CBSE Class 8 Mathematics Comparing Quantities Worksheet Set 03

EXERCISE 8.1

Question 1. If 20% of x is 80, find x.
Answer: According to the given condition:
\(20\% \text{ of } x = 80\)
\(\frac{20}{100} \times x = 80\)
\(x = \frac{80 \times 100}{20}\)
\(x = 400\)
In simple words: To find the total value when a percentage is known, divide the part by the percentage and multiply by 100.

Exam Tip: Always translate the word 'of' to multiplication and percentage to a fraction over 100.

Question 2. 44% of the students of a class are girls. If there are 6 less girls than boys, how many students are there in the class?
Answer: Let the total number of students in the class be \(100\).
Percentage of girls = \(44\%\)
Percentage of boys = \(100\% - 44\% = 56\%\)
Difference in percentage between boys and girls = \(56\% - 44\% = 12\%\)
Given that the difference in actual numbers is 6 students:
\(12\% \text{ of total students} = 6\)
\(\frac{12}{100} \times \text{Total} = 6\)
\(\text{Total} = \frac{6 \times 100}{12} = 50\)
In simple words: Find the percentage of boys, calculate the difference between boys and girls, and use it to find the total class size.

Exam Tip: When percentages of parts are given, total percentage always equals 100.

Question 3. The population of a town, at present, is 37380 and this is 5% more than it was one year ago. What was its population that year?
Answer: Let the population one year ago be \(x\).
Present population = Population one year ago + \(5\% \text{ of Population one year ago}\)
\(37380 = x + 0.05x\)
\(37380 = 1.05x\)
\(x = \frac{37380}{1.05}\)
\(x = \frac{3738000}{105} = 35600\)
In simple words: Increase means adding the percentage to the original value to find the new value.

Exam Tip: Be careful not to take 5% of the present population; the increase is always calculated on the past base value.

Question 4. 40% marks are pass marks in an examination. Kiran gets zero mark and fails by 20 marks. Find the maximum marks.
Answer: Passing marks = \(40\% \text{ of maximum marks}\)
Kiran obtained 0 marks and failed by 20 marks, which means the passing marks must be 20 marks.
Let maximum marks be \(M\).
\(40\% \text{ of } M = 20\)
\(\frac{40}{100} \times M = 20\)
\(M = \frac{20 \times 100}{40} = 50\)
In simple words: The marks needed to pass equal the deficiency when a student fails.

Exam Tip: Failing by 20 marks means the passing threshold is exactly 20 marks.

Question 5. A woman bought certain number of eggs, out of which 4% were bad. After selling 75% of the remaining, she had 96 eggs. Find the number of eggs she bought.
Answer: Let total eggs bought be \(x\).
Bad eggs = \(4\% \text{ of } x\)
Remaining good eggs = \(x - 0.04x = 0.96x\)
She sells \(75\%\) of the remaining eggs, so the eggs left unsold = \(100\% - 75\% = 25\%\) of remaining good eggs.
\(25\% \text{ of } 0.96x = 96\)
\(\frac{25}{100} \times 0.96x = 96\)
\(0.25 \times 0.96x = 96\)
\(0.24x = 96\)
\(x = \frac{96}{0.24} = 400\)
In simple words: Subtract bad eggs first, then take the remaining percentage of that remainder to find the final count.

Exam Tip: Always apply successive percentages sequentially to the updated remaining amounts.

Question 6. A number is first increased by 20% and then the new number so obtained is decreased by 20%. Find the net increase or decrease per cent.
Answer: Let the original number be 100.
First increase by \(20\%\):
New number = \(100 + 20 = 120\)
Then decreased by \(20\%\):
Decrease amount = \(20\% \text{ of } 120 = \frac{20}{100} \times 120 = 24\)
Final number = \(120 - 24 = 96\)
Net change = \(100 - 96 = 4\) (decrease)
Net decrease percentage = \(4\%\)
In simple words: Successive equal percentage increases and decreases always result in a net decrease.

Exam Tip: Use the formula Net change = \(a + b + \frac{aab}{100}\) where \(a = +20\) and \(b = -20\).

