Direct and inverse variation worksheet

Direct means y = kx, inverse means y = k over x. Most questions start from one pair: use it to find the constant, then write the equation or use it on a new value. The rest give an equation or a table and ask whether it is direct, inverse or neither.

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Sigma Prep · sigmaprep.io/worksheets Name Direct and Inverse Variation Date Period Answer each question. Solve direct variation problems $y$ varies directly as $x$ . When $x = 8$ , $y = 116$ . Find the constant of variation. Suppose $y$ is directly proportional to $x$ , and $y = 15$ when $x = 6$ . Find $x$ when $y = 35$ . If $y$ varies directly with $x$ and $y = 35$ when $x = 10$ , find $x$ when $y = 14$ . $y$ varies directly as $x$ . When $x = 14$ , $y = 217$ . Find $y$ when $x = 2$ . Suppose $y$ is directly proportional to $x$ , and $y = 264$ when $x = 16$ . Find $x$ when $y = 198$ . Solve inverse variation problems If $y$ varies inversely with $x$ and $y = 18.5$ when $x = 18$ , find $y$ when $x = 3$ . $y$ varies inversely as $x$ . When $x = 16$ , $y = 1.5$ . Find the constant of variation. Suppose $y$ is inversely proportional to $x$ , and $y = 7.5$ when $x = 24$ . Find $y$ when $x = 20$ . If $y$ varies inversely with $x$ and $y = 23.5$ when $x = 8$ , find the constant of variation. Suppose $y$ is inversely proportional to $x$ , and $y = 25.5$ when $x = 4$ . Find the constant of variation. Classify as direct, inverse or neither Does $y$ vary directly with $x$ , inversely with $x$ , or neither? $\begin{array}{cccc} x & 2 & 4 & 6 \\ y & 54 & 27 & 18 \end{array}$ Does $y$ vary directly with $x$ , inversely with $x$ , or neither? $y = \sqrt{x} + 108$ Does $y$ vary directly with $x$ , inversely with $x$ , or neither? $\frac{y}{175} = x$ Does $y$ vary directly with $x$ , inversely with $x$ , or neither? $\frac{x}{162} = \frac{1}{y}$ Does $y$ vary directly with $x$ , inversely with $x$ , or neither? $y - 100 x = 0$ Write the variation equation Given that $y$ varies inversely as $x$ , use $x = 14$ and $y = 14.5$ to write the equation. $y$ is directly proportional to $x$ ; $y = 82.5$ when $x = 11$ . What is the equation? Write an equation for $y$ in terms of $x$ , given that $y$ varies inversely with $x$ and $y = 5.5$ when $x = 6$ . When $x = 5$ , $y = 32.5$ . If $y$ varies directly with $x$ , write the equation. Suppose $y$ is directly proportional to $x$ . Find the equation of variation if $x = 7$ gives $y = 10.5$ . Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Direct and Inverse Variation

Date Period

Answer each question.

Solve direct variation problems
$y$ varies directly as $x$ . When $x = 8$ , $y = 116$ . Find the constant of variation.
Suppose $y$ is directly proportional to $x$ , and $y = 15$ when $x = 6$ . Find $x$ when $y = 35$ .
If $y$ varies directly with $x$ and $y = 35$ when $x = 10$ , find $x$ when $y = 14$ .
$y$ varies directly as $x$ . When $x = 14$ , $y = 217$ . Find $y$ when $x = 2$ .
Suppose $y$ is directly proportional to $x$ , and $y = 264$ when $x = 16$ . Find $x$ when $y = 198$ .

Solve inverse variation problems
If $y$ varies inversely with $x$ and $y = 18.5$ when $x = 18$ , find $y$ when $x = 3$ .
$y$ varies inversely as $x$ . When $x = 16$ , $y = 1.5$ . Find the constant of variation.
Suppose $y$ is inversely proportional to $x$ , and $y = 7.5$ when $x = 24$ . Find $y$ when $x = 20$ .
If $y$ varies inversely with $x$ and $y = 23.5$ when $x = 8$ , find the constant of variation.
Suppose $y$ is inversely proportional to $x$ , and $y = 25.5$ when $x = 4$ . Find the constant of variation.

