CBSE Class 7 Mathematics Comparing Quantities Worksheet Set 04

Question. Ram bought 5 kg potatoes and Vineet bought 3 kg 500 g potatoes. Find the ratio of Ram’s and Vineet’s potatoes.
Answer : 10 : 7

Question. If 3 : x : : 9 : 15, find x.
Answer : 5

Question. In what time will Rs. 2,700 will double itself if rate of interest is 4% p.a?
Answer : 25 years

Question. If the angles of a triangle are in the ratio 1 : 2 : 3, find the value of each angle.
Answer : 30º, 60º, 90º

Question. Find interest and amount when Principal = Rs. 1000, Rate = 5% p.a., Time = 3 years.
Answer : I = Rs. 150, A = Rs. 1150

Question. A school team won 7 games this year against 5 games won last year. What is the percent increase?
Answer : 40%

Question. Out of 400 oranges, 80 oranges are found rotten. Find the ratio of rotten oranges to good oranges.
Answer : 1 : 4

Question. Express the following percentage as fraction in its lowest form :
(a) 75%
(b) 20%
(c) 6%
Answer : (a) = 3/4 (b) = 1/5 (c) = 3/50

Question. What percent of Rs. 150 is Rs. 30?
Answer : 20%

Question. Ram buys an old house for Rs. 3,00,000. He spends Rs. 20% of the cost price on its repairs and sells it for Rs. 4,50,000. Find his gain or loss percent.
Answer : Gain 25%

Question. Chalk contains calcium, carbon and oxygen in the ratio of 10 : 3 : 12. Find the percentage of carbon in chalk.
Answer : 12%

Question. An electronics dealer buys a radio for Rs. 225 and spends Rs. 15 on transporting it to his shop. If he sells the radio for Rs. 300. Find his profit percent.
Answer : 25%

Question. Kapil bought a sofa set for Rs. 4500 and sold it for Rs. 4,200. Find his loss percent.
Answer : 6(2/3)%

Question. The selling price of an article is 4/5 of its cost price. Find the loss percent.
Answer : 20%

Question. Find interest and amount when Principal = Rs. 200, Rate = 12% p.a., Time = 6 months.
Answer : I = Rs. 12; A = Rs. 212

Question. Ram buys a coloured TV for Rs. 10,000 and sells it for Rs. 12,000. Find his gain %.
Answer : 20%

Question. There are 13 boys and 12 girls in the class. What is the percentage of girls in the class?
Answer : 48%

Question. Find simple interest on the sum of Rs. 800 for 2.5 years at 4.5% p.a.
Answer : Rs. 90

Question. In a fruit basket, 50% are apples, 25% are oranges and remaining are bananas. Find the % of bananas.
Answer : 25%

Question. Calculate the time if Principal = Rs. 2500, Simple Interest = Rs. 625, Rate of Interest = 5% p.a.
Answer : 5 years

Question. Calculate the rate of interest if Principal = Rs. 5000, S.I. = Rs. 700, Time = 2 years.
Answer : 7% p.a.

Question. Rs. 600 are invested at 5% simple interest p.a. In how much time will it double itself ?
Answer : 20 years

Question. A farmer bought a cow for Rs. 12,000 and sold it at a gain of 10%. Find the S.P. of the cow.
Answer : Rs. 13,200

Question. A shop has 500 bulbs out of which 10 are defective. What percent of bulbs are defective?
Answer : 2%

Question. At what rate percent per annum will Rs. 800 amount to Rs. 1000 in 5 years?
Answer : 5%

Exercise 8.1

Question 1. Write each of the following fractions as a per cent:
(a) \( \frac{4}{5} \)
(b) \( \frac{3}{10} \)
(c) \( \frac{1}{4} \)
(d) \( \frac{3}{4} \)
(e) \( \frac{3}{25} \)
(f) \( \frac{18}{20} \)
(g) \( \frac{11}{50} \)
(h) \( \frac{1}{20} \)
Answer:
To change any fraction into a percentage, multiply it by 100%:
(a) \( \frac{4}{5} \times 100\% = 4 \times 20\% = 80\% \)
(b) \( \frac{3}{10} \times 100\% = 3 \times 10\% = 30\% \)
(c) \( \frac{1}{4} \times 100\% = 25\% \)
(d) \( \frac{3}{4} \times 100\% = 3 \times 25\% = 75\% \)
(e) \( \frac{3}{25} \times 100\% = 3 \times 4\% = 12\% \)
(f) \( \frac{18}{20} \times 100\% = 18 \times 5\% = 90\% \)
(g) \( \frac{11}{50} \times 100\% = 11 \times 2\% = 22\% \)
(h) \( \frac{1}{20} \times 100\% = 5\% \)
In simple words: Multiply the fraction by 100 to find its percentage.

