Joint and combined variation worksheet

Joint variation multiplies: z = kxy. Combined variation mixes direct and inverse: z = kx over y. Either way the method is the same as direct variation, use the first set of values to find k, then use k with the second set.

Select your difficulty

Pick more than one for a sheet that mixes them.

The preview can look cramped on a phone. The PDF and printed sheet come out normal.



Sigma Prep · sigmaprep.io/worksheets Name Joint and Combined Variation Date Period Answer each question. Solve joint variation problems The number of bricks $N$ in a wall varies jointly with its length $l$ and its height $h$ , and $N = 189$ when $l = 3$ and $h = 6$ . Find $N$ when $l = 4$ and $h = 3$ . $y$ is jointly proportional to $a$ and $b$ . If $y = 310.5$ when $a = 9$ and $b = 3$ , what is $b$ when $y = 253$ and $a = 2$ ? The total wages $W$ of a crew vary jointly with the number of workers $n$ and the days worked $d$ . If $W = 810$ when $n = 10$ and $d = 6$ , what is $n$ when $W = 1404$ and $d = 8$ ? The cost $P$ of painting a wall varies jointly with its length $l$ and its height $h$ , and $P = 180$ when $l = 6$ and $h = 4$ . Find $P$ when $l = 12$ and $h = 13$ . Solve combined variation problems The cost $c$ per person of a trip varies directly with the miles $m$ and inversely with the number of people $p$ . If $c = 31$ when $m = 10$ and $p = 5$ , what is $m$ when $c = 46.5$ and $p = 4$ ? $w$ varies directly as $u$ and inversely as $v$ , and $w = 18$ when $u = 40$ and $v = 10$ . Find $w$ when $u = 6$ and $v = 3$ . The time $t$ a job takes varies directly with its size $s$ and inversely with the number of workers $n$ , and $t = 16.5$ when $s = 15$ and $n = 5$ . Find $n$ when $t = 11$ and $s = 16$ . $z$ varies directly with $x$ and inversely with $y$ . If $z = 17.5$ when $x = 40$ and $y = 8$ , what is $y$ when $z = 21$ and $x = 42$ ? Classify as direct, inverse or neither Does $y$ vary directly with $x$ , inversely with $x$ , or neither? $\frac{y}{116} = x$ Does $y$ vary directly with $x$ , inversely with $x$ , or neither? $y = \sqrt{x} + 173$ Does $y$ vary directly with $x$ , inversely with $x$ , or neither? $\frac{x}{103} = \frac{1}{y}$ Does $y$ vary directly with $x$ , inversely with $x$ , or neither? $y - 178 x = 0$ Does $y$ vary directly with $x$ , inversely with $x$ , or neither? $x y - 113 = 0$ Does $y$ vary directly with $x$ , inversely with $x$ , or neither? $\begin{array}{cccc} x & 2 & 3 & 4 \\ y & 219 & 329 & 439 \end{array}$ Does $y$ vary directly with $x$ , inversely with $x$ , or neither? $y = {(x + 170)}^{2}$ Does $y$ vary directly with $x$ , inversely with $x$ , or neither? $\begin{array}{cccc} x & 2 & 3 & 4 \\ y & 60 & 40 & 30 \end{array}$ Solve variation problems with squares and cubes If $y$ varies directly with the cube of $x$ and $y = 320$ when $x = 4$ , find $y$ when $x = 2$ . If $y$ varies inversely with the square of $x$ and $y = 304$ when $x = 3$ , find $y$ when $x = 4$ . If $y$ varies inversely with the cube of $x$ and $y = 162$ when $x = 2$ , find $y$ when $x = 6$ . If $y$ varies directly with the cube of $x$ and $y = 2058$ when $x = 7$ , find $y$ when $x = 3$ . Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Joint and Combined Variation

Date Period

Answer each question.

Solve joint variation problems
The number of bricks $N$ in a wall varies jointly with its length $l$ and its height $h$ , and $N = 189$ when $l = 3$ and $h = 6$ . Find $N$ when $l = 4$ and $h = 3$ .
$y$ is jointly proportional to $a$ and $b$ . If $y = 310.5$ when $a = 9$ and $b = 3$ , what is $b$ when $y = 253$ and $a = 2$ ?
The total wages $W$ of a crew vary jointly with the number of workers $n$ and the days worked $d$ . If $W = 810$ when $n = 10$ and $d = 6$ , what is $n$ when $W = 1404$ and $d = 8$ ?
The cost $P$ of painting a wall varies jointly with its length $l$ and its height $h$ , and $P = 180$ when $l = 6$ and $h = 4$ . Find $P$ when $l = 12$ and $h = 13$ .

