Exercise 8.2

1. Convert the given fractions to percentages:

a) The fraction is \[\dfrac{1}{8}\].

answer: To convert a fraction to a percent, the given fraction is multiplied by $100\%$.

The given fraction is $\dfrac{1}{8}$.

Multiply the given fraction by $100\%$ and simplify to get the corresponding percentage.

$\dfrac{1}{8} \times 100\% = \dfrac{{25}}{2}\% $

$\dfrac{1}{8} \times 100\% = 12.5\% $

Therefore, the fraction $\dfrac{1}{8}$ is equal to $12.5\% $.

b) The fraction is \[\dfrac{5}{4}\].

answer: To convert a fraction to a percent, the given fraction is multiplied by $100\%$.

The given fraction is \[\dfrac{5}{4}\].

Multiply the given fraction by $100\%$ and simplify to get the corresponding percentage.

$\dfrac{5}{4} \times 100\% = 5 \times 25\% $

$\dfrac{5}{4} \times 100\% = 125\% $

Therefore, the fraction \[\dfrac{5}{4}\] is equal to $125\% $.

c) The fraction is \[\dfrac{3}{{40}}\].

answer: To convert a fraction to a percent, the given fraction is multiplied by $100\%$.

The given fraction is \[\dfrac{3}{{40}}\].

Multiply the given fraction by $100\%$ and simplify to get the corresponding percentage.

$\dfrac{3}{{40}} \times 100\% = \dfrac{{3 \times 5}}{2}\% $

$\dfrac{3}{{40}} \times 100\% = 7.5\% $

Therefore, the fraction \[\dfrac{3}{{40}}\] is equal to $7.5\% $.

d) The fraction is \[\dfrac{2}{7}\].

answer: To convert a fraction to a percent, the given fraction is multiplied by $100\%$.

The given fraction is \[\dfrac{2}{7}\].

Multiply the given fraction by $100\%$ and simplify to get the corresponding percentage.

$\dfrac{2}{7} \times 100\% = \dfrac{{200}}{7}\% $

$\dfrac{2}{7} \times 100\% = 28\dfrac{4}{7}\% $

$\dfrac{2}{7} \times 100\% = 28.571428...$

So the fraction \[\dfrac{2}{7}\] is equal to $28\dfrac{4}{7}\% $ or $28.571428..\% $.

2. Convert the given decimal fractions to a percent:

a) The number is $0.65

answer: The given decimal is $0.65$.

The specified decimal number can also be written in the form $\dfrac{p}{q}$ as $\dfrac{{65}}{{100}}$.

Now, when converting a fraction to a percent, multiply the given fraction by 100$\%$.

Multiply the given fraction by $100\%$ and simplify to get the corresponding percentage.

$\dfrac{{65}}{{100}} \times 100\% = 65\% $

So the decimal number \[0.65\] is equal to $65\%$.

b) The number is $2.1$

answer: The specified decimal is $2.1$.

The specified decimal number can also be written in the form $\dfrac{p}{q}$ as $\dfrac{210}{{100}}$.

Now, when converting a fraction to a percent, multiply the given fraction by 100$\%$.

Multiply the given fraction by $100\%$ and simplify to get the corresponding percentage.

$\dfrac{210}{{100}} \times 100\% = 210\% $

So the decimal \[2.1\] is equal to $210\%$.

(C) The number is $0.02$

answer: The specified decimal is $0.02$.

The specified decimal number can also be written in the form $\dfrac{p}{q}$ as $\dfrac{2}{100}$.

Now, when converting a fraction to a percent, multiply the given fraction by $100%$.

Multiply the given fraction by $100%$ and simplify to get the corresponding percentage.

$\dfrac{2}{100}\times 100%=2\%$

Therefore, the decimal $0.02$ is equal to $2\%$.

(D) The number is $12.35.

answer: the given decimal is $12.35$.

The specified decimal number can also be written in the form $\dfrac{p}{q}$ as $\dfrac{1235}{100}$.

