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A ratio is a mathematical term used for comparing two or more quantities that are measured in the same units. We are using ratios in our daily lives without even realizing it. For example, in grocery shopping, cooking, and getting from place to place, we constantly use ratios.

The word ratio goes back to ancient Greece. The Latin translation of ratio means to be rationale or to reason. This reasoning pertains to understanding the relationship between two values.

For example, if there are $12$ notebooks and $4$ books in a bag, which means the ratio of notebooks to books is $3$ to $1$. We can also say that the ratio of books to notebooks is $1$ to $3$.

Let’s understand what is ratio and how it is calculated.

## What is Ratio?

The ratio is defined as the comparison of two quantities of the same units that indicates their sizes(or values) in relation to each other. This relation gives us how many times one quantity is equal to the other quantity. In simple words, the ratio is the number that can be used to express one quantity as a fraction of the other ones.

Looking back to the above example, where the ratio of notebooks to books in a bag is $3$ to $1$ indicates that the number of notebooks is three times the number of books. $\left(4 \left(\text{Number of books} \right) \times 3 = 12 \left(\text{Number of notebooks} \right) \right)$.

Similarly, the ratio of books to notebooks in a bag is $1$ to $3$, which means that the number of books in a bag is one-third the number of notebooks. $\left(12 \left(\text{Number of notebooks} \right) \times \frac{1}{3} = 4 \left(\text{Number of books} \right) \right)$.

## Writing Ratios

Ratios can be written in three different ways using ratio symbols or words while keeping the same meaning. The ways in which ratios can be represented are

- using fraction
- using the word ‘to’
- using the ratio symbol

The ratio in the above example, the ratio between notebooks to books can be written as

- using fraction: $\frac{3}{1}$
- using the word ‘to’: $3$ to $1$
- using the ratio symbol: $3 : 1$

**Note:** ‘:’ symbol is called ratio symbol.

The two numbers in a ratio are referred to as antecedent and consequent respectively.

**Antecedent:**The first number in a ratio is called antecedent, i.e., $a$ in $\frac{a}{b}$, or, $a$ to $b$, or $a : b$ is called antecedent.**Consequent:**The second number in a ratio is called consequent, i.e., $b$ in $\frac{a}{b}$, or, $a$ to $b$, or $a : b$ is called antecedent.

## Types of Ratios

Depending on the information needed, the ratios are broadly divided into two types:

- part-to-part ratio
- part-to-whole ratio

### Part-to-part Ratio

The part-to-part ratio denotes how two distinct entities or groups are related. For example, the ratio of notebooks to books in a bag is $3 : 1$.

### Part-to-whole Ratio

The part-to-whole ratio denotes the relationship between a specific group to a whole. For example, the ratio of notebooks to the total number of items in a bag is $3 : 4$.

**Note:** Total number of items in a bag is $12 \left( \text{notebooks} \right) + 4\left(\text{books} \right) = 16$.

## Calculating Ratios

A ratio expresses how much of one quantity is required as compared to another quantity. The two terms in the ratio can be simplified and expressed in their lowest form. Ratios when expressed in their lowest terms are easy to understand and can be simplified in the same way as we simplify fractions.

Let’s consider some examples to understand how to simplify and express numbers in the form of ratios.

### Examples

**Ex 1:** Express $18$ to $10$ in the form of a ratio.

$18$ to $10$ = $18 : 10 = \frac {18}{10} = \frac {9 \times 2}{5 \times 2} = \frac {9}{5} = 9 : 5$.

**Ex 2:** A certain daycare facility has 15 infants and 20 toddlers. What is the ratio of infants to toddlers?

Number of infants in a daycare facility = $15$

Number of toddlers in a daycare facility = $20$

Ratio of infants to toddlers = $15 : 20 = \frac{15}{20} = \frac{5 \times 3}{5 \times 4} = \frac{3}{4} = 3 : 4$.

**Ex 3:** There are $15$ girls and $10$ boys in a class. Find

Ratio of the number of boys to the number of girls

Ratio of the number of girls to the total number of students

Ratio of the number of boys to the total number of students

Number of girls in a class = $15$

Number of boys in a class = $10$

Total number of students in a class = $15 + 10 = 25$

Ratio of the number of boys to the number of girls = $10 : 15 = \frac{10}{15} = \frac{5 \times 2}{5 \times 3} = \frac{2}{3} = 2 : 3$.

Ratio of the number of girls to the total number of students = $15 : 25 = \frac{15}{25} = \frac{5 \times 3}{5 \times 5} = \frac{3}{5} = 3 : 5$.