Exam Tip: Simplify by dividing 100 by the denominator first, then multiply by the numerator.

Question 2. Write each of the following per cents as a fraction in the simplest form:
(a) 6%
(b) 84%
(c) 15%
(d) 44%
(e) 67%
(f) 95%
(g) 45%
(h) 35%
Answer:
To convert a percentage into a fraction, place the number over 100 and reduce to lowest terms:
(a) \( 6\% = \frac{6}{100} = \frac{3}{50} \)
(b) \( 84\% = \frac{84}{100} = \frac{21}{25} \)
(c) \( 15\% = \frac{15}{100} = \frac{3}{20} \)
(d) \( 44\% = \frac{44}{100} = \frac{11}{25} \)
(e) \( 67\% = \frac{67}{100} \) (Since 67 is a prime number, it cannot be reduced further)
(f) \( 95\% = \frac{95}{100} = \frac{19}{20} \)
(g) \( 45\% = \frac{45}{100} = \frac{9}{20} \)
(h) \( 35\% = \frac{35}{100} = \frac{7}{20} \)
In simple words: Write the number over 100 and divide top and bottom by their common factor.

Exam Tip: Always check if both numbers can be divided by 2 or 5 to reduce the fraction fully.

Question 3. Convert each of the following ratios into a per cent:
(a) 3 : 10
(b) 19 : 20
(c) 19 : 25
(d) 2 : 5
(e) 1 : 2
Answer:
First express each ratio as a fraction, then multiply by 100%:
(a) \( 3 : 10 = \frac{3}{10} \times 100\% = 30\% \)
(b) \( 19 : 20 = \frac{19}{20} \times 100\% = 19 \times 5\% = 95\% \)
(c) \( 19 : 25 = \frac{19}{25} \times 100\% = 19 \times 4\% = 76\% \)
(d) \( 2 : 5 = \frac{2}{5} \times 100\% = 2 \times 20\% = 40\% \)
(e) \( 1 : 2 = \frac{1}{2} \times 100\% = 50\% \)
In simple words: Turn the ratio into a fraction and multiply by 100.

Exam Tip: A ratio \( a : b \) simply means the fraction \( \frac{a}{b} \).

Question 4. Convert each of the following decimals into a per cent:
(a) 0.37
(b) 0.15
(c) 0.156
(d) 7.2
(e) 1.05
Answer:
To change a decimal into a percentage, multiply by 100% (shift the decimal point two places to the right):
(a) \( 0.37 \times 100\% = 37\% \)
(b) \( 0.15 \times 100\% = 15\% \)
(c) \( 0.156 \times 100\% = 15.6\% \)
(d) \( 7.2 \times 100\% = 720\% \)
(e) \( 1.05 \times 100\% = 105\% \)
In simple words: Move the decimal point two steps to the right and put a percent sign.

Exam Tip: Add a zero as a placeholder when moving the decimal point two places in numbers like 7.2.

Question 5. Write each of the following per cents as a decimal:
(a) 11.4%
(b) 36.0%
(c) 75%
(d) 0.4%
(e) 0.2%
(f) 18.4%
(g) 90.5%
(h) 313.02%
Answer:
To change a percentage into a decimal, divide by 100 (move the decimal point two places to the left):
(a) \( 11.4\% = \frac{11.4}{100} = 0.114 \)
(b) \( 36.0\% = \frac{36.0}{100} = 0.36 \)
(c) \( 75\% = \frac{75}{100} = 0.75 \)
(d) \( 0.4\% = \frac{0.4}{100} = 0.004 \)
(e) \( 0.2\% = \frac{0.2}{100} = 0.002 \)
(f) \( 18.4\% = \frac{18.4}{100} = 0.184 \)
(g) \( 90.5\% = \frac{90.5}{100} = 0.905 \)
(h) \( 313.02\% = \frac{313.02}{100} = 3.1302 \)
In simple words: Drop the percent sign and slide the decimal point two places to the left.

