CHAPTER 3 Ratios, Rates and Proportions 86 E Conversion problems in rate EXAMPLE 21 (a) Mutton is sold at the rate of $15/kg. Convert the price rate to cents/g. (b) The speed of a car is 90 km/h. Change the speed to m/s. Solution: (a) 1 kg of mutton costs $15. \ Cost of 1000 g of mutton = (15 × 100) cents = 1500 cents Cost of 1 g of mutton = 1500 1000 = 1.5 cents/g \ Price rate = 1.5 cents/g (b) The car travels 90 km in 1 hour. \ Distance travelled in 3600 s = (90 × 1000) m = 90 000 m Distance travelled in 1 s = 90 000 3600 = 25 m/s \ Speed = 25 m/s EXAMPLE 22 (a) A reptile crawls 3 m across a road at the speed of 2 m/s. How much time does it take in seconds? (b) A car 40 m away from the reptile is travelling at a speed of 72 km/h. How long does it take to travel 40 m in seconds? Will the reptile cross the road safely? Speed = Distance Time Solution: (a) Time = Distance Speed = 3 m 2 m/s = 1.5 s (b) Speed = 72 km/h = 72 × 1000 m 3600 s = 20 m/s Time = 40 m 20 m/s = 2 s The time taken for the reptile to cross 3 m of road is less than the time taken for the car to travel 40 m. \ The reptile will cross the road safely.
Ratios, Rates and Proportions CHAPTER 3 87 EXAMPLE 23 In a fitness test, Nurul’s heartbeat rate was 630 times in 6 minutes. If Nurul repeated the test for 10 minutes, estimate the number of heartbeats that will be recorded. F Problem-solving Solution: Stage 1: Understand the problem List the facts and the question. Facts: Nurul's heartbeat rate = 640 times 6 minutes Question: Estimate the number of heartbeats recorded in 10 minutes. Stage 2: Think of a plan • Let the number of heartbeats be y times in 10 minutes. • Use the cross-multiplication method to find the value of y. Stage 3: Carry out the plan 630 6 = y 10 630 × 10 = 6 × y y = 630 × 10 6 = 1050 The number of Nurul's hearbeats in 10 minutes was 1050 times. Stage 4: Look back Work backwards to check. The number of heartbeats in 6 minutes = 1050 ÷ 10 × 6 = 630 Duration Rate ($/h) First 1 hour or part of it 2.00 Every additional 1 2 hour or part of it 0.50 The table shows the parking rates at a car park. Siti parked her car for 3 1 2 hours. How much she has to pay?
CHAPTER 3 Ratios, Rates and Proportions 88 Discuss with your teammates. Explain how ratios and rates are different. Describe a situation where you might use a ratio. Describe a situation where you might use a rate. INTERACTIVE ZONE Practice 3.2 Basic Intermediate Advanced A Find the rate of each of the following. (a) Water temperature increases from 30°C to 54°C in 12 min. (b) A toy car moves 300 cm in 5 s. (c) 10 l of milk cost $45. (d) 8 loaves of white bread cost $12. B 12 kg of potatoes cost $66. (a) What is the price rate of potatoes? (b) Find the selling price of 3 kg of potatoes. C The speed of an aircraft is 756 km/h. Express the speed of the aircraft in m/s. 4 Determine the unit rate for each of the following situations. (a) The price for 12 oranges is $9. (b) Aman walks 800 metres in 10 minutes. (c) Rima pays $15 for 5 kg of rice. (d) John types 135 words in 3 minutes. 5 Convert (a) price rate of $18/m to $/cm, (b) speed of 72 km/h to m/s, (c) density of 4 g/cm3 to kg/m3 , (d) pressure of 12 N/m2 to N/cm2 , (e) acceleration of 0.2 m/s2 to km/h2 . 6 (a) The temperature of a cup of coffee dropped constantly from 85°C to 60°C in 20 minutes. Find the cooling rate of the coffee in °C/min. (b) Water flow out from a tank at a constant rate. If the height of water decreases from 50 cm to 41 cm in one and a half hour, what is the flow rate of the water in cm/min. 7 (a) A bakery used 350 kg of flour in 7 days. Find the consumption rate of the flour in kg/day. (b) A car travelled a distance of 364 km on 20 litres of petrol. What is its average rate of consumption in km per l? 8 A factory worker received a total wage of $600 per week. Find (a) the worker’s daily wage rate if he works 6 days a week, (b) the worker’s hourly wage rate if he works 45 hours a week. 9 The population of a town was 250 000 in 2011. In 2021, the population increased to 400 000. (a) Find the size of the population increased. (b) What is the average rate of increase per year? (a) Teddy won 42 out of 54 tennis matches. Find his average win rate. (b) Karan won 20 out of 30 tennis matches. Find his average win rate. (c) Who do you consider as the better player? Why? K Shop A B C Quantity (kg) 2 4.5 5 Price ($) 7.00 14.40 19.00 The table above shows the price paid for the same item X at three shops. Arrange the shops from the lowest to the highest price rates. L Car Distance travelled (km) Petrol consumption ( ) P 54 4.5 Q 42 3 R 63 6 The table above shows the quantity of petrol consumed and the corresponding distance travelled by three cars P, Q and R. Determine the car that has the best rate and the worst rate of petrol consumption.
