# Direct Variation Worksheet

Direct Variation Worksheet
• Page 1
1.
A soap manufacturing company packs 52 soaps per carton. Find the number of soaps that can be packed in 10 cartons, using a direct variation model.
 a. 468 b. 624 c. 572 d. 150

#### Solution:

Let y be the number of soaps and x be the number of cartons.

Since, the number of soaps and the number of cartons vary directly, the equation is of the form y = kx.

As each carton contains 52 soaps, the constant of variation is 52.

y = 52x
[Substitute the values.]

y = 52 × 10
[Substitute the values.]

y = 150
[Multiply.]

So, 150 soaps can be packed in 10 cartons.

2.
Which one of the following is not a direct variation?
 a. The amount of gas a person buys and the money he pays b. Profit and sales c. Circumference of a circle and its diameter d. A person′s age and his height

#### Solution:

"A personÃ¢â‚¬â„¢s age and his height" is not a direct variation, because people do not grow at constant rate.

3.
If the circumference of a circle varies directly as its diameter, then the constant of variation is
 a. 3π b. 2π c. 4π d. π

#### Solution:

Circumference of a circle = 2π r, where r - radius of the circle.

= π d
[ d = 2 r = diameter.]

So, the constant of variation is π.

4.
Determine whether the graph represents a direct variation or not.

 a. no b. yes

#### Solution:

The line is passing through the origin. So the equation of the line will be of the form y = mx.

So, the graph represents a direct variation.

5.
For the given function, find its equation if $y$ varies directly as $x$.
 $x$ $y$ 9 - 27 - 10 30 - 11 33
 a. $y$ = - 3$x$ b. $y$ = $\frac{1}{3}$$x$ c. $y$ = 5$x$ d. $y$ = 31$x$

#### Solution:

yx = - 27 / 9 = 30 / - 10 = 33 / - 11 = - 3

So, y varies directly as x. That is y = kx

The constant of variation is - 3.

So, the equation is y = - 3x.

6.
Determine whether $y$ varies directly as $x$, if so, write its equation.
 $x$ $y$ 6 8 8 6 11 2
.
 a. $y$ = ($\frac{6}{8}$)$x$ b. $y$ = 0.6$x$ c. $y$ does not vary directly as $x$ d. $y$ = 1.33 $x$

#### Solution:

yx = 8 / 6 = 1.33 . But 6 / 8 ≠ 1.33 and 2 / 11 ≠ 1.33 .

Since the ratio yx is not a constant, y does not vary directly as x.

7.
Find the slope of the direct variation $y$ = ($\frac{2}{3}$)$x$.
 a. - $\frac{3}{2}$ b. 2 c. 3 d. $\frac{2}{3}$

#### Solution:

y = 2 / 3 x

Constant of variation = k = 2 / 3

The slope of a direct variation has the same value as its constant of variation k.

Therefore, the slope of the given direct variation is 2 / 3 .

8.
A car runs 18 mi. on 2 gallons of gas. The number of gallons of gas varies directly with the number of miles the car runs. How many miles will the car run on 4.4 gallons of gas?
 a. 39.6 mi. b. 2 mi. c. 9 mi. d. 18 mi.

#### Solution:

Let x be the number of miles the car runs and let y be the number of gallons of gas used to travel x miles.

The relation is y = kx.
[Direct variation.]

Let (x1, y1) = (18, 2) and let (x2, y2) = (x2, 4.4)

y2x2 = y1x1
[Write a proportion.]

4.4x2 = 2 / 18
[Replace x1 = 18, y1 = 2, y2 = 4.4.]

x2 = (18)(4.4) / 2 = (9)(4.4) = 39.6 mi.
[Simplify.]

So, the car runs 39.6 mi. on 4.4 gallons of gas.

9.
For the function, $y$ varies directly as $x$. Find the constant of variation and write the equation.
 $x$ $y$ 7 56 9 72 10 80
 a. - 8, $y$ = - 8$x$ b. $\frac{1}{8}$, $x$ = 8$y$ c. 8, $y$ = 8$x$ d. - $\frac{1}{8}$, $y$ = ($\frac{1}{8}$)$x$

#### Solution:

yx = 56 / 7 = 72 / 9 = 80 / 10 = 8

So, y varies directly as x, that is y = kx.

The constant of variation is 8.

The equation is y = 8x.

10.
For the function, $y$ varies directly as $x$. Find the constant of variation and write the equation.
 $x$ $y$ 66 11 42 7 36 6
 a. - $\frac{1}{6}$, $y$ = - ($\frac{1}{6}$)$x$ b. $\frac{1}{6}$, $x$ = 6$y$ c. 6, $y$ = 6$x$ d. 6, $x$ = $\frac{1}{6}$$y$

#### Solution:

yx = 11 / 66 = 7 / 42 = 6 / 36 = 1 / 6

So, y varies directly as x, that is y = kx.

Constant of proportionality is 1 / 6 .

The equation is y = (1 / 6) x (or) x = 6y.