Direct Variation Examples: 47 Problems with Answers

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Direct Variation.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A proportional relationship $y = kx$ that always passes through the origin — when one quantity doubles, so does the other.

Distance varies directly with time at constant speed: $d = 60t$ .

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: In direct variation $y=kx$ , zero input gives zero output and the two quantities scale together by a fixed factor.

Common stuck point: The procedure for direct variation is the easy part; the trap is calling any straight line direct variation. Asking "When $x=0$ is $y=0$ , and does doubling $x$ double $y$ ?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: When $x=0$ is $y=0$ , and does doubling $x$ double $y$ ?

Worked Examples

Example 1

easy

y

varies directly with

x

, and

y = 18

when

x = 3

. Find the constant

k

and write the direct variation equation.

Answer

k = 6

;

y = 6x

First step

Direct variation:

y = kx

Full solution

2
Find $k$ : $k = y/x = 18/3 = 6$ .
3
Equation: $y = 6x$ .
4
Check: when $x=3$ , $y = 6 \times 3 = 18$ ✓

In direct variation

y = kx

k

is found by dividing

y

x

. Here

k = 18/3 = 6

Example 2

medium

The cost of fabric varies directly with length. 5 meters costs $35. How much do 8 meters cost? Set up a proportion.

Example 3

medium

Sarah's pay is a direct variation of hours worked: $45 for 3 hours. How much does she earn in 11 hours?

Example 4

medium

Convert 60 miles per hour to a direct variation

d = kt

in miles for time

t

in hours.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

y = kx

and

y = 24

when

x = 4

, find

y

when

x = 7

Example 2

medium

A machine produces 150 units in 5 hours. Assuming direct variation, how many units in 9 hours?

Example 3

easy

y = 7x

a direct variation?

Example 4

easy

y = 3x + 1

a direct variation?

Example 5

easy

In the direct variation

y = kx

with

k = 6

, find

y

when

x = 3

Example 6

easy

In a direct variation,

y = 20

when

x = 4

. Find

k

Example 7

easy

Does

y = kx

always pass through

(0,0)

Example 8

easy

Distance varies directly with time:

d = 60t

. Find

d

t = 4

Example 9

easy

x = 0

gives

y = 3

, can the relation be a direct variation?

Example 10

easy

Write the direct variation where

y

is always

4

times

x

Example 11

medium

A table

(x,y) = (2,10),(4,20),(6,30)

. Is it a direct variation? Give

k

Example 12

medium

A table

(x,y) = (1,4),(2,7),(3,10)

. Is it a direct variation?

Example 13

medium

y

varies directly with

x

and

y = 15

when

x = 5

, find

y

when

x = 8

Example 14

medium

Which is a direct variation: (A)

y = \frac{x}{2}

or (B)

y = \frac{x}{2} + 1

Example 15

medium

The cost of gas varies directly with gallons:

\$12

for

4

gallons. Find the cost for

9

gallons.

Example 16

medium

In a direct variation, when

x

triples, what happens to

y

Example 17

medium

A line passes through

(0,0)

and

(6, 9)

. Write it as a direct variation.

Example 18

challenge

A relation has

(x,y) = (3,12)

and

(7, 28)

. Decide if it is a direct variation and justify with

k

Example 19

challenge

Two boxes of nails weigh proportionally. If

5

boxes weigh

W

and

8

boxes weigh

W + 9

kg, find the weight of one box.

Example 20

challenge

Explain why all direct variations are linear but not all linear relationships are direct variations.

Example 21

medium

y

varies directly with

x

and

y = 9

when

x = 6

, find

x

when

y = 21

Example 22

medium

Wages vary directly with hours:

\$54

for

6

hours. How many hours earn

\$90

Example 23

easy

y = -2x

a direct variation?

Example 24

easy

y

varies directly with

x

and

y = 50

when

x = 10

, find

y

when

x = 25

Example 25

easy

An apple costs the same amount each. 4 apples cost $3. Write a direct variation for cost

c

in terms of number of apples

n

Example 26

easy

y = 0

a direct variation?

Example 27

easy

A line

y = kx

passes through

(2, -8)

. Find

k

Example 28

medium

y

varies directly with

x

and

y

becomes 4 times as large, what happens to

x

Example 29

medium

Convert: 2 dozen eggs cost $7.20. Find the direct variation cost per egg.

Example 30

medium

A graph of

y

x

shows a straight line through

(0, 0)

and

(5, 12)

. Write the direct variation.

Example 31

medium

y

varies directly with

x

. When

x = 12

y = 30

. Find

x

when

y = 75

Example 32

medium

A spring obeys Hooke's law: stretch is directly proportional to force. A 4 N force stretches it 6 cm. How far does a 10 N force stretch it?

Example 33

medium

Decide whether each describes a direct variation: (i) perimeter of a square vs side length, (ii) area of a square vs side length.

Example 34

medium

y = kx

and changing

x

from 4 to 7 changes

y

from 12 to 21, verify the direct variation.

Example 35

medium

Is the relationship modeled by

y = 2x^2

a direct variation?

Example 36

hard

A direct variation passes through

(8, 14)

. Find

y

when

x = -6

Example 37

hard

A graph of

y

x

has slope 4 but passes through

(0, 3)

. Is this a direct variation? Why?

Example 38

hard

Two direct variations

y = k_1 x

and

z = k_2 x

share the same input. If

k_1 = 4

and

k_2 = -7

, what direct variation relates

y

and

z

Example 39

hard

y

varies directly with

x

and the ordered pair

(a, 18)

is on the graph

y = 2x

, find

a

Example 40

hard

Quantity

y

doubles each time

x

doubles. Is this a direct variation? Why or why not?

Example 41

hard

y

varies directly with

x

. Given

y = 12

when

x = 8

, find the equation and use it to compute

y

when

x = 50

Example 42

challenge

Three jars of jam cost $11.40. Use direct variation to find the cost of 17 jars.

Example 43

challenge

y

varies directly with

x

, and

y

varies directly with

z

. If

y = 12

when

x = 4

and

y = 20

when

z = 5

, write

y

explicitly as a function of

x

and

z

separately. Then determine the relationship between

x

and

z

when

y = 60

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Related Concepts

Proportional Relationships Linear Relationship Linear Functions Inverse Variation

Background Knowledge

These ideas may be useful before you work through the harder examples.

proportionalitylinear relationship