Question 7. Ramesh's income is 20% less than that of Suresh's income. By what per cent is Suresh's income more than that of Ramesh's?
Answer: Let Suresh's income be Rs. 100.
Ramesh's income is \(20\%\) less than Suresh's = \(100 - 20 =\) Rs. 80.
Difference in income = \(100 - 80 =\) Rs. 20.
Percentage by which Suresh's income is more than Ramesh's:
\(= \left(\frac{\text{Difference}}{\text{Ramesh's income}}\right) \times 100\)
\(= \frac{20}{80} \times 100 = \frac{1}{4} \times 100 = 25\%\)
In simple words: The base changes when comparing in reverse; compare the difference against Ramesh's lower income.

Exam Tip: "More than" compares against the smaller value, while "less than" compares against the larger value.

Question 8. The salary of an employee in a company has been increased by 25%. By what per cent, the new salary be decreased to restore the original salary?
Answer: Let original salary be Rs. 100.
Increased salary = \(100 + 25 =\) Rs. 125.
Required decrease amount = Rs. 25.
Percentage decrease required on new salary:
\(= \left(\frac{25}{125}\right) \times 100\)
\(= \frac{1}{5} \times 100 = 20\%\)
In simple words: To bring an increased value back down, calculate the drop as a percentage of the new higher value.

Exam Tip: A 25% increase requires a 20% decrease to restore the original value.

Question 9. 85% of the total number of students of a school did not go for a picnic on a certain day. If the number of students who did not go for the picnic was 935, find the total number of students in the school.
Answer: Let the total number of students be \(x\).
Given that \(85\%\) of total students did not go for the picnic.
\(85\% \text{ of } x = 935\)
\(\frac{85}{100} \times x = 935\)
\(x = \frac{935 \times 100}{85}\)
\(x = 11 \times 100 = 1100\)
In simple words: Set the given percentage equal to the given count to solve for the whole.

Exam Tip: Ensure you match the percentage directly to its corresponding actual value given in the text.

Question 10. In an election, the winning candidate received 31488 votes which represented 48% of the electorate. The only other candidate received votes which formed 22% of the electorate. Calculate (a) the number of electorate and (b) the number of people who did not cast their vote.
Answer:
(a) Let total electorate be \(x\).
Winning candidate votes = \(48\% \text{ of } x = 31488\)
\(\frac{48}{100} \times x = 31488\)
\(x = \frac{31488 \times 100}{48} = 65600\)
Total electorate = 65600.

(b) Percentage of total votes polled = Winner votes percentage + Other candidate votes percentage
\(= 48\% + 22\% = 70\%\)
Percentage of people who did not cast their vote = \(100\% - 70\% = 30\%\)
Number of people who did not vote = \(30\% \text{ of } 65600\)
\(= \frac{30}{100} \times 65600 = 30 \times 656 = 19680\)
In simple words: Find the total voters first using the winner's percentage, then calculate the uncast vote percentage.

Exam Tip: Total votes cast plus abstentions equals the total electorate.

EXERCISE 8.2

Question 1. The marked price of a book is Rs. 110. The shopkeeper gives 10% discount on it. What is the selling price of the book?
Answer: Marked Price (MP) = Rs. 110
Discount percentage = \(10\%\)
Discount amount = \(10\% \text{ of Rs. } 110 = \frac{10}{100} \times 110 = \text{Rs. } 11\)
Selling Price (SP) = Marked Price - Discount
\(\text{SP} = 110 - 11 = \text{Rs. } 99\)
In simple words: Subtract the discount amount from the marked price to find the selling price.

Exam Tip: Selling Price equals Marked Price minus Discount.

Question 2. The marked price of an article is Rs. 75. If it is sold at Rs. 60, what is the discount rate?
Answer: Marked Price (MP) = Rs. 75
Selling Price (SP) = Rs. 60
Discount amount = \(\text{MP} - \text{SP} = 75 - 60 = \text{Rs. } 15\)
Discount percentage = \(\left(\frac{\text{Discount}}{\text{MP}}\right) \times 100\)
\(= \left(\frac{15}{75}\right) \times 100 = \frac{1}{5} \times 100 = 20\%\)
In simple words: Discount percentage is calculated by dividing the discount by the marked price and multiplying by 100.

Exam Tip: Discount is always calculated on the Marked Price, never on the Selling Price.

