Ratio, Proportion and Percentage

Ratio, Proportion and Percentage

In this topic, you will learn about ratios, proportions, and percentages. These concepts are essential in many areas, including finance, cooking, and science. Understanding them allows you to compare quantities, solve problems, and make informed decisions in everyday life.

3.1 Ratio

3.1.1 Ratios Involving Two Quantities

A ratio compares one quantity to another. It can be written in words, using the ratio sign (:), or as a fraction. For example, if there are 2 lecturers for 143 students, the ratio can be expressed as:

• In words: 2 lecturers to 143 students
• Using the ratio sign: 2 : 143
• As a fraction: 2/143

It is important to understand that the first number represents the first quantity, and the second number represents the second quantity.

Example 1

A school has a pupil-teacher ratio of 40 to 1. This means:

• In words: 40 pupils for every 1 teacher
• Using the ratio sign: 40 : 1
• As a fraction: 40/1

To simplify a ratio, divide both numbers by their greatest common divisor. For example, the ratio 10 to 5 can be simplified to 2 to 1:

10 : 5 = 2 : 1

3.1.2 Ratios Involving More Than Two Quantities

Ratios can also involve more than two quantities. For example, in a recipe for salad dressing, if you use 1/6 cup of sugar, 1/2 cup of vinegar, and 1/3 cup of oil, the ratio of these ingredients can be found as follows:

Example 2

Find the ratio of sugar to vinegar to oil:

• 1/6 cup of sugar
• 1/2 cup of vinegar = 3/6
• 1/3 cup of oil = 2/6

The ratio of sugar to vinegar to oil is:

1 : 3 : 2

This means for every 1 part of sugar, you use 3 parts of vinegar and 2 parts of oil.

3.2 Proportion

3.2.1 Definition of Proportion

A proportion is an equation that states that two ratios are equal. For example, if we have:

a/b = c/d

This means that the ratio of a to b is equal to the ratio of c to d.

Example 3

If 5 eggs are needed for 2 cakes, how many eggs are needed for 14 cakes?

Let x be the number of eggs needed for 14 cakes. The proportion is:

x/5 = 14/2

Cross-multiply to solve for x:

2x = 14 * 5

x = 70/2

x = 35

You need 35 eggs for 14 cakes.

3.3 Percentage

3.3.1 Understanding Percentages

A percentage is a way of expressing a number as a fraction of 100. For example, 25% means 25 out of 100. It is useful for comparing different quantities.

Example 4

If a student scores 15 out of 25 marks, to find the percentage:

Percentage = (15/25) * 100 = 60%

3.3.2 Calculating Percentages

To calculate a percentage of a number, use the following formula:

Percentage of a number = (Percentage/100) * Number

Example 5

To find 20% of R200:

20% of 200 = (20/100) * 200 = R40

3.3.3 Converting Between Percentages, Fractions, and Decimals

You can convert percentages to fractions by placing the percentage over 100 and simplifying. For example, 25% can be written as:

25/100 = 1/4

To convert a decimal to a percentage, multiply by 100. For example, 0.75 as a percentage is:

0.75 * 100 = 75%

Conversely, to convert a percentage to a decimal, divide by 100. For example, 80% as a decimal is:

80/100 = 0.80

Summary

• A ratio compares two or more quantities.
• A proportion states that two ratios are equal.
• A percentage expresses a number as a fraction of 100.
• You can convert between percentages, fractions, and decimals.

Check your understanding

1. Express the ratio of 3 apples to 5 oranges in three different ways.
2. If a recipe calls for 4 cups of flour and you want to use 6 cups, what is the new ratio of flour to the original amount?
3. Calculate 15% of R300.
4. Convert 0.85 to a percentage.