Ratios, Rates and Proportions CHAPTER 3 89 3.3 Proportions A Understanding proportions From the findings of Activity 2, you have use proportions to find the number of words that you and your friend copied in 5 minutes. Proportion is a relationship between two ratios that are equal. Two pairs of quantities are in proportions if they are equivalent ratios. It can be written in a : b = c : d or a b = c d . B Direct proportions (I) Direct proportions When two quantities are in direct proportion, they increase or decrease in the same rate. In other words, the ratio of the quantities is equivalent. For example, the more hours you work, the more you earn. A quantity x is proportional to another quantity y if the ratios of two pairs of quantities are the same. For example, x1 : x2 = y1 : y2 or x1 x2 = y1 y2 where x1 and x2 are values of quantity x whereas y1 and y2 are values of quantity y. Objective : To determine the relationship between ratio and proportion. Instruction : Do this activity in pairs. 1. Choose one storybook. Select a page and copy the text. Time allocation for copying is 5 minutes. 2. Your partner will act as a timekeeper, telling you when to start copying and when to stop copying. 3. Copy pages comfortably to minimise spelling mistakes. 4. Stop copying when the time is up. Mark the place at where you stopped copying. 5. Count the total number of words copied in the given 5 minutes. 6. Exchange roles with your partner and repeat Steps 1 to 5. and obtain data for your friend. Make sure to use the same page when copying. 7. Record your findings in the following table. Name of student Number of words Time (minutes) 5 5 2
CHAPTER 3 Ratios, Rates and Proportions 90 EXAMPLE 24 Determine whether the following pairs of ratios are a proportion. (a) $15 : $75 and 2 kg : 10 kg (b) 1 l : 3 l and 16 km : 50 km Solution: (a) $15 : $75 = 15 75 1 5 = 1 5 2 kg : 10 kg = 2 10 1 5 = 1 5 ∴ The two pairs of ratios are in proportion. (b) 1 l : 3 l = 1 3 16 km : 50 km = 16 50 8 25 = 8 25 ∴ The two pairs of ratios are not in proportion. EXAMPLE 25 The table shows the distance travelled and the amount of petrol used by a car. Distance travelled (km) 51 255 Amount of petrol (l) 3 15 Determine whether the distance travelled is directly proportional to the amount of petrol used. Solution: 51 km : 255 km and 3 l : 15 l 51 km : 255 km = 51 255 1 5 = 1 5 3 l : 15 l = 3 15 = 1 5 ∴ The distance travelled is directly proportional to the amount of petrol used. EXAMPLE 26 The length of a rope is directly proportional to its mass. If a 5 m rope is 1.2 kg, what is the mass of 15 m rope? Solution: Let the mass of 15 m rope = x kg x kg : 1.2 kg = 15 m : 5 m x 1.2 = 15 5 5x = 1.2 × 15 x = 1.2 × 15 5 = 3.6 kg ∴ The mass of 15 m rope is 3.6 kg.
Ratios, Rates and Proportions CHAPTER 3 91 EXAMPLE 27 If 3 kg of avocados cost $111, (a) find the cost of 1 kg of avocados. (b) what is the cost of 10 kg of avocados? (c) how many kilograms of avocados can be bought for $185? Solution: Using unitary method (a) 3 kg = $111 ∴ 1 kg = $ 111 3 = $37 Using proportion method (c) Let the quantity of avocados that can be bought = x kg x kg : 1 kg = $185 : $37 x 1 = 185 37 37x = 185 x = 185 37 = 5 kg ∴ 5 kg of avocados can be bought for $185. (b) Cost of 10 kg of avocados = $37 × 10 = $370 (II) Determining whether a quantity is directly proportional to another quantity There are many instances in our daily lives which involve direct proportion. For example, the price of sugar sold in a supermarket depends on the mass of the sugar. The price of sugar and the mass of sugar are quantities which varies directly with each other. When one quantity increases, the other quantity also increases. One quantity y is directly proportional to another quantity x if and only if y x = k, where k is called the constant of proportion. y directly proportional to x is written as y ∝ x. When y is directly proportional to x, the graph of y against x is a straight line which passes through the origin. EXAMPLE 28 Number of apples, N 3 4 5 6 Total cost ($), P 12 16 20 24 The table shows the number of apples and the total cost of apples that are sold at a fruit stall. Determine whether P is directly proportional to N and draw a graph to show the relation between N and P. x 0 y
CHAPTER 3 Ratios, Rates and Proportions 92 Solution: Number of apples, N 3 4 5 6 Total cost ($), P 12 16 20 24 P N 4 4 4 4 The value of P N is a constant. Therefore, P is directly proportional to N. N P 20 24 28 16 8 4 0 1 2 3 4 5 6 12 EXAMPLE 29 x (years) 5 6 7 8 y (kg) 18 20 21 24 The table shows the relationship between the mass, y kg, of a boy and his age, x years. Determine whether y is directly proportional to x. Solution: x (years) 5 6 7 8 y (kg) 18 20 21 24 y x 3.6 3.33 3 3 y x is not a constant. So, y is not directly proportional to x. When two quantities are in direct proportion, they increase or decrease at the same rate. The ratio of quantities is equivalent. For example, the more hours you work, the more you earn. When one quantity decrease at the same rate as the other quantity increase, the two quantities are in indirect proportion with each other. For example, time to complete a task and number of workers are in inverse proportion. (III) Expressing a direct proportion in the form of an equation involving two variables When a quantity y is directly proportional to another quantity x, we can express the proportion in the equation form y = kx where k is a constant which can be determined. EXAMPLE 30 The price, $ y, of a watermelon is directly proportional to the mass, m kg, of the watermelon. Given that y = 25 when m = 5, write an equation in terms of y and m.