Now, when converting a fraction to a percent, multiply the given fraction by $100%$.

Multiply the given fraction by $100%$ and simplify to get the corresponding percentage.

$\dfrac{1235}{100}\times 100%= 1235\%$

So the decimal $12.35$ is equal to $1235\%$.

3. Estimate how much of the numbers is colored in to find the percentage that is colored in:

i) The given number is,

(image will be updated soon)

answer: The given figure has a circle with a shaded part.

From the given figure, it can be estimated that the colored part of the circle is the ${\dfrac{1}{4}^{th}}$ part.

So ${\text{The colored part}} = \dfrac{1}{4}$.

Now, to find the percentage of fraction colored, convert the fraction obtained from fraction colored to percent. This can be done by multiplying the fraction by $100\%$ and then simplifying.

Therefore,

\[{\text{Percentage of colored part}} = \dfrac{1}{4} \times 100\% \]

\[{\text{Percentage of colored portion}} = \dfrac{{100}}{4}\% \]

\[{\text{The percentage of the colored part}} = 25\% \]

Therefore, the percentage of the colored part of the given figure is 25$\%$.

ii) The given number is,

(image will be updated soon)

answer: The given figure has a circle with a shaded part.

The figure shows that the circle is divided into 5 equal parts.

Therefore, the colored part of the circle can be estimated to be the ${\dfrac{3}{5}^{th}}$ part.

So ${\text{The colored part}} = \dfrac{3}{5}$.

Now, to find the percentage of fraction colored, convert the fraction obtained from fraction colored to percent. This can be done by multiplying the fraction by $100\%$ and then simplifying.

Therefore,

\[{\text{Percentage of colored part}} = \dfrac{3}{5} \times 100\% \]

\[{\text{Percentage of colored part}} = \dfrac{{300}}{5}\% \]

\[{\text{Percentage of colored part}} = 60\% \]

Therefore, the percentage of the colored part of the given figure is 60$\%$.

iii) The given number is,

(image will be updated soon)

answer: The given figure has a circle with a shaded part.

The figure shows that the circle is divided into 8 equal parts.

Therefore, the colored part of the circle can be estimated to be the ${\dfrac{3}{8}^{th}}$ part.

So ${\text{The colored part}} = \dfrac{3}{8}$.

Now, to find the percentage of fraction colored, convert the fraction obtained from fraction colored to percent. This can be done by multiplying the fraction by $100\%$ and then simplifying.

Therefore,

\[{\text{Percentage of colored part}} = \dfrac{3}{8} \times 100\% \]

\[{\text{Percentage of colored portion}} = \dfrac{{300}}{8}\% \]

\[{\text{Percentage of colored part}} = 37.5\% \]

Therefore, the percentage of the colored part of the given figure is 37.5$\%$.

4. Find the percentage of the following values:

a) $ 15\% {\text{ }}$ de 250

answer: The specified number is 250.

The percentage of any number $n$, that is, $n\%$, is written as $\dfrac{n}{{100}}$.

$15\%$ is also represented in fractional form as $\dfrac{{15}}{{100}}$.

Therefore, $15\%$ of 250 is represented as $\dfrac{{15}}{{100}} \times 250$.

Evaluate the previous expression.

$\dfrac{{15}}{{100}} \ times 250 = 15 \ times $2.5

$\dfrac{{15}}{{100}} \ times 250 = $37.5

So it turns out that $15\% {\text{ of 250}}$ is $37.5$.

b) $1\%$ of 1 hour.

answer: It is known that 1 hour has 60 minutes.

Also, 60 minutes equals $\left( {60 \times 60} \right)$ seconds.

The percentage of any number $n$, that is, $n\%$, is written as $\dfrac{n}{{100}}$.

So if you find $1%$ of 1 hour, you will find $1%$ of $\left( {60 \times 60} \right)$ seconds. $1\%$ is also represented as $\dfrac{1}{{100}}$.