Exam Tip: When moving two places left on small numbers like 0.4%, place extra zeroes before the digits.

Question 6. Find the value of n for each of the following:
(a) \( \frac{16}{100} = \frac{8}{n} \)
(b) \( \frac{36}{n} = \frac{9}{25} \)
(c) \( \frac{n}{100} = \frac{1}{50} \)
(d) \( \frac{25}{100} = \frac{1}{n} \)
(e) 12% = \( \frac{n}{25} \)
(f) n% = \( \frac{19}{25} \)
(g) 13% = \( \frac{26}{n} \)
(h) 2% = \( \frac{1}{n} \)
Answer:
Use cross-multiplication to solve for \( n \):
(a) \( \frac{16}{100} = \frac{8}{n} \)

\( \implies 16 \times n = 8 \times 100 \)

\( \implies n = \frac{800}{16} = 50 \)
(b) \( \frac{36}{n} = \frac{9}{25} \)

\( \implies 9 \times n = 36 \times 25 \)

\( \implies n = \frac{36 \times 25}{9} = 4 \times 25 = 100 \)
(c) \( \frac{n}{100} = \frac{1}{50} \)

\( \implies 50 \times n = 100 \times 1 \)

\( \implies n = \frac{100}{50} = 2 \)
(d) \( \frac{25}{100} = \frac{1}{n} \)

\( \implies \frac{1}{4} = \frac{1}{n} \)

\( \implies n = 4 \)
(e) \( 12\% = \frac{12}{100} = \frac{n}{25} \)

\( \implies n = \frac{12 \times 25}{100} = \frac{12}{4} = 3 \)
(f) \( n\% = \frac{n}{100} = \frac{19}{25} \)

\( \implies n = \frac{19 \times 100}{25} = 19 \times 4 = 76 \)
(g) \( 13\% = \frac{13}{100} = \frac{26}{n} \)

\( \implies 13 \times n = 26 \times 100 \)

\( \implies n = \frac{2600}{13} = 200 \)
(h) \( 2\% = \frac{2}{100} = \frac{1}{n} \)

\( \implies \frac{1}{50} = \frac{1}{n} \)

\( \implies n = 50 \)
In simple words: Cross-multiply both sides of the equal sign to find the unknown number n.

Exam Tip: Remember that any percentage \( x\% \) can be rewritten as the fraction \( \frac{x}{100} \) before solving.

Question 7. There are 50 students in a class. Eighteen of them are girls and the rest are boys. What per cent of the students are girls? What per cent of the students are boys?
Answer:
Total number of students in the class = 50.
Number of girls = 18.
Number of boys = \( 50 - 18 = 32 \).
1. Percentage of girls:
\( \frac{18}{50} \times 100\% = 18 \times 2\% = 36\% \).
2. Percentage of boys:
\( \frac{32}{50} \times 100\% = 32 \times 2\% = 64\% \).
Therefore, 36% of the students are girls and 64% are boys.
In simple words: Divide the number of girls or boys by 50 and multiply by 100.

Exam Tip: Verify your answer by checking that the percentages add up to 100% (\( 36\% + 64\% = 100\% \)).

Question 8. Mala has a collection of 50 bangles. She has 20 gold bangles and 30 silver bangles. Find the percentage of gold and silver bangles?
Answer:
Total number of bangles = 50.
Number of gold bangles = 20.
Number of silver bangles = 30.
1. Percentage of gold bangles:
\( \frac{20}{50} \times 100\% = 20 \times 2\% = 40\% \).
2. Percentage of silver bangles:
\( \frac{30}{50} \times 100\% = 30 \times 2\% = 60\% \).
Hence, Mala has 40% gold bangles and 60% silver bangles.
In simple words: Put each count over 50 and multiply by 100 to get 40% gold and 60% silver.

Exam Tip: Since the total is 50, you can double each quantity to quickly find its percentage out of 100.