Ratios, Rates and Proportions CHAPTER 3 93 Solution: y ∝ m y = km When m = 5, y = 25 25 = k(5) k = 25 5 = 5 Therefore, y = 5m. (IV) Finding the value of a variable in a direct proportion When y is directly proportional to x, the value of an unknown variable can be determined by writing the proportion in the equation form, then substituting the given value of a variable into the equation to find the value of the other variable. EXAMPLE 31 Given that p is directly proportional to q, and p = 18 when q = 6. Find the value of p when q = 5. Solution: p is directly proportional to q. So, the equation is p = kq. When q = 6, p = 18, 18 = k(6) k = 18 6 = 3 Hence, p = 3q. When q = 5, p = 3(5) = 15 When y ∝ x, the value of x or y can be found by using the relation (a) y = kx (b) y1 x1 = y2 x2 Using the proportion method, p1 q1 = p2 q2 Let p1 = 18, q1 = 6, q2 = 5 Then, 18 6 = p2 5 p2 = 3 × 5 = 15 (V) Solving problems involving direct proportion for cases y ∝ xn, where n = 1 2 , 2, 3 If y ∝ x n , where n = 1 2 , 2, 3, then the equation is y = kx n where k is a constant. The graph of y against x n is a straight line which passes through the origin. If y ∝ x n and sufficient information is given, the values of variable x or variable y can be determined.
CHAPTER 3 Ratios, Rates and Proportions 94 EXAMPLE 32 The area, A, of a circle is directly proportional to the square of its radius, r. Given that the area of the circle is 154 cm2 when its radius is 7 cm. Find (a) the equation for A in terms of r, (b) the value of r when A = 28 2 7 cm2 . Solution: (a) A is directly proportional to r2 . So, the equation is A = kr2 . When r = 7, A = 154, 154 = k × 72 k = 154 49 = 22 7 Therefore, A = 22 7 r 2 . (b) When A = 28 2 7 , 28 2 7 = 22 7 r 2 r = 198 22 = 9 r = 3 cm EXAMPLE 33 Given that y is directly proportional to the cube root of x, and y = 6 when x = 8. Find the value of x when y = 12. Solution: y ∝ 3 x, so y = k 3 x. When y = 6, x = 8 6 = k 3 8 6 = k(2) k = 3 y = 33 x The value of x or y for the proportion y ∝ x n can be obtained by using the relation (a) y = kxn (b) y1 x1 n = y2 x2 n When y = 12, 12 = 33 x 3 x = 4 x = 43 = 64
Ratios, Rates and Proportions CHAPTER 3 95 C Inverse proportions (I) Inverse proportion When one quantity decreases in the same rate as the other quantity increases, the two quantities are in indirect proportion with each other. For example, time to complete a task and number of workers are in inverse proportion. EXAMPLE 34 It takes 12 workers to plaster a building in 25 days. How long will 15 workers take to complete the same job if they work at the same rate? Solution: Using unitary method 12 workers = 25 days 1 worker = 12 × 25 = 300 days 15 workers = 300 15 = 20 days \ 15 workers will take 20 days to complete the same job. (II) Determining whether a quantity is inversely proportional to another quantity In a situation where one quantity increases, the other quantity decreases, such a situation involves inverse proportion. For example, the period of time taken to build a house would increase if the number of workers building the house is decreased. Conversely, if the number of workers increases, then the period of time to build the house decreases. If the variable y is inversely proportional to the variable x, then xy is a constant, that is, xy = k where k is called the constant of proportion. The symbol for ‘y is inversely proportional to x’ is y ∝ 1 x . When y is directly proportional to 1 x , the graph of y against 1 x is a straight line. EXAMPLE 35 Number of pupils, N 1 2 3 4 Time taken, T (minutes) 72 36 24 18 The table shows the number of pupils needed to clear some rubbish in a school and the time taken to clear the rubbish. Determine whether T is inversely proportional to N. x – 0 1 y
CHAPTER 3 Ratios, Rates and Proportions 96 Solution: Number of pupils, N 1 2 3 4 Time taken, T (minutes) 72 36 24 18 TN 72 72 72 72 The value of TN is a constant. Therefore, T is inversely proportional to N. (III) Expressing an inverse proportion in the form of an equation involving two variables When a quantity y is inversely proportional to another quantity x, we can express the proportion in the equation form y = k x where k is a constant which can be determined. EXAMPLE 36 Given that y is inversely proportional to x, and y = 12 when x = 5. Write an equation which relates x and y. Solution: y is inversely proportional to x, So, y = k x . When x = 5, y = 12 12 = k 5 k = 60 Therefore, the equation is y = 60 x . (IV) Finding the value of a variable in an inverse proportion When y is inversely proportional to x, the value of an unknown variable can be determined by writing the proportion in the equation form, then substituting the given value of a variable into the equation to find the value of the other variable. EXAMPLE 37 Given that J is inversely proportional to V, and J = 6 when V = 8. Find the value of J when V = 12. Solution: J is inversely proportional to V. So, the equation is J = k V . When V = 8, J = 6 6 = k 8 k = 6 × 8 = 48 Hence, J = 48 V . When V = 12, J = 48 12 = 4 Using the proportion method, J1 V1 = J2 V2 Let J1 = 6, V1 = 8, V2 = 12 Then, 6 × 8 = J2 × 12 ∴ J2 = 6 × 8 12 = 4 When y ∝ 1 x , the value of x or y can be found using the relation (a) y = k x (b) x1 y1 = x2 y2
