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Basics Of Ratios

Introduction and Representation of Ratio

Ratio:

  • The ratio is the comparison of two quantities.
  • The quantities used in ratio can be anything, like pens, fruits, bikes, books, shirts etc. or two groups of things.
  • These quantities are called terms.

For example:

Alex is making coffee for his friends. He needs 3 cups of milk and 2 spoons of sugar. Here, a quantity of milk and sugar are compared.

There are three ways to represent the ratio of cups of milk and number of spoons of sugar.

(i) Using the word 'to'

It can be represented as,

 \(3\) cups of milk to \(2\) spoons of sugar.

(ii) Using colon \((:)\)

It can be represented as,

\(3:2\)

(iii) Using fraction bar

It can be represented as,

\(\dfrac {3}{2}\)

NOTE: While writing a ratio, the order is very important, i.e., \(3:2\neq2:3\)

The ratio is also a multiplicative comparison of two numbers.

Example: Mr. Jones has \(3\) pens and \(2\) pencils. 

It means he has \(\dfrac {3}{2}\) as many pens as he has pencils.

Illustration Questions

Which one of the following ratios DOES NOT represent \(3\) cups of flour and \(2\) cups of milk?

A \(3:2\)

B \(\dfrac {3}{2}\)

C \(3\) cups of flour to \(2\) cups of milk

D \(2:3\)

×

Given: 

\(3\) cups of flour and \(2\) cups of milk.

Number of cups of flour \(=3\)

Number of cups of milk \(=2\)

The ratio between \(3\) cups of flour and \(2\) cups of milk can be written as-

(i) \(3\)  cups of flour to \(2\)cups of milk.

(ii) \(3:2\)

(iii) \(\dfrac {3}{2}\)

Thus, \(2:3\) does not represent the ratio of \(3\) cups of flour and \(2\) cups of milk.

Hence, option(D) is correct.

Which one of the following ratios DOES NOT represent \(3\) cups of flour and \(2\) cups of milk?

A

\(3:2\)

.

B

\(\dfrac {3}{2}\)

C

\(3\) cups of flour to \(2\) cups of milk

D

\(2:3\)

Option D is Correct

Types of Ratio

Ratio

  • The ratio is the comparison between two quantities.
  • There are three types of ratios:
  1. Part to part ratio
  2. Part to whole ratio
  3. Whole to part ratio

1. Part to part ratio

  • Part to part ratio represents a comparison between two parts of the same whole.
  • For example: Consider a hexagonal plate as shown in the figure.
  • The hexagonal plate is divided into six equal parts out of which two are shaded.
  • Thus, we will write part to part ratio i.e. shaded part to unshaded part for the hexagonal plate.

Number of shaded parts \(=2\)

Number of unshaded parts \(=4\)

Thus, ratio of shaded parts to unshaded parts \(=\dfrac{\text{Number of shaded parts}}{\text{Number of unshaded parts}}\)

\(=\dfrac{2}{4}\)

\(=2:4\)

  • Thus, shaded part to unshaded part ratio is part to part ratio.
  • Part to part ratio is not considered to be a fraction.

2. Part to Whole Ratio

  • Part to whole ratio represents the comparison between a part and its whole.
  • In the above example,

Number of shaded parts \(=2\)

Total number of parts \(=6\)

Part to whole ratio \(=\dfrac{\text{Number of shaded parts}}{\text{Total number of parts}}\)

\(=\dfrac{2}{6}\)

\(\dfrac{2}{6}\) can be expressed as \(2:6\)

3. Whole to Part Ratio

  • Whole to part ratio represents the comparison between a whole and a particular part.
  • In the above example:

Number of unshaded parts \(=4\)

Total number of parts \(=6\)

Whole to part ratio \(=\dfrac{\text{Total number of parts}}{\text{Number of unshaded parts}}\)

\(=\dfrac {6}{4}\)

\(\dfrac {6}{4}\) can also be expressed as \(6:4\)

Illustration Questions

There are \(10\) boys and \(13\) girls in the class. Which one of the following options is true?

A Ratio of girls to boys is \(10:13\)

B Ratio of boys to girls is \(13:10\)

C Ratio of girls to total students is \(13:23\)

D Ratio of total students to boys is \(10:23\)

×

Given:

Total number of boys \(=10\)

Total number of girls \(=13\)

Total number of students = Total number of girls + Total number of boys

\(=13+10=23\)

Ratio of girls to boys \(=\dfrac{\text{Total number of girls}}{\text{Total number of boys}}\)

\(=\dfrac{13}{10}\)

\(=13:10\)

Hence, option (A) is incorrect.

Ratio of boys to girls \(=\dfrac{\text{Total number of boys}}{\text{Total number of girls}}\)

\(=\dfrac{10}{13}\)

\(=10:13\)

Hence, option (B) is incorrect.

