Lesson 12: Solving Inverse Variation Problems

New Concept

If the product of two variables is a constant, then the equation is an inverse variation.

Why it matters

Algebra is the language of relationships, and inverse variation describes a fundamental pattern where one quantity shrinks as another grows. Mastering this concept allows you to model complex real-world systems, from gravitational forces to economic principles of supply and demand.

What’s next

Next, you’ll learn to identify this relationship in data tables and use it to solve for unknown values.

If the product of two variables is a constant, the equation represents an inverse variation. The relationship is described by the equations:

If a construction project requires 10 workers 12 days to complete, how long would it take 8 workers? The constant is $10 \cdot 12 = 120$ worker-days. So, $8 \cdot d = 120 \implies d = 15$ days.
For the data set with x-values {2, 4, 8} and y-values {16, 8, 4}, we check the products: $2 \cdot 16 = 32$, $4 \cdot 8 = 32$, and $8 \cdot 4 = 32$. This is an inverse variation with $k=32$.
If a car travels at 50 miles per hour for 4 hours, the distance is 200 miles. If the speed changes to 40 miles per hour, the time becomes $t = \frac{200}{40} = 5$ hours.

Think of an inverse variation like sharing a pizza! The more friends (x) you invite, the smaller the slice (y) each person gets. But the total amount of pizza (k) stays the same. The product of friends and slice size is always one whole pizza. It’s a perfect see-saw balance where one goes up, the other must go down.

A joint variation involves three variables, where one variable depends on the product of the other two. It can be written as:

The area of a triangle is $A = \frac{1}{2}bh$. Area (A) varies jointly as base (b) and height (h) with a constant of variation $k = \frac{1}{2}$.
If z varies jointly as x and y, and z=40 when x=2 and y=5, find k. $\frac{40}{(2)(5)} = \frac{40}{10} = 4$. The constant of variation is $k=4$.
The volume of a cylinder is $V = \pi r^2 h$. The volume (V) varies jointly as the height (h) and the square of the radius ($r^2$), with $k=\pi$.

Joint variation is all about teamwork! Imagine you're earning money (z) by washing cars. Your earnings depend jointly on the number of cars you wash (x) and the price per car (y). If you wash more cars or charge a higher price, your total earnings go up. The ratio of your earnings to the product of cars and price is constant.

To determine if a data set represents an inverse variation, multiply the corresponding x and y values for each ordered pair. If the product, $xy$, is the same nonzero constant (k) for all pairs of data, it is an inverse variation. The equation is then $y = \frac{k}{x}$.

Data: (x, y) pairs are (2, 15), (5, 6), (10, 3). Check products: $2 \cdot 15 = 30$, $5 \cdot 6 = 30$, $10 \cdot 3 = 30$. It's an inverse variation with $k=30$.
Data: (x, y) pairs are (1, 12), (2, 10), (3, 8). Check products: $1 \cdot 12 = 12$, $2 \cdot 10 = 20$. The products are not constant, so this is not an inverse variation.

Are two variables playing on a see-saw? To find out, you need to be a data detective! For every pair of (x, y) values you have, multiply them together. If you keep getting the exact same number, that's your constant, k, and you've proven it's an inverse variation. If the products are different, then they aren't playing by inverse rules.

The gravitational force of attraction ($F_g$) is both a joint variation and an inverse variation, combining to form one equation:

If you triple the mass of one object ($m_1$), the gravitational force ($F_g$) triples. $F_{new} = g\frac{(3m_1)m_2}{d^2} = 3F_g$.
If you triple the distance (d) between two objects, the force becomes nine times weaker. $F_{new} = g\frac{m_1m_2}{(3d)^2} = g\frac{m_1m_2}{9d^2} = \frac{1}{9}F_g$.

Think of gravity as a cosmic force with two rules. Rule one (joint variation): the bigger the objects (masses $m_1$ and $m_2$), the stronger the pull between them. Rule two (inverse square variation): the farther apart they get, the weaker the pull becomes, and it weakens really fast! This single, elegant law perfectly describes this dual behavior across the universe.