GSEB Class 7 Maths Solutions Chapter 8 Comparing Quantities Exercise 8.2

Question 1. Convert the given fractional numbers to per cents.
(a) \( \frac { 1 }{ 8 } \)
(b) \( \frac { 5 }{ 4 } \)
(c) \( \frac { 3 }{ 40 } \)
(d) \( \frac { 2 }{ 7 } \)
Answer:
(a) We have \( \frac { 1 }{ 8 } = \frac {1}{8} \times \frac {100}{ 100 } \)
\( = \frac { 100 }{ 8 }% \)
\( = \frac { 25 }{ 2 }% \) or 12.5%
So, \( \frac { 1 }{ 8 } \) equals 12.5%.
(b) We have \( \frac { 5 }{ 4 } = \frac { 5 }{ 4 } \times \frac { 100 }{ 100 } \)
\( = \frac {5 \times 100 }{ 4 }% \)
\( = (5 \times 25)% = 125% \)
Thus, \( \frac { 5 }{ 4 } \) is 125%.
(c) We have \( \frac { 3 }{ 40 } = \frac { 3 }{ 40 } \times \frac { 100 }{ 100 } \)
\( = \frac {3 \times 100 }{ 40 }% \)
\( = \frac {3 \times 25 }{ 10 }% \)
\( = \frac { 75 }{ 10 }% \) or 7.5%
Therefore, \( \frac { 3 }{ 40 } \) is 7.5%.
(d) We have \( \frac { 2 }{7} = \frac { 2 }{ 7 } \times \frac { 100 }{ 100 } \)
\( = \frac { 200 }{ 7 }% \)
\( = 28\frac { 4 }{7}% \)
Thus, \( \frac {2}{7} \) is equal to \( 28\frac { 4 }{ 7 }% \).
In simple words: To change a fraction to a percentage, multiply it by 100%. This shows what part of a hundred the fraction represents.

Exam Tip: Remember to simplify the fraction to its lowest terms before converting to a percentage for easier calculation, and ensure the percentage symbol is included in your final answer.

Question 2. Convert the given decimal fractions to percents.
(a) 0.65
(b) 2.1
(c) 0.02
(d) 12.35
Answer:
(a) We have \( 0.65 = \frac { 65 }{ 100 } = \frac { 65 }{ 100 } \times 100\% = 65\% \)
(b) We have: \( 2.1 = \frac { 21 }{ 10 } = \frac { 21 \times 10 }{ 10 \times 10 } = \frac { 210 }{ 100 } = \frac { 210 }{ 100 } \times 100\% = 210\% \)
(c) We have: \( 0.02 = \frac { 2 }{ 100 } = \frac { 2 }{ 100 } \times 100\% = 2\% \)
(d) We have: \( 12.35 = \frac { 1235 }{ 100 } = \frac { 1235 }{ 100 } \times 100\% = 1235\% \)
In simple words: To change a decimal to a percentage, multiply the decimal by 100. This is like moving the decimal point two places to the right and adding a percent sign.

Exam Tip: Shifting the decimal point two places to the right and adding a percent sign is a quick way to convert decimal numbers to percentages. Always double-check your decimal point placement.

Question 3. Estimate what part of the figures is coloured and hence find the per cent which is coloured.
Answer:
(i) This figure shows 1 part out of 4 parts shaded. So, \( \frac {1}{ 4 } \) part is shaded.
And \( \frac { 1 }{ 4 } = \frac { 1 }{ 4 } \times 100\% = \frac { 100 }{ 4 }\% = 25\% \)
Thus, the coloured part is 25%.
(ii) This figure shows 3 parts out of 5 parts shaded. So, \( \frac { 3 }{ 5 } \) part is shaded.
And \( \frac { 3 }{ 5 } \times 100\% = 3 \times 20\% = 60\% \)
Thus, 60% part is shaded.
(iii) Here, 3 parts out of 8 parts are shaded. So, \( \frac { 3 }{ 8 } \) part is shaded.
And \( \frac { 3 }{8} \times 100\% = \frac { 3 }{ 2 } \times 25\% = \frac { 75 }{ 2 }\% = 37.5\% \)
Thus, 37.5% part is shaded.
In simple words: To find the percentage of a shaded part, first figure out the fraction of the shape that is shaded. Then, change this fraction into a percentage by multiplying it by 100.

Exam Tip: When estimating shaded parts, count the total number of equal sections and the number of shaded sections to form a fraction. Convert this fraction to a percentage by multiplying by 100.