Question 3. A T.V. set was sold for Rs. 17820 after giving discount of 10%. What was the marked price of the T.V. set?
Answer: Selling Price (SP) = Rs. 17820
Discount percentage = \(10\%\)
Let Marked Price be \(x\).
\(\text{SP} = \text{MP} - \text{Discount}\)
\(17820 = x - 10\% \text{ of } x\)
\(17820 = 0.90x\)
\(x = \frac{17820}{0.90} = \text{Rs. } 19800\)
In simple words: When selling price and discount are known, divide SP by the remaining percentage to get the marked price.

Exam Tip: SP = \(90\%\) of MP since there is a 10% discount.

Question 4. A shopkeeper was selling all his items at 20% discount. If Rohit bought a coat which was marked at Rs. 2400, how much did he pay for the coat?
Answer: Marked Price (MP) = Rs. 2400
Discount percentage = \(20\%\)
Discount amount = \(20\% \text{ of } 2400 = \frac{20}{100} \times 2400 = \text{Rs. } 480\)
Amount paid by Rohit (Selling Price) = \(2400 - 480 = \text{Rs. } 1920\)
In simple words: Subtract 20% of the marked price from the marked price to find the final price paid.

Exam Tip: Alternatively, find 80% of the marked price directly: \(0.80 \times 2400 = 1920\).

Question 5. The marked price of a frock is Rs. 440. A discount of 15% is allowed on sales. What is the amount of discount on it and what is its selling price?
Answer: Marked Price (MP) = Rs. 440
Discount percentage = \(15\%\)
Discount amount = \(\frac{15}{100} \times 440 = \frac{6600}{100} = \text{Rs. } 66\)
Selling Price (SP) = \(\text{MP} - \text{Discount} = 440 - 66 = \text{Rs. } 374\)
In simple words: Calculate the 15% discount value first, then subtract it from Rs. 440.

Exam Tip: Double check your subtraction: \(440 - 66 = 374\).

EXERCISE 8.3

Question 1. An article is bought for Rs. 250 and sold for Rs. 375. Find the gain percentage.
Answer: Cost Price (CP) = Rs. 250
Selling Price (SP) = Rs. 375
Gain = \(\text{SP} - \text{CP} = 375 - 250 = \text{Rs. } 125\)
Gain percentage = \(\left(\frac{\text{Gain}}{\text{CP}}\right) \times 100\)
\(= \left(\frac{125}{250}\right) \times 100 = \frac{1}{2} \times 100 = 50\%\)
In simple words: Profit percentage is profit divided by cost price multiplied by 100.

Exam Tip: Both profit and loss are always calculated with respect to the Cost Price.

Question 2. An article is bought for Rs. 400 and sold at a gain of 20%. Find the selling price of the article.
Answer: Cost Price (CP) = Rs. 400
Gain percentage = \(20\%\)
Gain amount = \(20\% \text{ of } 400 = \frac{20}{100} \times 400 = \text{Rs. } 80\)
Selling Price (SP) = \(\text{CP} + \text{Gain} = 400 + 80 = \text{Rs. } 480\)
In simple words: Add the profit amount to the cost price to get the selling price.

Exam Tip: SP = \(\text{CP} \times \left(1 + \frac{\text{Gain}\%}{100}\right)\).

Question 3. An article is sold for Rs. 600 at a loss of 20%. Find the cost price of the article.
Answer: Selling Price (SP) = Rs. 600
Loss percentage = \(20\%\)
Let Cost Price be \(x\).
\(\text{SP} = \text{CP} - \text{Loss}\)
\(600 = x - 20\% \text{ of } x\)
\(600 = 0.80x\)
\(x = \frac{600}{0.80} = \frac{60000}{80} = \text{Rs. } 750\)
In simple words: Since there is a loss, the selling price is 80% of the cost price. Divide SP by 0.80 to find CP.

Exam Tip: Never calculate loss percentage on the selling price.

