# Exercise 8.1 Comparing Quantities- NCERT Solutions Class 8

## Chapter 8 Ex.8.1 Question 1

Find the ratio of the following.

(i) Speed of a cycle $$15\,\rm{ km}$$ per hour to the speed of scooter $$30\,\rm{ km}$$ per hour.

(ii) $$5\,\rm{ m}$$ to $$10\,\rm{ km}$$

(iii) $$50$$ paise to $$\rm{Rs}\, 5$$

### Solution

What is known?

Value of two quantities, which needs to be compared.

What is unknown?

Ratio

Reasoning:

A relationship between two quantities is normally expressed as the quantity of one divided by the other.

Steps:

(i):

Speed of a cycle $$= 15 \;\rm{km/hr}$$

Speed of a scooter $$= 30\,\rm{ km/hr}$$

Speed of cycle: Speed of scooter  \begin{align}=\frac{{{15}}}{{{30}}}=\frac{{1}}{{2}} \end{align}

The answer is $$1:2$$

(ii)

Given data: $$5\,\rm{ m}$$ to $$10 \,\rm{km}$$

Quantities can be compared only when the units are same.

$$1 \rm{km} = 1000 \rm{m}$$

Therefore, $$10\,\rm{ km} = 10 \times 1000 = 10000\,\rm{ m}$$

$$5\, \rm{m}$$ to $$10\,\rm{ km}$$ $$= 5\,\rm{m}$$  to $$10000 \,\rm{m}$$ \begin{align}=\frac{{5}}{{{10000}}}{ = }\frac{{1}}{{{2000}}}\end{align}

The answer is $$1:2000$$

(iii)

Given data: $$50$$ paise to $$\rm{Rs}\, 5$$

Quantities can be compared only when the units are same.

\begin{align}\rm{Rs}\, 1 &= 100\,\text{ paise}\\\rm{Rs} \,5 &= 5 \times 100\,\text{paise}\\ &= 500\, \text{paise}\end{align}

$$50$$ paise to $$\rm{Rs}\, 5 = 50$$ paise to $$500$$ paise

\begin{align}\frac{{50}}{{500}} = \frac{1}{{10}} \end{align}

The answer is $$1:10$$

## Chapter 8 Ex.8.1 Question 2

Convert the following ratios to percentages

(i) $$3:4$$

(ii) $$2:3$$

### Solution

What is known?

Ratios

What is unknown?

Percentages of given ratios

Reasoning:

A ratio is a comparison of any two quantities by division. A percent is a special ratio that compares any quantity to $$100,$$ with $$100$$ representing one whole.

Steps:

(i) $$3:4$$

Given data: $$3:4$$

\begin{align}\frac{3}{{4}}&{ \times 100}\\&= 3 \times 25\\&= 75\%\end{align}

The answer is  $$75\%$$

(ii) $$2:3$$

Given data: $$2:3$$

Type 1: Decimal Form

\begin{align}\frac{2}{3} &\times 100\\&= 2 \times 33.33\\&= 66.67\% \end{align}

The answer is $$66.67\%$$ (Decimal form)

Type 2: Fractions Form

\begin{align}\frac{{{2 \times 100}}}{{3}}&= \frac{{{200}}}{{3}}\\& = 66\frac{{2}}{{3}}{\% }\end{align}

The answer is \begin{align}{66}\frac{{2}}{{3}}\% \end{align} (Mixed fraction form)

## Chapter 8 Ex.8.1 Question 3

$$72\%$$ of $$25$$ students are interested in mathematics. How many are not interested in mathematics?

### Solution

What is known?

Total number of students $$= 25$$

Percentage of students who are interested in Mathematics $$= 72\%$$

What is unknown?

Number of students who are not interested in Mathematics

Reasoning:

Percentage is a special ratio that compares any quantity to $$100,$$ with $$100$$ representing one whole.

