs Rs 42 Rs 840 × 100% = 5% Therefore, Rajkumar's profit is Rs 42 and profit percent is 5%.Example 2 : Ramji bought a motorbike for Rs 82,500. Due to some defects in it, he sold it for Rs 82,000. Find his loss and loss percent. Solution : Cost price of the motorbike (C.P.) = Rs 82500 Selling price of the motorbike (S.P.) = Rs 82000 ∴ Loss = C.P. – S.P. = Rs (82500 – 82000) = Rs 500 And loss % = actual loss C.P. × 100% = Rs 500 Rs 82500 × 100%= 20 33%= 0.606% Therefore, Ramji's loss is Rs 500 and loss percent is 0.606%.Example 3 : Pramod bought a computer for Rs 33,500. Due to some defects in it, he had to pay Rs 1250 for repair. He sold it for Rs 35,792.50 find his loss or profit percent. Solution : C.P. of computer = Rs 33500Repairing charge = Rs 1250 C.P. + repairing charge = Rs 34750

70 Acme Mathematics 8Now, C.P. of computer = Rs 34750 S.P. of computer = Rs 35792.50 Here, S.P. > C.P. Therefore profit is made Profit = S.P. – C.P. = Rs (35792.50 – 34750) = Rs 1042.5 and profit % = actual profit C.P. × 100% = 1042.5 34750 × 100% = 3% Therefore, Pramod made 3% profit. Example 4 : A watch is sold for Rs 1700 at a loss of 10%,(a) What is its cost price ? (b) At what price should it be sold to gain 20% ? Solution : (a) S.P. of watch = Rs 1700 Loss percent = 10% Let, C.P. of watch = Rs 100 then.S.P. = Rs (100 – 10) = Rs 90. since, loss is 10%Now, If the S.P. is Rs 90 then C.P. is Rs 100 If the S.P. is Rs 1 then C.P. is Rs 100 90If the S.P. is Rs 1700 then C.P. is Rs 100 90 × 1700 = Rs 1888.90 Therefore, C.P. of watch is Rs 1888.90Alternative method:S.P. of watch = Rs 1700Loss % = 10% Let, C.P. of watch = Rs x. then, C.P. of watch = 100 100 – Loss% × S.P. of watch or, x = 100 100 – 10 × Rs 1700or, x = Rs 100 90 × 1700or, x = Rs 1888.90Therefore, C.P. of watch is Rs 1888.90

Acme Mathematics 8 71(b) Now, Cost Price (C.P.) of watch = Rs 1888.90C.P. of the watch = Rs 1888.90 Profit percent = 20%Let, C.P. of the watch = Rs 100 then, S.P. of the watch = Rs (100 + 20) = Rs 120 Now, when C.P. is Rs 100 then S.P. is Rs 120When C.P. is Rs 1 then S.P. is Rs 120100When C.P. is Rs 1888.90, then S.P. is Rs 120 100 × 1888.90 = Rs 2266.68 Alternative method:Profit percent = 20% S.P. of watch = 100 + profit % 100 × C.P. of watch = 100 + 20 100 × Rs 1888.90 = Rs 2,266.68 Hence, the watch should be sold at Rs 2266.68 to gain 20%.Classwork1. Fill in the blanks.(a) C.P. = .............(b) S.P. = .............(c) profit % = profit....... × 100%(d) S.P. = C.P. + ........2. Tick the correct option.(a) The profit is male when, .............(i) S.P. > C.P. (ii) C.P. > S.P.(iii) S.P. = C.P. (iv) C.P. – S.P.(b) The loss is made, when .....(i) C.P. > S.P. (ii) S.P. > C.P. (iii) C.P. – S.P. (iv) S.P. = C.P.

72 Acme Mathematics 83. Choose whether the following statements are 'True' or 'False'.(a) Profit = C.P. – S.P.(b) Loss + C.P. = S.P.(c) S.P. = C.P. + profit(d) S.P. + loss = C.P.(e) Loss = C.P. – S.P.(f) Profit % = lossS.P. × 100(g) If there is no profit and no loss then C.P. = S.P.(h) If a gas cylinder is purchased for Rs. 1780 and overhead expenses (transportation) is Rs. 25 then cost price of the gas cylinder is Rs. 1855.(i) The price of an item can go up by 200%(j) Loss percentage is calculated as LossC.P. × 1004. If C.P. of the calculator is Rs. 10,000 and S.P. of the calculator is Rs. 9500; then its loss percent is ............(a) 10% (b) 5% (c) 9.5% (d) 6%5. If C.P. of the calculator is Rs. 700 and S.P. of the calculator is Rs. 1400; then its loss percent is ............(a) 100% (b) 50% (c) 200% (d) 140%Exercise 2.91. Find the gain percent in the following: (a) Cost Price = Rs 1140 and Selling Price = Rs 1145 (b) Cost Price = Rs 1432 and Selling Price = Rs 1543 2. Find the loss percent in the following: (a) Selling Price = Rs 1350 and Cost Price = Rs 1370 (b) Selling Price = Rs 1124 and Cost Price = Rs 12403. Find the cost price, when (a) S.P. = Rs 4500 and Gain = 15% (b) S.P. = Rs 1000 and Loss = 2%