Exercise 8.2

Question 1. Find:
(a) 10% of 60
(b) 20% of 315
(c) 10% of 42.5
(d) 20% of Rs. 500
(e) 15% of 50 l
(f) 6% of 400 kg
Answer:
(a) \( \frac{10}{100} \times 60 = \frac{1}{10} \times 60 = 6 \)
(b) \( \frac{20}{100} \times 315 = \frac{1}{5} \times 315 = 63 \)
(c) \( \frac{10}{100} \times 42.5 = \frac{42.5}{10} = 4.25 \)
(d) \( \frac{20}{100} \times \text{Rs. } 500 = \frac{1}{5} \times 500 = \text{Rs. } 100 \)
(e) \( \frac{15}{100} \times 50\text{ l} = \frac{15}{2}\text{ l} = 7.5\text{ l} \)
(f) \( \frac{6}{100} \times 400\text{ kg} = 6 \times 4\text{ kg} = 24\text{ kg} \)
In simple words: Change the percent to a fraction over 100 and multiply by the given number.

Exam Tip: Always write the correct unit of measurement (such as kg, l, or Rs.) with your final answer.

Question 2. What per cent of:
(a) 10 is 2?
(b) 40 is 8?
(c) 200 is 25?
(d) Rs. 50 is Rs. 3.50?
(e) 9 km is 6 km?
(f) 500 is 35?
Answer:
Use the formula: \( \text{Required percentage} = \frac{\text{Part}}{\text{Total}} \times 100\% \)
(a) \( \frac{2}{10} \times 100\% = 2 \times 10\% = 20\% \)
(b) \( \frac{8}{40} \times 100\% = \frac{1}{5} \times 100\% = 20\% \)
(c) \( \frac{25}{200} \times 100\% = \frac{25}{2}\% = 12.5\% \)
(d) \( \frac{3.50}{50} \times 100\% = 3.50 \times 2\% = 7\% \)
(e) \( \frac{6}{9} \times 100\% = \frac{2}{3} \times 100\% = \frac{200}{3}\% = 66\frac{2}{3}\% \) (or \( 66.67\% \))
(f) \( \frac{35}{500} \times 100\% = \frac{35}{5}\% = 7\% \)
In simple words: Divide the smaller part by the base number and multiply by 100.

Exam Tip: In phrases like "What percent of A is B", the number after "of" is always the denominator.

Question 3. Find the quantity of which:
(a) \( 6\frac{1}{4}\% \) is 2.
(b) 80% is 704.
(c) 7% is 343.
Answer:
Let the unknown total quantity be \( x \):
(a) \( 6\frac{1}{4}\% \text{ of } x = 2 \)

\( \implies \frac{25}{4}\% \text{ of } x = 2 \)

\( \implies \frac{25}{4 \times 100} \times x = 2 \)

\( \implies \frac{1}{16} \times x = 2 \)

\( \implies x = 2 \times 16 = 32 \)
(b) \( 80\% \text{ of } x = 704 \)

\( \implies \frac{80}{100} \times x = 704 \)

\( \implies x = \frac{704 \times 100}{80} = \frac{704 \times 5}{4} = 176 \times 5 = 880 \)
(c) \( 7\% \text{ of } x = 343 \)

\( \implies \frac{7}{100} \times x = 343 \)

\( \implies x = \frac{343 \times 100}{7} = 49 \times 100 = 4900 \)
In simple words: Divide the given value by the percentage and multiply by 100 to find the whole quantity.

Exam Tip: Convert mixed numbers like \( 6\frac{1}{4}\% \) to improper fractions (\( \frac{25}{4}\% \)) before multiplying.

Question 4. Find the number whose \( 12\frac{1}{2}\% \) is 64.
Answer:
Let the required number be \( x \).
Convert the mixed fraction into an improper fraction: \( 12\frac{1}{2}\% = \frac{25}{2}\% \).
According to the question:
\( \frac{25}{2}\% \text{ of } x = 64 \)

\( \implies \frac{25}{2 \times 100} \times x = 64 \)

\( \implies \frac{1}{8} \times x = 64 \)

\( \implies x = 64 \times 8 = 512 \).
Hence, the required number is 512.
In simple words: \( 12\frac{1}{2}\% \) equals one-eighth of a number. Multiplying 64 by 8 gives 512.

Exam Tip: Remember that \( 12\frac{1}{2}\% \) is equivalent to the simple fraction \( \frac{1}{8} \).

Question 5. Anjana scored 64 marks out of 80. What percentage of marks did she score?
Answer:
Marks scored by Anjana = 64.
Total possible marks = 80.
Percentage of marks scored:
\( \frac{64}{80} \times 100\% \)
Divide 64 and 80 by 16:
\( = \frac{4}{5} \times 100\% = 4 \times 20\% = 80\% \).
Therefore, Anjana scored 80% marks.
In simple words: Divide 64 by 80 and multiply by 100. That gives 80%.