Classify as direct, inverse or neither
Does $y$ vary directly with $x$ , inversely with $x$ , or neither? $\begin{array}{cccc} x & 2 & 4 & 6 \\ y & 54 & 27 & 18 \end{array}$
Does $y$ vary directly with $x$ , inversely with $x$ , or neither? $y = \sqrt{x} + 108$
Does $y$ vary directly with $x$ , inversely with $x$ , or neither? $\frac{y}{175} = x$
Does $y$ vary directly with $x$ , inversely with $x$ , or neither? $\frac{x}{162} = \frac{1}{y}$
Does $y$ vary directly with $x$ , inversely with $x$ , or neither? $y - 100 x = 0$

Write the variation equation
Given that $y$ varies inversely as $x$ , use $x = 14$ and $y = 14.5$ to write the equation.
$y$ is directly proportional to $x$ ; $y = 82.5$ when $x = 11$ . What is the equation?
Write an equation for $y$ in terms of $x$ , given that $y$ varies inversely with $x$ and $y = 5.5$ when $x = 6$ .
When $x = 5$ , $y = 32.5$ . If $y$ varies directly with $x$ , write the equation.
Suppose $y$ is directly proportional to $x$ . Find the equation of variation if $x = 7$ gives $y = 10.5$ .



Sigma Prep · sigmaprep.io/worksheets Name Direct and Inverse VariationAnswer keyVersion 1 Date Period Answer each question. Solve direct variation problems $y$ varies directly as $x$ . When $x = 8$ , $y = 116$ . Find the constant of variation. $14.5$ Suppose $y$ is directly proportional to $x$ , and $y = 15$ when $x = 6$ . Find $x$ when $y = 35$ . $14$ If $y$ varies directly with $x$ and $y = 35$ when $x = 10$ , find $x$ when $y = 14$ . $4$ $y$ varies directly as $x$ . When $x = 14$ , $y = 217$ . Find $y$ when $x = 2$ . $31$ Suppose $y$ is directly proportional to $x$ , and $y = 264$ when $x = 16$ . Find $x$ when $y = 198$ . $12$ Solve inverse variation problems If $y$ varies inversely with $x$ and $y = 18.5$ when $x = 18$ , find $y$ when $x = 3$ . $111$ $y$ varies inversely as $x$ . When $x = 16$ , $y = 1.5$ . Find the constant of variation. $24$ Suppose $y$ is inversely proportional to $x$ , and $y = 7.5$ when $x = 24$ . Find $y$ when $x = 20$ . $9$ If $y$ varies inversely with $x$ and $y = 23.5$ when $x = 8$ , find the constant of variation. $188$ Suppose $y$ is inversely proportional to $x$ , and $y = 25.5$ when $x = 4$ . Find the constant of variation. $102$ Classify as direct, inverse or neither Does $y$ vary directly with $x$ , inversely with $x$ , or neither? $\begin{array}{cccc} x & 2 & 4 & 6 \\ y & 54 & 27 & 18 \end{array}$ Inverse, $k = 108$ Does $y$ vary directly with $x$ , inversely with $x$ , or neither? $y = \sqrt{x} + 108$ Neither Does $y$ vary directly with $x$ , inversely with $x$ , or neither? $\frac{y}{175} = x$ Direct, $k = 175$ Does $y$ vary directly with $x$ , inversely with $x$ , or neither? $\frac{x}{162} = \frac{1}{y}$ Inverse, $k = 162$ Does $y$ vary directly with $x$ , inversely with $x$ , or neither? $y - 100 x = 0$ Direct, $k = 100$ Write the variation equation Given that $y$ varies inversely as $x$ , use $x = 14$ and $y = 14.5$ to write the equation. $y = \frac{203}{x}$ $y$ is directly proportional to $x$ ; $y = 82.5$ when $x = 11$ . What is the equation? $y = 7.5 x$ Write an equation for $y$ in terms of $x$ , given that $y$ varies inversely with $x$ and $y = 5.5$ when $x = 6$ . $y = \frac{33}{x}$ When $x = 5$ , $y = 32.5$ . If $y$ varies directly with $x$ , write the equation. $y = 6.5 x$ Suppose $y$ is directly proportional to $x$ . Find the equation of variation if $x = 7$ gives $y = 10.5$ . $y = 1.5 x$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Direct and Inverse VariationAnswer keyVersion 1

Date Period

Answer each question.

Solve direct variation problems
$y$ varies directly as $x$ . When $x = 8$ , $y = 116$ . Find the constant of variation. $14.5$
Suppose $y$ is directly proportional to $x$ , and $y = 15$ when $x = 6$ . Find $x$ when $y = 35$ . $14$
If $y$ varies directly with $x$ and $y = 35$ when $x = 10$ , find $x$ when $y = 14$ . $4$
$y$ varies directly as $x$ . When $x = 14$ , $y = 217$ . Find $y$ when $x = 2$ . $31$
Suppose $y$ is directly proportional to $x$ , and $y = 264$ when $x = 16$ . Find $x$ when $y = 198$ . $12$