Question 4. A man sold two pens at Rs. 99 each. He sold one at a loss of 10% and the other at a gain of 10%. Find his loss or gain per cent.
Answer: For Pen 1 (sold at \(10\%\) loss):
\(\text{SP}_1 = \text{Rs. } 99\)
\(\text{Loss}\% = 10\%\)
\(\text{CP}_1 = \frac{\text{SP}}{1 - 0.10} = \frac{99}{0.90} = \text{Rs. } 110\)

For Pen 2 (sold at \(10\%\) gain):
\(\text{SP}_2 = \text{Rs. } 99\)
\(\text{Gain}\% = 10\%\)
\(\text{CP}_2 = \frac{\text{SP}}{1 + 0.10} = \frac{99}{1.10} = \text{Rs. } 90\)

Total Cost Price = \(110 + 90 = \text{Rs. } 200\)
Total Selling Price = \(99 + 99 = \text{Rs. } 198\)
Since Total CP (\(200\)) > Total SP (\(198\)), there is a loss.
Loss = \(200 - 198 = \text{Rs. } 2\)
Loss percentage = \(\left(\frac{2}{200}\right) \times 100 = 1\%\)
In simple words: Find individual cost prices first, sum them up, compare with total selling price, and calculate overall percentage.

Exam Tip: When two identical items are sold at equal profit and loss percentages, there is always a net loss given by \(\frac{x^2}{100}\%\).

Question 5. By selling an article for Rs. 1530, Ram loses 10%. For how much should he sell it to gain 20%?
Answer: First case: SP = Rs. 1530, Loss = \(10\%\)
\(\text{CP} = \frac{\text{SP}}{1 - \text{Loss}\%} = \frac{1530}{0.90} = \text{Rs. } 1700\)
Second case: Desired gain = \(20\%\)
New Selling Price for \(20\%\) gain = \(\text{CP} \times \left(1 + \frac{20}{100}\right)\)
\(= 1700 \times 1.20 = \text{Rs. } 2040\)
In simple words: Find the original cost price first using the first selling price, then apply the new target profit percentage to find the new selling price.

Exam Tip: Always find the Cost Price as a bridge between two selling scenarios.

Question 6. If the cost price of 10 articles is equal to the selling price of 8 articles, find the loss or gain percentage.
Answer: Let the cost price of 1 article be Re. 1.
Cost price of 10 articles = Rs. 10.
Given that selling price of 8 articles = cost price of 10 articles = Rs. 10.
Selling price of 10 articles = \(\frac{10}{8} \times 10 = \text{Rs. } 12.50\)
Since SP (12.50) > CP (10), there is a gain.
Gain = \(12.50 - 10 = \text{Rs. } 2.50\)
Gain percentage = \(\left(\frac{2.50}{10}\right) \times 100 = 25\%\)
In simple words: Assume a convenient value for unit cost, find total CP and SP for the same number of items, and compute profit.

Exam Tip: Shortcut formula: \(\text{Gain}\% = \left(\frac{\text{CP count} - \text{SP count}}{\text{SP count}}\right) \times 100 = \left(\frac{10 - 8}{8}\right) \times 100 = 25\%\).

Question 7. If the cost price of 8 articles is equal to the selling price of 10 articles, find the loss or gain per cent.
Answer: Let the cost price of 1 article be Re. 1.
Cost price of 8 articles = Rs. 8.
Selling price of 10 articles = Rs. 8.
Selling price of 1 article = \(\frac{8}{10} = \text{Rs. } 0.80\)
Cost price of 1 article = Re. 1.
Since CP (1) > SP (0.80), there is a loss.
Loss per article = \(1 - 0.80 = \text{Rs. } 0.20\)
Loss percentage = \(\left(\frac{0.20}{1}\right) \times 100 = 20\%\)
In simple words: When selling more items to recover the cost of fewer items, a loss occurs.

Exam Tip: Use formula \(\text{Loss}\% = \left(\frac{10 - 8}{10}\right) \times 100 = 20\%\).

Question 8. Jaipal purchased articles for Rs. 3200 and sold one-fourth of it at a loss of 10% and one-half of it at a gain of 5%. Find the selling price of the remaining articles to gain 20% on the whole.
Answer: Total Cost Price = Rs. 3200
Target total profit = \(20\%\) on the whole.
Total desired Selling Price = \(3200 \times \left(1 + \frac{20}{100}\right) = 3200 \times 1.20 = \text{Rs. } 3840\)

First part sold: One-fourth of articles = \(\frac{1}{4} \times 3200 = \text{Rs. } 800\) (CP)
Sold at \(10\%\) loss: \(\text{SP}_1 = 800 \times 0.90 = \text{Rs. } 720\)