Ratios, Rates and Proportions CHAPTER 3 97 (V) Solving problems involving inverse proportion for cases y ∝ 1 xn , where n = 1, 2, 3, 1 2 If y ∝ 1 xn , where n = 1, 2, 3, 1 2 , then the equation is y = k xn where k is a constant. The graph of y against 1 xn is a straight line which passes through the origin. If y ∝ 1 xn and sufficient information is given, the value of variable x or variable y can be determined. EXAMPLE 38 Given that V is inversely proportional to the square root of x, and V = 12 when x = 36. (a) Form an equation expressing V in terms of x. (b) Calculate the value of V when x = 9. (c) Calculate the value of x when V = 144. Solution: (a) V is inversely proportional to the square root of x. So, V = k x . When x = 36, V = 12, 12 = k 3 6 12 = k 6 k = 12 × 6 = 72 Therefore, V = 72 x . (b) When x = 9, (c) When V = 144, 144 = 72 x 144x = 72 x = 72 144 = 1 2 x = 1 1 2 2 2 = 1 4 V = 72 9 = 72 3 = 24 The value of x or y for the proportion y ∝ 1 xn can be found using the relation (a) y = k xn (b) x1 n y1 = x2 n y2
CHAPTER 3 Ratios, Rates and Proportions 98 D Solving problems related to proportions EXAMPLE 39 The ratio of the number of Mathematics books to the number of Science books in a bookshop is 9 : 4. If there are 24 Science books in the bookshop, how many Mathematics books are there in the bookshop? Solution: Using unitary method There are 9 parts Mathematics books and 4 parts Science books. Thus, 4 parts are equivalent to 24 books. 1 part is equivalent to 24 4 = 6 books. 9 parts are equivalent to 9 × 6 = 54 books. There are 54 Mathematics books. Using proportion method Let x be the number of Mathematics books. So, x : 24 = 9 : 4 x 24 = 9 4 x = 9 4 × 24 = 54 There are 54 Mathematics books. EXAMPLE 40 If the average speed of an airplane is 600 km/h, the flight time from Town P to Town Q is 8 hours. If the average speed of the airplane increases to 640 km/h, how much time will be saved? Solution: Distance travelled = Average speed × Time = 600 × 8 = 4800 km New flight time = Distance travelled Average speed = 4800 km 640 km/h = 7.5 h ∴ Time saved = 8 h – 7.5 h = 0.5 h
Ratios, Rates and Proportions CHAPTER 3 99 EXAMPLE 41 The ratio of the base to the height of a right-angled triangle is 3 : 4. If its perimeter is 6 cm and the length of the other side is 2.5 cm, find the length of the base and the height of the triangle. Solution: Let the base be b cm and the height be h cm. b : h = 3 : 4 b : b + h = 3 : 3 + 4 b : 3.5 = 3 : 7 b 3.5 = 3 7 b = 3 7 × 3.5 = 1.5 h = 3.5 – 1.5 = 2 Hence, base = 1.5 cm, height = 2 cm. Practice 3.3 Basic Intermediate Advanced h cm b cm 2.5 cm b + h + 2.5 = 6 b + h = 6 – 2.5 = 3.5 cm A Determine whether the following pairs of ratios are in proportion. (a) $70 : $168 and 5 kg : 12 kg (b) 2 l : 10 l and 36 km : 180 km (c) 160 km : 220 km and 2 h : 3 h (d) $200 : $75 and 8 l : 3 l B The table shows the number of eggs needed for baking a certain number of cakes. Number of cake baked 5 16 Number of eggs used 30 96 Determine whether the number of cakes baked is directly proportional to the number of eggs needed. C The length of a rod is directly proportional to its mass. If a 5 m rod is 8 kg, what is the mass of a 9 m rod? D A car used 1 litre of fuel to travel 18 km. How far could the car travel on 30 litres of fuel at the same rate? E A tray of 12 eggs costs $3. (a) Find the cost of an egg. (b) What is the cost of 30 eggs? (c) How many eggs can be bought for $270? F The time taken, t hours, to do a paint job is inversely proportional to the number of workers N. Given that 16 workers take 15 days to paint a factory. (a) How long will 24 workers take to paint the same factory? (b) How many workers are needed to paint the same factory in 16 days? G An airplane usually flies from City A to City B in 5 hours at an average speed of 680 km/h. Due to bad weather, the airplane decreased its average speed to 600 km/h. (a) How long would the same journey take at average speed of 600 km/h? (b) Find the time difference for the flight at two different average speeds.
