[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

-25o 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