How to Use This Page
1. Solve each exercise problem on your own first — attempt every step before looking here. 2. Compare your method with the step-by-step solution shown. 3. If your answer differs, rework from the step where you diverged. 4. Pay special attention to the Common Mistakes section — these are the errors that cost marks in exams. 5. Use the Exam Checklist before your test to ensure you have covered every topic.
Key Concepts (Quick Recap)
Percentage
- Percentage = (Part / Whole) × 100
- x% of y = (x / 100) × y
- Percentage increase = (Increase / Original) × 100
- Percentage decrease = (Decrease / Original) × 100
- If A is x% more than B, then A = B × (1 + x/100)
- If A is x% less than B, then A = B × (1 - x/100)
Profit and Loss
- Profit = SP - CP (when SP > CP)
- Loss = CP - SP (when CP > SP)
- Profit % = (Profit / CP) × 100
- Loss % = (Loss / CP) × 100
- SP when profit = CP × (1 + Profit%/100)
- SP when loss = CP × (1 - Loss%/100)
- CP when profit = SP × 100 / (100 + Profit%)
- CP when loss = SP × 100 / (100 - Loss%)
Discount
- Discount = Marked Price (MP) - Selling Price (SP)
- Discount % = (Discount / MP) × 100
- SP = MP × (1 - Discount%/100)
Simple Interest (SI)
- SI = (P × R × T) / 100
- Amount A = P + SI
- P = Principal, R = Rate (% per annum), T = Time (years)
Compound Interest (CI)
- Amount A = P(1 + R/100)^n
- CI = A - P
- Half-yearly: A = P(1 + R/200)^(2n)
- Quarterly: A = P(1 + R/400)^(4n)
- When interest is compounded annually but time is in fractional years: A = P(1 + R/100)^whole × (1 + fractional_part × R/100)
Exercise 4.1 — Percentage
Q1. Find 25% of 360.
25% of 360 = (25/100) × 360
= 0.25 × 360
= 90
Answer: 90
Q2. What percentage is 45 of 180?
Percentage = (45/180) × 100
= 0.25 × 100
= 25%
Answer: 25%
Q3. If 30% of a number is 75, find the number.
Let the number be x.
30% of x = 75
(30/100) × x = 75
x = 75 × 100/30
x = 7500/30
x = 250
Answer: 250
Q4. A student scored 360 marks out of 480. What is the percentage?
Percentage = (360/480) × 100
= (3/4) × 100
= 75%
Answer: 75%
Q5. The price of a commodity increased from Rs 80 to Rs 96. Find the percentage increase.
Increase = 96 - 80 = 16
Percentage increase = (16/80) × 100
= (1/5) × 100
= 20%
Answer: 20%
Q6. The population of a town decreased from 25,000 to 22,500. Find the percentage decrease.
Decrease = 25,000 - 22,500 = 2,500
Percentage decrease = (2,500/25,000) × 100
= 10%
Answer: 10%
Q7. If A is 20% more than B, and B = 250, find A.
A = B × (1 + 20/100)
= 250 × (120/100)
= 250 × 1.2
= 300
Answer: A = 300
Q8. If a number is reduced by 15% to 425, what was the original number?
Let the original number be x.
x × (1 - 15/100) = 425
x × (85/100) = 425
x = 425 × 100/85
x = 500
Answer: 500
Q9. The price of a shirt increased by 10% and then decreased by 10%. Is the final price the same as the original?
Let original price = Rs 100
After 10% increase: 100 × (110/100) = Rs 110
After 10% decrease: 110 × (90/100) = Rs 99
Final price = Rs 99, which is less than Rs 100.
Answer: No. The final price is less than the original. The net change is a 1% decrease.
Q10. In an examination, 92% of students passed and 48 students failed. How many students appeared?
If 92% passed, then 8% failed.
8% of total = 48
(8/100) × total = 48
total = 48 × 100/8
total = 600
Answer: 600 students appeared for the examination.
Q11. A fruit seller had 300 oranges. He sold 40% on the first day and 35% of the remaining on the second day. How many oranges are left?
First day: 40% of 300 = (40/100) × 300 = 120 sold
Remaining after first day = 300 - 120 = 180
Second day: 35% of 180 = (35/100) × 180 = 63 sold
Remaining after second day = 180 - 63 = 117
Answer: 117 oranges are left.
Q12. If the price of petrol is increased by 25%, by what percentage should consumption be reduced so that expenditure remains the same?
Let original price = Rs 100 per litre, consumption = 1 litre.
Original expenditure = 100 × 1 = Rs 100
New price = 125 per litre
For same expenditure: 125 × new consumption = 100
New consumption = 100/125 = 0.8 litres
Reduction in consumption = 1 - 0.8 = 0.2 litres
Percentage reduction = (0.2/1) × 100 = 20%
Answer: 20% reduction in consumption.
Q13. The ratio of boys to girls in a class is 5:3. If there are 40 boys, find the total number of students.
