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A proportion states that two ratios are equal and can solve scale, percent, rate, and similarity problems.
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:
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:
A ratio table makes the scale factor visible and is often easier than an equation.
Cross products
For valid denominators,
if and only if
Cross multiplication is shorthand for multiplying both sides by $bd$; it is not a mysterious diagonal rule.
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:
For $20$ servings:
The unit-rate and proportion methods express the same relationship.
Percent proportions
Percent means per hundred:
If $18$ of $24$ students submit an assignment,
so $p=75$. Therefore $75\%$ submitted.
Scale drawings and maps
A scale relates drawing length to actual length. Keep units compatible before setting a proportion.
If $1$ cm represents $5$ km, then $7.2$ cm represents $36$ km. Area scales by the square of a length scale, not by the same linear factor.
Similar figures
Corresponding sides of similar figures are proportional. Match sides opposite equal angles or in the same relative position.
Do not compare a short side to a long side in one ratio and reverse that order in another.
Direct-proportion graph
A relationship $y=kx$ is directly proportional. Its graph is a straight line through the origin, and $k=y/x$ is constant for nonzero $x$.
A linear relationship with nonzero intercept is not a direct proportion, even though it has a constant additive rate.
When a proportion is not appropriate
Not every situation scales multiplicatively. A taxi fare with a fixed starting charge, temperature conversion, and bulk discount may not follow $y=kx$.
Check the context, table ratios, and graph before assuming proportionality.
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?
Related topics
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
A recipe uses 3 cups of flour for 8 servings. How many cups are needed for 20 servings?
- 8x = 60.
- x = 7.5 cups.
End of lesson
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