CHAPTER 3 Ratios, Rates and Proportions 100 H E 2 4 F 16 m The table shows some values of the variables E and F. Given that F ∝ E2 , find the value of m. I G 16 128 y H 2 x 6 The table shows some values of the variables G and H. Given that G is directly proportional to the cube of H, calculate (a) the value of x, (b) the value of y. J G 16 36 F 12 m The table shows some values of the variables F and G such that F is inversely proportional to the square root of G. Find the value of m. K Given that b ∝ 1 a3 , and b = 3 when a = 2. Find the value of b when a = 4. 3.4 Relationship between Ratios, Rates and Proportions with Percentages, Fractions and Decimals A Determining the relationship between percentages and ratios Percentage is a fraction with 100 as its denominator. (a) Convert fraction to percentage. For example: 1 2 = 1 × 50 2 × 50 = 50 100 = 50% or 1 2 × 100% = 50% (b) Convert percentage to fraction. For example: 20% = 20 100 = 20 100 = 1 5 Fractions Percentages ×100% ÷100% A decimal can be converted to a fraction with power of 10 as the denominator. 0.1 = 1 10 , 0.01 = 1 100 Hence, decimals and percentages are interchangeable like fractions. Decimals Percentages ×100% ÷100% Percentage is a ratio comparing a part to the whole with the value of the whole set as 100. Percentage can be expressed in the fraction or decimal form. 1 5
Ratios, Rates and Proportions CHAPTER 3 101 EXAMPLE 42 The number of red radishes to the number of white radishes in a basket is in the ratio of 3 : 7. Find the percentage of white radishes in the basket. Solution: The ratio of the number of white radishes to the total number of radishes = 7 : 10 7 10 = 7 × 10 10 × 10 = 70 100 = 70% Hence, percentage of white radishes in the basket is 70%. EXAMPLE 43 Mr Zaid has a piece of land. He used 45% of the area of the land to plant durian trees. Find the ratio of the area of land used to plant durian trees to the total area of his land. Solution: Percentage of land used = 45% = 45 100 The ratio of the area of land used to the total area of the land = 45 : 100 = 9 : 20 B Determine the percentage of a quantity by applying the concept of proportions EXAMPLE 44 Determine the percentage of each of the following quantities by applying the concept of proportion. (a) A class has 25 students. 10 of the students are boys. What is the percentage of boys in the class? (b) Rahim has $400. He used $220 to buy pants and T-shirts. What is the percentage of money spent? Solution: (a) Let the percentage of boys in the class be m. Number of boys Total number of students = m 100 Proportion
CHAPTER 3 Ratios, Rates and Proportions 102 ×4 10 25 = m 100 ×4 m = 40 The percentage of boys in the class is 40%. (b) Let the percentage of money spent be s. Money spent Sum of the money = s 100 220 400 = s 100 220 × 100 = 400 × s s = 220 × 100 400 = 55 The percentage of money spent by Rahim is 55%. EXAMPLE 45 Determine the percentage of changes in each of the following quantities. (a) The population of a village increases from 35 600 to 37 380 people in 6 months. Find the percentage of the population increase of the village. (b) The temperature of some rice decreases from 80°C to 48°C after several minutes. Find the percentage of the temperature decrease of the rice. Solution: (a) The increase of the number of the population = 37 380 – 35 600 = 1780 Let the percentage increase in the population be p. The increase in population Total initial population = p 100 1780 35 600 = p 100 1780 × 100 = 35 600 × p p = 1780 × 100 35 600 = 5 The percentage increase in the population is 5%.
Ratios, Rates and Proportions CHAPTER 3 103 (b) Decrease in temperature = 80 – 48 = 32°C Let the percentage decrease in temperature be s. Decrease in temperature Initial temperature = s 100 32 80 = s 100 32 × 100 = 80 × s s = 32 × 100 80 = 40 The percentage decrease in the temperature of the rice is 40%. C Problem- solving EXAMPLE 46 Rafi sold his bicycle for $180 and made a loss of 10%. Calculate the original price of the bicycle. Solution: Stage 1: Understand the problem List the facts and the question. Facts: Rafi sold his bicycle at $180. He made a loss of 10% Question: What is the original price of his bicycle? Stage 2: Think of a plan • He sold the bicycle at $180 but gain a loss of 10%. • Loss means not giving any profit to Rafi. • Calculate the percentage of the Rafi's bicycle selling price. Stage 3: Carry out the plan Percentage of the selling price = 100% – 10% = 90% 90% of the original price = $180 1% of the original price = 180 90 = $2 100% of the original price = 100 × $2 = $200
CHAPTER 3 Ratios, Rates and Proportions 104 Stage 4: Look back Work backwards to check. Selling percentage = 100% – 10% = 90% Original price = Selling price Selling percentage = $180 90 100 = $200 EXAMPLE 47 Nadia bought 90 pieces of biscuits. She gave 3 5 of the biscuits to her friends. Then, she gave 18 pieces of the biscuits to her brother. Find the percentage of the biscuits remained with Nadia. Solution: Stage 1: Understand the problem List the facts and the question. Facts: Nadia has 90 pieces of biscuits. She gave to her friends 3 5 of her biscuits. She gave 18 pieces of biscuits to her brother. Question: What is the percentage of Nadia's remaining biscuits? Stage 2: Think of a plan • How many pieces of biscuits is 3 5 out of 90 pieces? • Then, subtract 8 pieces when she gave them to her brother. • Calculate the percentage of Nadia's remaining biscuits. Stage 3: Carry out the plan Number of remaining biscuits = 90 – 1 3 5 × 902 – 18 = 90 – 54 – 18 = 18 Percentage of the remaining biscuits = 18 90 × 100% = 20%
