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.

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Pick more than one for a sheet that mixes them.



Sigma Prep · sigmaprep.io/worksheets Name Joint and Combined Variation Date Period Answer each question. $z$ varies jointly with $x$ and $y$ , and $z = 64$ when $x = 2$ , $y = 4$ . Find $z$ when $x = 9$ and $y = 9$ $z$ varies directly with $x$ and inversely with $y$ , and $z = 33$ when $x = 18$ , $y = 6$ . Find $z$ when $x = 49$ and $y = 7$ Direct variation, inverse variation, or neither: $y = x + 5$ $z$ varies jointly with $x$ and $y$ , and $z = 48$ when $x = 2$ , $y = 4$ . Find $z$ when $x = 2$ and $y = 5$ $z$ varies directly with $x$ and inversely with $y$ , and $z = 16$ when $x = 4$ , $y = 4$ . Find $z$ when $x = 9$ and $y = 3$ Direct variation, inverse variation, or neither: $y = \frac{x}{6}$ $z$ varies jointly with $x$ and $y$ , and $z = 160$ when $x = 4$ , $y = 5$ . Find $z$ when $x = 3$ and $y = 7$ $z$ varies directly with $x$ and inversely with $y$ , and $z = 24$ when $x = 16$ , $y = 8$ . Find $z$ when $x = 54$ and $y = 9$ Direct variation, inverse variation, or neither: $y = \frac{3}{x}$ $z$ varies jointly with $x$ and $y$ , and $z = 120$ when $x = 5$ , $y = 4$ . Find $z$ when $x = 3$ and $y = 4$ $z$ varies directly with $x$ and inversely with $y$ , and $z = 16$ when $x = 12$ , $y = 6$ . Find $z$ when $x = 56$ and $y = 8$ Direct variation, inverse variation, or neither: $y = x + 7$ $z$ varies jointly with $x$ and $y$ , and $z = 270$ when $x = 6$ , $y = 5$ . Find $z$ when $x = 9$ and $y = 8$ $z$ varies directly with $x$ and inversely with $y$ , and $z = 108$ when $x = 12$ , $y = 2$ . Find $z$ when $x = 6$ and $y = 2$ Direct variation, inverse variation, or neither: $y = x + 9$ $z$ varies jointly with $x$ and $y$ , and $z = 180$ when $x = 5$ , $y = 6$ . Find $z$ when $x = 3$ and $y = 9$ $z$ varies directly with $x$ and inversely with $y$ , and $z = 27$ when $x = 15$ , $y = 5$ . Find $z$ when $x = 6$ and $y = 2$ Direct variation, inverse variation, or neither: $x y = 8$ $z$ varies jointly with $x$ and $y$ , and $z = 126$ when $x = 6$ , $y = 3$ . Find $z$ when $x = 3$ and $y = 7$ $z$ varies directly with $x$ and inversely with $y$ , and $z = 14$ when $x = 4$ , $y = 4$ . Find $z$ when $x = 2$ and $y = 2$ 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.

$z$ varies jointly with $x$ and $y$ , and $z = 64$ when $x = 2$ , $y = 4$ . Find $z$ when $x = 9$ and $y = 9$
$z$ varies directly with $x$ and inversely with $y$ , and $z = 33$ when $x = 18$ , $y = 6$ . Find $z$ when $x = 49$ and $y = 7$
Direct variation, inverse variation, or neither: $y = x + 5$
$z$ varies jointly with $x$ and $y$ , and $z = 48$ when $x = 2$ , $y = 4$ . Find $z$ when $x = 2$ and $y = 5$
$z$ varies directly with $x$ and inversely with $y$ , and $z = 16$ when $x = 4$ , $y = 4$ . Find $z$ when $x = 9$ and $y = 3$
Direct variation, inverse variation, or neither: $y = \frac{x}{6}$
$z$ varies jointly with $x$ and $y$ , and $z = 160$ when $x = 4$ , $y = 5$ . Find $z$ when $x = 3$ and $y = 7$
$z$ varies directly with $x$ and inversely with $y$ , and $z = 24$ when $x = 16$ , $y = 8$ . Find $z$ when $x = 54$ and $y = 9$
Direct variation, inverse variation, or neither: $y = \frac{3}{x}$
$z$ varies jointly with $x$ and $y$ , and $z = 120$ when $x = 5$ , $y = 4$ . Find $z$ when $x = 3$ and $y = 4$
$z$ varies directly with $x$ and inversely with $y$ , and $z = 16$ when $x = 12$ , $y = 6$ . Find $z$ when $x = 56$ and $y = 8$
Direct variation, inverse variation, or neither: $y = x + 7$
$z$ varies jointly with $x$ and $y$ , and $z = 270$ when $x = 6$ , $y = 5$ . Find $z$ when $x = 9$ and $y = 8$
$z$ varies directly with $x$ and inversely with $y$ , and $z = 108$ when $x = 12$ , $y = 2$ . Find $z$ when $x = 6$ and $y = 2$
Direct variation, inverse variation, or neither: $y = x + 9$
$z$ varies jointly with $x$ and $y$ , and $z = 180$ when $x = 5$ , $y = 6$ . Find $z$ when $x = 3$ and $y = 9$
$z$ varies directly with $x$ and inversely with $y$ , and $z = 27$ when $x = 15$ , $y = 5$ . Find $z$ when $x = 6$ and $y = 2$
Direct variation, inverse variation, or neither: $x y = 8$
$z$ varies jointly with $x$ and $y$ , and $z = 126$ when $x = 6$ , $y = 3$ . Find $z$ when $x = 3$ and $y = 7$
$z$ varies directly with $x$ and inversely with $y$ , and $z = 14$ when $x = 4$ , $y = 4$ . Find $z$ when $x = 2$ and $y = 2$