With that,

$1\% {\text{ of }}\left( {60 \times 60} \right){\text{ seconds}} = \dfrac{1}{{100}}\left( {60 \times 60} \ right){\text{seconds}}$.

$1\% {\text{ de }}\left( {60 \times 60} \right){\text{ seconds}} = \dfrac{{3600}}{{100}}{\text{ seconds}}$ .

$1\% {\text{ of }}\left( {60 \times 60} \right){\text{ seconds}} = 36{\text{ seconds}}$.

c) $20% $ of ₹2500

answer: The value given is ₹2,500.

The percentage of any number $n$, that is, $n\%$, is written as $\dfrac{n}{{100}}$.

$20\%$ is also represented in fractional form as $\dfrac{{20}}{{100}}$.

Therefore, $20\%$ of 2500 is represented as $\dfrac{{20}}{{100}} \times 2500$.

Evaluate the previous expression.

$\dfrac{{20}}{{100}} \times 2500 = 20 \times 25$

$\dfrac{{20}}{{100}} \times 2500 = 500$

So $20%$ of ₹2500 is ₹500.

d) $75% $ of 1kg

answer: It is known that 1 kg is 1000 g.

The percentage of any number $n$, that is, $n\%$, is written as $\dfrac{n}{{100}}$.

$75\%$ is also represented in fractional form as $\dfrac{{75}}{{100}}$.

Therefore, $75\%$ of 1000 is represented as $\dfrac{{75}}{{100}} \times 1000$.

Evaluate the previous expression.

$\dfrac{{75}}{{100}} \times 1000 = 750{\text{g}}$

Therefore,

$750{\text{ g}} = \dfrac{{750}}{{1000}}{\text{ kg}}$

$750{\text{g}} = 0,75{\text{kg}}$

So $75\%$ of 1 kg is $0.75{\text{ kg}}$.

5. Find the total if:

a) $5\%$ of which are 600

answer: Let $x$ be the assumed size.

The percentage of any number $n$, that is, $n\%$, is written as $\dfrac{n}{{100}}$.

$5\%$ of $x$ can be represented as $\dfrac{5}{{100}}$ of $x$, which equals 600.

This can be formulated as follows,

$5\% {\text{ of }}x = $600

Therefore,

$\dfrac{5}{{100}} \times x = $600

evaluate more,

$x = \dfrac{{600 \times 100}}{5}$

$x = \dfrac{{60000}}{5}$

$ x = 12000 $

Therefore, the quantity is determined to be 12,000.

b) $12\%$ of which are ₹1080.

Respondedor:Let $x$ be the amount assumed.

The percentage of any number $n$, that is, $n\%$, is written as $\dfrac{n}{{100}}$.

$12\%$ of $x$ can be represented as $\dfrac{{12}}{{100}}$ of $x$, which is ₹1080.

This can be formulated as follows,

$12\% {\text{ of }}x = 1080$

Therefore,

$\dfrac{{12}}{{100}} \times x = 1080$

evaluate more,

$x = \dfrac{{1080 \times 100}}{{12}}$

$x = \dfrac{{108000}}{{12}}$

$ x = 9000 $

Therefore, the amount is £9,000.

c) $ 40\% $ of which are 500 km.

answer: The default amount is $x$.

The percentage of any number $n$, that is, $n\%$, is written as $\dfrac{n}{{100}}$.

$40\%$ of $x$ can be represented as $\dfrac{{40}}{{100}}$ of $x$, which equals 500 km.

This can be formulated as follows,

$ 40\% {\text{ de }}x = 500{\text{ km}}$

Therefore,

$\dfrac{{40}}{{100}} \times x = $500

evaluate more,

$x = \dfrac{{500 \times 100}}{{40}}$

$x = \dfrac{{50000}}{{40}}$

$x = 1,250{\text{km}}$

Therefore, the amount is 1250 km.

d) $70% $ of which is 14 minutes

Respondedor:The assumed amount is $x$.