Acme Mathematics 8 734. Find the selling price, when(a) C.P = Rs 5000 and Loss = 5% (b C.P. = Rs 625 and Gain = 7% 5. (a) A calculator is sold for Rs 230 at a profit of 15%, find its cost price.(b) A watch is sold for Rs 1700 at a loss of 10%. What is its cost price ?6. (a) An article bought for Rs 500 is sold at a loss of 15%. What is the selling price ? (b) A 'Hand Book of Mathematics' is bought for Rs 60 and sold at a profit of 20%. What is the selling price ? (c) Narayan sold an article at a loss of 20%. If he had bought it for Rs 500, find the selling price of the article.7. (a) A computer costing Rs 33500 was sold for Rs 34000, find the loss or profit percentage. (b) A shopkeeper buys 12 dozens of pen paying Rs 4.25 for each pen and sells each pen for Rs 5. Find his total profit and profit percentage. 8. When an article is sold for Rs 23, there is profit of 15%. (a) What is the cost price ? (b) At what price must it be sold to make a profit of 10% ? 9. By selling a watch for Rs 2400, a dealer gains 25%. (a) What is the cost price ? (b) How much percent would be profit by selling it for Rs 2040 ? 10. A watch seller loss 20% by selling a watch for Rs 1260. (a) What was its cost price ? (b) If he had to gain 30% at what price should have he sold it ? 11. Ashok lost 5% by selling an article for Rs 30.60. If he sold it for Rs 50.20, find his gain or loss percent.12. (a) Rajan bought a book for Rs 180 and sold it to Sajan at a profit of 20%. Sajan sold that book to Nirajan at a loss of 20%. At what price Nirajan should sell the book to receive 5% profit ? (b) Megha bought a cycle for Rs 4000 and sold it to Mahima at a loss of 10% . Mahima sold the cycle to Madhab at a loss of 5%. Find the cost price of Madhab. (c) Prakash sold an article to Shital at 10% profit. Shital sold the same article for Rs 759 with 15% profit to Surya. At what price did Prakash buy it?

74 Acme Mathematics 8(d) Dipika sold an article at a profit of 5% to Dipak. Dipak sold the article at a profit of 5% to Rupak for Rs 4851. What is the(i) cost price of Dipak ? (ii) cost price of Dipika ? (e) Rajaram sold an article to Raghuram at a profit of 25%. Raghuram sold the same article to Tikaram at a loss of 10%. If Tikaram paid Rs 990 for it, how much did it cost to Rajaram ? 13. (a) Kashi sells two computers for Rs 28000 each. On one he gains 8% and on the other he goes on loss of 8%. Find his loss or gain percent. (b) A fruit seller purchased 300 oranges for Rs 1500 and sold 90 of them at the rate of Rs 5.50 per orange and the remaining at Rs 5.25 each. What profit or loss percent has he made? 14. (a) Janak bought an old car for Rs 550000 and spent Rs 90000 on its repairing. He sold that car to Prakash at a gain of 10%. How much did Prakash pay for the car ? (b) A set of sofa was bought for Rs 7000 and Rs 100 was spent on its transportation. At what price should it be sold to have a gain of 9%? 15. (a) Saroj bought 40 books for Rs 1600. He sold 10 books for Rs 40 each and remaining books for Rs 45, what is his total profit or loss ?(b) Suntali bought 4000 oranges at the rate of 70 paisa each. But 400 of them were rotten and she sold the remaining oranges at the rate of 90 paisa each. Find her total profit or loss.16. A retailer purchased 1000 LED bulbs at Rs. 80 each. 100 bulbs were damaged and he sold remaining bulbs at 21.5% profit.(a) Find the cost price of 1000 bulbs. (b) Find the number of remaining bulbs.(c) Find the selling price of all bulbs. (d) Find the selling price of each bulb.Project WorkCollect the price of four different goods from three different shop and chose the shop to buy the goods.

Acme Mathematics 8 75B. Discount Generally businessmen fix prices and sell at lower prices after deducting some amount of money as commission or discount. The price fixed on them is called the labelled price or the marked price. Thus, a discount is the deduction of the money deducted from the labelled price, given to a customer by a seller. The printed price of an article is called the marked price. The difference between the marked price and discount is called the net price. Discount is always calculated on marked price. Therefore, the total price after deduction (discount) = (100 – discount) % of labelled price.Solved ExamplesExample 1 : An article is marked Rs 160. During 'Dashain' festival 10% discount is allowed on it. What will be its net selling price ? Solution : Marked selling price = Rs 160 Discount = 10% of Rs 160 = Rs 10 × 160100= Rs 16 ∴ Net selling Price = Rs (160 – 16) = Rs 144. Therefore, the net selling price of an article is Rs 144. Example 2 : A correction pen is marked at Rs 150. During New Year a discount is given and it is sold for Rs 130. Find the discount percentage. Solution : Marked Price = Rs 150 Net Price = Rs 130 Discount = Rs (150 – 130) = Rs 20 Discount % = 20150 × 100% = 13.33% Therefore, the discount is 13.33%. Example 3 : A shopkeeper marked the price of a radio as Rs 800. He sold it allowing a discount of 10% and made a profit Rs 150. Find the cost price of the radio. Solution : The marked price of the radio (M.P.) = Rs 800 The percentage of discount = 10% Profit = Rs 150 Cost price = ?

