Math

Ratios, Rates, Proportional Relationships, and Units

2 min readEasy5-question drill

Ratios and proportions show up on nearly every Math section, and unit conversions are some of the most reliable points you can grab. Master the setup and you'll never lose time second-guessing whether to multiply or divide.

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A ratio compares two quantities. If a recipe uses 2 cups of flour for every 3 cups of sugar, the ratio of flour to sugar is 2:3 (also written 2/3). Order matters — 2:3 is not the same as 3:2.

A rate is a ratio that compares two different units, like miles per hour or dollars per pound. A unit rate has a 1 in the denominator: 60 miles per 1 hour, $4 per 1 pound.

A proportion is an equation saying two ratios are equal, like 2/3 = 8/12. This is your main tool. Whenever two quantities scale together (more of one means proportionally more of the other), set up a proportion and cross-multiply.

The cross-multiply rule: if a/b = c/d, then a × d = b × c.

Example: "3 apples cost $2. How much do 12 apples cost?" Set up:

3 apples / $2 = 12 apples / $x

Keep the same units in the same position — apples on top, dollars on bottom, on BOTH sides. Cross-multiply: 3x = 2 × 12 = 24, so x = 8. Twelve apples cost $8.

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Setting up a proportion (3 apples = $2)

Quantity	Left side	Right side
Apples (top)	3	12
Dollars (bottom)	2	x
Cross-multiply	3·x = 2·12	x = 8

Keep apples on top and dollars on bottom on both sides, then cross-multiply.

Unit conversion uses the same logic. To convert, multiply by a fraction equal to 1 — a conversion factor — arranged so the unit you're getting rid of cancels.

There are 5,280 feet in 1 mile. To convert 5 miles to feet:

5 miles × (5,280 feet / 1 mile) = 26,400 feet

The word "miles" cancels (top and bottom), leaving feet. Always write units in your fractions and cancel them like numbers — if the wrong unit cancels, you flipped the factor.

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Unit cancellation

Step	Expression	Result
Start	5 miles	—
Multiply by factor	5 mi × (5280 ft / 1 mi)	miles cancel
Compute	5 × 5280	26,400 ft

The unit 'miles' cancels, leaving feet.

A proportional relationship between x and y means y = kx, where k is the constant of proportionality (the unit rate). Its graph is a straight line through the origin (0,0). If you double x, you double y. This is different from a general linear relationship y = mx + b, which only passes through the origin when b = 0.

A proportional relationship y = 2x: a straight line through the origin where slope = unit rate.

The whole topic boils down to one habit: label your units, set quantities in matching positions, and cancel or cross-multiply. Do that and these become free points.

Quick check

Check your understanding with a question from this topic:

There are 5,280 feet in one mile. How many feet are in 5 miles?

Your answer:

Enter a whole number, fraction (e.g. 3/4), or decimal (e.g. .75).

Worked examples

Example 1

There are 5,280 feet in one mile. How many feet are in 7 miles?

Example 2

A car travels 150 miles using 5 gallons of gas. At this rate, how many gallons are needed to travel 240 miles?

Example 3

A factory produces 18 widgets every 4 minutes. Working at this constant rate, how many widgets does it produce in 2 hours?

Common pitfalls

Flipping the ratio order

A ratio of "flour to sugar" of 2:3 is not 3:2. When you set up a proportion, the same unit must be in the same position (top or bottom) on both sides — mixing them up gives the reciprocal of the right answer.

Multiplying instead of dividing in conversions

Don't guess whether to multiply or divide. Write the conversion factor as a fraction and check that the unwanted unit cancels. If "miles" doesn't cancel, flip the factor.

Confusing proportional with merely linear

Proportional relationships satisfy y = kx and pass through (0,0). A line like y = 2x + 5 is linear but NOT proportional — doubling x does not double y because of the +5.

Ignoring mismatched units in the question

If a rate is per minute but the question asks per hour (or gives hours), convert first. Plugging in 2 (hours) where the rate expects minutes is a classic wrong answer.

Key takeaways

A proportion sets two equal ratios; cross-multiply (a/b = c/d → ad = bc) to solve.
Keep matching units in matching positions on both sides of a proportion.
Convert units by multiplying by a fraction equal to 1, arranged so the unwanted unit cancels.
A proportional relationship is y = kx: a straight line through the origin, where k is the unit rate.
When a problem mixes rates and units, convert everything to the same units before computing.

Watch & learn

Curated Khan Academy walkthroughs on Ratios, Rates, Proportional Relationships, and Units. They're complementary to this lesson — watch one if a written explanation isn't clicking, or after to reinforce.

Tracks your progress across lessons.

Try it yourself

5 practice questions on Ratios, Rates, Proportional Relationships, and Units, drawn from the question bank. The tutor is one click away if you get stuck.

Where this lesson fits

Ratios, Rates, Proportional Relationships, and Units

You are here

→Percentages →Linear Equations in 2 Variables →Linear Functions