Ratios, Rates and Proportions CHAPTER 3 105 Stage 4: Look back Work backwards to check. Percentage of remaining biscuits = 90 – 1 3 5 × 902 – 18 90 × 100% = 20% Practice 3.4 Basic Intermediate Advanced A The number of students who wear watches to the number of students who do not wear watches in a class is 9 : 16. Find the percentage of students who wear watches. B The number of female workers in a factory decreased from 72 to 54. Find the percentage decrease in the number of female workers. C The number of members of a badminton club has declined 16% to 42 people. Determine the initial number of members of the club. 4 Jun Hong answered 65% of the questions correctly in a Science quiz. He answered 7 questions wrongly. If Jun Hong answered all the questions in the quiz, how many questions are there in the Science quiz? 5 The initial price of a box of chocolate is $22.00. The selling price of the chocolate is 4 5 of the initial price. Maria used 40% of her pocket money to buy the chocolate. How much pocket money does Maria have? 6 The mass of John is 20% heavier than the mass of Marie. The mass of Wong is 12% lighter than Marie’s mass. If John’s mass is 66 kg, find the mass of Wong. 7 The money collected for an excursion is divided into two portions. 0.4 of the money collected is used to pay for transport. 3 5 of the second portion is used to pay for food and drinks. The balance of the second portion which is $480 is used to buy souvenirs. Determine the percentage and the total amount of money used to pay for food and drinks. 3.5 Scale A scale is a ratio. It can be expressed as the length on a drawing (map / plan / model) to the length of the actual object. A scale is used to show a real object with sizes reduced or enlarged by a certain ratio / number. Map scales are good examples of ratios in everyday life. The scale of a map is usually expressed in the form 1 : n. For example, a scale of 1 : 250 000 means that 1 cm on the map represents 250 000 cm on the ground. They must be in the same unit. Scale of a drawing = Size of the drawing Size of the object or Length of side of the drawing Length of corresponding side of the object
CHAPTER 3 Ratios, Rates and Proportions 106 For example, A 3 cm 6 cm B 4 cm 8 cm Drawing Object If A is the scale drawing of B, then the scale is 3 : 4. The scale of a scale drawing is usually expressed in the form 1 : n which means 1 unit in the drawing represents n units in the actual object. For example, the scale of 3 : 4 is expressed as 1 : 4 3 . In the scale of 1 : n, if (a) n = 1, then the drawing and object have the same size. (b) n . 1, then the drawing has a smaller size than the object. (c) n , 1, then the drawing has a larger size than the object. If a drawing has a scale of 1 : n, then the drawing is 1 n times the size of the object. The picture above (not drawn to scale) is a 1 : 48 000 000 scale map of the sailing route from Kalimantan to Java. The sailing route shown on the map is about 1.5 cm. What is the distance of the boat trip in kilometres? Give the answer correct to 3 significant figures. 4 INDONESIA Java Kalimantan 480 km 1 cm
Ratios, Rates and Proportions CHAPTER 3 107 EXAMPLE 48 ? 4 cm Drawing Actual The diagram shows a drawing of an actual horse. If the scale is 1 : 50, what is the height of the actual horse? Solution: Scale = 1 : 50 Drawing length : Actual length = 1 cm : 50 cm = (4 × 1) cm : (4 × 50) cm = 4 cm : 200 cm = 4 cm : 2 m ∴ The height of the actual horse is 2 m. EXAMPLE 49 Write the following scales in the form 1 : n. (a) 1 cm represents 20 m (b) 2 cm represents 5 km Solution: (a) 1 cm : 20 m = 1 cm : (20 × 100) cm = 1 cm : 2000 cm ∴ The scale is 1 : 2000. (b) 2 cm : 5 km = 2 cm : (5 × 1000 × 100) cm = 2 cm : 500 000 cm = 1 cm : 250 000 cm ∴ The scale is 1 : 250 000. EXAMPLE 50 A map has a scale of 1 : 10 000. If the distance between two buildings on a map is 6 cm, what is the actual distance on the ground between two buildings? A scale is usually expressed in the form of 1 : n. For example, 1 cm to 1 km. Actual length = Drawing length × Scale Drawing length = Actual length ÷ Scale Convert: m to cm km to cm
CHAPTER 3 Ratios, Rates and Proportions 108 Solution: Scale = 1 : 10 000 1 cm : 10 000 cm = 1 cm : 100 m Map distance : Actual distance = (6 × 1) cm : (6 × 100) m = 6 cm : 600 m ∴ The actual distance between two buildings on the ground is 600 m. EXAMPLE 51 A length of 3 cm on the map represents a distance of 6 km. (a) Find the scale of the map in the form of 1 : n. (b) Find the length on the map that represents a distance of 900 m on the ground. Solution: (a) Given that 3 cm represents 6 km. 6 km = 6 × 1000 m = 6000 m = 6000 × 100 = 600 000 cm Scale = 3 cm : 600 000 cm = 3 3 : 600 000 3 = 1 : 200 000 ∴ The scale of the map is 1 : 200 000. (b) Let the map distance = x cm 200 000 cm = 200 000 ÷ 100 = 2000 m Scale = 1 cm : 2000 m x cm : 900 m = 1 cm : 2000 m x 1 = 900 2000 x = 0.45 cm ∴ 0.45 cm on the map represents 900 m. Scale enables comparison between the actual size of an object with the size of its drawing. A scale can be expressed as ratio (for example, 1 : 15 000) or fraction (for example, 1 15 000 ). Explain how would you know whether a drawing drawn to scale 5 : 2 is bigger or smaller than the actual object.