Exam Tip: Simplify the fraction first by canceling common factors before multiplying by 100.

Question 6. If the height of a plant has increased from 8 cm to 10 cm, what is the percentage increase in the height?
Answer:
Original height of the plant = 8 cm.
New height of the plant = 10 cm.
Increase in height = \( 10\text{ cm} - 8\text{ cm} = 2\text{ cm} \).
Percentage increase formula:
\( \text{Percentage increase} = \frac{\text{Increase}}{\text{Original height}} \times 100\% \)
\( = \frac{2}{8} \times 100\% = \frac{1}{4} \times 100\% = 25\% \).
Thus, the plant's height increased by 25%.
In simple words: The plant grew 2 cm. Divide 2 by the original 8 cm to get 25%.

Exam Tip: Always divide the increase by the original starting value, not by the new increased value.

Question 7. The monthly rent of a house has increased from Rs. 2500 to Rs. 3000. What is the percentage increase in the rent?
Answer:
Original rent = Rs. 2500.
New rent = Rs. 3000.
Increase in rent = \( 3000 - 2500 = \text{Rs. } 500 \).
Percentage increase:
\( \frac{\text{Increase}}{\text{Original rent}} \times 100\% \)
\( = \frac{500}{2500} \times 100\% = \frac{1}{5} \times 100\% = 20\% \).
Therefore, the rent increased by 20%.
In simple words: The rent went up by Rs. 500. Divide 500 by the original 2500 to get a 20% rise.

Exam Tip: Base your percentage calculation strictly on the starting rent of Rs. 2500.

Question 8. The price of an electric motor was Rs. 5500. Its price has increased by 2%. What is the new price of the motor?
Answer:
Original price of the motor = Rs. 5500.
Rate of increase = 2%.
Amount of increase:
\( 2\% \text{ of } 5500 = \frac{2}{100} \times 5500 = 2 \times 55 = \text{Rs. } 110 \).
New price of the motor:
\( \text{Original price} + \text{Increase} = 5500 + 110 = \text{Rs. } 5610 \).
Hence, the new price of the electric motor is Rs. 5610.
In simple words: Find 2% of Rs. 5500, which is Rs. 110. Add that to get Rs. 5610.

Exam Tip: Remember to add the price hike back to the original price to find the final new price.

Question 9. A pool has 10 lanes. During a swim meet, 80% of the lanes are used. How many lanes are not used?
Answer:
Total number of lanes = 10.
Percentage of lanes used = 80%.
Percentage of lanes not used = \( 100\% - 80\% = 20\% \).
Number of lanes not used:
\( 20\% \text{ of } 10 = \frac{20}{100} \times 10 = 2 \text{ lanes} \).
Therefore, 2 lanes are not used.
In simple words: If 80% of the 10 lanes are in use, 20% are free, which means 2 lanes are unused.

Exam Tip: Subtracting 80% from 100% first makes the calculation quicker and saves time.

Question 10. 10% of the 440 people who came to see a circus show had free passes to get in. How many people had free passes? How many people had to pay to get in?
Answer:
Total number of spectators = 440.
Percentage with free passes = 10%.
1. Number of people with free passes:
\( 10\% \text{ of } 440 = \frac{10}{100} \times 440 = 44 \text{ people} \).
2. Number of people who had to pay:
\( 440 - 44 = 396 \text{ people} \).
Thus, 44 people entered with free passes and 396 people paid for entry.
In simple words: 10% of 440 is 44 free visitors. Subtract 44 from 440 to find that 396 people paid.

Exam Tip: Be sure to write down two separate answers since the question asks two things.

Question 11. Find the percentage increase or decrease:
(a) Price of a shirt decreased from Rs. 80 to 60.
(b) Marks in a test increased from 20 to 30.
Answer:
(a) Original price = Rs. 80, New price = Rs. 60.
Decrease in price = \( 80 - 60 = \text{Rs. } 20 \).
Percentage decrease:
\( \frac{20}{80} \times 100\% = \frac{1}{4} \times 100\% = 25\% \).
So, the price decreased by 25%.
(b) Original marks = 20, New marks = 30.
Increase in marks = \( 30 - 20 = 10 \).
Percentage increase:
\( \frac{10}{20} \times 100\% = \frac{1}{2} \times 100\% = 50\% \).
So, the test marks increased by 50%.
In simple words: Find the change first, divide it by the starting value, and multiply by 100.

