Introduction to Equivalent Ratios

Today, we’re going to talk about something super interesting: equivalent ratios. Ratios are a way to compare two or more quantities. When we say “equivalent ratios,” we mean different ratios that represent the same relationship between quantities.

Understanding Ratios

A ratio shows how much of one thing there is compared to another. For example, if there are 2 apples and 3 oranges, we can write the ratio of apples to oranges as 2:3.

What Are Equivalent Ratios?

Equivalent ratios are ratios that can be simplified to the same form. For example:

  • The ratio 2:3 is equivalent to 4:6 because if you double both numbers in 2:3, you get 4:6.
  • Similarly, 6:9 is also equivalent to 2:3 because if you divide both numbers by 3, you again get 2:3.

Key Rules for Ratios

  1. Simplifying Ratios: You can simplify a ratio by dividing both parts by the same number. This helps you find equivalent ratios.
  2. Finding Equivalent Ratios: You can multiply or divide both parts of the ratio by the same number to create equivalent ratios.
  3. Using Ratios in Word Problems: Read the problem carefully, identify the ratios involved, and then apply the key rules to find the solution.

Tips and Tricks

  • Visual Aids: Drawing pictures or using objects can help you understand ratios better.
  • Practice with Real-Life Examples: Think about recipes or sports scores; these often involve ratios.
  • Cross-Multiplication: If you’re comparing two ratios, use cross-multiplication to check if they are equivalent. For example, for the ratios (a:b) and (c:d), check if (a \times d = b \times c).

Examples of Word Problems

Let’s look at a couple of examples to see how we can solve word problems using equivalent ratios.

Example 1

Problem: In a fruit basket, there are 4 apples and 6 oranges. What is the ratio of apples to oranges? Are there any equivalent ratios?

Solution: The ratio is 4:6. To simplify, divide both numbers by 2. So, we get 2:3. Equivalent ratios could be 8:12 or 10:15.

Example 2

Problem: A recipe requires 3 cups of flour for every 2 cups of sugar. If you use 9 cups of flour, how much sugar do you need?

Solution: The ratio of flour to sugar is 3:2. If we have 9 cups of flour, that’s 3 times the original amount (3 x 3 = 9). So, we also need 2 x 3 = 6 cups of sugar.

Questions

Easy Level Questions

  1. If there are 2 cats and 3 dogs, what is the ratio of cats to dogs?
  2. Write the equivalent ratio for 1:4 by multiplying both parts by 2.
  3. If a car travels 30 miles in 1 hour, what is the ratio of miles to hours?
  4. There are 5 boys and 10 girls in a class. What is the simplified ratio of boys to girls?
  5. If you mix 2 parts lemonade to 3 parts water, what is the ratio of lemonade to water?
  6. A recipe calls for 4 eggs to 2 cups of sugar. What is the simplified ratio?
  7. If there are 6 red balls and 4 blue balls, what is the ratio of red to blue?
  8. Write the equivalent ratio for 2:5 by multiplying both parts by 3.
  9. If a box contains 8 chocolates and 12 candies, what is the ratio of chocolates to candies?
  10. What is the simplified version of the ratio 10:15?

Medium Level Questions

  1. A car uses 3 litres of petrol for every 50 miles. How much petrol is used for 150 miles?
  2. If a recipe requires 2 cups of rice to 5 cups of water, what is the ratio of rice to water?
  3. If there are 12 boys and 8 girls in a school, what is the ratio of boys to girls in simplest form?
  4. A fruit seller has 15 apples and 25 oranges. What is the equivalent ratio of apples to oranges?
  5. If you have a 4:5 ratio of chocolate to vanilla ice cream, how much vanilla is needed for 12 scoops of chocolate?
  6. A car travels 120 miles using 4 gallons of petrol. What is the ratio of miles to gallons?
  7. If a box contains 20 red balls and 10 blue balls, what is the simplified ratio of red to blue?
  8. A recipe calls for 3 cups of flour for every 4 cups of sugar. If you use 9 cups of flour, how much sugar do you need?
  9. What is the equivalent ratio for 5:12 if you multiply both parts by 2?
  10. If there are 30 students in class, and the ratio of boys to girls is 2:3, how many girls are there?

Hard Level Questions

  1. A recipe requires 2 cups of sugar and 3 cups of flour. If you have 12 cups of sugar, how much flour do you need?
  2. If the ratio of cats to dogs in a pet shop is 5:2 and there are 35 cats, how many dogs are there?
  3. In a class of 40 students, the ratio of boys to girls is 3:2. How many girls are there?
  4. A fruit basket contains apples and bananas in the ratio of 4:5. If there are 36 bananas, how many apples are there?
  5. The ratio of red marbles to blue marbles is 7:3. If there are 21 red marbles, how many blue marbles are there?
  6. If the ratio of pencils to pens in a box is 5:4 and there are 60 pencils, how many pens are there?
  7. A car uses 1 litre of petrol for every 15 miles. How much petrol will it use for 120 miles?
  8. If the ratio of chocolate to strawberry ice cream is 9:5 and you have 45 scoops of chocolate, how many scoops of strawberry do you have?
  9. In a recipe, the ratio of water to juice is 2:3. How much juice do you need if you have 8 cups of water?
  10. The ratio of boys to girls in a sports team is 4:5. If there are 36 players, how many girls are on the team?

Answers

Easy Level Answers

  1. 2:3
  2. 2:8
  3. 30:1
  4. 1:2
  5. 2:3
  6. 2:1
  7. 3:2
  8. 6:15
  9. 2:3
  10. 2:3

Medium Level Answers

  1. 18 litres
  2. 2:5
  3. 3:2
  4. 3:5
  5. 15 cups
  6. 30:1
  7. 2:1
  8. 6 cups
  9. 10:24
  10. 18 girls

Hard Level Answers

  1. 18 cups of flour
  2. 14 dogs
  3. 16 girls
  4. 24 apples
  5. 9 blue marbles
  6. 48 pens
  7. 8 litres
  8. 25 scoops of strawberry
  9. 12 cups of juice
  10. 20 girls

Feel free to ask any questions if you’re unsure about anything! Let’s practice more on equivalent ratios and become experts!