Ratio and Proportion

What Is Ratio?

A ratio compares two or more quantities of the same kind by division. It shows how many times one value contains or is contained in another.

For example, if there are 2 apples and 3 oranges, the ratio is written as \( 2 : 3 \).

Simplifying Ratios

To simplify a ratio, divide each term by their highest common factor (HCF). For example, \( 10 : 15 \) simplifies to \( 2 : 3 \) because both terms can be divided by 5.

Expressing Ratios in a Given Form

Sometimes, you are asked to write a ratio in the form \( 1 : n \) or \( n : 1 \). Divide both parts by the appropriate number to achieve this.

Dividing a Quantity in a Given Ratio

To divide an amount in a given ratio, first add the parts of the ratio. Then, divide the total by that sum and multiply each ratio part by the result.

Example: Divide \$60 in the ratio \( 2 : 3 \): Total parts = \( 2 + 3 = 5 \) Each part = \( \frac{60}{5} = 12 \) Share: \( 2 \times 12 = \$24 \), \( 3 \times 12 = \$36 \)

What Is Proportion?

Proportion means that two ratios are equal. For example, if \( \frac{2}{3} = \frac{4}{6} \), then these are in proportion.

Proportional Reasoning

You use proportional reasoning when solving problems involving scaling, recipes, maps, and similar real-life contexts. If one quantity increases, the other increases or decreases in the same ratio.

Example: If 4 pencils cost \$20, then 1 pencil costs \$5. So, 7 pencils cost \( 7 \times 5 = \$35 \). This uses direct proportion.

Direct and Inverse Proportion

Direct Proportion

Two quantities are said to be in direct proportion when an increase in one results in a proportional increase in the other (and vice versa). This means: if one doubles, the other doubles.

Mathematically, if \( x \) is directly proportional to \( y \), we write: \( x \propto y \) or \( x = ky \), where \( k \) is a constant.

Example: The cost of apples is directly proportional to the weight bought. If 2 kg cost \$5, then 4 kg will cost \$10.

Inverse Proportion

Two quantities are in inverse proportion when one increases and the other decreases in such a way that their product remains constant.

Mathematically, if \( x \) is inversely proportional to \( y \), we write: \( x \propto \frac{1}{y} \) or \( x = \frac{k}{y} \), where \( k \) is a constant.

Example: If 4 people can complete a job in 6 hours, then 8 people (working at the same rate) will complete it in 3 hours. More people → less time = inverse proportion.

How to Identify the Type of Proportion:

• If quantities grow or shrink together at the same rate → direct proportion
• If one increases while the other decreases → inverse proportion

Important:

• Use the same units when working with ratios and proportions.
• Always simplify final answers where possible.
• Ensure the context makes sense (e.g., money to nearest cent or dollar, people must be whole numbers).

Ratio and Proportion Rules and Tricks