Ratio = 5:3
Boys = 5 parts = 40, so 1 part = 40/5 = 8
Girls = 3 parts = 3 × 8 = 24
Total = 40 + 24 = 64
Answer: 64 students
Q14. A person earns Rs 18,000 per month. He spends 30% on rent, 25% on food, 15% on education, and saves the rest. Find his savings.
Total expenditure = 30% + 25% + 15% = 70%
Savings = 100% - 70% = 30%
Savings = 30% of 18,000 = (30/100) × 18,000 = Rs 5,400
Answer: Rs 5,400
Q15. If the length of a rectangle is increased by 20% and the breadth is decreased by 10%, find the percentage change in area.
Let original length = L, original breadth = B
Original area = L × B
New length = L × (120/100) = 1.2L
New breadth = B × (90/100) = 0.9B
New area = 1.2L × 0.9B = 1.08LB
Percentage change = (1.08LB - LB)/LB × 100 = 0.08 × 100 = 8%
Answer: Area increases by 8%.
Exercise 4.2 — Profit, Loss, and Discount
Q1. A shopkeeper bought an article for Rs 500 and sold it for Rs 600. Find the profit percentage.
CP = Rs 500, SP = Rs 600
Profit = SP - CP = 600 - 500 = Rs 100
Profit % = (Profit/CP) × 100 = (100/500) × 100 = 20%
Answer: 20% profit
Q2. A man bought a watch for Rs 800 and sold it for Rs 680. Find the loss percentage.
CP = Rs 800, SP = Rs 680
Loss = CP - SP = 800 - 680 = Rs 120
Loss % = (Loss/CP) × 100 = (120/800) × 100 = 15%
Answer: 15% loss
Q3. A trader sold an article for Rs 924 at a profit of 10%. Find the cost price.
SP = Rs 924, Profit % = 10%
CP = SP × 100/(100 + Profit%)
CP = 924 × 100/110
CP = 92400/110
CP = Rs 840
Answer: CP = Rs 840
Q4. A dealer sold a machine for Rs 4,140 at a loss of 8%. Find the cost price.
SP = Rs 4,140, Loss % = 8%
CP = SP × 100/(100 - Loss%)
CP = 4,140 × 100/92
CP = 414,000/92
CP = Rs 4,500
Answer: CP = Rs 4,500
Q5. By selling a book for Rs 270, a bookseller gains 20%. For how much should he sell it to gain 35%?
Step 1: Find CP.
SP = Rs 270, Profit % = 20%
CP = 270 × 100/120 = Rs 225
Step 2: Find new SP for 35% profit.
New SP = CP × (100 + 35)/100
= 225 × 135/100
= 225 × 1.35
= Rs 303.75
Answer: Rs 303.75
Q6. The CP of 15 articles is equal to the SP of 12 articles. Find the profit percentage.
Let CP of 1 article = Rs x
CP of 15 articles = Rs 15x
SP of 12 articles = Rs 15x (given)
SP of 1 article = 15x/12 = Rs 5x/4
Profit on 1 article = SP - CP = 5x/4 - x = x/4
Profit % = (Profit/CP) × 100 = (x/4)/x × 100 = 25%
Answer: 25% profit
Q7. A shopkeeper marks an article at Rs 1,200 and sells it at a discount of 15%. Find the selling price and the discount amount.
Marked Price (MP) = Rs 1,200
Discount % = 15%
Discount = (15/100) × 1,200 = Rs 180
SP = MP - Discount = 1,200 - 180 = Rs 1,020
Alternatively: SP = MP × (1 - 15/100) = 1,200 × 0.85 = Rs 1,020
Answer: SP = Rs 1,020, Discount = Rs 180
Q8. A shirt is marked at Rs 850 and sold for Rs 765. Find the discount percentage.
MP = Rs 850, SP = Rs 765
Discount = MP - SP = 850 - 765 = Rs 85
Discount % = (85/850) × 100 = 10%
Answer: 10% discount
Q9. A shopkeeper offers a discount of 20% and still makes a profit of 25%. If the CP is Rs 400, find the marked price.
CP = Rs 400, Profit % = 25%, Discount % = 20%
Step 1: Find SP.
SP = CP × (1 + Profit%/100) = 400 × 125/100 = Rs 500
Step 2: Find MP.
SP = MP × (1 - Discount%/100)
500 = MP × 80/100
MP = 500 × 100/80 = Rs 625
Answer: Marked Price = Rs 625
Q10. After giving a discount of 10% on a marked price, a shopkeeper earns a profit of 20%. If the CP is Rs 450, find the MP.
CP = Rs 450, Profit % = 20%, Discount % = 10%
Step 1: Find SP.
SP = 450 × 120/100 = Rs 540
Step 2: Find MP.
SP = MP × (100 - 10)/100
540 = MP × 90/100
MP = 540 × 100/90 = Rs 600
Answer: MP = Rs 600
Q11. A shopkeeper buys 50 pens for Rs 400 and sells them at Rs 10 each. Find the profit or loss percentage.
Total CP = Rs 400
CP per pen = 400/50 = Rs 8
SP per pen = Rs 10
Profit per pen = 10 - 8 = Rs 2
Total profit = 50 × 2 = Rs 100
Profit % = (100/400) × 100 = 25%
Answer: 25% profit
Q12. A man bought two TVs for Rs 15,000 each. He sold one at a profit of 10% and the other at a loss of 10%. Find his overall profit or loss.