Second part sold: One-half of articles = \(\frac{1}{2} \times 3200 = \text{Rs. } 1600\) (CP)
Sold at \(5\%\) gain: \(\text{SP}_2 = 1600 \times 1.05 = \text{Rs. } 1680\)

Remaining articles value (CP) = \(3200 - (800 + 1600) = \text{Rs. } 800\)
Let selling price of remaining articles be \(S\).
Total Selling Price = \(\text{SP}_1 + \text{SP}_2 + S\)
\(3840 = 720 + 1680 + S\)
\(3840 = 2400 + S\)
\(S = 3840 - 2400 = \text{Rs. } 1440\)
In simple words: Calculate total target revenue, subtract parts already earned, and the remainder gives the needed selling price for the rest.

Exam Tip: Break down transactions into parts and ensure the sum of individual selling prices equals the total target selling price.

Question 9. A piano is marked for sale at Rs. 13500. The dealer allows 6% discount and then makes a profit of 25% on the cost price. Find the cost price of the piano.
Answer: Marked Price (MP) = Rs. 13500
Discount percentage = \(6\%\)
Discount amount = \(\frac{6}{100} \times 13500 = \text{Rs. } 810\)
Selling Price (SP) = \(\text{MP} - \text{Discount} = 13500 - 810 = \text{Rs. } 12690\)
Let Cost Price be \(CP\).
Given profit = \(25\%\)
\(\text{SP} = \text{CP} \times \left(1 + \frac{25}{100}\right)\)
\(12690 = \text{CP} \times 1.25\)
\(\text{CP} = \frac{12690}{1.25} = \text{Rs. } 10152\)
In simple words: Find the selling price using the marked price and discount, then use that selling price to work backward to the cost price using the profit percentage.

Exam Tip: The Selling Price is the common bridge connecting Marked Price (via Discount) and Cost Price (via Profit/Loss).

Question 10. Vineeta sells sarees. She allows 5% discount on the marked price and still gains 10%. Find the marked price of a saree which costs her Rs. 950.
Answer: Cost Price (CP) = Rs. 950
Gain percentage = \(10\%\)
Selling Price (SP) = \(\text{CP} \times \left(1 + \frac{10}{100}\right) = 950 \times 1.10 = \text{Rs. } 1045\)
Let Marked Price be \(MP\).
Discount = \(5\%\) on \(MP\)
\(\text{SP} = \text{MP} - 5\% \text{ of MP}\)
\(1045 = 0.95 \times \text{MP}\)
\(\text{MP} = \frac{1045}{0.95} = \text{Rs. } 1100\)
In simple words: First calculate the selling price from the cost price and profit, then find the marked price using the discount and selling price.

Exam Tip: Never calculate the discount percentage on the cost price.

EXERCISE 8.4

Question 1. Kiran purchases a sewing machine marked for Rs. 3270. If the rate of VAT is 10%, find the amount which she has to pay for buying the machine.
Answer: Marked Price = Rs. 3270
VAT rate = \(10\%\)
VAT amount = \(10\% \text{ of } 3270 = \frac{10}{100} \times 3270 = \text{Rs. } 327\)
Total amount to be paid = Marked Price + VAT
\(= 3270 + 327 = \text{Rs. } 3597\)
In simple words: VAT is a tax added to the marked price, so calculate the tax and add it to the price.

Exam Tip: Total Bill Amount = Marked Price + VAT amount.

Question 2. The cost of a chair at a shop was Rs. 450. The VAT charged was 5%. Find the bill amount.
Answer: Cost of chair = Rs. 450
VAT rate = \(5\%\)
VAT amount = \(\frac{5}{100} \times 450 = \text{Rs. } 22.50\)
Bill amount = \(450 + 22.50 = \text{Rs. } 472.50\)
In simple words: Add 5% tax to the original cost to get the final bill.

Exam Tip: Always include paise decimals properly for money calculations.

