**Introduction**:

**Definition of equivalent ratio:**

The ratio which we get by multiplying or dividing by the same non-zero number to the two terms of the given ratio is called an equivalent ratio. Equivalent ratios are ratios that express the same relationship.

**Obtaining an equivalent fraction:**

To get a ratio equivalent to the given ratio, we begin by representing the ratio in the form of a fraction. Then multiplying and dividing the first term and second term by the same non-zero number, we get an equivalent ratio. At last, it is represented in the ratio form.

**Example:**

Find an equivalent ratio of 3/2

.

**Solution:** The fractional form of the ratio given is 3/2

.

**Step 1:** Multiply by a non-zero number to both numerator and denominator. In this case, let us take.

3 × 4 / 2 × 4= 12 / 8

**Step 2:** Write the equivalent fraction in the ratio form, i.e., 12:8.

Therefore, 12:8 is the equivalent ratio of 3:2.

**5.2.1 Finding equivalent ratios using multiplications**

**Example 1:** Write three equivalent ratios to 5:6.

**Solution:** The fractional form of the ratio given is 5/6

.

**Step 1:** Multiply by a non-zero number to both numerator and denominator. In this case, let us take 2, 3 and 4 since we are finding three equivalent ratios.

5 × 2 / 6 × 2=10/12

5 × 3 / 6 × 3=15/18

5 × 4 / 6 × 4=20/24

**Step 2:** Write the equivalent fraction in the ratio form, i.e., 10:12, 15:18 and 20:24.

Therefore 10:12, 15:18 and 20:24 are the equivalent ratios of 5:6.

**Example 2:** A team of disaster response force consists of 6 doctors for every 25 nurses. If the ratio remains constant and there are 30 doctors, find the numbers of nurses the team must have.

**Solution:**

**First method:** Make a table with equivalent ratios. We know that ratio of doctors to that of nurses is 6:25.

**Step 1:** Multiply both terms of ratio by same non-zero number till we obtain 30 doctors.

**Step 2:** Check the corresponding nurses when the doctors are 30. Hence, 125 nurses have to be arranged for 30 doctors.

**Second method: **

**Solution:**

Ratio of doctors to that of nurses is 6:25. Guess the non-zero number by which 6 is to be multiplied to get 30. Use the same number to multiply 25 as well. In this case, take 5.

**Step 1:** Write the ratio as a fraction and multiply 5 to both terms.

6 / 25 =6 × 5 / 25 × 5 =30/125

**5.2.2 Finding equivalent ratios using divisions**

**Example 1: **Jane is sharing pens. The table below shows you the ratio in which they are shared. If Jane keeps the ratio same and gives 7 black pens to his friend, then find how many blue pens he must also share.

**Solution:**

**Step 1:** Divide 56 and 32 by the same non-zero number until we get 7 black pens; let’s start by 4.

We get 56 ÷ 4 = 14 and 32 ÷ 4 = 8.

**Step 2:** Divide 56 and 32 by the same non-zero number until we get 7 black pens; let’s take 8.

We get 56 ÷ 8 = 7 and 32 ÷ 8 = 4.

Therefore, we understand that Jane gives 4 blue pens and 7 black pens.

**Example 2: **Shawn needs 180 kilograms of meat to feed his 8 pet dragons. Find the quantity of meat Shawn needs to feed 2 dragons only.

**Solution: **

**Step 1:** Divide 180 and 8 by the same non-zero number until we get 2 dragons; let’s start by 2.

We get 180 ÷ 2 = 90 and 8 ÷ 2 = 4.

**Step 2:** Divide 90 and 4 by the same non-zero number until we get 2 dragons; let’s start by 2.

We get 90 ÷ 2 = 45 and 4 ÷ 2 = 2.

Therefore, we understand that Shawn needs 45 kilograms of meat.

**5.2.3 Finding equivalent ratios**

**Example 1:** Which of the following ratios are equivalent to 16:20:

2:3, 4:5, 18:22, 20: 25

**Solution:**

**Step 1:** Make a table of equivalent ratios, take the given ratio and reduce it to the lowest possible value. Take 4 and divide both the terms.

**Step 2:** Make equivalent ratios with the now reduced form, i.e., 4:5.

Therefore, 4:5, 20:25 are the equivalent ratios to 16:20.

**Example 2:** Which of the following ratios are equivalent to 45:81:

30:54, 14:5, 18:20, 35: 25

**Solution:**

**Step 1:** Make a table of equivalent ratios, take the given ratio and reduce it to the lowest possible value. Take 9 and divide both the terms.

**Step 2:** Make equivalent ratios with the now reduced form, i.e., 5:9.

Therefore, 5:9, 30:54 are the equivalent ratios to 45:81.

### What have we learned?

• Understanding equivalent ratios

• Find equivalent ratios using multiplications

• Find equivalent ratios using divisions

• Find equivalent ratios

### Mind Map :

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