Question 4. Find:
(a) 15% of 250
(b) 1% of 1 hour
(c) 20% of Rs. 2500
(d) 75% of 1 kg
Answer:
(a) 15% of \( 250 = \frac {15}{ 100 } \) of 250
\( = \frac { 15 \times 250 }{ 100 } = \frac { 75 }{2} \)
\( = 37.5 \)
(b) 1% of 1 hour \( = \frac {1}{ 100 } \) of 60 minutes
(Since 1 hour = 60 minutes)
\( = \frac { 1 }{ 100 } \times 60 \) minutes
\( = \frac { 60 }{ 100 } \) minutes
\( = \frac { 3 }{ 5 } \) minutes
To express this in seconds: \( \frac { 3 }{ 5 } \times 60 \) seconds = 36 seconds.
(c) 20% of Rs. \( 2500 = \frac {20}{ 100 } \) of Rs. 2500
\( = \frac { 20 \times 2500 }{ 100 } \)
\( = 20 \times 25 = 500 \)
Thus, 20% of Rs. 2500 = Rs. 500.
(d) 75% of 1 kg = 75% of 1000 g
(Since 1 kg = 1000 g)
\( = \frac {75}{ 100 } \times 1000 \) g
\( = \frac {75 \times 1000 }{ 100 } \)
\( = 750 \) g
Thus, 75% of 1 kg = 750 g.
In simple words: To find a percentage of a quantity, multiply the quantity by the percentage written as a fraction (percentage divided by 100) or a decimal. Remember to convert units if needed.

Exam Tip: When calculating percentages of quantities, ensure units are consistent (e.g., convert hours to minutes, kg to grams) before multiplying. Always simplify fractions for easier calculations.

Question 5. Find the whole quantity if
(a) 5% of it is 600.
(b) 12% of it is 1080.
(c) 40% of it is 500 km.
(d) 70% of it is 14 minutes.
(e) 8% of it is 40 litres.
Answer:
(a) 5% of a quantity is 600.
Let the quantity be \( x \).
So, 5% of \( x = 600 \)
\( \frac { 5 }{ 100 } \times x = 600 \)
\( \implies x = \frac { 600 \times 100 }{ 5 } \)
\( \implies x = 120 \times 100 \)
\( \implies x = 12000 \)
Thus, the needed quantity is 12000.
(b) 12% of a quantity is 1080.
Suppose the required quantity = \( x \).
So, 12% of \( x = 1080 \)
\( \frac { 12 }{ 100 } \times x = 1080 \)
\( \implies x = \frac { 1080 \times 100 }{ 12 } \)
\( \implies x = 90 \times 100 \)
\( \implies x = 9000 \)
Thus, the needed amount = Rs. 9000.
(c) 40% of a quantity is 500 km.
Let the quantity be \( x \).
So, 40% of \( x = 500 \) km
\( \frac { 40 }{ 100 } \times x = 500 \) km
\( \implies x = \frac { 500 \times 100 }{ 40 } \)
\( \implies x = 125 \times 10 \) km
\( \implies x = 1250 \) km
Thus, the needed quantity is 1250 km.
(d) 70% of a quantity is 14 minutes.
Let the required quantity be \( x \).
So, 70% of \( x = 14 \) minutes
\( \frac { 70 }{ 100 } \times x = 14 \) minutes
\( \implies x = \frac { 14 \times 100 }{ 70 } \)
\( \implies x = 2 \times 10 \)
\( \implies x = 20 \) minutes
Thus, the needed quantity is 20 minutes.
(e) 8% of a quantity is 40 litres.
Let the quantity be \( x \).
So, 8% of \( x = 40 \) litres
\( \frac { 8 }{ 100 } \times x = 40 \) litres
\( \implies x = \frac { 40 \times 100 }{ 8 } \)
\( \implies x = 5 \times 100 \)
\( \implies x = 500 \) litres.
Thus, the needed quantity is 500 litres.
In simple words: To find the total amount when you know a part of it as a percentage, you can divide the given part by the percentage (written as a decimal or fraction). This helps you work backward to the whole.

Exam Tip: When finding the whole quantity from a given percentage, always set up an equation where 'x' represents the unknown whole. Remember that the percentage should be written as a fraction over 100.