Sigma Prep · sigmaprep.io/worksheets Name Joint and Combined VariationAnswer keyVersion 1 Date Period Answer each question. $z$ varies jointly with $x$ and $y$ , and $z = 64$ when $x = 2$ , $y = 4$ . Find $z$ when $x = 9$ and $y = 9$ $648$ $z$ varies directly with $x$ and inversely with $y$ , and $z = 33$ when $x = 18$ , $y = 6$ . Find $z$ when $x = 49$ and $y = 7$ $77$ Direct variation, inverse variation, or neither: $y = x + 5$ Neither $z$ varies jointly with $x$ and $y$ , and $z = 48$ when $x = 2$ , $y = 4$ . Find $z$ when $x = 2$ and $y = 5$ $60$ $z$ varies directly with $x$ and inversely with $y$ , and $z = 16$ when $x = 4$ , $y = 4$ . Find $z$ when $x = 9$ and $y = 3$ $48$ Direct variation, inverse variation, or neither: $y = \frac{x}{6}$ Direct variation, $k = \frac{1}{6}$ $z$ varies jointly with $x$ and $y$ , and $z = 160$ when $x = 4$ , $y = 5$ . Find $z$ when $x = 3$ and $y = 7$ $168$ $z$ varies directly with $x$ and inversely with $y$ , and $z = 24$ when $x = 16$ , $y = 8$ . Find $z$ when $x = 54$ and $y = 9$ $72$ Direct variation, inverse variation, or neither: $y = \frac{3}{x}$ Inverse variation, $k = 3$ $z$ varies jointly with $x$ and $y$ , and $z = 120$ when $x = 5$ , $y = 4$ . Find $z$ when $x = 3$ and $y = 4$ $72$ $z$ varies directly with $x$ and inversely with $y$ , and $z = 16$ when $x = 12$ , $y = 6$ . Find $z$ when $x = 56$ and $y = 8$ $56$ Direct variation, inverse variation, or neither: $y = x + 7$ Neither $z$ varies jointly with $x$ and $y$ , and $z = 270$ when $x = 6$ , $y = 5$ . Find $z$ when $x = 9$ and $y = 8$ $648$ $z$ varies directly with $x$ and inversely with $y$ , and $z = 108$ when $x = 12$ , $y = 2$ . Find $z$ when $x = 6$ and $y = 2$ $54$ Direct variation, inverse variation, or neither: $y = x + 9$ Neither $z$ varies jointly with $x$ and $y$ , and $z = 180$ when $x = 5$ , $y = 6$ . Find $z$ when $x = 3$ and $y = 9$ $162$ $z$ varies directly with $x$ and inversely with $y$ , and $z = 27$ when $x = 15$ , $y = 5$ . Find $z$ when $x = 6$ and $y = 2$ $27$ Direct variation, inverse variation, or neither: $x y = 8$ Inverse variation, $k = 8$ $z$ varies jointly with $x$ and $y$ , and $z = 126$ when $x = 6$ , $y = 3$ . Find $z$ when $x = 3$ and $y = 7$ $147$ $z$ varies directly with $x$ and inversely with $y$ , and $z = 14$ when $x = 4$ , $y = 4$ . Find $z$ when $x = 2$ and $y = 2$ $14$ 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.

