Direct and Inverse Proportions

Two quantities are in proportion when they change in a fixed, predictable way. There are two main kinds you need at Checkpoint.

• In direct proportion, as one quantity grows, the other grows at the same rate. If you double one, you double the other.
• In inverse proportion, as one quantity grows the other shrinks. If you double one, you halve the other.

A quick test:

• Direct: pick any pair of values $(x, y)$ and the ratio $\dfrac{y}{x}$ is the same every time.
• Inverse: pick any pair of values $(x, y)$ and the product $x \times y$ is the same every time.

For example, $1$ pencil costs $30$ cents, $2$ pencils cost $60$ cents, $3$ pencils cost $90$ cents. The ratio cost ÷ pencils is always $30$, so the cost is in direct proportion to the number of pencils.

Now suppose $2$ workers take $12$ hours to paint a wall, $3$ workers take $8$ hours, and $4$ workers take $6$ hours. The product workers × hours is always $24$, so the number of hours is in inverse proportion to the number of workers.

When $y$ is directly proportional to $x$, you can write the relationship as an equation: $y = kx,$ where $k$ is a constant of proportionality — a fixed number that does not change. We write the relationship in words as $y \propto x$ ("$y$ is proportional to $x$").

To find $k$, substitute a matching pair of values for $x$ and $y$:

• "$y = 18$ when $x = 6$" gives $18 = k \times 6$, so $k = 3$.
• The full rule is $y = 3x$, and you can use it to find $y$ for any $x$.

The graph of a direct proportion is always a straight line through the origin. The steeper the line, the larger the constant $k$. The line passes through $(0, 0)$ because if you have zero of one, you have zero of the other.

You can also write direct proportion as $\dfrac{y}{x} = k$, which is sometimes more useful in a problem.

When $y$ is inversely proportional to $x$, doubling one halves the other and the product $xy$ stays the same. As an equation: $y = \frac{k}{x}, \qquad \text{or equivalently} \qquad xy = k.$

We write this in words as $y \propto \dfrac{1}{x}$ ("$y$ is proportional to $1/x$").

To find $k$, multiply any matching pair:

• "$y = 5$ when $x = 8$" gives $k = 5 \times 8 = 40$.
• The rule is $y = \dfrac{40}{x}$, so $y = 4$ when $x = 10$.

The graph of an inverse proportion is a smooth curve in the first quadrant (when both quantities are positive). It hugs the axes more and more closely as $x$ or $y$ gets large, but it never actually touches them.

Many real-world situations are inverse proportions: speed and time for a fixed distance, the number of workers and the hours to finish a job, the pressure and volume of a fixed amount of gas at constant temperature.

Given a table of values, you can decide between direct, inverse and neither by checking every row.

$x$	$y$	$y \div x$	$x \times y$
2	6	3	12
4	12	3	48
5	15	3	75

Here $y \div x$ is the same in every row, so $y$ is directly proportional to $x$, with $y = 3x$.

$x$	$y$	$y \div x$	$x \times y$
2	12	6	24
4	6	1.5	24
6	4	$\tfrac{2}{3}$	24

Here $x \times y$ is the same in every row, so $y$ is inversely proportional to $x$, with $y = \dfrac{24}{x}$.

If neither column is constant, the relationship is not a simple proportion. Many real-world rules look proportional at first glance, so always check more than one pair of values.

A short checklist:

1. Try $y \div x$ — same in every row? Direct, with $k$ equal to that value.
2. Try $x \times y$ — same in every row? Inverse, with $k$ equal to that value.
3. If both fail, try other rules (linear, square, square root) or fit no simple proportion at all.

Two graph shapes are worth knowing at sight.

• A direct proportion graph is a straight line that passes through the origin $(0, 0)$. The steeper the line, the larger $k$.
• An inverse proportion graph is a smooth decreasing curve in the first quadrant that gets closer and closer to the $x$-axis on the right and the $y$-axis at the top, but never touches either.

To find $k$ from a graph:

• For direct proportion, pick any clear point on the line and use $k = \dfrac{y}{x}$.
• For inverse proportion, pick any clear point on the curve and use $k = x \times y$.

A graph also tells you what happens at the edges. A direct line keeps growing forever in both directions. An inverse curve grows without bound near the axes and flattens out far from them.

Proportion ideas are everywhere in maths and science.

• Ratios and unit pricing — finding the best value at a supermarket uses direct proportion.
• Speed, time and distance — at constant speed, distance is directly proportional to time; at constant distance, time is inversely proportional to speed.
• Physics formulae — gas pressure and volume are inversely proportional, current and resistance are inversely proportional at constant voltage.
• Scale drawings and maps — every distance on the map is directly proportional to the real distance.
• Currency conversion — the amount you receive in another currency is directly proportional to the amount you start with.

If a later topic talks about quantities "in proportion" or "varying", you can return to the same two ideas — $y = kx$ and $y = \dfrac{k}{x}$ — and the rest becomes much easier.