The percentage of any number $n$, that is, $n\%$, is written as $\dfrac{n}{{100}}$.

$70\%$ of $x$ can be represented as $\dfrac{{70}}{{100}}$ of $x$, which is 14 minutes.

This can be formulated as follows,

$ 70\% {\text{ de }}x = 14{\text{ minutos}}$

Therefore,

$\dfrac{{70}}{{100}} \times x = $14

evaluate more,

$x = \dfrac{{14 \times 100}}{{70}}$

$x = \dfrac{{1400}}{{70}}$

$x = 20{\text{minutes}}$

Therefore, the value is 20 minutes.

e) $8\%$ of which are 40 liters

answer: The default amount is $x$.

The percentage of any number $n$, that is, $n\%$, is written as $\dfrac{n}{{100}}$.

$8\%$ of $x$ can be represented as $\dfrac{8}{{100}}$ of $x$, which is 40 liters.

This can be formulated as follows,

$8\% {\text{ de }}x = 40{\text{ litros}}$

Therefore,

$\dfrac{8}{{100}} \times x = $40

evaluate more,

$x = \dfrac{{40 \times 100}}{8}$

$x = \dfrac{{4000}}{8}$

$ x = 500 $

This results in a quantity of 500 liters.

6. Convert the given percentages to decimal fractions and in the simplest forms to fractions as well:

a) The percentage is $25\%$.

Respondedor:The percentage of any number $n$, that is, $n\%$, is written as $\dfrac{n}{{100}}$.

The specified percentage $25\%$ can also be represented as $\dfrac{{25}}{{100}}$.

Therefore, the fractional form of the given percentage is $\dfrac{{25}}{{100}}$.

Simplify the fraction form obtained to determine the simplest form of the given percentage.

$\dfrac{{25}}{{100}} = \dfrac{1}{4}$

So the simplest form is $\dfrac{1}{4}$.

Now the decimal form of the given percentage is obtained by dividing the simpler form obtained above.

$\dfrac{1}{4} = 0,25$

So, for $25\%$, the fractional form is $\dfrac{{25}}{{100}}$, the simplest form is $\dfrac{1}{4}$, and the decimal form is $0.25 $.

b) The percentage is $150\%$.

Respondedor:The percentage of any number $n$, that is, $n\%$, is written as $\dfrac{n}{{100}}$.

The specified percentage $150\%$ can also be represented as $\dfrac{{150}}{{100}}$.

Therefore, the fractional form of the given percentage is $\dfrac{{150}}{{100}}$.

Simplify the fraction form obtained to determine the simplest form of the given percentage.

$\dfrac{{150}}{{100}} = \dfrac{3}{2}$

So the simplest form is $\dfrac{3}{2}$.

Now the decimal form of the given percentage is obtained by dividing the simpler form obtained above.

$\dfrac{3}{2} = 1,5$

So, for 150$\%$, the fractional form is $\dfrac{{150}}{{100}}$, the simplest form is $\dfrac{3}{2}$, and the decimal form is 1, $5.

c) The percentage is $20\%$.

Respondedor:The percentage of any number $n$, that is, $n\%$, is written as $\dfrac{n}{{100}}$.

The specified percentage $20\%$ can also be represented as $\dfrac{{20}}{{100}}$.

Therefore, the fractional form of the given percentage is $\dfrac{{20}}{{100}}$.

Simplify the fraction form obtained to determine the simplest form of the given percentage.

$\dfrac{{20}}{{100}} = \dfrac{1}{5}$

So the simplest form is $\dfrac{1}{5}$.

Now the decimal form of the given percentage is obtained by dividing the simpler form obtained above.

$\dfrac{1}{5} = 0,2$

So, for $20\%$, the fractional form is $\dfrac{{20}}{{100}}$, the simplest form is $\dfrac{1}{5}$, and the decimal form is $0.2 $.

d) The percentage is $5\%$.

Respondedor:The percentage of any number $n$, that is, $n\%$, is written as $\dfrac{n}{{100}}$.

