Proportions

A proportion states that two ratios are equal and can solve scale, percent, rate, and similarity problems.

Equivalent ratios scale both quantities by the same factor. A ten-by-ten grid, a double number line, and paired arrows show the same proportional relationship in percent, fraction, and rate form.

A proportion preserves a multiplicative relationship: both ratios compare corresponding quantities in the same order.

Ratios and proportions

A ratio compares quantities by division. A proportion is an equation of ratios:

\frac ab=\frac cd,qquad b,d\ne0.

The units and order of comparison must match. If the first ratio is students per class, the second must also be students per class.

Equivalent ratios

Multiplying or dividing both terms of a ratio by the same nonzero factor creates an equivalent ratio:

\frac35=\frac{3\cdot4}{5\cdot4}=\frac{12}{20}.

A ratio table makes the scale factor visible and is often easier than an equation.

Worked example

Since $20$ is $2.5$ times $8$, the flour amount should also be $2.5$ times $3$.

Unit-rate method

Find the amount per one unit, then scale:

\frac38=0.375\text{ cup per serving}.

For $20$ servings:

$$ 0.375(20)=7.5. $$

The unit-rate and proportion methods express the same relationship.

Common mistakes

Reversing one ratio. Keep corresponding quantities in the same positions.

Mixing units. Convert before comparing.

Adding the same amount instead of multiplying by the same factor. Proportions are multiplicative.

Assuming every straight line is proportional. It must pass through the origin.

Applying a length scale directly to area or volume. Powers of the scale factor matter.

Quick self-check

What two quantities does each ratio compare?
Are their order and units consistent?
Is the relationship truly multiplicative?
Can a scale factor or unit rate solve it more clearly?
Does the cross-product equation match the written proportion?
Is the answer reasonable under the same scale factor?