Simplifying Ratios and Equivalent Ratios

Let’s look at simplifying ratios and finding equivalent ratios. These are useful skills that can make problems involving ratios easier to solve and understand.

Simplifying Ratios

Simplifying a ratio means breaking it down to its simplest form. You can think of it as reducing a fraction. It makes the numbers smaller and easier to work with, but the relationship between the numbers stays the same.

To simplify a ratio, you divide both numbers by their greatest common divisor (GCD). For example, if you have a ratio of 8:12, the GCD of 8 and 12 is 4. So, you’d divide both numbers by 4, giving you a simplified ratio of 2:3.

Let’s look at two more examples:

Example:

In the ratio 15:25, the GCD is 5.

Dividing both 15 and 25 by 5 results in a simplified ratio of 3:5.

Example:

For the ratio 7:14, the GCD is 7.

Dividing both terms by 7 gives a simplified ratio of 1:2.

Why simplify ratios?

Simplifying ratios makes them easier to understand and work with. It also allows you to see the most basic relationship between the quantities involved.

Equivalent Ratios

Equivalent ratios are just ratios that represent the same relationship between quantities, even if the numbers themselves are different. If you take a simplified ratio and multiply or divide both numbers by the same amount, you’ll get an equivalent ratio.

For example:

  • 4:6, 6:9 and 8:12 are all equivalent ratios because they simplify to 2:3
  • 5:15 and 1:3 are equivalent since both simplify to 1:3

How to Find Equivalent Ratios

To find equivalent ratios, you can either multiply or divide both terms in the ratio by the same non-zero number.

Example 1:

Starting with a simplified ratio of 2:3, if you multiply both terms by 2, you will get a new ratio of 4:6. This is an equivalent ratio because it represents the same relationship as 2:3.

Example 2:

Similarly, if you have a ratio of 3:4 and you multiply both terms by 3, you get 9:12. This is equivalent to the original ratio, as both will simplify to 3:4.

Example 3:

You can also find equivalent ratios by dividing both terms by the same number, as long as that number is not zero. For example, 8:12 can be reduced to 4:6 by dividing both terms by 2.

Practice Questions

Question 1:

Simplify the ratio 18:27

The GCD of 18 and 27 is 9

Therefore, 18 \div 9 = 218 \div 9 = 2 and 27 \div 9 = 327 \div 9 = 3

So the simplified ratio is 2:32:3


Question 2:

What is the equivalent ratio of 5:7 if both terms are multiplied by 3?

Multiply both terms by 3

So, 5 \times 3 = 155 \times 3 = 15 and 7 \times 3 = 217 \times 3 = 21

The equivalent ratio is 15:2115:21


Question 3:

A cake recipe requires ingredients in the ratio 2:3:4 for butter, sugar and flour respectively. If you have 120 g of flour, how much butter and sugar will you need?

The total ratio is 2+3+4 = 92+3+4 = 9. Each part is 120 \div 4 = 30g120 \div 4 = 30g

For butter, it’s 2 \times 30 = 60g2 \times 30 = 60g and for sugar, it’s 3 \times 30 = 90g3 \times 30 = 90g


Question 4:

Two towns A and B are in the ratio of 3:4 in population. If the total population of both towns is 21,000, find the population of each town.

The total ratio is 3+4 = 73+4 = 7. Each part is 21,000 \div 7 = 3,00021,000 \div 7 = 3,000

For town A, it’s 3 \times 3,000 = 9,0003 \times 3,000 = 9,000 and for town B, it’s 4 \times 3,000 = 12,0004 \times 3,000 = 12,000

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