$z$ varies jointly with $x$ and $y$ , and $z = 64$ when $x = 2$ , $y = 4$ . Find $z$ when $x = 9$ and $y = 9$ $648$
$z$ varies directly with $x$ and inversely with $y$ , and $z = 33$ when $x = 18$ , $y = 6$ . Find $z$ when $x = 49$ and $y = 7$ $77$
Direct variation, inverse variation, or neither: $y = x + 5$ Neither
$z$ varies jointly with $x$ and $y$ , and $z = 48$ when $x = 2$ , $y = 4$ . Find $z$ when $x = 2$ and $y = 5$ $60$
$z$ varies directly with $x$ and inversely with $y$ , and $z = 16$ when $x = 4$ , $y = 4$ . Find $z$ when $x = 9$ and $y = 3$ $48$
Direct variation, inverse variation, or neither: $y = \frac{x}{6}$ Direct variation, $k = \frac{1}{6}$
$z$ varies jointly with $x$ and $y$ , and $z = 160$ when $x = 4$ , $y = 5$ . Find $z$ when $x = 3$ and $y = 7$ $168$
$z$ varies directly with $x$ and inversely with $y$ , and $z = 24$ when $x = 16$ , $y = 8$ . Find $z$ when $x = 54$ and $y = 9$ $72$
Direct variation, inverse variation, or neither: $y = \frac{3}{x}$ Inverse variation, $k = 3$
$z$ varies jointly with $x$ and $y$ , and $z = 120$ when $x = 5$ , $y = 4$ . Find $z$ when $x = 3$ and $y = 4$ $72$
$z$ varies directly with $x$ and inversely with $y$ , and $z = 16$ when $x = 12$ , $y = 6$ . Find $z$ when $x = 56$ and $y = 8$ $56$
Direct variation, inverse variation, or neither: $y = x + 7$ Neither
$z$ varies jointly with $x$ and $y$ , and $z = 270$ when $x = 6$ , $y = 5$ . Find $z$ when $x = 9$ and $y = 8$ $648$
$z$ varies directly with $x$ and inversely with $y$ , and $z = 108$ when $x = 12$ , $y = 2$ . Find $z$ when $x = 6$ and $y = 2$ $54$
Direct variation, inverse variation, or neither: $y = x + 9$ Neither
$z$ varies jointly with $x$ and $y$ , and $z = 180$ when $x = 5$ , $y = 6$ . Find $z$ when $x = 3$ and $y = 9$ $162$
$z$ varies directly with $x$ and inversely with $y$ , and $z = 27$ when $x = 15$ , $y = 5$ . Find $z$ when $x = 6$ and $y = 2$ $27$
Direct variation, inverse variation, or neither: $x y = 8$ Inverse variation, $k = 8$
$z$ varies jointly with $x$ and $y$ , and $z = 126$ when $x = 6$ , $y = 3$ . Find $z$ when $x = 3$ and $y = 7$ $147$
$z$ varies directly with $x$ and inversely with $y$ , and $z = 14$ when $x = 4$ , $y = 4$ . Find $z$ when $x = 2$ and $y = 2$ $14$

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, direct and inverse variation or constant of variation.

What you can put on this worksheet

Direct variation: If $y$ varies directly with $x$ and $y = 30$ when $x = 5$ , find $y$ when $x = 1$ → $6$
Inverse variation: If $y$ varies inversely with $x$ and $y = 4$ when $x = 5$ , find $y$ when $x = 2$ → $10$
Joint variation: $z$ varies jointly with $x$ and $y$ , and $z = 80$ when $x = 4$ , $y = 5$ . Find $z$ when $x = 2$ and $y = 5$ → $40$
Combined variation: $z$ varies directly with $x$ and inversely with $y$ , and $z = 4$ when $x = 4$ , $y = 4$ . Find $z$ when $x = 15$ and $y = 5$ → $12$
Direct, inverse or neither: Direct variation, inverse variation, or neither: $y = \frac{6}{x}$ → Inverse variation, $k = 6$

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 5: direct variation, inverse variation, joint variation, combined variation and direct, inverse or neither. 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

Joint and Combined Variation

Joint and Combined VariationAnswer keyVersion 1

How to do these

Where students go wrong

What you can put on this worksheet

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