Some quantities depend on more than one thing at a time. The cost of carpeting a room depends on its length and its width; the interest on a loan depends on how much you borrow, the rate and how long you take. When a quantity is a constant times the product of others, it varies jointly with them. It’s the natural next step after direct variation and inverse variation, and solving a joint variation problem uses exactly the same four steps.
What Is Joint Variation?
for some nonzero constant
The Four Kinds of Variation
| Type | Equation | In words |
|---|---|---|
| direct | ||
| inverse | ||
| joint | ||
| combined |
The wording in a problem tells you which one to write. “Jointly” means multiply; “inversely” puts that variable in the denominator; “the square of” squares it.
How to Solve a Joint Variation Problem
Example 1: Finding k and Using It
Step 1 — write the equation.
Step 2 — substitute the known values.
Step 3 — solve for
Step 4 — use it.
Example 2: Joint Variation With a Square
Step 1.
Step 2.
Step 3.
The constant came out as about
Combined Variation
When a quantity varies directly with some inputs and inversely with others, it’s combined variation. The direct ones go on top and the inverse ones underneath.
Example 3: Combined Variation
Step 1.
Step 2.
Step 3.
Joint Variation in Familiar Formulas
Once you know what to look for, joint variation is everywhere:
| Formula | Varies jointly with | Constant |
|---|---|---|
| area of a triangle, | base and height | |
| area of a rectangle, | length and width | 1 |
| simple interest, | principal, rate and time | 1 |
| volume of a cylinder, | height and the square of the radius | |
| kinetic energy, | mass and the square of speed |
That last one explains a lot about driving: double your speed and the energy a crash has to absorb goes up by a factor of four, because speed is squared.
A Joint Variation Word Problem
The cost of painting a wall varies jointly with its length and height. A wall 12 ft long and 8 ft high costs
Step 1.
Step 2.
Step 3.
The bigger wall costs
How to Recognise Joint Variation in a Table
Given a table of values, test the ratio
Common Mistakes to Avoid
- Adding instead of multiplying. Joint variation is
, never . - Forgetting a power. “The square of
” means in the equation. - Putting an inverse variable on top. “Inversely with
” means goes in the denominator. - Skipping
. You can’t use the equation until you’ve found the constant from the given values. - Reusing the old values in step 4. Substitute the new values once
is known.
Practice Problems
Work each one before opening the answer.
Problem 1.
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Answer:
Problem 2. Using the equation from Problem 1, find
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Answer:
Problem 3.
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Answer: it is multiplied by 6
Problem 4.
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Answer:
Problem 5.
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Answer:
Problem 6. Simple interest varies jointly with principal and time.
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Answer:
Joint variation completes the family: direct, inverse, joint and combined variation together describe most of the simple proportional relationships you’ll meet in science and everyday formulas.