76 Acme Mathematics 8We have, Marked price – Discount = C.P + Profit or, Rs 800 – 10% of Rs 800 = C.P. + Rs 150 or, Rs 800 – 10 100 × Rs 800 = C.P. + Rs 150 or, Rs 800 – Rs 80 = C.P. + Rs 150or, Rs 720 = C.P. + Rs 150 or, C.P. = Rs 720 – Rs 150 ∴ C.P. = Rs 570. Thus, the cost price of the ratio = Rs 570.Classwork1. Fill in the blanks.(a) List price of an article is also known as ...........(b) Discount = Discount % of .................(c) If discount percentage is zero, then M.P. ........... S.P.(d) If there is loss in a transaction then S.P. is ......... C.P.(e) M.P. – S.P. = ...............(f) Overhead expresses are added to ............. .2. Tick the correct statement.(a) Discount % is calculated as ...............(i) discount M.P. × 100% (ii) discountS.P. × 100(iii) discountC.P. × 100 (iv) None of all(b) Marked price is calculated as ........(i) M.P.discount × 100 (ii) S.P. × 100100–discount % (iii) discountdiscount% × 100 (iv) S.P.discount(c) Selling price is calculated as ..........(i) M.P.discount × 100 (ii) S.P.discount × 100 (iii) discountdiscount% × 100 (iv) M.P. 100 – discount100

Acme Mathematics 8 77(d) Marked price (M.P.) is calculated, when S.P. is give as....(i) 100 – S.P.(100 + discount%) (ii) S.P. × 100(100 – discount%)(i) 100 + S.P.100 + discount (ii) S.P. – 100discount % – 100Exercise 2.101. Find the S.P. and rate of discount in each of the following cases: (a) M.P = Rs 1000 and discount = Rs 200(b) M.P. = Rs 5000 and discount = Rs 1000 (c) M.P. = Rs 8000 and discount = Rs 2000 2. Find discount amount and discount percent.(a) M.P. = Rs 5000 and S.P = Rs 4000 (b) M.P. = Rs 80000 and S. P. = Rs 60000 (c) M.P. = Rs 10000 and S.P. = Rs 7500 3. FInd the M.P in each of the following cases. (a) Discount = Rs 1200 and S.P. = Rs 4000 (b) Discount = Rs 1500 and S.P. = Rs 5000 (c) Discount = Rs 6000 and S.P = Rs 150004. Find the M.P. in each of the following cases: (a) Discount rate = 15% and S.P. = Rs 4000 (b) Discount rate = 20% and S.P. = Rs 5000 (c) Discount rate = 25% and S.P. = Rs 7500 5. (a) After 12% discount, the price of an article is Rs 1320. Find the marked price. (b) The marked price of an article is Rs 1600. If it is sold for Rs 1472, what is the discount rate ?(c) The marked price of a radio is Rs 810. If 10% discount is allowed. Find the selling price.(d) Due to 'Dipawali', a T.V. is sold for Rs 14900 after allowing a discount of 23%. What is the marked price ? SP = MP – Discount

78 Acme Mathematics 86. (a) The marked price of a book is Rs 160. If it is sold at 15% discount, find the discount amount and selling price.(b) The marked price of a machine is Rs 4500. If it is sold at 8% discount, find the discount amount and selling price. 7. (a) A shopkeeper marked the price of a radio as Rs 5000. He sold it allowing a discount of 10% and made a profit Rs 500. Find the cost price of the radio. (b) Kosish marked the price of a radio as Rs 2000. He sold it allowing a discount of 15% and made a profit of Rs 500. Find the cost price of the radio. (c) The marked price of a book is Rs 500. If 30% discount is allowed then a loss of Rs 100 is made, what is the cost price of the book ? (d) The marked price of a camera is Rs 8000. If 10% discount is allowed and then the price is increased by 5%, what will be the final price ? 8. (a) The cost price of a mobile set is Rs 4000. Find the marked price of the set so that there will be a profit of Rs 800 after allowing a discount of 20%. (b) After allowing 4% discount on the marked price of an article there is a profit of Rs 160. If the cost price of the article is Rs 800. Calculate the marked price. 9. (a) A dealer marks a T-shirt with a price 20% more than the cost price. If he sells it at a discount of 15%, what will be his gain or loss percent ? [Take cost price as Rs 100] (b) A shopkeeper marks his goods 30% above the cost price and gives discount of 10% on the marked price. What gain percent does he make ? (c) A shopkeeper marks his goods 25% above the cost price and gives discount of 10% on the marked price. What gain percent does he make ? (d) A shopkeeper marks his goods 20% above the cost price and gives discount of 5% on the marked price. What gain percent does he make ? 10. (a) A dealer buys an article for Rs 520. The marked price is 10% above his cost price. He later sells the article at a discount of 7% on the marked price. Find the selling price. (b) Indra marks his goods 20% above the cost price. He sells it at a discount of 15%. If his selling price is Rs 102, calculate his cost price.Project WorkCollect the information from the different sources about discount. List the rate and its discount. Paste on your 'Math Corner'.Note : Discount are given in Festival and special events.

Acme Mathematics 8 791. Marked price of watch is Rs. 3000. 5% discount is given on its price.(a) If the marked price of the watch is MP and discount amount is D then, write the formula to find the selling price.(b) Find the discount amount given on the watch.(c) Find the selling price of the watch.2. Manoj visited a furniture store to get a table and some chairs. A pair of table and four chairs is available for Rs. 16,000.(a) If the price of a table is Rs. 6,000 then how much does a chair cost?(b) How much discount did Manoj get while purchasing a pair of table and chairs at 5% discount? Find the price after discount.(c) If the shopkeeper earned 10% profit even after offering 5% discount, at what price did the shopkeeper purchase a pair of a table and chairs?3. The marked price of a watch is Rs. 2000. The shopkeeper allows 15% discount in it.(a) Write the formula to find discount percent if discount amount A and marked price M are given.(b) Find the discount amount given in the watch.(c) Find profit or loss of the shopkeeper if he/she had bought the watch with Rs. 1500.4. A fruit seller bought 100 kg of apples for Rs. 14500. Out of them, 5 kg were rotten. He sold the remaining apples for Rs. 160 per kg. (a) What percentages of apples were rotten?(b) Find his percentage of profit or loss.(c) Write the cost of 10000 kg of apples in scientific notation.(d) If 12 oranges cost Rs. 72 and 11 mangoes cost Rs. 121, what is the cost of 3 mangoes?5. Prem went to a computer shop to buy a computer. The Marked price of a computer is Rs. 50,000.(a) If marked price and selling price are represented by MP and SP respectively. Write the formula to find discount D.Mixed Exercise

