Ratios Fundamentals

A ratio compares two or more quantities by division rather than by subtraction. Once a ratio is written in lowest terms, it can be scaled up or down to match any actual total, part, or even the difference between two parts — and it converts directly into a fraction or a percentage. This lesson builds the habit of thinking in "parts" that makes ratio problems fast and reliable.

What you will learn

• Interpret a ratio as a comparison between quantities
• Convert between ratios, fractions, and percentages
• Scale a ratio to match a given total or part

Keep paper and a pencil nearby. Try each problem before opening a hint, and mark sections reviewed as you work through them.

What a ratio compares

A ratio compares two quantities by division. Writing "the ratio of red to blue marbles is 3 to 5," or $3:5$, means that for every 3 red marbles there are 5 blue marbles — not that there are exactly 3 red and 5 blue marbles, but that the two counts stand in that proportion. A ratio has no units of its own; $3:5$ describes the same relationship whether there are 8 marbles or 800.

It matters whether a ratio compares part to part or part to whole. "Red to blue is $3:5$" is a part-to-part ratio, comparing two pieces of the same collection. "Red to total is $3:8$" is a part-to-whole ratio, since $3+5=8$ is the whole collection. Reading a problem carefully to see which comparison is being made is the single most common place ratio problems go wrong.

Ratios, fractions, and percentages

Every ratio can be rewritten as a fraction, and every fraction can be rewritten as a percentage, by multiplying by $100\%$. If red to blue marbles is $3:5$, then red is $\frac{3}{5}$ as numerous as blue — but red is $\frac{3}{8}$ of the whole collection, since the whole is $3+5=8$ parts. Converting a ratio to a percentage almost always means dividing by the sum of all the parts, not by one of the individual terms, because a percentage is inherently a part-to-whole comparison.

This conversion runs in both directions. A statement like "60% of the coins are pennies" is the same as the part-to-whole ratio $60:100$, which reduces to $3:5$; since pennies are 3 of every 5 coins, the remaining 2 of every 5 coins are non-pennies, giving a pennies-to-non-pennies part-to-part ratio of $3:2$.

Scaling a ratio to match reality

A ratio in lowest terms, such as $3:5$, is really a statement about "parts": 3 parts of one quantity for every 5 parts of the other. To match this ratio to an actual situation, find one scale factor that turns parts into real units, then apply that same scale factor to every term of the ratio. If the total is known, divide the total by the sum of the parts ($3+5=8$) to get the scale factor; if only one actual quantity is known, divide that quantity by its own number of parts instead.

The same idea handles a less obvious case: matching a ratio to a known difference between two parts. If red is to blue as $3:5$ and blue outnumbers red by a known amount, that amount corresponds to $5-3=2$ parts, so dividing the known difference by 2 gives the scale factor. Whichever quantity is known — a total, a single part, or a difference — the method is the same: figure out how many parts that quantity represents, divide to find the scale factor, then multiply every term of the ratio by it.

See the reasoning unfold

Pause after each step and predict what comes next.

Common mistakes to catch

• Dividing a part by another part instead of by the sum of all the parts when converting to a percentage.
• Scaling only one term of a ratio and forgetting to apply the same scale factor to the other terms.
• Treating a known difference between two quantities as if it were their total when finding the scale factor.

Try with support

Guided practice

Choose a strategy, take a first step, and use the hints when you need direction.

Put it into practice

Independent practice

Work on your own first. If you reveal a solution, work through it and mark it reviewed.

Bring it together

• A ratio $a:b$ compares two quantities by division; check whether a problem's comparison is part-to-part or part-to-whole.
• Convert a ratio to a percentage by dividing a part by the sum of all the parts, then multiplying by 100%.
• Scale a ratio to a real situation by finding what one part is worth from a known total, part, or difference, then multiplying every term by that scale factor.