Ratios, Rates and Proportions CHAPTER 3 109 A map designer measures the main street from the airport to the bank and get 5 cm. Cycling along the main street from the airport to the bank is known to be 1 km 200 m. What ratio is the map using? Give your answer in the form 1 : x. Critical Thinking EXAMPLE 52 The homeowner wishes to do some renovations to his bedroom. He wants to add a bookshelf, which is approximately 150 cm. He will place it on the floor plan marked with a X. Given that the floor plan of the house is 1 : 200. What length should the bookshelf be drawn to in the floor plans? Solution: He wanted to add a bookshelf of about 150 cm, which he would place on the floor plan marked with a X. The ratio for the scale drawing is 1 : 200. This means 200 cm in the real world is represented by 1 cm on the floor plan. 00:00 MIN 00:00 MIN 00:00 MIN 00:00 MIN Main Street
110 CHAPTER 3 Ratios, Rates and Proportions Let l be the length of the bookshelf on floor plan. The ratio 1 : 200 is equivalent to l 150 = 1 200 200 l = 150 l = 150 200 = 0.75 cm The length of the bookshelf that should be drawn on the floor plan is 0.75 cm. Practice 3.5 Basic Intermediate Advanced A 3.5 cm The diagram shows a drawing of an elephant. If the scale is 1 : 100, what is the actual height of the elephant? B Write the following scales in the form 1 : n. (a) 1 cm represents 10 m (b) 5 cm represents 100 m (c) 2 cm represents 1 km (d) 5 cm represents 2 km C Simplify each of the following scales. (a) 1 mm to 5 m (b) 2 cm to 3 km D A map of Mauritius has a scale of 1 : 500 000. If the distance between Port Louis and Quatre Bornes on the map is 3.2 cm, what is the actual distance (on the ground) between these two towns? E A map has a scale of 1 : 50 000. If the actual distance between two bus stations is 3 km, what is the distance between these two bus stations on the map? 6 A particular map shows a scale of 1 : 500. A rectangular pond measures 9 cm by 6.5 cm on the map. (a) Find the actual length and breadth of the pond in metres. (b) What is the actual area of the pond in hectares? (1 ha = 10 000 m2 ) 7 D E C B A 4 cm 12 cm 5 cm The diagram shows a rose garden ABE in a farm ABCDE. Using a scale of 1 cm to represent 2 m, (a) draw an accurate scale drawing of the farm, (b) find the length of AE in the diagram, (c) find the perimeter of the rose garden. 8 The map of Mauritius shows that the distance between Port Louis and Curepipe is 5.2 cm. Estimate the actual distance (on the ground) between these two towns. Port Louis Rose Hill Quatre Bornes Curepipe Vacoas Goodlands Bon Accueil Bel Air Mahebourg Chemin Grenier 0 10 20 30 40 50 Kilometres
111 Ratios, Rates and Proportions CHAPTER 3 9 A map has a scale of 1 : 500 000. What length on the map represents a 54 km long railway line? On the floor plan of an office with a scale of 1 : 200, the dimensions of rectangular meeting room are 11 m by 8 m. (a) Draw an accurate scale drawing of the meeting room. (b) Find the area of the meeting room on the map in cm2 . Summary Summary Summary Ratio of two quantities • Comparison of two quantities in the same unit and written as a : b or a b . • a : b and c : d are equivalent ratios if a b = c d . • Given a ratio a : b, where a > b – Sum of two quantities can be found by adding the quantities, a + b. – Difference of two quantities can be found by subtracting the quantities, a – b. • When a number is increased in the ratio p : q, where p > q, the resulting value is obtained by multiplying the value by the multiplying factor p q . • When a number is decreased in the ratio p : q, where p < q, the resulting value is obtained by multiplying the value by the multiplying factor p q . • A rate is usually expressed as one quantity per unit of another quantity. Rates must include the units of quantities. Scale • A scale can be expressed as ‘length on drawing : actual length’. It is usually given as a ratio form of 1 : n. Map scale = map distance : actual distance Ratio of three quantities • Comparison of three quantities in the same unit and written as a : b : c. • Given that the ratios a : b and b : c, the ratio a : b : c can be found by using the common quantity b. – If the value of b is the same in both ratios, then the two ratios can be combined and written as a : b : c. – If the values of b are different, then the common value for b can be found by determining their LCM. Direct proportion • If y is directly proportional to x, then (a) y ∝ x (b) y = kx, k is a constant • If y ∝ x n, then (a) y = kxn , k is a constant (b) y1 x1 n = y2 x 2 n Inverse proportion • If y is inversely proportional to x, then (a) y ∝ 1 x (b) y = k x , k is a constant • If y ∝ 1 xn , then (a) y = k xn , k is a constant (b) x1 n y1 = x2 n y2 Integers Ratios, Rates and Proportions Proportion • If two pairs of ratios are equivalent, they are proportional. For example, if a : b is equivalent to c : d, then a : b = c : d or a b = c d . • When quantities are in direct proportion, they increase or decrease at the same rate. • When quantities are inversely proportional, one increases as the other decreases.
