BRAIN WORKS Direct Proportion

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1 BRAIN WORKS 16. California, the third largest state in the United States, has an east-west dimension of 480 km. (a) A map of California, drawn with a scale of 1 : 1,500,000, fits eactl from north to south inside a rectangle of length 84 cm. What is the north-south dimension of California? (b) If ou sketch a map of California on a piece of paper 1 cm b 8 cm, what is an appropriate scale ou would choose such that the map fits inside a rectangle with a uniform border of 1 cm around the sides of the paper? Direct Proportion When two quantities are related to each other, their changes ma follow a certain pattern. For instance, if a sandwich costs $1 at our school cafeteria, ou will have to pa $ for two sandwiches, $3 for three sandwiches, and so on. Thus, the amount ou pa is related to the number of sandwiches ordered. We shall eplore this special relationship further in the following class activit. Objective: To understand the idea of direct proportion. 1 Questions The following table shows the relationship between the number of books bought () and the total cost of the books ($). Number of books () Total cost ($) Chapter 14 proportions 17

2 (a) Cop and complete the following table (b) On a sheet of graph paper, plot the corresponding points (, ) in (a) using the scale for both aes as shown. (c) What can ou sa about the points ou have plotted in (b)? (d) Write down an equation connecting and. (e) What is the value of when = 8? (f) Does the graph of the equation in (d) pass through the origin (0, 0)? Total cost ($) O Number of books When two quantities, and, var in such a wa that is a constant, the are said to be in direct proportion, or is said to be directl proportional to. REMARKS Do ou know that a distance on a map and its actual distance are in direct proportion? In Class Activit 1, we have which is a constant. = 5 1 = 10 =... = 30 6 = 5, Hence, the number of books bought (), and the total cost ($) are in direct proportion. Also, we see that the points (, ) lie on a straight line passing through the origin. The equation of the straight line is = 5. 18

3 When two quantities, and, are in direct proportion, we have = k, where k is a constant, that is = k In Class Activit 1, the constant k = 5. Thus, = 5. In general, we have the following properties: When two quantities, and, are in direct proportion: = k, where k is a constant the graph of = k is a straight line passing through the origin Discuss In the graph in Class Activit 1, (a) what is the gradient of the straight line? (b) what does the gradient represent? Eample 7 The following table shows the traveling time (t hr) and the distance covered ( mi) b a car. Traveling time (t hr) Distance covered ( mi) (a) Show that t and are in direct proportion. (b) Draw the graph of against t. (c) Find the equation connecting t and. (d) Find the value of when t = 1.5. (e) Find the value of t when = 7. Solution (a) t t Since t is a constant, t and are in direct proportion. Chapter 14 proportions 19

4 (b) The graph of against t is shown below. 150 Distance covered (mi) = 60t O 1 3 Traveling time (hr) t (c) Since t = 60t. = 60, the equation connecting t and is (d) When t = 1.5, = Substitute t = 1.5 into the = 75. equation in (c). (e) When = 7, 7 = 60t t = 7 60 = 1. Substitute = 7 into the equation in (c). Tr It! 7 The following table shows the number of hours (t) worked and the corresponding wages ($w) of a worker. Number of hours worked (t) Wages ($w) (a) Show that t and w are in direct proportion. (b) Draw the graph of w against t. (c) Find the equation connecting t and w. (d) If the worker worked 35 hours, find his wages. (e) If the wages of a worker were $675, find the number of hours he worked. 130

5 When two quantities, and, are in direct proportion, = k, where k is a constant. Suppose ( 1, 1 ) and (, ) are an two pairs of values of and, where 0 and 0, then 1 = k 1 and = k. 1 Hence, 1 that is k k = 1 = 1. When two quantities, and, are in direct proportion, given an two pairs of values of and, ( 1, 1 ) and 1 1 (, ), we have =, where 0 and 0. The equalit of two ratios is called a proportion. We sa that 1,, 1, and are in proportion. Eample 8 The height, H cm, of a pile of books is directl proportional to the number of books, n, in the pile. The height of a pile of 10 books is 15 cm. What is the height of a pile of 4 books? Solution Method 1 Since n and H are in direct proportion, H = kn, where k is a constant. When n = 10, H = = k 3 10 k = = 1.5 Hence, H = 1.5n When n = 4, H = = 36 The height of a pile of 4 books is 36 cm. REMARKS The value of k is actuall the thickness of each book in centimeters. Estimate: Is the answer reasonable? 4 books are slightl more than twice of 10 books. Is the answer 36 cm slightl more than twice of 15 cm? Chapter 14 proportions 131