Solve combined variation problems
The cost $c$ per person of a trip varies directly with the miles $m$ and inversely with the number of people $p$ . If $c = 31$ when $m = 10$ and $p = 5$ , what is $m$ when $c = 46.5$ and $p = 4$ ?
$w$ varies directly as $u$ and inversely as $v$ , and $w = 18$ when $u = 40$ and $v = 10$ . Find $w$ when $u = 6$ and $v = 3$ .
The time $t$ a job takes varies directly with its size $s$ and inversely with the number of workers $n$ , and $t = 16.5$ when $s = 15$ and $n = 5$ . Find $n$ when $t = 11$ and $s = 16$ .
$z$ varies directly with $x$ and inversely with $y$ . If $z = 17.5$ when $x = 40$ and $y = 8$ , what is $y$ when $z = 21$ and $x = 42$ ?

Classify as direct, inverse or neither
Does $y$ vary directly with $x$ , inversely with $x$ , or neither? $\frac{y}{116} = x$
Does $y$ vary directly with $x$ , inversely with $x$ , or neither? $y = \sqrt{x} + 173$
Does $y$ vary directly with $x$ , inversely with $x$ , or neither? $\frac{x}{103} = \frac{1}{y}$
Does $y$ vary directly with $x$ , inversely with $x$ , or neither? $y - 178 x = 0$
Does $y$ vary directly with $x$ , inversely with $x$ , or neither? $x y - 113 = 0$
Does $y$ vary directly with $x$ , inversely with $x$ , or neither? $\begin{array}{cccc} x & 2 & 3 & 4 \\ y & 219 & 329 & 439 \end{array}$
Does $y$ vary directly with $x$ , inversely with $x$ , or neither? $y = {(x + 170)}^{2}$
Does $y$ vary directly with $x$ , inversely with $x$ , or neither? $\begin{array}{cccc} x & 2 & 3 & 4 \\ y & 60 & 40 & 30 \end{array}$

Solve variation problems with squares and cubes
If $y$ varies directly with the cube of $x$ and $y = 320$ when $x = 4$ , find $y$ when $x = 2$ .
If $y$ varies inversely with the square of $x$ and $y = 304$ when $x = 3$ , find $y$ when $x = 4$ .
If $y$ varies inversely with the cube of $x$ and $y = 162$ when $x = 2$ , find $y$ when $x = 6$ .
If $y$ varies directly with the cube of $x$ and $y = 2058$ when $x = 7$ , find $y$ when $x = 3$ .