80 Acme Mathematics 8(b) How much discount did Prem get while buying a computer on discount of 10%?(c) The shopkeeper got 20% profit after selling them at 10% discount. What was the cost price of computer?6. Gopal went to market to buy a watch. The marked price of the watch is Rs. 3500.(a) If marked price is represented by MP, discount amount by D and selling price by SP, write the formula of SP.(b) How much discount did Gopal get if marked price and discount is in ratio 22:3? (c) The shopkeeper got 25% profit after selling it at Rs. 3022.73 discount. What was the cost price of the watch?7. Hari Shrestha went to the utensil shop to buy thermos. The marked price of a thermos is Rs. 1250 and discount is 10%.(a) Write down the ratio between selling price and discount.(b) How much discount did Hari Shrestha get while buying a thermos on discount of 12%.(c) The shopkeeper gets 10% loss after selling at 12% discount. What was the cost price of the thermos?8. Surendra went to the bicycle shop to buy a bicycle. The marked price of bicycle is Rs. 7500 and discount is Rs. 750.(a) If marked price is MP and discount rate D%. Write the formula to find the discount amount.(b) How much percent is given?(c) The shopkeeper made 5% loss after selling the bicycle. What was the cost price of the cycle?9. A shopkeeper marked the price of a television Rs. 40,000. He/She sold the television after allowing 10% discount and made a profit of Rs. 4,000.(a) Write the formula to find discount percentage (D%) when marked price (MP) and discount amount (DA) is given.(b) Find the selling price of the television.(c) How much money did the shopkeeper pay for the television? Find(d) If Rs. 6400 is the profit in a television, how many televisions should be sold to make at least Rs. 1,50,000 profit?

Acme Mathematics 8 8110. A fruit seller bought 500 oranges for Rs. 4000. 50 of them were rotten. He sold the remaining oranges for Rs. 10 each.(a) Find the cost of 15 oranges. (b) How many oranges are good for sell? Calculate it. (c) What percentage of oranges were rotten ? Find it. (d) Find his profit or loss percent. 11. Ramesh went to an electric shop to buy an induction oven. The marked price of the oven is Rs. 7600.(a) If discount is D%, write the relation between MP, SP and D%. (b) What price Ramesh has to pay for induction oven if he gets 12% of discount? (c) If shopkeeper sold it allowing a discount of 20%, the actual loss is Rs. 70, find the cost price of the article. 12. A shopkeeper marked the price of a radio as Rs. 5000. He sold it allowing a discount of 10% and made a profit Rs. 500(a) Write the formula for finding rate of discount when discounted amount and marked price are given.(b) What is the selling price of the radio?(c) Find the cost price of the radio.(d) If the discount was not allowed, then what would be the profit.

82 Acme Mathematics 8EvaluationTime: 50 minutes Full Marks: 211. Deeyana and Aarushi are two sisters, and their ages are 8 years and 12 years respectively. One day, their mother gives Rs. 500 and asked to divide it in the ratio of their ages.(a) Define ratio. [1](b) How much money each of them receives? [2](c) How much is the percentage Aarushi receives more than that Deeyana? [1](d) Find the ratios of money each of them receives. [1]2. A shopkeeper marks the price of a football Rs. 1250.(a) If he gives 12% discount, calculate the discount amount. [1](b) If the cost price of the football is Rs. 1000, find his profit percentage. [2]3. Prem went to market to buy a TV.The marked price of a TV is Rs.50,000. (a) If marked price and selling price are represented by MP and SP respectively, write the formula to find discount D. [1](b) How much discount did Prem get while buying a TV on discount of 10% ? [1](c) The shopkeeper got 20% profit after selling them at 10% discount. What was the cost price of TV ? [2]4. The marked price of a machine is Rs. 45000 and it is sold on 25% profit after 8% discount . (a) Define profit. [1](b) Find the discount amount. [1](c) Find the selling price of the machine. [1](d) Compare the cost price and selling price of the machine. [1]5. The marked price of television is Rs. 50,000. The shopkeeper allows 10% discount on it. (a) Define marked price. [1](b) Find discount amount. [2](c) The shopkeeper got 10% profit after selling it 10% discount, what is the cost price ? [2]

Acme Mathematics 8 832.5 Unitary MethodThe method of finding the value of unit (one) quantity from the value of the given quantities first and then the value of required quantities is called the unitary method. Unitary method means \"The method of One\" or \"to find the rate\". Solved ExamplesExample 1 : The weight of 72 books is 9 kg. What is the weight of 100 such books ?Solution :Number of books Weight (in kg)72 9100 ?Here, The weight of 72 books is 9 kg. The weight of 1 book is 972 kg. The weight of 100 books is 972 × 100 kg. = 12.5 kg ∴ The weight of 100 such books is 12.5 kg.Now, we shall use the concept of proportion and unitary method in the development of the concept of variation. Now, consider the following examples.A. Direct variation Study the given table carefully,Petrol (in litre) 5 10 15 8 xCost (in Rs) 800 1600 2400 1260 ?In the table we can see that as the amount of petrol increases, cost also increases and vice versa. We can find the relation between petrol and its cost as follows: Cost of 5 litre of petrol = Rs 800Cost of 1 litres of petrol = Rs 800 5 = Rs 160 Cost of 10 litres of petrol = Rs 160 × 10 = Rs 1600 Cost of 15 litres of petrol = Rs 160 × 15 = Rs 2400 Cost of 8 litres of petrol = Rs160 × 8 = Rs 1260 ∴ Cost of x litres of petrol = Rs 160 × x = Rs 160x