Question 3. Find the buying price of each of the following items when 5% VAT is imposed on the purchase of the following:
(a) A towel of Rs. 110
(b) A pack of three soaps at Rs. 135
(c) 5 kg of sugar at Rs. 25 per kg
Answer:
(a) Cost of towel = Rs. 110
VAT (5%) = \(\frac{5}{100} \times 110 = \text{Rs. } 5.50\)
Buying price = \(110 + 5.50 = \text{Rs. } 115.50\)

(b) Cost of soaps = Rs. 135
VAT (5%) = \(\frac{5}{100} \times 135 = \text{Rs. } 6.75\)
Buying price = \(135 + 6.75 = \text{Rs. } 141.75\)

(c) Cost of 5 kg of sugar at Rs. 25 per kg = \(5 \times 25 = \text{Rs. } 125\)
VAT (5%) = \(\frac{5}{100} \times 125 = \text{Rs. } 6.25\)
Buying price = \(125 + 6.25 = \text{Rs. } 131.25\)
In simple words: Compute total base price first, then add 5% VAT to each item.

Exam Tip: For part (c), calculate total cost for 5 kg before adding tax.

Question 4. Kiran purchased an article for Rs. 5400 including 8% VAT. Find the price of the article excluding VAT.
Answer: Total price including VAT = Rs. 5400
VAT rate = \(8\%\)
Let price excluding VAT be \(x\).
\(x + 8\% \text{ of } x = 5400\)
\(1.08x = 5400\)
\(x = \frac{5400}{1.08} = \text{Rs. } 5000\)
In simple words: Divide the tax-included price by 1.08 to find the original list price.

Exam Tip: Price including VAT = Price excluding VAT \(\times (1 + \frac{\text{VAT}\%}{100})\).

EXERCISE 8.5

Question 1. Find the number of conversion periods and the rate for each conversion period for each of the following:
(a) A sum taken for \(1\frac{1}{2}\) years at \(10\%\) per annum compounded half yearly.
(b) A sum taken for 2 years at \(12\%\) per annum compounded half yearly.
Answer:
(a) Time \(n = 1\frac{1}{2}\) years = \(\frac{3}{2}\) years.
Since interest is compounded half yearly, conversion periods per year = 2.
Number of conversion periods = \(\frac{3}{2} \times 2 = 3\).
Rate per conversion period (half-yearly rate) = \(\frac{10\%}{2} = 5\%\) per half-year.

(b) Time \(n = 2\) years.
Compounded half yearly, conversion periods per year = 2.
Number of conversion periods = \(2 \times 2 = 4\).
Rate per conversion period = \(\frac{12\%}{2} = 6\%\) per half-year.
In simple words: When compounding half-yearly, double the number of years to get periods and halve the annual rate.

Exam Tip: Number of conversion periods = \(\text{Years} \times \text{Compounding frequency per year}\).

Question 2. A sum of Rs. 20000 is borrowed by Meena for 2 years at 8% compounded annually. Find the amount she has to pay at the end of 2 years.
Answer: Principal (\(P\)) = Rs. 20000
Rate (\(R\)) = \(8\%\) per annum compounded annually
Time (\(n\)) = 2 years
Amount (\(A\)) = \(P \left(1 + \frac{R}{100}\right)^n\)
\(A = 20000 \left(1 + \frac{8}{100}\right)^2\)
\(A = 20000 \left(\frac{108}{100}\right)^2\)
\(A = 20000 \times 1.08 \times 1.08\)
\(A = 20000 \times 1.1664 = \text{Rs. } 23328\)
In simple words: Apply the compound amount formula directly with annual compounding.

Exam Tip: Always write the formula \(A = P(1 + \frac{R}{100})^n\) before substituting values.

Question 3. What amount is to be paid on a loan of Rs. 12000 for \(1\frac{1}{2}\) years at 10% per annum compounded half yearly?
Answer: Principal (\(P\)) = Rs. 12000
Time \(n = 1\frac{1}{2}\) years = \(3\) half-years (conversion periods)
Annual rate = \(10\%\), so half-yearly rate \(R = \frac{10}{2} = 5\%\) per half-year
Amount (\(A\)) = \(P \left(1 + \frac{R}{100}\right)^n\)
\(A = 12000 \left(1 + \frac{5}{100}\right)^3\)
\(A = 12000 \left(\frac{105}{100}\right)^3 = 12000 \times (1.05)^3\)
\(A = 12000 \times 1.157625 = \text{Rs. } 13891.50\)
In simple words: Halve the rate and double the number of years for half-yearly compounding.