6 Method Since n and H are in direct proportion, H1 H = n 1 n Let n 1 = 10, H 1 = 15, and n = 4. Substituting these values into the proportion, we have: 15 H = 10 4 H = = 36 Hence, the height of a pile of 4 books is 36 cm. Tr It! 8 The number of pieces, n, of chocolate and their total mass, M grams, are in direct proportion. The mass of 15 pieces of chocolate is 180 g. Find the mass in grams of 0 pieces of chocolate. In real life, two quantities, and, ma not be in direct proportion but n and m, where n and m are rational numbers, ma be in direct proportion. The following eample illustrates such a situation. Eample 9 The weight of a solid metal sphere and the cube of its radius are in direct proportion. When the radius of the sphere is in., its weight is 19 oz. Find the weight in ounces of a sphere of radius of 1.5 in. Solution Tr It! 9 Let the weight of a sphere of radius r in. be m oz. Then, we have m = kr 3, where k is a constant. When r =, m = 19. \ 19 = k 3 3 k = 19 8 = 4 \ m = 4r 3 When r = 1.5, m = = 81 The weight of a sphere of radius 1.5 in. is 81 oz. The mass of a square piece of glass panel is directl proportional to the square of the length of its side. When its side is 0 cm, the mass is 1,000 g. Find the mass in grams of a glass panel of side 30 cm. 13

7 EXERCISE 14.3 BASIC PRACTICE 1. In each of the following tables, determine whether and are in direct proportion. (a) (b) (c) In each of the following equations, determine whether and are in direct proportion. (a) = 4 (b) = + (c) = (d) = 1 4. If two quantities, and, are in direct proportion, find the values of p and q in the following table q 8 p 4 (d) It is given that w is directl proportional to t. When t = 4, w = 0. Find (a) the value of w when t = 6, (b) the value of t when w = 45.. In each of the following graphs, determine whether and are in direct proportion. (a) It is given that A is directl proportional to r and r 0. When r = 5, A = 75. Find (a) the value of A when r = 4, (b) the value of r when A = 147. FURTHER PRACTICE 7. The following table shows the total price ($P) for copies of books. O 4 6 Copies of books () (b) Total price ($P) O (a) Show that and P are in direct proportion. (b) Draw the graph of P against. (c) Describe the graph in (b). (d) Find the equation connecting and P. (e) Hence, find the total price for 8 copies of books. Chapter 14 proportions 133

8 8. The following table shows the mass (m g) of a pinewood cube of side cm. Length of a side ( cm) Mass (m g) (a) Is m proportional to? (b) Is m proportional to 3? (c) Find an equation connecting m and. (d) Hence, find the mass in grams of a pinewood cube of side 9 cm. 1. The period (the time taken for one complete oscillation) of a simple pendulum is directl proportional to the square root of its length. When its length is 1.0 m, its period is.01 seconds. Find (a) the period of the pendulum when its length is 0.8 m, (b) the length of the pendulum when its period is 1.0 second. Give our answers correct to decimal places. MATH@ WORK 9. The cost of renting a car is directl proportional to the number of das the car is being rented for. The cost of renting a car for 4 das is $40. Find the cost of renting a car for 7 das. 10. The mass of a metal plate is directl proportional to its volume. When its volume is 0 cm 3, its mass is 10 g. If the volume of the metal plate is 50 cm 3, what is its mass? 11. When a car is traveling steadil along a highwa, its consumption of gasoline is directl proportional to the distance traveled. A car travels 100 mi on.7 gal of gasoline. Find, giving our answer correct to 1 decimal place, (a) the gasoline consumption of the car for a distance of 74 mi, (b) the maimum distance that the car can travel with 1 gal of gasoline. length 13. The vertical falling distance of a ball is directl proportional to the square of the time of falling. The ball falls 80 m in 4 s. (a) Find the vertical falling distance of the ball when the time taken is 6 s. (b) If the ball is dropped from a height of 45 m, find the time it takes to hit the ground. BRAIN WORKS 14. (a) In our dail life, we often encounter a wide variet of quantities involving direct proportion. Describe two such quantities. (b) Draw a graph to show their relationship. (c) Find an equation connecting the quantities. 15. Jordan s height and weight increase as he grows bigger. Do ou think his height and weight are in direct proportion? Give reasons for our answer. 134

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