Direct and Inverse Proportions

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.

• Direct: y ÷ x is constant.
• Inverse: x × y is constant.
• Double one in direct: the other doubles too.
• Double one in inverse: the other halves.

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.

• Direct proportion: $y = kx$ with constant $k$.
• Find $k$ from one matching pair of values.
• The graph is a straight line through the origin.
• Equivalent form: $\dfrac{y}{x} = k$.

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.

• Inverse proportion: $y = \dfrac{k}{x}$, equivalently $xy = k$.
• Find $k$ by multiplying one matching pair $x \times y$.
• Graph is a curve that approaches the axes but never touches them.
• Bigger $x$ gives smaller $y$; smaller $x$ gives bigger $y$.

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

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.

• Tabulate and check $y \div x$ for direct proportion.
• Check $x \times y$ for inverse proportion.
• Test every row — one match is not enough.
• If neither is constant, the rule is not a simple proportion.

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.

• Direct graph: straight line through the origin.
• Inverse graph: decreasing curve that approaches the axes.
• Find $k$ for direct: $k = y \div x$ at any point.
• Find $k$ for inverse: $k = x \times y$ at any point.

• Ratios and unit pricing are direct proportion problems.
• Speed, time and distance combine direct and inverse proportion.
• Many physics and chemistry laws are direct or inverse proportions.
• Scale and currency conversion both use $y = kx$.

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