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Table of Contents

In this lesson, we will understand the concept of a **ratio and of a proportion**, the relationship between the two, and how we can use ratio and proportion in solving mathematical problems and real world problems also.

**See the fact file below for more information on the ratios and proportions or alternatively, you can download our 32-page Ratios and Proportions worksheet pack to utilise within the classroom or home environment.**

## Key Facts & Information

### RATIO AND PROPORTION

- Before we start identifying what the relationship is between ratio and proportion, we have to first identify what each of them is.
**RATIO**– A comparison between two numbers, and is commonly written as x:y wherein x and y are the numbers being compared.- How do we use ratios?
- For example, Angel has 3 shelves and 7 books. What is the ratio of books to shelves?
- 7:3

- Since we have identified the pattern of ratios, which is x:y, the first thing we need to understand and identify is which number should we place as the βxβ and which one is the βy.β
- Based on the question, the ratio we are looking for is the ratio of books to shelves, therefore, we can identify the number of books as the βxβ while the number of shelves as the βy.β
**PROPORTION**– An equation wherein two ratios are being identified as equals.- An example of a proportion is given below.
- 3:4 = 6:8

### APPLYING RATIOS

- We can use ratio to solve mathematical problems and also real world problems.
- First, we can use ratios in filling out tables.
- We first have to understand what the table means. In the example, the table shows the ratio of boys to girls.
- The columns can be interpreted as:
- There are 3 boys per 5 girls.
- There are 6 boys per 10 girls.
- There are 9 boys pero 15 girls.

- Next, we have to understand the relationship between each column or each ratio.
- Since we already know that we can express ratios as divisions then we can rewrite the table.
- Notice that β is an equivalent ratio of 6/10. How?
- Going back to what we have learned about comparing fractions, 3 is a multiple of 6 and 5 is a multiple of 10.
- If we multiply β by 2/2, we will get 6/10.
- Therefore, we can say that β is a multiple of 6/10.
- Moving forward, β is also a multiple of 9/15. Itβs the same situation as the previous ratio, the only difference this time is that we multiply β by 3/3 in order to get 9/15.
- Therefore, we can already identify that the missing ratio on the table is β multiplied by 4.
- Now, let us try solving a more complicated problem.
- Take the problem below as an example.
- If it takes 7 hours to complete 4 tasks, how many tasks would be completed in 35 hours?
- First, we have to identify the ratios.
- βIt takes 7 hours to complete 4 tasksβ
- 7:4

- βIt takes 35 hours to complete X tasksβ
- 35:X

- Our goal is to find the value of X.
- Now, since we have two ratios, we can solve this problem using proportions.
- One way to solve this is to use cross multiplication. We have learned about cross multiplication when we were solving for fractions.
- Then, since we are dealing with proportions. We can express them as:
- 7x = 140

- And we can perform the basic operations. First, divide 140 by 7, then we will be able to get the value of X, which is 20.
- Therefore, to answer the question – βIt takes 35 hours to complete 20 tasks.β
- There are also other ways to solve this problem:
- Since it takes 7 hours to complete 4 tasks, how many hours would it take to complete a task?
- We have to divide 7 by 4.
- This would give us 1.75.
- It takes 1.75 hours to complete a task.
- Now that we know this, let us answer this question. In 35 hours, how many tasks would be completed?
- Since we already know that a task can be accomplished in 1.75 hours, we need to divide 35 by 1.75.
- This would give us 20.
- Therefore, 20 tasks can be accomplished in 35 hours.

### FINDING PERCENTAGE USING RATIO

- In this section, we will try to understand how we can connect ratio to percent.
**PERCENT**– Percent is the ratio of a number and 100.- Remember that we can express ratios like fractions A/B, and with that we can also divide A by B.
- Since percent is the ratio of a number and 100, we can express this as X/100.
- Therefore, we can divide X by 100.
- 1% = 1/100

- With this in mind, we can solve problems related to finding or showing percentages.
- Take the problem below as an example.
- In a box of apples, there are 8 red apples and 2 green apples. What percent of apples in the box are green apples?
- First, we have to identify the given.
- 8 red apples
- 2 green apples

- Then, we can express it as a ratio 8:2 or 8/2, but remember that we are looking for the percent of green apples, therefore we have to express it as 2:8 or 2/8.
- We know that in order to express 2/8 as a percentage, we need to equate it to a percent.
- We also know that percentages can be expressed as X/100.
- Now that we have the equation above, we need to find X.
- We can do the same methods we used in the previous section.
- 8x = 200

- With this equation, we will get the value of X as 25.
- Therefore, the percentage of green apples in the box is 25%
- Now, let us go back to the ratio we got, 2/8.
- Based on our knowledge about fractions, we can divide 2 by 8. If we do this, we will get 0.25.
- Since we already have this, we can just multiply it with 100 since percentages are X/100, which will give us the same answer that
- we got and that is 25%.

### CONVERTING UNITS

- We will find out how we can use ratios to convert units in this section.
- There are 60 seconds in a minute. How many seconds are there in 3 minutes?
- This is easy for us by now, we know that we can solve it using the equation below:
- 60 x 3 = 180

- But, how do we get seconds from minutes?
- We are applying what we have learned about ratios and proportions.
- From the equation above, we need to solve for X.
- Therefore, we will get this equation:
- x seconds = 60 x 3

- Solving for x, we need to multiply 60 and 3, which will give us 180.
- After that, putting β180β to the position of x, we will get 180 seconds.
- Therefore, the answer is 180 seconds.

**Ratios and Proportions Worksheets**

This is a fantastic bundle which includes everything you need to know about the ratios and proportions across 32 in-depth pages. These are** ready-to-use Ratios and Proportions worksheets that are perfect for teaching students about the the concept of a ratio and of a proportion, the relationship between the two, and how we can use ratio and proportion in solving mathematical problems and real world problems also. **

### Complete List Of Included Worksheets

- Lesson Plan
- Ratios and Proportions
- Word to Ratio
- ER
- Table
- Express
- Pictures
- Wording
- Are They?
- Boxes
- Mix N Match
- Problems

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