Exam Tip: Calculate powers carefully: \(1.05^3 = 1.157625\).

Question 4. Find the amount to be paid at the end of 2 years on Rs. 2400 at 5% per annum: (a) Compounded annually (b) Compounded half yearly
Answer:
(a) Compounded annually:
\(P = 2400, R = 5\%, n = 2\)
\(A = 2400 \left(1 + \frac{5}{100}\right)^2 = 2400 \times (1.05)^2\)
\(A = 2400 \times 1.1025 = \text{Rs. } 2646\)

(b) Compounded half yearly:
\(P = 2400, \text{Half-year rate} = \frac{5}{2}\% = 2.5\%, n = 2 \text{ years} = 4 \text{ half-years}\)
\(A = 2400 \left(1 + \frac{2.5}{100}\right)^4 = 2400 \times (1.025)^4\)
\(A = 2400 \times 1.10381289 = \text{Rs. } 2649.15\)
In simple words: Compounding more frequently (half-yearly) yields slightly more interest than annually.

Exam Tip: Notice that half-yearly compound amount is always higher than annual compounding for the same nominal rate and time.

Question 5. Find the difference between the compound interest and the simple interest for 2 years on Rs. 8000 at 6% per annum compounded annually.
Answer: \(P = 8000, R = 6\%, n = 2 \text{ years}\)
Simple Interest (SI) = \(\frac{P \times R \times n}{100} = \frac{8000 \times 6 \times 2}{100} = \text{Rs. } 960\)
Compound Amount (A) = \(8000 \left(1 + \frac{6}{100}\right)^2 = 8000 \times (1.06)^2\)
\(A = 8000 \times 1.1236 = \text{Rs. } 8988.80\)
Compound Interest (CI) = \(A - P = 8988.80 - 8000 = \text{Rs. } 988.80\)
Difference = \(\text{CI} - \text{SI} = 988.80 - 960 = \text{Rs. } 28.80\)
In simple words: Calculate simple interest and compound interest separately, then find their difference.

Exam Tip: For 2 years, the difference between CI and SI can also be checked using the direct formula: \(\text{Difference} = P \left(\frac{R}{100}\right)^2 = 8000 \times \left(\frac{6}{100}\right)^2 = 8000 \times 0.0036 = 28.80\).

SHARPEN YOUR UNDERSTANDING (MCQ)

Question 1. An article marked at Rs. 50 is sold for Rs. 45. What is the rate of discount?
(a) 5%
(b) 10%
(c) 15%
(d) 20%
Answer: (b) 10%
Discount = \(50 - 45 = \text{Rs. } 5\)
Discount percentage = \(\left(\frac{5}{50}\right) \times 100 = 10\%\)
In simple words: Discount is Rs. 5 out of Rs. 50, which is 10%.

Exam Tip: Divide discount by marked price and multiply by 100.

Question 2. Swati paid Rs. 214 for a pair of shoes worth Rs. 200. What is the rate of VAT charged?
(a) 14%
(b) 7%
(c) 28%
(d) 3.5%
Answer: (b) 7%
VAT amount = \(214 - 200 = \text{Rs. } 14\)
VAT rate = \(\left(\frac{14}{200}\right) \times 100 = 7\%\)
In simple words: The extra tax paid is Rs. 14 on a base price of Rs. 200.

Exam Tip: VAT percentage is calculated on the original list price.

Question 3. The compound interest on Rs. 3000 at 5% per annum, compounded annually for 2 years is:
(a) Rs. 10
(b) Rs. 305
(c) Rs. 307.50
(d) Rs. 3307.50
Answer: (c) Rs. 307.50
\(A = 3000 \left(1 + \frac{5}{100}\right)^2 = 3000 \times (1.05)^2 = 3000 \times 1.1025 = \text{Rs. } 3307.50\)
\(\text{CI} = 3307.50 - 3000 = \text{Rs. } 307.50\)
In simple words: Subtract the principal from the total compound amount to get the interest.

Exam Tip: Option (d) is the total amount, while option (c) is the compound interest.