Sigma Prep · sigmaprep.io/worksheets Name Joint and Combined VariationAnswer keyVersion 1 Date Period Answer each question. Solve joint variation problems The number of bricks $N$ in a wall varies jointly with its length $l$ and its height $h$ , and $N = 189$ when $l = 3$ and $h = 6$ . Find $N$ when $l = 4$ and $h = 3$ . $126$ $y$ is jointly proportional to $a$ and $b$ . If $y = 310.5$ when $a = 9$ and $b = 3$ , what is $b$ when $y = 253$ and $a = 2$ ? $11$ The total wages $W$ of a crew vary jointly with the number of workers $n$ and the days worked $d$ . If $W = 810$ when $n = 10$ and $d = 6$ , what is $n$ when $W = 1404$ and $d = 8$ ? $13$ The cost $P$ of painting a wall varies jointly with its length $l$ and its height $h$ , and $P = 180$ when $l = 6$ and $h = 4$ . Find $P$ when $l = 12$ and $h = 13$ . $1170$ Solve combined variation problems The cost $c$ per person of a trip varies directly with the miles $m$ and inversely with the number of people $p$ . If $c = 31$ when $m = 10$ and $p = 5$ , what is $m$ when $c = 46.5$ and $p = 4$ ? $12$ $w$ varies directly as $u$ and inversely as $v$ , and $w = 18$ when $u = 40$ and $v = 10$ . Find $w$ when $u = 6$ and $v = 3$ . $9$ The time $t$ a job takes varies directly with its size $s$ and inversely with the number of workers $n$ , and $t = 16.5$ when $s = 15$ and $n = 5$ . Find $n$ when $t = 11$ and $s = 16$ . $8$ $z$ varies directly with $x$ and inversely with $y$ . If $z = 17.5$ when $x = 40$ and $y = 8$ , what is $y$ when $z = 21$ and $x = 42$ ? $7$ Classify as direct, inverse or neither Does $y$ vary directly with $x$ , inversely with $x$ , or neither? $\frac{y}{116} = x$ Direct, $k = 116$ Does $y$ vary directly with $x$ , inversely with $x$ , or neither? $y = \sqrt{x} + 173$ Neither Does $y$ vary directly with $x$ , inversely with $x$ , or neither? $\frac{x}{103} = \frac{1}{y}$ Inverse, $k = 103$ Does $y$ vary directly with $x$ , inversely with $x$ , or neither? $y - 178 x = 0$ Direct, $k = 178$ Does $y$ vary directly with $x$ , inversely with $x$ , or neither? $x y - 113 = 0$ Inverse, $k = 113$ Does $y$ vary directly with $x$ , inversely with $x$ , or neither? $\begin{array}{cccc} x & 2 & 3 & 4 \\ y & 219 & 329 & 439 \end{array}$ Neither Does $y$ vary directly with $x$ , inversely with $x$ , or neither? $y = {(x + 170)}^{2}$ Neither Does $y$ vary directly with $x$ , inversely with $x$ , or neither? $\begin{array}{cccc} x & 2 & 3 & 4 \\ y & 60 & 40 & 30 \end{array}$ Inverse, $k = 120$ Solve variation problems with squares and cubes If $y$ varies directly with the cube of $x$ and $y = 320$ when $x = 4$ , find $y$ when $x = 2$ . $40$ If $y$ varies inversely with the square of $x$ and $y = 304$ when $x = 3$ , find $y$ when $x = 4$ . $171$ If $y$ varies inversely with the cube of $x$ and $y = 162$ when $x = 2$ , find $y$ when $x = 6$ . $6$ If $y$ varies directly with the cube of $x$ and $y = 2058$ when $x = 7$ , find $y$ when $x = 3$ . $162$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Joint and Combined VariationAnswer keyVersion 1

Date Period

Answer each question.

Solve joint variation problems
The number of bricks $N$ in a wall varies jointly with its length $l$ and its height $h$ , and $N = 189$ when $l = 3$ and $h = 6$ . Find $N$ when $l = 4$ and $h = 3$ . $126$
$y$ is jointly proportional to $a$ and $b$ . If $y = 310.5$ when $a = 9$ and $b = 3$ , what is $b$ when $y = 253$ and $a = 2$ ? $11$
The total wages $W$ of a crew vary jointly with the number of workers $n$ and the days worked $d$ . If $W = 810$ when $n = 10$ and $d = 6$ , what is $n$ when $W = 1404$ and $d = 8$ ? $13$
The cost $P$ of painting a wall varies jointly with its length $l$ and its height $h$ , and $P = 180$ when $l = 6$ and $h = 4$ . Find $P$ when $l = 12$ and $h = 13$ . $1170$

Solve combined variation problems
The cost $c$ per person of a trip varies directly with the miles $m$ and inversely with the number of people $p$ . If $c = 31$ when $m = 10$ and $p = 5$ , what is $m$ when $c = 46.5$ and $p = 4$ ? $12$
$w$ varies directly as $u$ and inversely as $v$ , and $w = 18$ when $u = 40$ and $v = 10$ . Find $w$ when $u = 6$ and $v = 3$ . $9$
The time $t$ a job takes varies directly with its size $s$ and inversely with the number of workers $n$ , and $t = 16.5$ when $s = 15$ and $n = 5$ . Find $n$ when $t = 11$ and $s = 16$ . $8$
$z$ varies directly with $x$ and inversely with $y$ . If $z = 17.5$ when $x = 40$ and $y = 8$ , what is $y$ when $z = 21$ and $x = 42$ ? $7$