Ratio of the number of boys to the total number of students = $10 : 25 = \frac{10}{25} = \frac{5 \times 2}{5 \times 5} = \frac{2}{5} = 2 : 5$.

**Ex 4:** Find the ratio of $75 cm$ to $1.5 m$.

The given numbers are not in the same units.

So, by converting them into the same units, we get

$1.5 m = 1.5 \times 100 cm = 150 cm \left( 1 m = 100 cm \right)$

The required ratio is $75 cm : 150 cm = 75 : 150 = \frac{75}{150} = \frac {75 \times 1}{75 \times 2} = \frac{1}{2} = 1:2$.

**Ex 5:** Ramesh deposited ₹$2050$ in a bank and in the month of January he withdrew ₹ $410$ from his account on the last date of the month. Find the ratio of

(a) Money withdrawn to the total money deposited.

(b) Money withdrawn to the remaining amount in the bank.

Amount deposited = ₹$2050$

Amount withdrew = ₹$410$

Balance amount = $2050 – 410 =$₹$1640$

Ratio of money withdrawn to the total money deposited = $410 : 2050 = \frac{410}{2050} = \frac{41 \times 10}{41 \times 5 \times 10} = \frac{1}{5} = 1:5$.

Ratio of money withdrawn to the remaining amount in the bank = $410 :1640 = \frac{410}{1640} = \frac{41 \times 10}{41 \times 4 \times 10} = \frac{1}{4} = 1:4$.

**Ex 6:** From the figure, find the ratio of

(a) The number of squares to the number of triangles.

(b) The number of circles to the number of rectangles.

(c) The number of blue coloured shapes to the total number of shapes.

Number of squares = $2$

Number of triangles = $3$

Number of circles = $3$

Number of rectangles = $3$

Total number of shapes = $2 + 3 + 3 + 3 = 11$

The ratio of the number of squares to the number of triangles = $2: 3$.

The ratio of the number of circles to the number of rectangles = $3 : 3 = \frac{3}{3} = 1 = 1 : 1$.

Number of blue coloured shapes = $3$

The ratio of the number of blue-coloured shapes to the total number of shapes =$3: 11$.

## Points to Remember

Following are the important points to remember while solving problems on ratios:

- In case both the numbers $a$ and $b$ are equal in the ratio $a: b$, then $a: b = 1$
- If $a \gt b$ in the ratio $a : b$, then $a : b \gt 1$
- If $a \lt b$ in the ratio $a : b$, then $a : b \lt 1$
- It is to be ensured that the units of the two quantities are similar before comparing them.

## Conclusion

The ratio is a comparison of two quantities of the same units that indicates their sizes in relation to each other. One can find a ratio of two quantities only when they are in the same units.

## Practice Problems

- Find the ratio of the following numbers
- $12$ to $16$
- $4.2$ to $16.6$
- $\frac{2}{5}$ to $\frac{4}{15}$

- There are $8$ pencils and $12$ pens in a box. Find the ratio of
- pencils to pens
- pens to pencils
- pens to the total number of items
- pencils to the total number of items

## Recommended Reading

- Factors and Multiples (With Methods & Examples)
- Fractions On Number Line – Representation & Examples
- Reducing Fractions – Lowest Form of A Fraction
- Comparing Fractions (With Methods & Examples)
- Like and Unlike Fractions
- Improper Fractions(Definition, Conversions & Examples)
- How To Find Equivalent Fractions? (With Examples)
- 6 Types of Fractions (With Definition, Examples & Uses)
- What is Fraction? – Definition, Examples & Types
- Mixed Fractions – Definition & Operations (With Examples)
- Multiplication and Division of Fractions
- Addition and Subtraction of Fractions (With Pictures)

## FAQs

### How do you calculate the ratio?

The following steps are used to find the ratio of two numbers $a$ and $b$

(a) Write it in the form $a:b = \frac{a}{b}$

(b) Reduce the fraction to its lowest form

(c) Express the result by using the colon symbol ‘:’

### What are the different ways of writing the ratio?

There are three different ways of writing a ratio. These are

(a) using fraction $\left(\frac{a}{b} \right)$

(b) using the word ‘to’ $\left( a \text{to} b\right)$

(c) using the ratio symbol $a: b$

### How many types of ratios are there?

The ratios are broadly divided into two types:**Part-to-part Ratio:** The part-to-part ratio denotes how two distinct entities or groups are related. **Part-to-whole Ratio: **The part-to-whole ratio denotes the relationship between a specific group to a whole.