Ratios

Ratios compare quantities multiplicatively and power proportions, scale drawings, mixtures, probability, rates, and percent.

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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 ratio compares two quantities by division. It answers “how many times as much?” or “how much of one quantity for another?”

Reading a ratio

If a box contains $6$ blue tiles and $4$ gold tiles, the blue-to-gold ratio is $6:4$, “$6$ to $4$,” or $6/4$. The order carries meaning: gold to blue is $4:6$, not $6:4$.

A ratio may compare part to part, part to whole, or whole to part. For the same box:

blue to gold is $6:4$;
blue to total is $6:10$;
total to gold is $10:4$.

Name the quantities before writing numbers so the order remains clear.

Equivalent ratios

Multiplying or dividing both parts by the same nonzero number preserves the comparison:

$$ 6:4=3:2=12:8. $$

This is the same equivalence principle used with fractions. A ratio in simplest whole-number form is useful, but an unsimplified ratio may reveal the original counts.

Ratio tables

A ratio table organizes equivalent pairs without hiding their meaning.

Batches	Flour (cups)	Water (cups)
$1$	$3$	$2$
$2$	$6$	$4$
$5$	$15$	$10$

Each row preserves the flour-to-water ratio $3:2$. Scaling both entries together is essential.

Finding a missing quantity

You can also write a proportion:

\frac{2}{5}=\frac{x}{15}.

Cross multiplication gives $5x=30$, so $x=6$. The ratio-table reasoning explains why that algebra works.

Common mistakes

Reversing the order. Write labels such as blue:white above the numbers.

Scaling only one quantity. Equivalent ratios require the same multiplier or divisor on both parts.

Confusing part-to-part with part-to-whole. Add all parts before finding a fraction of the total.

Adding instead of multiplying. Moving from $2:3$ to $4:5$ adds $2$ to both entries but does not preserve the ratio.

Quick self-check

What two quantities am I comparing, and in what order?
Do they use the same units, or is this a rate?
Did both quantities receive the same scale factor?
If I divided a total, do the shares add back correctly?