Steps:

Percentage of students who are not interested in Mathematics

\begin{align}& = \left( {100-72} \right)\%\\&= 28\% \end{align}

Therefore, number of students who are not good in Mathematics

\begin{align}&= 28\%\,\text{of the total number of students}\\&= 28\% \,\text{of} \,25 \\&= \frac{{{28}}}{{{100}}}{ \times 25}\\&=\frac{{{28}}}{{4}}\\& = 7\end{align}

The total number of students who are not interested in Mathematics are $$7.$$

## Chapter 8 Ex.8.1 Question 4

A football team won $$10$$ matches out of the total number of matches they played. If their win percentage was $$40$$, then how many matches did they play in all?

### Solution

What is known?

Total number of matches $$= 10$$

Win percentage $$= 40\%$$

What is unknown?

Total number of matches played.

Reasoning:

Assuming the total number of matches played as $$x,$$ equating $$40\%$$ of $$x$$ is to $$10,$$ the value of $$x$$ can be found.

Steps:

Let the total number of matches played$$=x$$

\begin{align}40\% \;{\rm{ of}}\;x &= 10\\\frac{{40}}{{100}} \times x &= 10\\x &= \frac{{10 \times 100}}{{40}}\\&= 25\end{align}

The total number of matches played by football team is $$25.$$

## Chapter 8 Ex.8.1 Question 5

If Chameli had $$\rm{Rs}\, 600$$ left after spending $$75\%$$ of her money, how much did she have in the beginning?

### Solution

What is known?

Percentage of amount Chameli spent $$= 75\%$$

Amount left with her $$= \rm{Rs} \,600$$

What is unknown?

Amount Chameli had in the beginning

Reasoning:

Since the whole is considered as $$100\%,$$ Percentage of amount left with Chameli is

$\left( {100 - 75} \right)\%= 25\%$

Assuming the total amount in the beginning as $$x,$$ and equating $$25\%$$ of $$x$$ to $$600,$$ the value of $$x$$ can be found.

Steps:

Let the total amount Chameli had with her in the beginning $$= x$$

Percentage of amount left with Chameli $$= 100-75 = 25%$$

\begin{align}25\% {\rm\;{of}}\;x &= 600\\\frac{{{25}}}{{{100}}}{ \times }x&= 600\\ x &= \frac{{{600 \times 100}}}{{{25}}}\\&= 2,400\end{align}

The amount that Chameli had in the beginning is $$\rm{Rs} \,2,400$$

## Chapter 8 Ex.8.1 Question 6

If $$60\%$$ people in a city like cricket, $$30\%$$ like football and the remaining like other games, then what percent of the people like other games? If the total number of people is $$50$$ lakhs, find the exact number who like each type of game.

### Solution

What is known?

Percentage of people who like cricket $$= 60\%$$

Percentage of people who like football $$= 30\%$$

Total number of people: $$50$$ lakhs

What is unknown?

Percentage of people like other games

Exact number of people who like the game

Reasoning:

Since the whole is considered as $$100\%,$$ percentage of people who like other games is $$100\% – (60+30) \% = 10\%$$

Number of people who like each game can be found using percentage and total number of people.

Steps:

Percentage of people who like other games$$= 100\% – (60+30)\% = 10\%$$

Number of people who like cricket

\begin{align}&= 60\% \;{\rm{ of }}\;50\,{\rm{ lakhs}}\\&= \frac{{{60}}}{{{100}}}{ \times 50,00,000}\\&= 30,00,000\\&= 30\,{\rm{ lakhs}}\end{align}

Number of people who like football

\begin{align}&= {\rm{ }}30\% \,{\rm{ of }}\,50\,{\rm{ lakhs}}\\&= \frac{{{30}}}{{{100}}}{ \times 50,00,000}\\&= 15,00,000\\&= 15\,\rm{ Lakhs}\end{align}

Number of people who like other games

\begin{align}&= 10\% \,{\rm{ of}}\;50 \;{\rm{lakhs}}\\&= \frac{{{10}}}{{{100}}}{ \times 50,00,000}\\&= 5,00,000\\&= 5\;{\rm{lakhs}}\end{align}

Percentage of people who like other games $$= 10\%$$
Number of people who like cricket $$= 30$$ lakhs
Number of people who like football $$= 15$$ lakhs
Number of people who like other games $$= 5$$ lakhs