Exam Tip: State clearly whether the percentage represents an "increase" or a "decrease" in your final line.

Exercise 8.3

Question 1. Find the gain(profit)/loss percentage, when:
(a) C.P. = Rs. 1500 ; S.P. = Rs. 1650
(b) C.P. = Rs. 3650 ; S.P. = Rs. 3504
(c) C.P. = Rs. 400 ; S.P. = Rs. 360
(d) C.P. = Rs. 750 ; S.P. = Rs. 600
Answer:
(a) Here, S.P. > C.P., so there is a profit (gain):
\( \text{Gain} = 1650 - 1500 = \text{Rs. } 150 \)
\( \text{Gain}\% = \frac{150}{1500} \times 100\% = 10\% \).
(b) Here, S.P. < C.P., so there is a loss:
\( \text{Loss} = 3650 - 3504 = \text{Rs. } 146 \)
\( \text{Loss}\% = \frac{146}{3650} \times 100\% = \frac{146 \times 2}{73}\% = 2 \times 2\% = 4\% \).
(c) Here, S.P. < C.P., so there is a loss:
\( \text{Loss} = 400 - 360 = \text{Rs. } 40 \)
\( \text{Loss}\% = \frac{40}{400} \times 100\% = 10\% \).
(d) Here, S.P. < C.P., so there is a loss:
\( \text{Loss} = 750 - 600 = \text{Rs. } 150 \)
\( \text{Loss}\% = \frac{150}{750} \times 100\% = \frac{1}{5} \times 100\% = 20\% \).
In simple words: Compare S.P. and C.P. to see if it is a profit or loss. Then divide the difference by C.P. and multiply by 100.

Exam Tip: Both gain percentage and loss percentage are always calculated with C.P. in the denominator.

Question 2. Find the selling price (S.P.) when:
(a) C.P. = Rs. 620 ; Gain = 12%
(b) C.P. = Rs. 740 ; Loss = 7%
Answer:
(a) Calculate gain value:
\( \text{Gain} = 12\% \text{ of } 620 = \frac{12}{100} \times 620 = \frac{744}{10} = \text{Rs. } 74.40 \)
\( \text{S.P.} = \text{C.P.} + \text{Gain} = 620 + 74.40 = \text{Rs. } 694.40 \).
(b) Calculate loss value:
\( \text{Loss} = 7\% \text{ of } 740 = \frac{7}{100} \times 740 = \frac{518}{10} = \text{Rs. } 51.80 \)
\( \text{S.P.} = \text{C.P.} - \text{Loss} = 740 - 51.80 = \text{Rs. } 688.20 \).
In simple words: Work out the profit and add it to the cost price, or work out the loss and subtract it.

Exam Tip: Remember: S.P. = C.P. + Profit, but S.P. = C.P. - Loss.

Question 3. Find the cost price (C.P.) when:
(a) S.P. = Rs. 810 ; Loss = 10%
(b) S.P. = Rs. 756 ; Gain = 8%
Answer:
(a) When there is a loss:
\( \text{S.P.} = \text{C.P.} \times \frac{100 - \text{Loss}\%}{100} \)
\( 810 = \text{C.P.} \times \frac{90}{100} \)

\( \implies \text{C.P.} = \frac{810 \times 100}{90} = 9 \times 100 = \text{Rs. } 900 \).
(b) When there is a gain:
\( \text{S.P.} = \text{C.P.} \times \frac{100 + \text{Gain}\%}{100} \)
\( 756 = \text{C.P.} \times \frac{108}{100} \)

\( \implies \text{C.P.} = \frac{756 \times 100}{108} = 7 \times 100 = \text{Rs. } 700 \).
In simple words: Divide the selling price by the remaining percent for loss, or by the increased percent for gain, then multiply by 100.

Exam Tip: Never calculate 10% of S.P. to find the loss; loss percentage is only based on C.P.

