Equivalent and Simplified Ratios

Understanding Ratios

A ratio compares two or more quantities, showing how much of one thing there is compared to another. It is written using the notation a:b, where a and b represent the quantities being compared.

Ratios can compare parts of a whole or parts to each other:

• Part-to-part ratio compares one part directly to another part. For example, if a fruit basket has 3 apples and 2 oranges, the ratio of apples to oranges is 3:2.
• Part-to-whole ratio compares one part to the total amount. Using the same example, the ratio of apples to total fruit is 3:5 (since 3 apples + 2 oranges = 5 fruits).

Ratios help us understand the relative sizes of quantities without needing exact numbers.

For instance, if a recipe calls for 4 parts flour to 1 part sugar, the ratio of flour to sugar is 4:1, meaning flour is used four times as much as sugar.

Equivalent Ratios

Equivalent ratios express the same relationship between quantities, even if the numbers are different. You can find equivalent ratios by multiplying or dividing both terms by the same non-zero number (called the scale factor).

For example, the ratio 2:3 is equivalent to 4:6 because both terms have been multiplied by 2:

$$2 \times 2 = 4, \quad 3 \times 2 = 6$$

Similarly, dividing both terms of 10:15 by 5 gives the equivalent ratio 2:3:

$$\frac{10}{5} = 2, \quad \frac{15}{5} = 3$$

To check if two ratios are equivalent, you can:

• Multiply across and compare: For ratios a:b and c:d, check if $a \times d = b \times c$.
• Convert both ratios to fractions and see if they are equal: $\frac{a}{b} = \frac{c}{d}$.

For example, to check if 3:5 and 6:10 are equivalent:

$$3 \times 10 = 30, \quad 5 \times 6 = 30$$

Since both products are equal, the ratios are equivalent.

Using scale factors is especially useful when working with larger numbers or when adjusting recipes, maps, or models.

For example, if a map uses a scale factor of 1:50,000, a 2 cm distance on the map represents 100,000 cm in real life. This is calculated as $2 \times 50,000 = 100,000$ cm, which is equal to $1,000\,\mathrm{m}$ or $1\,\mathrm{km}$. Therefore, 2 cm on the map represents 2 km in real life.

Simplifying Ratios

Simplifying a ratio means expressing it in its simplest form, where the terms have no common factors other than 1. This is done by dividing both terms by their highest common factor (HCF).

For example, to simplify the ratio 12:18:

• Find the HCF of 12 and 18, which is 6.
• Divide both terms by 6:

$$\frac{12}{6} = 2, \quad \frac{18}{6} = 3$$

So, the simplified ratio is 2:3.

Simplifying ratios makes them easier to understand and compare.

When simplifying ratios involving decimals, first multiply both terms by a power of 10 to convert decimals to whole numbers, then simplify. For example, simplify 0.4 : 1.2 by multiplying both terms by 10 to get 4 : 12, then simplify by dividing both terms by 4 to get 1 : 3.

For example, simplify 0.6 : 1.2:

• Multiply both terms by 10 to remove decimals: 6 : 12
• Find HCF of 6 and 12, which is 6
• Divide both terms by 6: 1 : 2

So, 0.6 : 1.2 simplifies to 1 : 2.

Simplifying ratios is similar to simplifying fractions; both involve dividing by the greatest common factor.

For example, the ratio 50:80 can be simplified by dividing both terms by 10:

$$\frac{50}{10} = 5, \quad \frac{80}{10} = 8$$

So, the simplest form is 5:8.

Simplifying helps when sharing quantities or comparing proportions in real-life contexts such as mixing paints, dividing money, or sharing food.