Solve inverse variation problems
If $y$ varies inversely with $x$ and $y = 18.5$ when $x = 18$ , find $y$ when $x = 3$ . $111$
$y$ varies inversely as $x$ . When $x = 16$ , $y = 1.5$ . Find the constant of variation. $24$
Suppose $y$ is inversely proportional to $x$ , and $y = 7.5$ when $x = 24$ . Find $y$ when $x = 20$ . $9$
If $y$ varies inversely with $x$ and $y = 23.5$ when $x = 8$ , find the constant of variation. $188$
Suppose $y$ is inversely proportional to $x$ , and $y = 25.5$ when $x = 4$ . Find the constant of variation. $102$

Classify as direct, inverse or neither
Does $y$ vary directly with $x$ , inversely with $x$ , or neither? $\begin{array}{cccc} x & 2 & 4 & 6 \\ y & 54 & 27 & 18 \end{array}$ Inverse, $k = 108$
Does $y$ vary directly with $x$ , inversely with $x$ , or neither? $y = \sqrt{x} + 108$ Neither
Does $y$ vary directly with $x$ , inversely with $x$ , or neither? $\frac{y}{175} = x$ Direct, $k = 175$
Does $y$ vary directly with $x$ , inversely with $x$ , or neither? $\frac{x}{162} = \frac{1}{y}$ Inverse, $k = 162$
Does $y$ vary directly with $x$ , inversely with $x$ , or neither? $y - 100 x = 0$ Direct, $k = 100$

Write the variation equation
Given that $y$ varies inversely as $x$ , use $x = 14$ and $y = 14.5$ to write the equation. $y = \frac{203}{x}$
$y$ is directly proportional to $x$ ; $y = 82.5$ when $x = 11$ . What is the equation? $y = 7.5 x$
Write an equation for $y$ in terms of $x$ , given that $y$ varies inversely with $x$ and $y = 5.5$ when $x = 6$ . $y = \frac{33}{x}$
When $x = 5$ , $y = 32.5$ . If $y$ varies directly with $x$ , write the equation. $y = 6.5 x$
Suppose $y$ is directly proportional to $x$ . Find the equation of variation if $x = 7$ gives $y = 10.5$ . $y = 1.5 x$

How to do these

If $y$ varies directly with $x$ and $y = 12$ when $x = 3$ , find $y$ when $x = 7$

Direct variation means y = kx.
12 = k times 3, so k = 4.
Now y = 4 times 7.

The answer is $28$ .

Where students go wrong

Scaling y by the same factor as x without finding the constant. That works for direct variation and fails completely for inverse, where doubling x halves y. Finding k first works for both.

Also called constant of variation or y varies directly with x.

What you can put on this worksheet

Direct variation: $y$ varies directly as $x$ . When $x = 10$ , $y = 120$ . Find $x$ when $y = 24$ . → $2$
Inverse variation: Suppose $y$ is inversely proportional to $x$ , and $y = 20$ when $x = 7$ . Find $y$ when $x = 10$ . → $14$
Direct, inverse, or neither?: Does $y$ vary directly with $x$ , inversely with $x$ , or neither? $y = \frac{54}{x}$ → Inverse, $k = 54$
Write the equation: Write an equation for $y$ in terms of $x$ , given that $y$ varies directly with $x$ and $y = 80$ when $x = 8$ . → $y = 10 x$
A fraction for the constant: If $y$ varies directly with $x$ and $y = 7$ when $x = 4$ , find $y$ when $x = 17$ → $\frac{119}{4}$

Questions about these worksheets

Yes. Every direct and inverse variation sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Algebra 1 is usually taken around 8th or 9th grade, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the direct and inverse variation questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

You pick from 5: direct variation, inverse variation, direct, inverse, or neither?, write the equation and a fraction for the constant. Tick as many as you want and set how many of each, or let it spread them evenly.

Scaling y by the same factor as x without finding the constant. That works for direct variation and fails completely for inverse, where doubling x halves y. Finding k first works for both.

Direct and inverse variation worksheet

Select your difficulty

Solve direct variation problems

Solve inverse variation problems

Classify as direct, inverse or neither

Write the variation equation

Solve direct variation problems

Solve inverse variation problems

Classify as direct, inverse or neither

Write the variation equation

How to do these

Where students go wrong

What you can put on this worksheet

Questions about these worksheets

Are these direct and inverse variation worksheets free?+

Do the direct and inverse variation worksheets come with an answer key?+

Are these direct and inverse variation worksheets printable?+

What grade are these direct and inverse variation worksheets for?+

Can I get a different version of the same direct and inverse variation worksheet?+

Is this a free alternative to Kuta Software for direct and inverse variation?+

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What do students get wrong with direct and inverse variation?+