84 Acme Mathematics 8Similarly, the ratio of amount of petrol and cost of petrol is same or constant. i.e. 5 800 = 10 1600 = 15 2400 = 8 1260 = x 160x If there are two quantities and one increases as the other increases (or vice versa) and the ratio of the corresponding values of the quantities is constant, is called direct variation. B. Indirect Variation Study the given table carefully,Number of workers 5 10 16 8 xNumber of days 16 8 5 10 ?In the table we see that as the number of workers increases the number of days decreases and vice versa. We can find the relation between workers and their work as follows. 5 men can complete the work in 16 days. 1 man can complete the work in 16 × 5 days = 80 days 10 men can complete the work in 8010 days = 8 days16 men can complete the work in 8016 days = 5 days8 men can complete the work in 808 days = 10 days The ratio of number of workers and number of days is not same or constant in this variation i.e. 180 ≠ 108 ≠ 165 and so on. If there are two quantities and one increases as the other decreases (or vice versa) and the ratio of the corresponding values of the quantities is not constant, it is called indirect variation.Solved ExamplesExample 1 : The cost of 40 pens is Rs 800. Find the cost of 5 dozen of such pens.Solution : Here,Pen Cost (Rs)40 80060 (5 dozen) ?

Acme Mathematics 8 85Unitary Method: The cost of 40 pens = Rs 800 The cost of 1 pen = Rs 800 40The cost of 60 pens = Rs 800 × 60 40= Rs 1200∴ The cost of 5 dozens of pens is Rs 1200.Variation Method: Let: The cost of 5 dozen pens be Rs x. ThenNumber of pens Cost of pen (in Rs)40 8005 dozen = 60 xHere, the relation between number of pen and cost of pen is direct variation.So, 40 60 = 800xor, x = 800 × 60 40or, x = 1200 ∴ The cost of 5 dozen pens is Rs 1200. Example 2 : 15 men can do a piece of work in 10 days. In how many days can 18 men do that work ?Solution : Here,Men Work Days15 1 1018 1 ?Unitary method:15 men can do a work in 10 days. 1 men can do a work in 10 × 15 days. 18 men can do a work in 10 × 15 18 days = 25 3 days. ∴ 18 men can do that work in 81 3 days.

86 Acme Mathematics 8Variation Method: Let, the required days be x. Then,Number of Men Work No. of Days15 1 1018 1 xHere, the relation between number of men and number of days is indirect variation, So, 15 18 = x 10 or, 15 × 10 = x × 18or,15 × 10 18 = xor, x = 81 3∴ The required number of days is 8 1 3Example 3 : 15 men can reap 20 hectares of field in 12 days. How long will it take for 12 men to reap 30 hectares of field ?Solution : Here,Men Hectares Days15 20 1212 30 x = ?Let 12 men reap 30 hectares of field in x days. The number of men and days are in indirect proportion, the number of hectares of the field and the number of the days are in direct proportion, We have, 12x = 1215 × 2030or, 12 × 20 × x = 12 × 15 × 30or, x = 12 × 15 × 30 12 × 20∴ x = 22 1 2 days.Thus, 12 men can reap 30 hectares of field in 22 1 2 days.

Acme Mathematics 8 87Classwork1. Tick () the correct statement.(a) The method of finding the value of one when the value of many is given is called 'Unitary Method'.(b) The more the quantity, more will be the cost.(c) The lesser the quantity, lesser will be the price.(d) In inverse proportion when one goes up the other will go down.(e) The more the workers the more the time to complete the work.(f) There is inverse relation between the time and the work.2. Tick () the correct answer.(g) Cost of 4 oranges is Rs. 40. The cost of 1 orange is .......(i) Rs. 5 (ii) Rs. 10(iii) Rs. 15 (iv) Rs. 100(h) The cost of 10 litres of petrol is Rs. 1820. The cost of 1 litre of petrol is .......(i) Rs. 182 (ii) Rs. 821(iii) Rs. 1500 (iv) Rs. 812(i) 5 women can complete the work in 10 days. 1 woman can complete the some work in ........ days.(i) 40 days (ii) 10 days(iii) 20 days (iv) 50 daysExercise 2.111. (a) If the cost of 7 litres of petrol is Rs 434, what is the cost of 10 litres of petrol ? (b) The cost of 70 kg. of potato is Rs 2380. What is the cost of 30 kg of potato ? (c) If the cost of 5 dozen of sticker is Rs 60, what is the cost of 33 such sticker ? (d) The service charge of 7 motor-bikes is Rs 875, how much service charge will be of 2 motorbikes ? (e) The cost of painting the four walls of a room of area 240 m2 is Rs 200. What is the cost of painting 300 m2 ?

88 Acme Mathematics 8(f) Rs 5000 will support a family of 5 persons for 30 days. For how long will Rs 9000 support the same family ? (g) 12 men earn Rs 4800 in a certain time. How many men working at the same rate will earn Rs 13200 in the same time ?2. (a) 30 workers are required to complete a work in 30 days. How many workers are required to complete the same work in 20 days ? (b) A garrison of 4 men requires 200 kg of food for 30 days. How much food will be required for 40 days ? (c) A group of 150 students in a hostel requires 60 kg of provisions per day. How much provision will be required for 200 students per day ? (d) 15 men can reap 20 'Bighas' field in 10 days. How long will it take for 8 men to reap the same field ? (e) A motor bike is traveling at a speed of 40 km/hour and completes the journey in 312hours. How long will it take to complete the same journey if the motor bike travels at a speed of 55 km/hour ? (f) 30 men can load a truck of goods in 4 hours. How long will it take for: (i) 12 men (ii) 40 men to load the same amount of goods ? 3. (a) 5 women can complete a piece of work in 24 days working 8 hour per day. How many women can complete the same work in 12 days working 10 hours per day ? (b) 200 labourers need 50 days to construct a road of 4 km long. How many labourers is needed to construct the 8 km long road in 80 days ? (c) 10 m long wall can be constructed in 4 days by 6 workers. In how many days 2 workers can construct 15 m long wall ? 4. (a) 12 workers can do a piece of work in 20 days. How many workers should be added, to complete the work in the 16 days ?

