# Ratio and Proportion

Ratio

Ratio refers to a comparison between two similar things.

Ratio can be represented with a colon.
a:b

Ratio can also be represented as a fraction.
$\bf\displaystyle\frac{a}{b}$

Ratio of yellow beads to green beads = 1 to 2 = 1:2 = $\bf\displaystyle\frac{1}{2}$

Ratio of yellow beads to green beads = 2 to 4 = 2:4 = $\bf\displaystyle\frac{2}{4}$

Ratio of yellow beads to green beads = 3 to 6 = 3:6 = $\bf\displaystyle\frac{3}{6}$

Proportion
Proportion refers to two equal ratios.
a:b = c:d
$\bf\displaystyle\frac{a}{b}$$\bf\displaystyle\frac{c}{d}$

For ratios $\bf\displaystyle\frac{a}{b}$ and $\bf\displaystyle\frac{c}{d}$ to be in proportion ad = bc

To Prove Ratios are in Proportion, Cross-Multiply

In order to cross-multiply, it is helpful to understand algebraic equations.
It is also helpful to know your multiplication and division tables.

$\bf\displaystyle\frac{a}{b}$$\bf\displaystyle\frac{c}{d}$
For ratios $\bf\displaystyle\frac{1}{2}$ and $\bf\displaystyle\frac{6}{12}$

bc = 2 × 6
1 × 12 = 2 × 6
12 = 12
Because ad = bc, ratios $\bf\displaystyle\frac{1}{2}$ and $\bf\displaystyle\frac{6}{12}$ are in proportion.

To Solve for an Unknown Variable in Proportionate Ratios, Cross-Multiply
In order to cross-multiply, it is helpful to understand algebraic equations.
It is also helpful to know your multiplication and division tables.

$\bf\displaystyle\frac{a}{b}$$\bf\displaystyle\frac{c}{d}$
For ratios $\bf\displaystyle\frac{2}{4}$ and $\bf\displaystyle\frac{5}{d}$

bc = 4 × 5
2 × d = 4 × 5
2d = 20
$\bf\displaystyle\frac{2d}{2}$ = $\bf\displaystyle\frac{20}{2}$
d = 10

Question
Prove that the ratios of yellow to green beads in the image below are in proportion.

$\bf\displaystyle\frac{1}{2}$$\bf\displaystyle\frac{2}{4}$

$\bf\displaystyle\frac{a}{b}$$\bf\displaystyle\frac{c}{d}$
For ratios $\bf\displaystyle\frac{1}{2}$ and $\bf\displaystyle\frac{2}{4}$

bc = 2 × 2
1 × 4 = 2 × 2
4 = 4
Because ad = bc, ratios $\bf\displaystyle\frac{1}{2}$ and $\bf\displaystyle\frac{2}{4}$ are in proportion.

Question

In the image below, two ramps are in proportion.  Find the length L of the larger ramp.

6

$\bf\displaystyle\frac{a}{b}$$\bf\displaystyle\frac{c}{d}$
For ratios $\bf\displaystyle\frac{2}{3}$ and $\bf\displaystyle\frac{4}{L}$

bc = 3 × 4
2 × L = 3 × 4
2L = 12
$\bf\displaystyle\frac{2L}{2}$$\bf\displaystyle\frac{12}{2}$
L = 6

Question

In the image below, two ramps are in proportion.  Find the length Q of the smaller ramp.

10

$\bf\displaystyle\frac{a}{b}$$\bf\displaystyle\frac{c}{d}$
For ratios $\bf\displaystyle\frac{20}{40}$ and $\bf\displaystyle\frac{5}{Q}$

bc = 40 × 5
20 × Q = 40 × 5
20Q = 200
$\bf\displaystyle\frac{20Q}{20}$$\bf\displaystyle\frac{200}{20}$
Q = 10

Practice – Questions
1.  Prove that the ratios of yellow to green beads in the image below are in proportion.

2.  In the image below, two ramps are in proportion.  Find the length L of the larger ramp.

3.  In the image below, two ramps are in proportion.  Find the length Q of the smaller ramp.

4.  In the image below, two rectangular boxes are in proportion.  Find the length L of the larger box.

5.  In the image below, two rectangular boxes are in proportion.  Find the width W of the smaller box.

1.  For ratios $\bf\displaystyle\frac{1}{2}$ and $\bf\displaystyle\frac{3}{6}$
Because ad = bc, ratios $\bf\displaystyle\frac{1}{2}$ and $\bf\displaystyle\frac{3}{6}$ are in proportion.