# Ratio

In our daily lives, we come across various instances where different quantities of an object are compared.

Let's say that you visited a local grocery store to buy a packet of your favorite snack.

You ask the store owner to give you a packet of that snack.

The store owner will ask you the size of the packet you want to buy.

The comparison of different quantities of the snack in various sizes of packets and their respective prices come under the concept of ratio.

How would you find the best buy among the available options?

The best buy would mean the most quantity for the least price.

This can be calculated by finding the ratio of the same for each size.

Let us further explore this concept in detail.

In this chapter, we will learn about ratio scale, ratio meaning, ratio definition, ratio calculator, ratio table, ratio math, and ratio formula in the concept of ratio.

Check out the interactive simulation to know more about the lesson and try your hand at solving a few interesting practice questions at the end of the page.

## Lesson Plan

 1 What Is the Ratio? 2 Important Notes on Ratio 3 Challenging Questions on Ratio 4 Solved Examples on Ratio 5 Interactive Questions on Ratio

## What Is a Ratio?

### Ratio Definition

The ratio is defined as the relation between the quantities of two or more objects and it indicates the amount of one object contained in the other.

We will try to understand the meaning of ratio better using the following example.

Jane is helping her mother bake cookies for her school fair.

Her mother asked her to measure the ingredients as per the quantities given in the recipe and arrange them in different bowls.

How would you relate to this scenario while studying about ratios?

The recipe for making cookies include different ingredients and the quantities will vary with the number of cookies Jane plans to make.

If the recipe says "two cups of flour require one cup of milk," it indicates a ratio of 1:2

Similarly, if Jane's mother asks her to add 6 choco chips with 2 teaspoons of sugar for each cookie, that would mean a ratio of 6:2 or 3:1

We will learn about the simplification of ratios to understand these numbers.

In the grid given below, some squares are colored orange and others are colored green.

How would you find the ratio of their occurrence in the grid?

The number of orange-colored squares =  9

The number of green-colored squares = 11

The above squares can be represented in the ratio = 9:11

The above case depicts how the concept of the ratio is relatable in our day-to-day lives.

## Ratio Formula

We use the ratio formula while comparing the relationship between two numbers or quantities.

The general form of representing a ratio of two quantities say $$\text a$$ and $$\text b$$ is

$$\text a : \text b$$

which is read as $$\text a$$ ratio $$\text b$$.

The fraction form that represents this ratio is

$$\dfrac{a}{b}$$

To further simplify a ratio, we will follow the same procedure that we use for simplifying a fraction.

 $$( \text a : \text b) = \dfrac{a}{b}$$

Example

In a class of 50 students, 23 are girls and the remaining are boys.

Find the ratio of the total number of boys to the number of girls.

Solution:

The total number of girls = 23

The total number of boys can be calculated as

\begin{align*} &= \text {total number of students} - \text {total number of girls} \\ \\ &= 50 - 23 \\ \\ &= 27 \end{align*}

The desired ratio is

( Number of boys : Number of girls )

$$27:23$$

### Ratio scale

The ratio scale is a level of measurement that helps in comparing the intervals and differences.

## How Do You Calculate a Ratio?

Step 1 Find the quantities of both the objects for which we are determining the ratio.

Step 2 Write it in the form $$( \text a : \text b )$$.

Step 3 The sum of $$\text a$$ and $$\text b$$ would give the total quantities for both objects.

Step 4 Simplify the ratios further, if possible. The simplified ratio is the final result.

Example

Tim is making chocolate milkshakes for his friends.

He adds 1 glass of chocolate syrup to 5 glasses of milk.

How would you calculate the ratio of chocolate syrup to milk in the milkshake?

Solution:

The number of glasses of chocolate syrup = 1

The number of glasses of milk = 5

This simply gives us the ratio of chocolate syrup to milk as,

1:5

The total number of glasses of milkshake that will be prepared is 6.

Assume 6 more friends show up at the same time. What will be the new ratio for the milkshake?

Tim would add 1 glass of syrup and 5 glasses of milk to the already prepared milkshake.

This would give him 12 glasses of milkshake.

The number of glasses of syrup = 2

The number of glasses of milk = 10

New ratio = 2:10

Tim followed the same recipe; however, the obtained ratio is different.

Is it really different?

No, it is not different!

( 1:5 ) = ( 2:10 )

i.e., the new ratio also equals 1:5

How are these two ratios the same?

We will find out in the next section.

## How Do You Simplify a Ratio?

To simplify a ratio, we need to do the following.

Step 1 Write the given ratio $$( \text a : \text b )$$ in the form of a fraction $$\frac{\text a}{ \text b }$$.

Step 2 Find the highest common factor of $$\text a$$ and $$\text b$$.

Step 3 Divide the numerator and denominator of the fraction with that GCF to obtain the simplified fraction.

Step 4 Represent this fraction in the ratio form to get the result.

Example

Kate is solving a problem on ratios.

She gets (18:10) as the answer.

Can the answer be further simplified?

Solution:

1. We will write the given ratio in the form of a fraction.

(18:10) = $$\dfrac {18}{10}$$

2. The GCF of 10 and 18 will be 2. Dividing the numerator and denominator by 2, we get

$$\dfrac {18 \div 2}{10 \div 2}$$

=  $$\dfrac {9}{5}$$

3. Thus the simplified ratio is (9:5)

Important Notes
• In case both the numbers $$\text a$$ and $$\text b$$ are equal in the ratio $$( \text a : \text b )$$, then$$( \text a : \text b )$$ = 1.
• If $$\text a > \text b$$ in the ratio $$( \text a : \text b )$$, then $$( \text a : \text b ) > 1$$.
• If $$\text a < \text b$$ in the ratio$$( \text a : \text b )$$, then $$( \text a : \text b ) < 1$$.
• Check that the units of the two quantities are similar before comparing them.