Acme Mathematics 8 89 (b) 24 workers can do a piece of work in 10 days. How many workers should be added to complete the work in 8 days ? 5. (a) If 5 articles cost Rs 1750, how many more articles can be bought by Rs 2800 ? (b) If the cost of 32 copies is Rs 960, how much more is the cost of 3 dozen copies ? (c) If the cost of 10 pens is Rs 300, how much more is the cost of 2 dozen pens ? (d) If the cost of 2 dozen pens is Rs 600, how much more is the cost of 30 pens ? (e) If 25 pens cost Rs 246, how many more pens can be bought by Rs 984 ? 6. (a) In a hostel 30 students have food enough for 40 days. How many students should leave the hostel so that the food is enough for 100 days ? (b) In a hostel 50 students have food enough for 54 days. How many students should be added in the hostel so that the food is enough for only 45 days ? (c) In a hostel 150 students have food enough for 90 days. How many students should be added in the hostel so that the food is enough for only 75 days ? (d) In a hostel 80 students have food enough for 45 days. How many students should leave the hostel so that the food is enough for 60 days ?Project WorkList any works that are related to direct variation and indirect variation with reason. Present your list in the class.

90 Acme Mathematics 82.6 Simple InterestIf you have deposited Rs 1,00,000 in a bank, after one year the bank provides you Rs 1,06,500. Here,(a) The money deposited in the bank is called the principal. Here Rs 1,00,000 is the principal.(b) The additional money given by the bank is called the interest of money. Here, Rs 6500 is the interest of Rs 100000.(c) The sum of money Rs 106500 is called the amount, which is the sum of principal and interest. (d) If the principal remains the same throughout the loan period, the interest calculated on this principal is called the simple interest. If, Principal = pRate of interest per year (annum) = R Number of year = TSimple interest = I and Amount = A Then, (i) I = P × T × R100(ii) P = I × 100TR(iii) A = P + I (iv) R = I × 100PT(v) T = I × 100 P × RSolved ExamplesExample 1 : Find the principal which amounts A after T years at the rate of R% per year on simple interest I. Solution : Here, Amount = A, Rate of interest = R% Simple interest = I Number of year = T years Principal = Rs P (suppose)

Acme Mathematics 8 91Using formula, A = P + I or, A = P + PTR 100or, A = 100 P + PTR 100or, 100A = (100 + TR)P or, P = 100 A 100 + TR∴ Required principal is = Rs 100 A 100 + TRHence, if principal (P), Amount (A), Rate of interest (R), Simple Interest (I) and number of year (T) are given we can find the remaining quantity as:(i) A = P + I(ii) P = 100 A 100 + TR(iii) T = A – P PR × 100(iv) R = A – P PT × 100(v) A = P 100 + TR 100Example 2 : Calculate the simple interest on Rs 4000 for 3 years at the rate of 6% per annumSolution : Here, Principal (P) = Rs 4000 Number of year (T) = 3 years Rate of interest (R) = 6% Simple interest (I) = ? Using the formula: I = P × T × R 100or, I = 400 × 3 × 6 100or, I = 720 ∴ Simple interest on Rs 4000 is Rs 720

92 Acme Mathematics 8Example 3 : Find the amount when the simple interest on Rs 80000 for 2 years 6 months at the rate of 6.5% per annum is Rs 13000. Solution : Here, Principal (P) = Rs 80000Rate of interest (R) = 6.5% per annum Time (T) = 2 years 6 months = 2.5 years Simple Interest (I)= Rs 13000 Amount (A) = ? Now using formula, Amount = P + I = Rs 80000 + Rs 13000 = Rs 93000∴ The required amount is Rs 93000.Example 4 : Find the principal which will amount Rs 5500 in 2 years at the rate of 6.5% per year. Solution : Here, Amount (A) = Rs 5500Rate of interest (R) = 6.5% Time (T) = 2 years Principal (P) = ? Using the formula, P = A × 100100 + TR or, P = 5500 × 100100 + 2 × 6.5or, P = 550000113or, P = 4867.25 ∴ The required principal is Rs 4867.25Example 5 : Find the amount, on the principal Rs 82570 for 2 years 6 months at the rate of 6.5% per annum. Solution : Principal (P) = Rs 82570Time (T) = 2 years 6 months = 2.5 years Rate of interest (R) = 6.5 %

Acme Mathematics 8 93Using the formula : A = P 100 + TR 100We have, A = Rs 82570 100 + 2.5 × 6.5 100= Rs 82570 × 1.1625 = Rs 95987.63 ∴ The required amount is Rs 95987.63Example 6 : Calculate the amount, on the principal Rs. 10,00,00 for 5 years at the rate of 7% per annum if the bank takes 10% income tax in the interest.Solution : Here, Principal (P) = Rs. 10,00,000Time (T) = 5 yearsRate of interest (R) = 7%Interest (I) = P × T × R 100= Rs 1000000 × 5 × 7 100= Rs. 350000Now, Income Tax = 10% of Interest= 10 × 350000 100= Rs. 35000Amount (A) = Principal + Interest – Income Tax= Rs. 10,00,000 + Rs. 35000 – Rs. 35000= Rs. 1315000Classwork1. Tick () the correct statement.(a) Interest is paid when a person or form takes loan.(b) Amount is the difference between principal and interest.(c) Rate of interest is measured in interest of principal Rs 100 in a year.(d) Sum of principal and interest gives amount.