Ratio of girls to total students \(=\dfrac{\text{Total number of girls}}{\text{Total number of students}}\)

\(=\dfrac{13}{23}\)

\(=13:23\)

Hence, option (C) is correct.

Ratio of total students to boys \(=\dfrac{\text{Total number of students}}{\text{Total number of boys}}\)

\(=\dfrac{23}{10}\)

\(=23:10\)

Hence, option (D) is incorrect.

There are \(10\) boys and \(13\) girls in the class. Which one of the following options is true?

A

Ratio of girls to boys is \(10:13\)

.

B

Ratio of boys to girls is \(13:10\)

C

Ratio of girls to total students is \(13:23\)

D

Ratio of total students to boys is \(10:23\)

Option C is Correct

Ratio through Figures

Ratio

  • Ratio is the relationship between two quantities.
  • It can be represented through figures.

For example: What is the ratio of the number of pink triangles to the number of blue triangles?

Number of pink triangles \(=2\)

Number of blue triangles \(=4\)

Total number of triangles \(=6\)

\(\therefore\) Ratio of pink triangles to blue triangles

\(=\dfrac{\text{Number of pink triangles}}{\text{Number of blue triangles}}\)

\(=\dfrac{2}{4}\)

\(=2:4\)

Illustration Questions

What is the ratio of the number of pink cubes to the total cubes?

A \(7:8\)

B \(8:15\)

C \(8:7\)

D \(15:8\)

×

Given:

Number of pink cubes \(=8\)

Number of yellow cubes \(=7\)

Total cubes \(=8+7\)

\(=15\)

\(\therefore\) Ratio of pink cubes to total number of cubes \(=\dfrac{\text{Number of pink cubes}}{\text{Total number of cubes}}\)

\(=\dfrac{8}{15}\)

\(=8:15\)

Hence, option (B) is correct.

What is the ratio of the number of pink cubes to the total cubes?

image
A

\(7:8\)

.

B

\(8:15\)

C

\(8:7\)

D

\(15:8\)

Option B is Correct

Simplest Form of a Ratio

  • A ratio is in its simplest form if the numerator and the denominator do not have any common factor other than \(1.\)
  • The simplest form of a ratio is also known as reduced form.

How to simplify a ratio

  • We simplify a ratio written in fraction form in the same way as we simplify a fraction.
  • To simplify a ratio, find out the greatest common factor of numerator and denominator.
  • Divide both numerator and denominator by the greatest common factor.
  • We get the simplest form of ratio.

For example: \(\dfrac{2}{22}\)

Here, the GCF of  \(2\) and \(22\) is \(2.\) Thus, we will divide both numerator and denominator by \(2.\)

\(\dfrac{2\div2}{22\div2}=\dfrac{1}{11}\)

Hence, the simplest form of \(\dfrac{2}{22}\) is \(\dfrac{1}{11}\).

Note: The values of a ratio and of simplified form are always same.

Illustration Questions

Which one of the following is the simplest form of \(\dfrac{12}{8}\)?

A \(\dfrac{1}{8}\)

B \(\dfrac{2}{3}\)

C \(\dfrac{3}{2}\)

D \(\dfrac{12}{1}\)

×

The greatest common factor of \(12\) and \(8\) is \(4.\)

So, divide both numerator and denominator by \(4.\)

\(\dfrac{12\div4}{8\div4}=\dfrac{3}{2}\)

Thus, the simplest form of \(\dfrac{12}{8}\) is \(\dfrac{3}{2}\).

Hence, option (C) is correct.

Which one of the following is the simplest form of \(\dfrac{12}{8}\)?

A

\(\dfrac{1}{8}\)

.

B

\(\dfrac{2}{3}\)

C

\(\dfrac{3}{2}\)

D

\(\dfrac{12}{1}\)

Option C is Correct

Ratios in Decimal Form

  • The ratios can be expressed in decimal form.
  • To write a ratio in the decimal form, follow the steps given below:

For example:  \(4:5\)

Step 1:  Write the ratio in the fraction form.

\(4:5=\dfrac {4}{5}\)

Step 2: Divide the numerator by the denominator.

Thus, \(\dfrac {4}{5}=0.8\)

\(4:5=0.8\)

Illustration Questions

Which one of the following decimals is equivalent to \(1:2\)?

A \(0.5\)

B \(0.6\)

C \(0.3\)

D \(0.2\)

×

Given :- \(1:2\)

Writing \(1:2\) in fraction form

\(1:2=\dfrac {1}{2}\)

Dividing the numerator by the denominator,

image

Thus, \(\dfrac {1}{2}=0.5\)

Hence, option (A) is correct.

Which one of the following decimals is equivalent to \(1:2\)?

A

\(0.5\)

.

B

\(0.6\)

C

\(0.3\)

D

\(0.2\)

Option A is Correct

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