### Ratio: Mathematics

We have discussed the mathematical significance of ratio in our daily lives using different examples.

The following ratio calculator will help you observe these ratios mathematically for different values.

### Equivalent Ratios

The ratio obtained by multiplying or dividing the given ratio by a number is called its equivalent ratio.

### Ratio table

A ratio table is a list containing the equivalent ratios of any given ratio in a structured manner.

Example

The following ratio table gives the relation between the ratios and the numbers.

## What Are “ Part-to-Part” and “Part-to-Whole” Ratios?

Let's understand this concept using the following example.

A garden contains two varieties of plants, lilies, and roses. The number of lilies is 15 and the number of roses is 11

### Part-to-Part Ratios

The part-to-part ratio means the ratio of parts of a whole with each other.

For the above case, it would be the ratio of the lilies to that of the roses (Both lilies and roses are part of the garden.)

Desired ratio: 15:11

### Part-to-whole Ratios

The part-to-whole ratio is defined as the ratio of the individual parts to that of the total or whole quantity.

For the above case, it would be the ratio of the lilies to the total plants in the garden or the ratio of the roses to the total plants in the garden.

Desired ratio:

1. Lilies to the total plants = 15:26

2. Roses to the total plants = 11:26

Challenging Questions
• The ratio of two numbers is 11:6. If 3 is added to the greater number and 4 is subtracted from the smaller number, the greater number becomes twice the smaller one. Find the numbers.
• The ratio of A’s salary to B’s was 2:3. A’s salary is increased by 10% and B’s is decreased by 20%. What is the ratio of their salaries now?
• Divide 410 among 3 persons A, B, and C such that 3 times A’s share, 2 times B’s share, and 4 times C’s share are all equal. Find the shares of A, B, and C respectively. ## Solved Examples  Example 1 There are 49 boys and 28 girls in school. Express the ratio of the number of boys to that of girls. Solution We have, Number of boys = 49 Number of girls = 28 The ratio of the number of boys to that of girls = 49:28 Now, to simplify, divide the two terms by their GCF 7 Thus, the ratio of the number of boys to that of girls = 7:4  $$\therefore$$ Ratio = 7 : 4  Example 2 Paul has a measuring tape of length 20 $$\text{ft}$$ and its width is 2.4 $$\text{in}$$. What is the ratio of its length to width? Solution It is given that, Length of the measuring tape = 20 $$\text{ft}$$ Width of the measuring tape = 2.4 $$\text{in}$$ We know that, 1 ft = 12 $$\text{in}$$ Length of the measuring tape in inch is = 240 $$\text{in}$$ Ratio of its length to width = $$\frac{240 \text{ in}}{ 2.4 \text{ in}}$$ Dividing the numerator by denominator, we get, Ratio of its length to width = 100:1  $$\therefore$$ Ratio = 100:1  Example 3 Leonard wants to find the simplest form of 87:75. Can you help him? Solution The GCF of 87 and 75 is 3 We divide each term by 3 We get, $$\frac{87}{3}$$:$$\frac{75}{3}$$ = 29:25 Thus, the ratio 87:75 in the simplest form is 29:25  $$\therefore$$ Ratio = 29:25  Example 4 The ratio of marks obtained by Donald and Trevor is 4:7 Find the marks obtained by Donald if Trevor scored 77 marks in the exam. Solution Let us assume the marks obtained by Donald is 4$$x$$ and Trevor is 7$$x$$. We know that Trevor scored 77 marks in the exam. \begin{align*} \implies 7 \text x &= 77 \\ \implies \text x &= 11 \end{align*} Thus, marks obtained by Donald \begin{align*} &= 4 \text x \\ &= 11 \times 4 \\ &= 44 \end{align*}  $$\therefore$$ Marks obtained by Donald = 44  Example 5 Kylie wants to divide1700 into a ratio of 8:9

Can you help her with this task?

Solution

The sum of the terms of the ratio = 8 + 9 = 17

First part = $$$1700 \times \frac{8}{17}$$=$800

Second part = $$$1700 \times \frac{9 }{17}$$ =$900

 $$\therefore$$ First part = $800 Second part =$900

## Interactive Questions

Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.

## Let's Summarize

The mini-lesson targeted the fascinating concept of ratio. The math journey around ratio starts with what a student already knows, and goes on to creatively crafting a fresh concept in the young minds. Done in a way that not only it is relatable and easy to grasp, but also will stay with them forever. Here lies the magic with Cuemath.

At Cuemath, our team of math experts is dedicated to making learning fun for our favorite readers, the students!

Through an interactive and engaging learning-teaching-learning approach, the teachers explore all angles of a topic.

Be it worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in.

## 1. What is unit ratio?

A unit ratio is defined as the ratio of two numbers in which the second number is 1

For example, 5:1, 7:1, 3:1, etc.

## 2. What does 5 to 1 ratio mean?

5 to 1 ratio means the ratio of form 5:1, in which for every 5 quantities of the first part, there will be 1 part of the second quantity contained.

Also, it is a unit ratio.

For example: For every 5 parts of water in the solution, there will be 1 part of the solute.

Ratio, Proportion, and the Unitary Method