94 Acme Mathematics 82. Choose the correct answer(a) Interest is calculate as : (i) PTR1000 (ii) TR100(iii) AR100 (iv) PTR100(b) Amount is calculate as ....(i) P + I (ii) I – P(iii) P – I (iv) None(c) Principal is calculate as ....(i) I × 100AR (ii) 100 × ITR(iii) A × 100PTR (iv) All(d) The difference A – I gives, .....(i) Principal (ii) Rate(iii) Time (iv) Interest(e) The simple interest (I) when principal (P), number of years (N) and rate of interest (R) are given is .....(i) PTR100 (ii) PNR100(iii) 100NPTR (iv) A × I100(f) Interest for loan of Rs. 100000 at 9% per year for 2 years is ....(i) Rs. 25000 (ii) Rs. 9000(iii) Rs. 18000 (iv) Rs. 2000Exercise 2.121. Find the simple interest on: (a) Rs 3000 at 10% p.a. for 2 years (b) Rs 4000 at 10% p.a. for 3 years (c) Rs 50000 at 15% p.a. for 4 years. (d) Rs 10000 at 15% p.a. for 4 years (e) Rs 5000 for 6 months at 10% p.a.

Acme Mathematics 8 95(f) Rs 2000 for 3 months at 10% p.a. (g) Rs 7500 for 4 months at 12 1 2 % p.a. (h) Rs 4500 for 9 months at 6 1 2% p.a.2. Calculate the simple interest on: (a) Rs 4000 from 4th January 2021 to 3rd January 2023 at 4% p.a. (b) Rs 15000 from 2074 – 1 – 15 to 2078 – 1 – 14. at 6% p.a. (c) Rs 7500 for 73 days at 5% p.a. (d) Rs 800 for 3 years 4 months at 6% p.a.3. Find the principal and amount in the following cases:(a) Interest = Rs 200, Time = 2 yrs, Rate = 10% p.a.(b) Interest = Rs 2000, Time = 2 yrs, Rate = 25% p.a. (c) Interest = Rs 6000, Time = 2 yrs 6 month, Rate = 10% p.a.4. Find the rate of interest in each of the following cases. (a) Principal = Rs 1000, Time = 2 yrs. Interest = Rs 200(b) Principal = Rs 4000, Time = 2 yrs, Interest = Rs 2000 (c) Principal = Rs 24000, Time = 2 yrs 6 months, Interest = Rs 6000 5. Find the time in the following cases:(a) Principal = Rs 2000, Rate = 10%, Interest = Rs 400 (b) Principal = Rs 8000, Rate = 25%, Interest = Rs 4000(c) Principal = Rs 48000, Rate = 10%, Interest = Rs 24000. 6. (a) Mr. Ram Krishna took a loan of Rs 1,00,000 to buy a motorcycle. The loan was taken for 3 years at 10% per annum. What amount must he pay back ? (b) A man borrowed Rs 19200 from a bank at the rate of 7% p.a. How much amount should he pay at the end of 2 years ? (c) Krishna deposited a sum of Rs 8000 in a bank at the rate of 614 % p.a. Find the amount received by him at the end of 412 years. (d) A man borrowed a sum of Rs 25000 from a finance at the rate of 12% pre year. Calculate the amount received by the finance in 18 months.

96 Acme Mathematics 87. (a) On how much principal will be the simple interest Rs 610 in 4 years at the rate of 5% per year ? (b) Sagar borrowed a certain sum of money at the rate of 12% per annum. If he paid Rs 1200 as simple interest at the end of 3rd year, find the sum of money borrowed by him. 8. (a) At what rate percent per year will simple interest of Rs 500 amount Rs 650 in 2 years ? (b) Find the rate percent which will provide an amount of Rs 1450 for 2 years to the principal Rs 1250. 9. (a) Narayan got back the amount Rs 2232 instead of the sum of Rs 1800 which he had deposited in a bank at the rate of 6.5% per year. How long was the money deposited in the bank ? (b) Rs 44,000 amounts to Rs 49,500 at the rate of 2.5% p.a., find the time required. 10. (a) What is the interest on Rs 2160 at the rate of 20% for 4 years ? In how many years Rs 1800 earn the same interest at the rate of 10% ? (b) Find the simple interest on Rs 1080 for 4 years at the rate of 10%. In how many years will the interest on Rs 900 amount to same at the rate of 12% per year ? 11. (a) The interest for 3 years on Rs 2400 is Rs 576. Find the interest on Rs 4200 for 2 years at the same rate of interest.(b) If the interest for 5 years on Rs 3000 be Rs 525, find the interest on Rs 2200 for 6 years at the same rate of interest. 12. (a) If the interest for 4 years on Rs 350 be Rs 112, find the rate of interest. In what time will Rs 800 earn the interest Rs 320 at the same rate of interest ? (b) In what time will the simple interest on Rs 1800 at 6% p.a. be equal to the simple interest on Rs 1080 for 8 years at 5% p.a.(c) Find the simple interest on Rs 15,000 for 4 years at 8% per annum. In what time will the Rs 15,000 earn the same interest at 16% p.a. ? 13. (a) Dipendra deposited Rs 70,000 at fixed account in a Bank at 8.5% simple interest per year. He withdrew Rs 40,000 at the end of 6 years. How long the remainder should be put in the bank in order to receive Rs 60,000 as the total interest from the beginning ? (b) Ramkrishna deposited Rs 60000 at rate 8% in saving account. After 5 years he withdrew Rs 40000 and total interest of 5 years. How long should he keep the remaining amount to get total interest Rs 28800 from the beginning ?