Question 4. A cloth merchant buys cloth at Rs. 28 per metre and sells it at Rs. 35 per metre. Find his gain percentage.
Answer:
Cost Price per metre (C.P.) = Rs. 28.
Selling Price per metre (S.P.) = Rs. 35.
Since S.P. > C.P., there is a gain:
\( \text{Gain} = 35 - 28 = \text{Rs. } 7 \).
Gain percentage:
\( \text{Gain}\% = \frac{\text{Gain}}{\text{C.P.}} \times 100\% \)
\( = \frac{7}{28} \times 100\% = \frac{1}{4} \times 100\% = 25\% \).
Therefore, the merchant's gain is 25%.
In simple words: He makes Rs. 7 profit on every Rs. 28 spent. Dividing 7 by 28 gives a 25% gain.

Exam Tip: Since profit per metre is identical to profit on the whole roll, calculate directly per metre.

Question 5. A sofa set was bought for Rs. 1800 and sold at a loss of 25%. What was the selling price?
Answer:
Cost Price (C.P.) = Rs. 1800.
Loss percentage = 25%.
Amount of loss:
\( \text{Loss} = 25\% \text{ of } 1800 = \frac{25}{100} \times 1800 = \frac{1}{4} \times 1800 = \text{Rs. } 450 \).
Selling Price (S.P.):
\( \text{S.P.} = \text{C.P.} - \text{Loss} = 1800 - 450 = \text{Rs. } 1350 \).
Hence, the selling price of the sofa set was Rs. 1350.
In simple words: One-fourth of 1800 is 450. Subtract 450 from 1800 to get the selling price of Rs. 1350.

Exam Tip: A loss of 25% means the selling price is 75% of the cost price (\( 0.75 \times 1800 = 1350 \)).

Question 6. A shopkeeper buys an almirah for Rs. 850. The transport charges are Rs. 40. He sells it at a gain of 10%. What is the selling price?
Answer:
Purchase price = Rs. 850.
Overhead transportation cost = Rs. 40.
Total Cost Price (C.P.) = \( 850 + 40 = \text{Rs. } 890 \).
Gain percentage = 10%.
Amount of gain:
\( \text{Gain} = 10\% \text{ of } 890 = \frac{10}{100} \times 890 = \text{Rs. } 89 \).
Selling Price (S.P.):
\( \text{S.P.} = \text{Total C.P.} + \text{Gain} = 890 + 89 = \text{Rs. } 979 \).
Thus, the selling price of the almirah is Rs. 979.
In simple words: Add the Rs. 40 delivery cost to the price first. Then add 10% profit to get Rs. 979.

Exam Tip: Overhead expenses like transport or repairs must always be added to the purchase cost before finding profit.

Question 7. Harinam borrowed Rs. 14000 from a bank at 15% p.a. for 3 years. How much interest will he pay to the bank after 3 years?
Answer:
Principal (\( P \)) = Rs. 14000.
Rate of interest (\( R \)) = 15% per annum.
Time period (\( T \)) = 3 years.
Formula for simple interest:
\( \text{Simple Interest} = \frac{P \times R \times T}{100} \)
\( = \frac{14000 \times 15 \times 3}{100} \)
\( = 140 \times 45 = \text{Rs. } 6300 \).
Therefore, Harinam will pay Rs. 6300 as interest.
In simple words: Multiply 14000 by 15 and 3, then divide by 100 to get Rs. 6300.

Exam Tip: State the simple interest formula clearly before substituting the given values.

Question 8. Ramesh borrowed Rs. 2500 from his friend at 8% p.a. He returned the money after 1 year. How much did he pay back altogether?
Answer:
Principal (\( P \)) = Rs. 2500.
Rate (\( R \)) = 8% per annum.
Time (\( T \)) = 1 year.
Simple Interest:
\( \text{Interest} = \frac{P \times R \times T}{100} = \frac{2500 \times 8 \times 1}{100} = 25 \times 8 = \text{Rs. } 200 \).
Total amount repaid:
\( \text{Amount} = \text{Principal} + \text{Interest} = 2500 + 200 = \text{Rs. } 2700 \).
Hence, Ramesh paid back Rs. 2700 altogether.
In simple words: The interest is Rs. 200. Adding it to the borrowed Rs. 2500 gives Rs. 2700.

Exam Tip: The word "altogether" means you must find the total amount (Principal + Interest).