Question 6. Convert given per cents to decimal fractions and also to fractions in simplest forms:
(a) 25%
(b) 150%
(c) 20%
(d) 5%
Answer:
(a) We have 25% \( = \frac { 25 }{ 100 } \)
As a decimal: \( 0.25 \)
In simplest fraction form: \( \frac { 1 }{ 4 } \)
Thus, 25% = \( 0.25 = \frac { 1 }{ 4 } \).
(b) We have 150% \( = \frac { 150 }{ 100 } \)
As a decimal: \( 1.50 \) or \( 1.5 \)
In simplest fraction form: \( \frac { 3 }{ 2 } \)
Thus, 150% = \( 1.5 = \frac { 3 }{ 2 } \).
(c) We have 20% \( = \frac { 20 }{ 100 } \)
As a decimal: \( 0.20 \) or \( 0.2 \)
In simplest fraction form: \( \frac { 1 }{ 5 } \)
Thus, 20% = \( 0.2 = \frac { 1 }{ 5 } \).
(d) We have 5% \( = \frac {5}{ 100 } \)
As a decimal: \( 0.05 \)
In simplest fraction form: \( \frac { 1 }{ 20 } \)
Thus, 5% = \( 0.05 = \frac { 1 }{ 20 } \).
In simple words: To change a percentage to a decimal, divide it by 100. To change it to a simple fraction, write the percentage over 100 and then reduce the fraction to its smallest terms.

Exam Tip: Always remember that "per cent" means "out of one hundred." Dividing by 100 will give you the decimal, and then simplify the fraction \( \frac{\text{percentage}}{100} \) to get the simplest form.

Question 7. In a city, 30% are females, 40% are males and remaining are children. What per cent are children?
Answer:
Females make up 30% of the population and males make up 40%.
The remaining part of the population is children.
Total percentage of adults \( = 30\% + 40\% = 70\% \)
Percentage of children \( = 100\% - 70\% = 30\% \)
Thus, children represent 30% of the total population.
In simple words: To find the percentage of children, add the percentages of females and males, then subtract that sum from 100 percent. The remaining percentage is for children.

Exam Tip: Remember that the total population always represents 100%. If you are given percentages for some categories, subtract their sum from 100% to find the percentage of the remaining category.

Question 8. Out of 15,000 voters in a constituency, 60% voted. Find the percentage of voters who did not vote. Can you now find how many actually did not vote?
Answer:
Total voters = 15,000
Percentage of voters who voted = 60%
Percentage of voters who did not vote \( = 100\% - 60\% = 40\% \)
Now, to find how many actually did not vote, we calculate 40% of 15,000.
Number of voters who did not vote \( = \frac { 40 }{ 100 } \times 15000 \)
\( = 40 \times 150 \)
\( = 6000 \)
Thus, 6000 voters did not vote.
In simple words: First, subtract the voting percentage from 100 to get the percentage of non-voters. Then, use this percentage to calculate the actual number of people who did not vote from the total voter count.

Exam Tip: Always break down percentage problems into steps: first find the percentage, then use that percentage to calculate the actual quantity. Pay attention to what the question asks for (percentage, number, or both).

Question 9. Meeta saves Rs. 400 from her salary. If this is 10% of her salary, what is her salary?
Answer:
Meeta's saving = Rs. 400.
This saving is 10% of her total salary.
Let her total salary be \( x \).
So, 10% of \( x = \text{Rs. } 400 \)
\( \frac { 10 }{ 100 } \times x = 400 \)
\( \implies x = \frac { 400 \times 100 }{ 10 } \)
\( \implies x = 40 \times 100 \)
\( \implies x = 4000 \)
Thus, her salary = Rs. 4000.
In simple words: If you know a part of something and its percentage, you can find the whole amount by dividing the part by the percentage (as a decimal). Here, Rs. 400 is 10%, so her full salary is Rs. 4000.

Exam Tip: When a percentage of an unknown total is given, set up an equation where 'x' is the total. For example, if 'y' is 'p%' of 'x', then \( \frac{p}{100} \times x = y \). Solving for 'x' will give you the total amount.

Question 10. A local cricket team played 20 matches in one season. It won 25% of them. How many matches did they win?
Answer:
Total number of matches played = 20.
Percentage of matches won = 25%.
Number of matches won \( = 25\% \) of 20
\( = \frac { 25 }{ 100 } \times 20 \)
\( = \frac { 1 }{ 4 } \times 20 \)
\( = 5 \)
Therefore, the team won 5 matches.
In simple words: To find out how many matches were won, you calculate 25 percent of the total 20 matches played. This calculation shows the team won 5 matches.

Exam Tip: Clearly identify the total number and the percentage. Convert the percentage to a fraction or decimal, then multiply by the total to find the specific part. Always write out your steps clearly.