Inverse proportion

Two quantities are in inverse proportion if increasing one decreases the other in a precise way: their product stays constant.

Concept

Definition. $y$ varies inversely with $x$ if $xy = k$ for some positive constant $k$.

Equivalently: $y = \dfrac{k}{x}$. Doubling $x$ halves $y$. Tripling $x$ divides $y$ by $3$.

Check. When $x_1, y_1$ and $x_2, y_2$ are corresponding pairs,

$x_1 y_1 = x_2 y_2.$

So cross-multiplication looks slightly different from direct proportion. Where direct is $\dfrac{x_1}{y_1} = \dfrac{x_2}{y_2}$, inverse is $x_1 y_1 = x_2 y_2$.

Examples of inverse proportion.

• Workers and time. $6$ workers finish a job in $10$ days; $12$ workers finish the same job in $5$ days. ($6 \times 10 = 60 = 12 \times 5$.)
• Speed and time (for a fixed distance). Doubling speed halves the time.
• Pressure and volume of a gas at fixed temperature (Boyle's law from physics).
• Length and width of a rectangle of fixed area.

Examples that are NOT inverse.

• Speed and distance at fixed time , that's direct (faster $\to$ farther in same time).
• Number of items and total cost , direct (more items, higher total).
• Distance from a light and brightness , not simple inverse but inverse-square (a bigger effect).

Solving a typical problem. "$10$ workers can finish a project in $24$ days. How many days for $15$ workers?"

• Workers and time are in inverse proportion (assuming all workers work the same way).
• $x_1 y_1 = x_2 y_2 \Rightarrow 10 \cdot 24 = 15 \cdot y_2$.
• $y_2 = 240/15 = 16$ days.

Graph. Plot $y$ against $x$ for $y = k/x$. The graph is a hyperbola , a curve that approaches but never touches the axes. Note: not a straight line.

Worked examples

Example 1. $12$ taps fill a tank in $5$ hours. How long for $20$ taps?

• $12 \cdot 5 = 20 \cdot t$. $t = 60/20 = 3$ hours.

Example 2. A car at $60$ km/h covers a route in $4$ hours. How long at $80$ km/h?

• $60 \cdot 4 = 80 \cdot t$. $t = 240/80 = 3$ hours.

Example 3. $25$ books, each $200$ pages, fit in a shelf. If each book had $500$ pages, how many would fit?

• Books and pages-per-book are inversely related for fixed shelf depth.
• $25 \cdot 200 = N \cdot 500$. $N = 5000/500 = 10$ books.

Example 4. $x$ and $y$ satisfy $xy = 36$. If $x = 4$, find $y$; if $x = 9$, find $y$; sketch the graph.

• $x=4 \to y=9$; $x=9 \to y=4$; $x=6 \to y=6$. The points $(4,9), (6,6), (9,4)$ lie on a hyperbola.

Try it yourself

1. $20$ workers finish a job in $15$ days. How many days for $30$ workers?
2. A car at $50$ km/h takes $6$ hours to reach a city. At $75$ km/h, how long?
3. A pipe fills a tank in $8$ hours. With how many such pipes will the tank fill in $1$ hour?
4. Length and width of a rectangle of area $60$: list three valid pairs.
5. If $x$ and $y$ are inversely related and $x = 3$ gives $y = 12$, find $y$ when $x = 8$.
6. A bus at $40$ km/h takes $9$ hours. How fast must it go to do the trip in $6$ hours?
7. Are $x = 1, 2, 3, 4$ and $y = 24, 12, 8, 6$ in inverse proportion?
8. Sketch the graph of $y = 12/x$ for $x = 1, 2, 3, 4, 6, 12$.

Activity / Insight

Tug of war. Place $24$ identical books on a shelf. Stack them in different ways: $1$ row of $24$, $2$ rows of $12$, $3$ rows of $8$, $4$ rows of $6$, $6$ rows of $4$, $8$ rows of $3$. For each arrangement, "rows" and "books per row" multiply to $24$. Plot the pairs on graph paper to see the inverse-proportion hyperbola.