Question 4. If 72% of 250 students in a class are interested in English, how many students are not interested in English?
(a) 28 students
(b) 222 students
(c) 70 students
(d) 180 students
Answer: (a) 28 students
Percentage of students not interested = \(100\% - 72\% = 28\%\)
Number of students not interested = \(28\% \text{ of } 250 = \frac{28}{100} \times 250 = \frac{28 \times 5}{2} = 14 \times 2 = 28\)
In simple words: Find the percentage of students who are not interested and calculate their number.

Exam Tip: You can also find total interested students (\(180\)) and subtract from \(250\) to get \(70\)? Wait, let's recalculate: \(28\% \text{ of } 250 = 0.28 \times 250 = 70\). Let's re-evaluate option (c): 70 students.
Correct option is (c).

Exam Tip: Always double check percentage calculations with decimals.

Question 5. A fan was sold for Rs. 510 after allowing 15% discount. What was its actual selling price? (Marked Price)
(a) Rs. 600
(b) Rs. 525
(c) Rs. 586.50
(d) Rs. 433.50
Answer: (a) Rs. 600
\(\text{SP} = 0.85 \times \text{MP} = 510\)
\(\text{MP} = \frac{510}{0.85} = \text{Rs. } 600\)
In simple words: Divide the discounted price by 0.85 to find the marked price.

Exam Tip: Selling price is less than marked price after discount.

Question 6. A shirt was sold for Rs. 880 including 10% VAT. What was its price before VAT was added?
(a) Rs. 890
(b) Rs. 870
(c) Rs. 800
(d) Rs. 810
Answer: (c) Rs. 800
\(1.10 \times \text{Price} = 880\)
\(\text{Price} = \frac{880}{1.10} = \text{Rs. } 800\)
In simple words: Divide the total bill by 1.10 to remove the 10% VAT.

Exam Tip: Price before VAT is always less than the price including VAT.

Question 7. What is the difference between S.I. and C.I. on a sum of Rs. 1000 at 10% for 2 years?
(a) No difference
(b) Rs. 210
(c) Rs. 200
(d) Rs. 10
Answer: (d) Rs. 10
\(\text{SI} = \frac{1000 \times 10 \times 2}{100} = \text{Rs. } 200\)
\(\text{CI} = 1000 \left(1 + \frac{10}{100}\right)^2 - 1000 = 1000 \times 1.21 - 1000 = \text{Rs. } 210\)
Difference = \(210 - 200 = \text{Rs. } 10\)
In simple words: Compound interest earns interest on interest in the second year, making it higher than simple interest by Rs. 10.

Exam Tip: For 2 years, difference = \(P(\frac{R}{100})^2 = 1000 \times (\frac{10}{100})^2 = 1000 \times 0.01 = 10\).

Question 8. A sum is lent for one year at 16% p.a. If interest is compounded after every three months, how many times will interest be charged in one year?
(a) One time
(b) Two times
(c) Three times
(d) Four times
Answer: (d) Four times
1 year = 12 months.
Compounding interval = 3 months.
Number of times = \(\frac{12}{3} = 4\) times.
In simple words: Since there are four quarters in a year, interest is charged 4 times.

Exam Tip: Quarterly compounding means 4 conversion periods per year.

Question 9. A machine worth Rs. 12000 is depreciated by 10% per year. What will be its value after two years?
(a) Rs. 13200
(b) Rs. 12010
(c) Rs. 10800
(d) Rs. 9720
Answer: (d) Rs. 9720
Depreciated value \(A = 12000 \left(1 - \frac{10}{100}\right)^2 = 12000 \times (0.90)^2\)
\(A = 12000 \times 0.81 = \text{Rs. } 9720\)
In simple words: Apply successive percentage decreases for depreciation using minus instead of plus in the formula.

Exam Tip: Depreciation means reduction in value over time, so use \((1 - \frac{R}{100})\).

Question 10. By what per cent is 2000 less than 2400?
(a) 4%
(b) \(12\frac{1}{2}\%\)
(c) \(16\frac{2}{3}\%\)
(d) 20%
Answer: (c) \(16\frac{2}{3}\%\)
Difference = \(2400 - 2000 = 400\)
Percentage less = \(\left(\frac{400}{2400}\right) \times 100 = \frac{1}{6} \times 100 = 16.67\% = 16\frac{2}{3}\%\)
In simple words: Compare the difference against the reference value 2400.

Exam Tip: Always divide the difference by the base number that follows the word "than".