The specified percentage $5\%$ can also be represented as $\dfrac{5}{{100}}$.

Therefore, the fractional form of the given percentage is $\dfrac{5}{{100}}$.

Simplify the fraction form obtained to determine the simplest form of the given percentage.

$\dfrac{5}{{100}} = \dfrac{1}{{20}}$

So the simplest form is $\dfrac{1}{{20}}$.

$\dfrac{1}{{20}} = 0,05$

So, for $5\%$, the fractional form is $\dfrac{5}{{100}}$, the simplest form is $\dfrac{1}{{20}}$, and the decimal form is $0.05 $.

7. In a city, 30% are women, 40% are men, and the rest are children. What percentage are children?

Respondedor:The proportion of women is given as 30\%$ and the proportion of men in a city as 40\%$.

Therefore, the total percentage of men and women in a city is the sum of the two given percentages.

${\text{Overall percentage of men and women in a city}}$ = 30% + 40%

${\text{Total percentage of men and women in a city}} = 70\% $

The total percentage of children in the city is the difference between the total percentage of the city's population and the total percentage of men and women in the city.

${\text{Percentage of children in the city}} = {\text{Overall percentage}} - {\text{Percentage of men and women}}$${\text{Percentage of children in the city}} = 100 % - 70%$

Therefore,

${\text{The percentage of children in the city}} = 30\% $

Therefore, the percentage of children in the city is 30$\%$.

8. Of 15,000 voters in a constituency, $60\% voted for $. Find the percentage of voters who did not vote. Now find out how many really didn't vote.

answer: the total number of voters in a constituency is given as $60\%$.

Of the specified number of voters, $60\% $ chose candidates.

The percentage of candidates who did not vote is the difference between the total percentage of candidates and the percentage of candidates who voted.

${\text{The percentage of candidates who did not vote}}$ = 100% - 60%

${\text{Percentage of candidates who did not vote}} = 40\% $

Now, the actual number of candidates who did not vote is calculated as 40$\%$ of the total number of voters in the electoral district, that is, H. $40%$ of 15,000.

${\text{Number of candidates who did not vote}} = \dfrac{{40}}{{100}} \times 15000$

${\text{The number of candidates who did not vote}} = $6,000

Therefore, the number of candidates who did not vote is 6,000.

9. Meeta saves ₹4000 from her salary. If that is $10%$ of your salary. what is your salary

Respondedor:Let's say Meeta's total salary is ₹$x$.

You are supposed to save ₹4000 of your salary, which is ₹10% of your total salary.

This can be represented as follows,

$ 10\% {\text{ del salario total}} = 4000{\text{ Rs}}$

The percentage of any number $n$, that is, $n\%$, is written as $\dfrac{n}{{100}}$.

Therefore,

$ 10\% {\text{ de }}x = 4000{\text{ }}$

$\dfrac{{10}}{{100}} \times x = $4000

After further evaluation and cross multiplication,

$x = \dfrac{{4000 \times 100}}{{10}}$

$ x = 40000 $

Therefore, Meeta's total salary is Rs 40,000.

10. A local cricket team played 20 matches in one season. He earned $25%$ from them. How many games have they won?

answer: It is believed that the local cricket team played 20 matches in one season, earning $25\%$ in matches.

So the total number of games they won is $25\%$ of the number of games they played, represented as follows:

${\text{Total number of games won by the team}} = 25\% {\text{ of }}20$

Now the percentage of any number $n$, that is, $n\%$, is written as $\dfrac{n}{{100}}$.

Therefore,

${\text{Total number of games won by the team}} = \dfrac{{25}}{{100}} \times {\text{ }}20$

${\text{Total number of games won by the team}} = \dfrac{1}{4} \times {\text{ }}20$

${\text{Total number of games won by the team}} = $5

Therefore, the number of games won by the team is determined to be 5.

## NCERT Solutions for Grade 7 Mathematics Chapter 8 Comparing Sets Exercise 8.2

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