Equivalent Ratios and Ratio Tables

Scaling keeps a ratio true

A recipe uses $2$ cups of flour to $3$ cups of milk, a ratio of $2:3$. Double both amounts and the recipe still tastes the same:

$$2 : 3 \qquad 4 : 6 \qquad 6 : 9 \qquad 8 : 12$$

These are equivalent ratios. Each one is the first scaled up, and every one describes the same mixture.

$$2 : 3 \quad \xrightarrow{\;\times 2\;} \quad 4 : 6$$

Multiply both parts by the same number, or divide both by the same number. That is the only move allowed.

Why adding does not work

Multiplying both parts keeps the relationship. Adding to both parts does not.

Start	Operation	Result	Same ratio?
$2:3$	$\times 2$	$4:6$	yes
$2:3$	$\times 5$	$10:15$	yes
$2:3$	$+2$	$4:5$	no

Look at what happens to the sizes. In $2:3$ the second amount is $1.5$ times the first. In $4:6$ it still is. But in $4:5$ it is only $1.25$ times — the milk has been watered down.

Multiplying scales both quantities by the same factor, so their comparison is untouched. Adding gives the smaller number a bigger boost, so the gap between them shrinks in relative terms.

Ratio tables

A ratio table lists equivalent ratios in order. Each row is the same recipe at a different size.

Flour (cups)	Milk (cups)
$2$	$3$
$4$	$6$
$6$	$9$
$8$	$12$
$10$	$15$

Two patterns run through it, and either one can be used:

• Across each row, milk is always $1.5$ times the flour.
• Down each column, the numbers go up by $2$ and by $3$.

The across-the-row relationship is the more useful of the two, because it works for any row, including ones the table does not show.

Finding a missing value

A ratio table turns a question into a search for the scale factor.

$$2 : 3 = 14 : \; ?$$

Ask what took $2$ to $14$:

$$14 \div 2 = 7$$

Then do the same to the other part:

$$3 \times 7 = 21$$ $$2 : 3 = 14 : 21$$

Simplest form

Scaling down gives the simplest form of a ratio. Divide both parts by their greatest common factor:

$$12 : 18 \quad \xrightarrow{\;\div 6\;} \quad 2 : 3$$

Both $12$ and $18$ divide by $6$, and nothing bigger divides both. So $2:3$ is as small as this ratio gets in whole numbers.

Why the simplest form is worth finding

It makes two ratios comparable at a glance.

Are $18:24$ and $21:28$ the same? Written that way, nothing visibly connects them. Simplified, both become $3:4$, so they are equivalent.

Ratio	Divide by	Simplest form
$18:24$	$6$	$3:4$
$21:28$	$7$	$3:4$

This is the same reason equivalent fractions get reduced. Two comparisons written at different scales look unrelated until both are brought down to the smallest whole numbers that carry them.

Worked examples

Common mistakes

Practice problems