Ratios & Proportions

TipLearning Objectives

By the end of this lesson, you’ll be able to:

  • Interpret ratios in fraction, colon, and “to” forms.
  • Create equivalent ratios using ratio tables.
  • Solve proportions using equivalent ratios and cross-multiplication.
  • Apply ratios to real-world situations like recipes, scale models, and mixtures.

Key Ideas

  • A ratio compares two quantities.
    Examples: \(3:5\), \(\frac{3}{5}\), “3 to 5”.

  • Equivalent ratios show the same relationship:

    \[ \frac{3}{5} = \frac{6}{10} = \frac{9}{15} \]

Ratio Tables

A ratio table shows equivalent ratios created by multiplying or dividing both quantities by the same number.

Cats Dogs
4 7
8 14
12 21
16 28

Notice that every row represents the same ratio:

\[ \frac{4}{7}=\frac{8}{14}=\frac{12}{21}=\frac{16}{28} \]

Ratio tables are often a quick way to solve proportional reasoning problems without setting up an equation.

  • A proportion sets two ratios equal:

    \[ \frac{a}{b} = \frac{c}{d} \]

  • To solve a proportion, use cross-multiplication:

    \[ ad = bc \]

Important

Ratios do not automatically give you the total.
A ratio of \(2:3\) means “2 of one thing for every 3 of another,” not necessarily 5 total unless stated.

Common Problem Types

Identifying and Simplifying Ratios

Example: Simplify 12:18.

Since

\[ \frac{12}{18} = \frac{2}{3}, \]

the simplified ratio is \(2:3\).

Using Ratio Tables

Example: Cats:dogs = \(4:7\). There are 28 dogs. How many cats?

Cats Dogs
4 7
? 28

Since

\[ 7 \rightarrow 28 \text{ is } \times 4, \]

multiply the cats value by the same factor:

\[ 4 \times 4 = 16 \]

So there are 16 cats.

Solving Proportions

Example:

\[ \frac{x}{8} = \frac{3}{4} \]

Cross-multiply:

\[ 4x = 24 \]

So:

\[ x = 6 \]

Part–Whole vs Part–Part Ratios

Example: Boys:girls = \(3:5\), total students = 40

Total ratio parts:

\[ 3 + 5 = 8 \]

Each part:

\[ 40 \div 8 = 5 \]

Boys:

\[ 3 \times 5 = 15 \]

Scale Models / Maps

Example: Scale \(1\text{ inch} : 20\text{ miles}\)

For 3 inches:

\[ 1:20 = 3:x \]

So:

\[ x = 60 \]

The actual distance is 60 miles.

Strategies

  • Always determine if a ratio compares parts to parts or parts to whole.
  • Use ratio tables when scaling up or down.
  • Cross-multiply when solving proportions quickly.
  • Be careful with units—ratios ignore them unless explicitly included.
  • Keep the order of the ratio consistent. For example, if the ratio is cats:dogs, then every comparison should stay cats:dogs.

Worked Examples

Example 1

Question: Cats:dogs = \(4:7\). There are 28 dogs. How many cats?

Solution:

Cats Dogs
4 7
? 28

Since

\[ 7 \rightarrow 28 \text{ is } \times 4, \]

multiply the cats value by the same factor:

\[ 4 \times 4 = 16 \]

Answer: 16 cats

Example 2

Question:

\[ \frac{5}{x} = \frac{15}{18} \]

Solution:

Cross-multiply:

\[ 5 \cdot 18 = 15x \]

\[ 90 = 15x \]

\[ x = 6 \]

Answer: \(x = 6\)

Example 3

Question: A recipe uses flour and sugar in a \(3:2\) ratio. If you have 12 cups of flour, how much sugar?

Solution:

Flour Sugar
3 2
12 ?

Since

\[ 3 \rightarrow 12 \text{ is } \times 4, \]

multiply the sugar value by the same factor:

\[ 2 \times 4 = 8 \]

Answer: 8 cups of sugar

WarningCommon Mistakes
  • Mixing up part-to-part and part-to-whole ratios.
  • Cross-multiplying incorrectly or before setting up a valid proportion.
  • Scaling only one part of a ratio instead of both parts.
  • Ignoring units when comparing two ratios.
  • Treating ratios as algebraic expressions without considering their context.

Practice Problems

  1. Simplify 45:60.

  2. If the ratio of red to blue marbles is \(2:5\) and there are 35 total marbles, how many are red?

  3. Solve:

    \[ \frac{x}{12} = \frac{3}{4} \]

  4. A blueprint uses a scale of \(1:50\). What actual length does 6 cm represent?

1.
GCF of 45 and 60 is 15.

\[ 45 \div 15 = 3 \]

\[ 60 \div 15 = 4 \]

Answer: \(3:4\)

2.
The ratio of red to blue marbles is \(2:5\).

Total ratio parts:

\[ 2 + 5 = 7 \]

Each part:

\[ 35 \div 7 = 5 \]

Red marbles:

\[ 2 \times 5 = 10 \]

Answer: 10 red marbles

3.
Cross-multiply:

\[ 4x = 36 \]

\[ x = 9 \]

Answer: \(x = 9\)

4.
A scale of \(1:50\) means each 1 cm on the blueprint represents 50 cm in real life.

For 6 cm:

\[ 6 \times 50 = 300 \]

So the actual length is 300 cm.

Since

\[ 100\text{ cm} = 1\text{ m}, \]

\[ 300\text{ cm} = 3\text{ m} \]

Answer: 300 cm, or 3 m

Summary

  • Ratios compare quantities; proportions set two ratios equal.
  • Equivalent ratios are created by multiplying or dividing both terms by the same number.
  • Ratio tables are useful for scaling ratios up or down.
  • Simplify ratios by dividing both terms by their GCF.
  • Cross-multiplication is a standard method for solving proportions.
  • Part–whole vs part–part distinctions matter in word problems.
  • Scale models rely on equivalent ratios to convert measurements.
  • Ask: is this part–part or part–whole before setting up equations.
  • Keep ratios in the same order across a proportion, such as red:blue = red:blue.
  • Use ratio tables when the scaling factor is easy to see.
  • Only cross-multiply after you’ve confirmed you truly have two equal ratios.
  • When scaling, multiply both parts of the ratio by the same number.