Classify as direct, inverse or neither
Does $y$ vary directly with $x$ , inversely with $x$ , or neither? $\frac{y}{116} = x$ Direct, $k = 116$
Does $y$ vary directly with $x$ , inversely with $x$ , or neither? $y = \sqrt{x} + 173$ Neither
Does $y$ vary directly with $x$ , inversely with $x$ , or neither? $\frac{x}{103} = \frac{1}{y}$ Inverse, $k = 103$
Does $y$ vary directly with $x$ , inversely with $x$ , or neither? $y - 178 x = 0$ Direct, $k = 178$
Does $y$ vary directly with $x$ , inversely with $x$ , or neither? $x y - 113 = 0$ Inverse, $k = 113$
Does $y$ vary directly with $x$ , inversely with $x$ , or neither? $\begin{array}{cccc} x & 2 & 3 & 4 \\ y & 219 & 329 & 439 \end{array}$ Neither
Does $y$ vary directly with $x$ , inversely with $x$ , or neither? $y = {(x + 170)}^{2}$ Neither
Does $y$ vary directly with $x$ , inversely with $x$ , or neither? $\begin{array}{cccc} x & 2 & 3 & 4 \\ y & 60 & 40 & 30 \end{array}$ Inverse, $k = 120$

Solve variation problems with squares and cubes
If $y$ varies directly with the cube of $x$ and $y = 320$ when $x = 4$ , find $y$ when $x = 2$ . $40$
If $y$ varies inversely with the square of $x$ and $y = 304$ when $x = 3$ , find $y$ when $x = 4$ . $171$
If $y$ varies inversely with the cube of $x$ and $y = 162$ when $x = 2$ , find $y$ when $x = 6$ . $6$
If $y$ varies directly with the cube of $x$ and $y = 2058$ when $x = 7$ , find $y$ when $x = 3$ . $162$

How to do these

$z$ varies jointly with $x$ and $y$ , and $z = 24$ when $x = 2$ , $y = 3$ . Find $z$ when $x = 4$ and $y = 5$

z = kxy, so 24 = 6k and k = 4.
z = 4 times 4 times 5.

The answer is $80$ .

Where students go wrong

Adding where the variation multiplies. "Varies jointly with x and y" is kxy, never k(x + y).

Also called joint variation, combined variation or constant of variation.

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 $x = 10$ . → $14$
Joint variation: For boxes of one height, the volume $V$ varies jointly with the length $l$ and the width $w$ , and $V = 252$ when $l = 6$ and $w = 7$ . Find $V$ when $l = 2$ and $w = 5$ . → $60$
Combined variation: The pressure $P$ of a gas varies directly with its temperature $T$ and inversely with its volume $V$ . If $P = 16$ when $T = 6$ and $V = 3$ , what is $V$ when $P = 40$ and $T = 45$ ? → $9$
Direct, inverse or neither: Does $y$ vary directly with $x$ , inversely with $x$ , or neither? $y = \frac{54}{x}$ → Inverse, $k = 54$
With a square or a cube: If $y$ varies directly with the square of $x$ and $y = 925$ when $x = 5$ , find $y$ when $x = 2$ . → $148$
Direct, inverse, or neither?: Does $y$ vary directly with $x$ , inversely with $x$ , or neither? $y = 98 x$ → Direct, $k = 98$
Rearranged equations, k in a denominator: Direct, inverse or neither? If it varies, find $k$ : $y = \frac{7}{x}$ → Inverse, $k = 7$

Questions about these worksheets

Yes. Every joint and combined 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 2 is usually taken around 10th or 11th 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 joint and combined 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 8: direct variation, inverse variation, joint variation, combined variation, direct, inverse or neither, with a square or a cube, direct, inverse, or neither? and rearranged equations, k in a denominator. Tick as many as you want and set how many of each, or let it spread them evenly.

Adding where the variation multiplies. "Varies jointly with x and y" is kxy, never k(x + y).

Joint and combined variation worksheet

Select your difficulty

Solve joint variation problems

Solve combined variation problems

Classify as direct, inverse or neither

Solve variation problems with squares and cubes

Solve joint variation problems

Solve combined variation problems

Classify as direct, inverse or neither

Solve variation problems with squares and cubes

How to do these

Where students go wrong

What you can put on this worksheet

Questions about these worksheets

Are these joint and combined variation worksheets free?+

Do the joint and combined variation worksheets come with an answer key?+

Are these joint and combined variation worksheets printable?+

What grade are these joint and combined variation worksheets for?+

Can I get a different version of the same joint and combined variation worksheet?+

Is this a free alternative to Kuta Software for joint and combined variation?+

Can I choose how hard the joint and combined variation questions are?+

What kinds of question are on a joint and combined variation worksheet?+

What do students get wrong with joint and combined variation?+