Question 9. John deposits Rs. 4000 in a bank. The bank pays interest at 8% p.a. Find the interest and the amount that he will receive after two years?
Answer:
Principal (\( P \)) = Rs. 4000.
Rate (\( R \)) = 8% per annum.
Time (\( T \)) = 2 years.
1. Simple Interest:
\( \text{Interest} = \frac{P \times R \times T}{100} = \frac{4000 \times 8 \times 2}{100} = 40 \times 16 = \text{Rs. } 640 \).
2. Total Amount:
\( \text{Amount} = \text{Principal} + \text{Interest} = 4000 + 640 = \text{Rs. } 4640 \).
Therefore, John will receive an interest of Rs. 640 and a total amount of Rs. 4640.
In simple words: The interest earned is Rs. 640. Added to the deposit, the total amount is Rs. 4640.

Exam Tip: Clearly write both the interest and the total amount to answer both questions asked.

Question 10. Rashi borrowed Rs. 6000 from a bank. If the bank charges interest at the rate of 14% p.a., what amount will he pay to the bank at the end of 2 years?
Answer:
Principal (\( P \)) = Rs. 6000.
Rate of interest (\( R \)) = 14% per annum.
Time (\( T \)) = 2 years.
Calculate the interest charged:
\( \text{Interest} = \frac{P \times R \times T}{100} = \frac{6000 \times 14 \times 2}{100} = 60 \times 28 = \text{Rs. } 1680 \).
Total amount to be paid:
\( \text{Amount} = \text{Principal} + \text{Interest} = 6000 + 1680 = \text{Rs. } 7680 \).
Hence, an amount of Rs. 7680 will be paid back after 2 years.
In simple words: The bank interest is Rs. 1680. Adding that to Rs. 6000 gives a total of Rs. 7680.

Exam Tip: Always calculate the interest first, then add it to the principal to get the final amount.

Question 11. What principal will amount to Rs. 2640 in 2 years at 10% p.a. interest?
Answer:
Total Amount (\( A \)) = Rs. 2640.
Time (\( T \)) = 2 years.
Rate (\( R \)) = 10% per annum.
Let the principal be \( P \).
Formula connecting Amount and Principal:
\( A = P + \frac{P \times R \times T}{100} \)
\( 2640 = P + \frac{P \times 10 \times 2}{100} \)
\( 2640 = P + \frac{20P}{100} = P + \frac{P}{5} = \frac{6P}{5} \)

\( \implies P = \frac{2640 \times 5}{6} = 440 \times 5 = \text{Rs. } 2200 \).
Therefore, the required principal is Rs. 2200.
In simple words: Over 2 years, Rs. 2200 grows by 20% in interest to reach Rs. 2640.

Exam Tip: Take \( P \) common in the formula \( A = P\left(1 + \frac{RT}{100}\right) \) to solve quickly.

Question 12. In how many years will Rs. 900 amount to Rs. 1050 at 10% p.a.?
Answer:
Principal (\( P \)) = Rs. 900.
Total Amount (\( A \)) = Rs. 1050.
Rate (\( R \)) = 10% per annum.
Calculate interest earned:
\( \text{Interest} = A - P = 1050 - 900 = \text{Rs. } 150 \).
Use the time formula:
\( T = \frac{\text{Interest} \times 100}{P \times R} \)
\( T = \frac{150 \times 100}{900 \times 10} = \frac{15000}{9000} = \frac{15}{9} = \frac{5}{3}\text{ years} = 1\frac{2}{3}\text{ years} \).
To express in months:
\( 1\text{ year} + \left(\frac{2}{3} \times 12\right)\text{ months} = 1\text{ year and } 8\text{ months} \).
Thus, it will take \( 1\frac{2}{3} \) years (or 1 year 8 months).
In simple words: The interest needed is Rs. 150. At Rs. 90 per year, it takes 1 year and 8 months.

Exam Tip: Convert fractional years into years and months by multiplying the fraction by 12.

Question 13. What sum of money will get annual interest of Rs. 1800 at 6% p.a.?
Answer:
Annual interest (\( \text{Interest} \)) = Rs. 1800.
Rate of interest (\( R \)) = 6% per annum.
Time period (\( T \)) = 1 year.
Formula for Principal (\( P \)):
\( P = \frac{\text{Interest} \times 100}{R \times T} \)
\( P = \frac{1800 \times 100}{6 \times 1} \)
\( P = 300 \times 100 = \text{Rs. } 30000 \).
Therefore, the required sum of money is Rs. 30,000.
In simple words: Dividing the yearly interest of Rs. 1800 by 6% gives a principal of Rs. 30,000.

Exam Tip: "Annual interest" means the time period \( T \) is exactly 1 year.