[direct variation][section 23]

(1.) y varies directly as x. If y = 6 when x = 4, find

y when x is 10.

y = kx use this formula

6 = k(4) replace y with 6, replace x with 4
__ ___
4 4 divide each side by 4

1.5 = k divide and cancel

y = kx use this formula again

y = 1.5x replace k with 1.5

y = 1.5(10) replace x with 10

y = 15 multiply

result: y = 15

(2.) y varies directly as x. If y = -2 when x = -1, find

y when x is 5.

y = kx use this formula

-2 = k(-1) replace y with -2, replace x with -1
___ _____
-1 -1 divide each side by -1

2 = k divide and cancel

y = kx use this formula again

y = (2)(5) replace with 2, replace x with 5

y = 10 multiply

result: y = 10

(3.) y varies directly as x. If y = 12 when x = 3, find

y when x = 24.

y = kx use this formula

12 = k(3) replace y with 12, replace x with 3
____ _____
3 3 divide each side by 3

4 = k divide and cancel

y = kx use this formula again

y = (4)(24) replace k with 4 & replace x with 24

y = 96 multiply

result: y = 96


(4.) The amount of pay, A, that Sam earns varies directly with

the number of hours, h, that he works. Last week, he

earned $108.75 for 25 hours of work. How much pay will

he earn for 120 hours of work?

A = kh here is the problem

108.75 = k(25) replace A & h with 108.75 & 25
_______ _____
25 25 divide each side by 25

4.35 = k divide and cancel

A = kh use the formula again

A = (4.35)(120) replace k & h with 4.35 & 120

A = 522 multiply

result: $522

(5.) On a scale drawing, 2 cm represents 50 meters. How many

meters will 5 cm represent?

m = kc use this formula

50 = k(2) replace m with 50 and c with 2
___ ___
2 2 divide each side by 2

25 = k divide and cancel

m = kc use the formula again

m = (25)(5) replace k & c with 25 & 5

m = 125 multiply

result: 125 meters

(6.) The gas consumption of a car varies directly as the

distance travelled. If a certain car uses 20 liters of

gas to travel 200 km, how many liters of gasoline will be

used on a trip of 700 km?

g = kd use this formula

20 = k(200) replace g & d with 20 & 200
____ ______
200 200 divide each side by 200

.1 = k divide and cancel

g = kd use this formula again

g = (.1)(700) replace k & d with .1 & 700

g = 70 multiply

result: 70 liters of gasoline

(7.) The amount of time, T, that it takes to read an article

varies directly with the length of the article. Sam

takes 3 minutes to read an article of 315 words. How

long will it take him to read an article of 945 words?

T = kw use this formula

3 = k(315) replace T & w with 3 and 315
___ ______
315 315 divide each side by 315


1/105 = k reduce and cancel

T = kw use this formula again

T = (1/105)(945) replace w with 945

T = 9 multiply

result: 9 minutes

(8.) The distance between two towns on a map varies directly

with the actual distance between the two towns. If 2.5

inches on the map represents 150 miles what is the actual

distance represented by 7 inches on the map?

m = kd use this formula

2.5 = k(150) replace m and d with 2.5 and 150
_____ _______
150 150 divide each side by 150

1/60 = k cancel

m = kd use this formula again

m = (1/60)d replace k with 1/60

7 = (1/60)d replace m with 7

(60)(7) = (60)(1/60)d multiply each side by 60

420 = d multiply and cancel

result: 420 miles

(9.) The speed of the blade tips of a windmill varies

directly with the speed of the wind. In a 20 mph

wind, the speed of the blade tips is 180 mph. Find

the speed of the blade tips in a 5 mph wind.

b = kw use this formula

180 = k(20) replace b with 180 and w with 20
_____ _____
20 20 divide each side by 20

9 = k divide and cancel

b = kw use this formula again

b = (9)(5) replace k with 9 and replace w with 5

b = 45 multiply

result: 45 mph

(10.) The amount of antifreeze required to prevent freezing at

-25^o C varies directly with the capacity of the cooling

system. A 20 liter cooling system requires 6.2 liters

of antifreeze. How much antifreeze is needed in a 25

liter system?

A = kS use this formula

6.2 = k(20) replace S with 20 , replace A with 6.2
_____ ______
20 20 divide each side by 20

.31 = k divide and cancel

A = kS use this formula again

A = .31(25) replace k & S with .31 & 25

A = 7.75 multiply

result: 7.75 liters