112 CHAPTER 3 Ratios, Rates and Proportions Section A 1. Given that x : y = 35 : 27, find x – y : x + y . A 4 : 31 C 7 : 3 B 5 : 3 D 31 : 4 2. Given that s : t = 18 : 5, find the ratio of t : s – t. A 2 : 5 C 4 : 11 B 3 : 8 D 5 : 13 3. Given that p : q = 9 : 13 and p = 135, what is the value of q? A 180 C 200 B 195 D 205 4. Given that m : n = 15 : 7 and m + n = 143, what is the value of m – n? A 52 C 64 B 54 D 68 5. 2 m for $25 7 m for $86 K 4 m for $48 10 m for $120 N 3 m for $39 10 m for $139 L 5 m for $68 7 m for $94 M The diagram shows the prices of cloth sold by four shops, K, L, M and N. In which shop was the cloth sold at a price that is in proportion to the length of cloth? A K C M B L D N 6. The ratio of the time Jalil spent on reading storybooks to the time he spent on watching television was 65 : 72. If the total time spent was 6.85 hours, calculate the time spent on reading storybooks. A 2 hours 50 minutes B 2 hours 55 minutes C 3 hours 10 minutes D 3 hours 15 minutes 7. A few months ago, the ratio of Suzie’s mass to Ming’s mass was 22 : 23 and their total mass was 90 kg. Now, Suzie is 0.5 kg lighter and Ming is 1 kg heavier. Find the new ratio of Suzie’s mass to Ming’s mass. 3 A 45 : 47 C 87 : 94 B 46 : 47 D 90 : 93 8. For a mathematics test in March, Alan and Sim scored a total mark of 190. The ratio of Alan’s marks to Sim’s marks was 49 : 46. If Alan scored 90 marks in another mathematics test in April, how many marks must Sim get so that his total mark for March and April is the same as Alan’s? A 94 C 96 B 95 D 97 9. 14 cm 4 cm 3 cm 3.2 cm 2.8 cm 1 cm Stall 1 Stall 2 Stall 3 Stall 6 Stall 5 Stall 4 The diagram shows the plan of a hawker centre. Each stall occupies a rectangular floor space. The plan is drawn to a scale of 1 : 200. Calculate the area of the floor space, in m2 , which stall 5 occupies. A 44.8 C 58.4 B 49.0 D 78.4 10. 5 cm 8 cm S T 9 cm The diagram shows two parallelograms, S and T. The length of the base and the height of S are in proportion to the length of the base and the height of T. Given that the height of S is 6 cm, calculate the height, in cm, of T. A 12 C 24 B 18 D 30 Section B 1. A van travels a distance of 175 km in 2 1 2 hours. Calculate the average speed of the van in (a) km/h, (b) m/s.
113 Ratios, Rates and Proportions CHAPTER 3 2. The table shows the mass of three pupils. Name of pupils Mass (kg) Jonah 54 Karan 60 Ryan 48 Express each of the following ratios in its simplest form. (a) Ryan’s mass : Karan’s mass (b) Jonah’s mass : Karan’s mass : Ryan’s mass 3. A small airplane flies a distance of 100 km on 25 litres of fuel. (a) Find the rate of fuel consumption in km/l. (b) If 1 litre of fuel costs 60 cents, what is the expected cost of fuel for a journey of 350 km? 4. Ritesh is paid $780 for 29 hours of work. (a) If he works for 24 hours, find his wage. (b) How many hours does he need to work in order to earn $1350? 5. (a) A 9.6 MB internet file takes 12 seconds to download. What is the rate at which file is downloaded? Give answer in kilobytes per second. (1 megabyte = 1024 kilobytes) (b) Anita's computer is downloading a 4.6 MB internet file at the rate of 340 KB/s. How long it take to download the file? Give your answer to the nearest second. 6. (a) The scale of a street map is 1 : 50 000. If the actual length of a street is 4 km, what is the length of the street on the map in cm? (b) A map has a scale of 1 : 500 000. If the distance between 2 towns is 6.4 cm on the map, what is the actual distance between the two towns in km? (c) The floor plan of a flat is drawn with a scale of 1 : 50. On the floor plan, the length of the flat is 30 cm. Find the actual length of the flat in m. 7. The price of a queen-size mattress is $950 and that of a king-size mattress is $1197. (a) The price of the queen - size mattress decreases to $800. In what ratio has the price decreased? (b) If the price of the king-size mattress decreases in the same ratio, what is its new price? 8. In an online sale, the prices of certain kitchenware were reduced in the ratio 4 : 5. (a) Find the multiplying factor. (b) What is the new price of an article which cost $450 initially? (c) Aurelie bought an article and paid $220, what was the original price? 9. The ratio of ingredients to make butter cookies is one part of sugar to two parts of butter and three parts of flour. (a) Find the ratio of sugar, butter and flour. (b) How much flour would Mrs Smith need if she used 25 kg of butter to bake butter cookies? 10. (a) Joy earned a total of $75 by selling 5 glasses of mixed fruit juice. After selling a total of 24 glasses of mixed fruit juice, how much money will she earn? Assume the relationship is directly proportional. (b) Anne made 14 finger puppets in 4 hours. (i) Find the number of finger puppets made by Anne in 18 hours. (ii) How many hours did she spend to make 56 finger puppets? Assume the relationship is directly proportional. 11. Given that F = k T , and T = 4 + S, where k is the constant of the proportion. (a) Find the value of k when F = 4 and S = 5. (b) State the equation that relates F and T. (c) Calculate the value of F when S = 32. 12. Given that R is inversely proportional to S and the cube of T. R = 1 4 when S = 5 and T = 2. Find the value of R when S = 10 and T = 3.
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