Acme Mathematics 8 9714. (a) The difference between interests in 3 years. at 7% and 5% p.a. of a sum of money is Rs 30. Find the sum.(b) The difference between interests in 3 years. at 15% and 10% p.a. of a sum of money is Rs 750. Find the sum. 15. Kashyap took a bank loan of Rs. 5000000 to build a house. The loan was taken for 6 years at 15% per annum, on the condition that at the end of 6 years he pays back the loan with the interest.(a) Calculate the interest for six years.(b) Calculate amount after 6 years.(c) What sum must he pay bank to clear the loan?16. Ram had borrowed a loan of Rs. 6000 from a bank 4 years ago. If he had paid total amount of Rs. 9000 and got rid of the loan.(a) Write R in terms of I, P and T, where the symbol have their usual meanings.(b) Find the simple interest.(c) What was the rate of interest?17. Mrs. Chapagain borrowed Rs. 9,900 from a bank at the rate of 8% per annum. Answer the following questions.(a) What do you mean by the rate of 85 per annum?(b) Write the formula to calculate the simple interest.(c) How much interest does she pay at the end of 2 years?(d) How much amount should she pay at the end of 2 years if she want to clear the loan.Project WorkCollect the following information from the bank.i. Rate of interest on fixed deposit on 30 days.ii. Rate of interest on fixed deposit on 110 days.iii. Rate of interest on fixed deposit on 6 months.iv. Rate of interest on fixed deposit on 1 year.v. Interest rate on saving accountvi. Interest rate on current account

98 Acme Mathematics 81. A worker can do a piece of work in 14 days.(a) How much work does he do in 1 day?(b) How much work does he do in 7 day?(c) If he works for 2 days and leaves, how much work is left to finish?2. A bank gave Rs. 12,480 to Manish as the interest after 3 years for the deposit of Rs. 64,000.(a) Find the rate of interest.(b) Find the total amount he received from the bank.(c) If Manish divided Rs. 64,000 to Subodh and Arun in the ratio of 3:5 then how much money did each get? Find.3. Solve the following problems.(a) 15 men can do a piece of work in 20 days. In how many days 25 men can do the same work? Find it.(b) Anjali deposited Rs. 50,000 in a bank at the rate of 12% p.a. for 3 years. Find the amount received by her after 3 years.4. Hari deposits Rs. 5000 for 3 years at the rate of 10% p.a. simple interest and Harka deposits Rs. 4000 for 4 years at the same rate of interest.(a) What is the meaning of 10% rate of interest?(b) Find the simple interest of Harka.(c) If Harka has to pay 5% of interest as interest tax then find the interest received by him.5. Dipak has deposited Rs. 3,00,000 in a bank for 3 years at the rate of Rs. 8 interest per annum for Rs. 100.(a) At what percentage of interest rate per annum had Dipak deposited the amount of money?(b) How much interest will Dipak get in 3 years?(c) The age of two sons are 8 and 2 years respectively. If Dipak divides his Rs. 2,00,000 to his son based on the ratio of the age, how much more money will the elder son get than the younger son?Mixed Exercise

Acme Mathematics 8 996. Aashika deposited some amount of money in banks. She has money in bank A and bank B in the ratio of 3:2, if she deposited Rs. 60,000 in bank A.(a) How much did Aashika deposited in bank B?(b) How much interest will Aashika obtain in 2 years at the rate of 5% per annum from bank A?(c) For how many years will Aashika have to deposit the amount in bank B with same rate to obtain the same amount of interest as from bank A.7. Sita pays Rs. 12,600 as interest at the rate of 12% p.a. in 3 years for her loan.(a) Express R in terms of I, T and P.(b) How much loan has she taken? Calculate it.(c) Divide interest into two parts in the ratio of 2:3.8. Rachit took a loan of Rs. 20,00,000 from a bank to go for abroad study at the rate of 12% p.a. simple interest. If he cleared the loan after 10 year then,(a) Define amount.(b) What is the interest paid by her in 10 years?(c) If the interest rate was 10% p.a., what amount of money has she paid in total?9. Rohan deposited the money he had, in Global IME bank and Citizens bank in the ration of 5 : 4. He deposited Rs. 80, 000 in the bank Global IME.(a) How much money did Rohan deposit in the Citizens bank? Find it.(b) How much simple interest will Rohan obtain in 2 years at the rate of 10% per annum from Global IME bank? Find it.(c) For what rate will Rohan have to deposit the amount in bank Citizens to get the equal interest, obtained from Global IME bank on 2.5 years? Write with calculation.10. The amount of Rs. 8000 becomes Rs. 10,400 in 3 years then.(a) Define rate of interest. (b) Write formula to calculate principal. (c) Find the interest. (d) Find the rate of interest.

100 Acme Mathematics 811. Mr. Thapa bought 5 packets of milk for Rs. 350 from a grocery store.(a) At what rate of cost did he buy the milk? (b) Find the cost of 15 packets of milk at the same rate. (c) How many packets of milk can be bought for Rs. 2100?12. Rampresad bought 4 kg of rice and 3 kg of wheat at Rs. 430 from a shop. Also shopkeeper told Ramprasad that 3 kg of rice cost Rs. 210.(a) Find the cost of 1 kg of rice.(b) Find the cost of 1 kg of wheat. (c) Prapti deposit Rs. 5600 in a bank and she gets 35 of the principal as interest after few years.(i) Find the interest. (ii) Find the amount.