Total CP = 15,000 + 15,000 = Rs 30,000
SP of first TV = 15,000 × 110/100 = Rs 16,500
SP of second TV = 15,000 × 90/100 = Rs 13,500
Total SP = 16,500 + 13,500 = Rs 30,000
Profit = SP - CP = 30,000 - 30,000 = 0
Answer: No profit, no loss. (Break-even)
Q13. By selling an article for Rs 1,500, a trader incurs a loss of 10%. At what price must he sell it to earn a profit of 15%?
Step 1: Find CP.
SP = Rs 1,500, Loss % = 10%
CP = 1,500 × 100/90 = Rs 1,666.67
Step 2: Find new SP for 15% profit.
New SP = CP × 115/100 = 1,666.67 × 1.15 = Rs 1,916.67
Answer: Rs 1,916.67 (approximately Rs 1,917)
Q14. A dealer buys goods for Rs 8,000 and spends Rs 500 on transport and Rs 200 on packing. He sells the goods for Rs 10,500. Find the profit percentage.
Total CP = Purchase price + Transport + Packing
= 8,000 + 500 + 200 = Rs 8,700
Profit = SP - CP = 10,500 - 8,700 = Rs 1,800
Profit % = (1,800/8,700) × 100 = 20.69%
Answer: Approximately 20.69% profit
Q15. The marked price of a refrigerator is Rs 16,000. A dealer offers two discount schemes: (i) 20% discount, (ii) successive discounts of 10% and 12%. Which is better for the buyer?
Scheme (i): 20% single discount
SP = 16,000 × 80/100 = Rs 12,800
Scheme (ii): Successive discounts of 10% and 12%
After 10% discount: 16,000 × 90/100 = Rs 14,400
After 12% discount: 14,400 × 88/100 = Rs 12,672
Rs 12,672 < Rs 12,800
Answer: Scheme (ii) with successive discounts is better for the buyer (lower price of Rs 12,672).
Exercise 4.3 — Simple Interest
Q1. Find the simple interest on Rs 5,000 at 8% per annum for 3 years.
P = Rs 5,000, R = 8%, T = 3 years
SI = (P × R × T)/100
= (5,000 × 8 × 3)/100
= 120,000/100
= Rs 1,200
Amount = P + SI = 5,000 + 1,200 = Rs 6,200
Answer: SI = Rs 1,200, Amount = Rs 6,200
Q2. Find the simple interest on Rs 12,000 at 6.5% per annum for 2 years.
P = Rs 12,000, R = 6.5%, T = 2 years
SI = (12,000 × 6.5 × 2)/100
= (12,000 × 13)/100
= 156,000/100
= Rs 1,560
Answer: SI = Rs 1,560
Q3. At what rate of interest will Rs 4,000 yield Rs 800 as simple interest in 4 years?
P = Rs 4,000, SI = Rs 800, T = 4 years
SI = (P × R × T)/100
800 = (4,000 × R × 4)/100
800 = 160R
R = 800/160 = 5%
Answer: Rate = 5% per annum
Q4. In how many years will Rs 6,000 amount to Rs 7,800 at 5% per annum simple interest?
P = Rs 6,000, Amount = Rs 7,800, R = 5%
SI = Amount - P = 7,800 - 6,000 = Rs 1,800
SI = (P × R × T)/100
1,800 = (6,000 × 5 × T)/100
1,800 = 300T
T = 1,800/300 = 6 years
Answer: 6 years
Q5. What sum will yield Rs 1,440 as simple interest in 3 years at 12% per annum?
SI = Rs 1,440, R = 12%, T = 3 years
SI = (P × R × T)/100
1,440 = (P × 12 × 3)/100
1,440 = 36P/100
P = (1,440 × 100)/36
P = Rs 4,000
Answer: Principal = Rs 4,000
Q6. A sum of money doubles itself in 10 years at simple interest. Find the rate of interest.
If money doubles, SI = P (since Amount = 2P, so SI = 2P - P = P)
SI = P, T = 10 years
P = (P × R × 10)/100
100P = 10PR
100 = 10R
R = 10%
Answer: 10% per annum
Q7. A sum of money triples itself in 15 years at simple interest. Find the rate.
If money triples, SI = 3P - P = 2P
SI = 2P, T = 15 years
2P = (P × R × 15)/100
200P = 15PR
200 = 15R
R = 200/15 = 13.33%
Answer: 13 1/3% per annum (or approximately 13.33%)
Q8. Rs 8,000 is invested at 5% per annum simple interest. What is the amount after 4 years?
P = Rs 8,000, R = 5%, T = 4 years
SI = (8,000 × 5 × 4)/100 = Rs 1,600
Amount = 8,000 + 1,600 = Rs 9,600
Answer: Rs 9,600
Q9. The simple interest on a certain sum at 6% per annum for 5 years is Rs 900. Find the sum.
SI = Rs 900, R = 6%, T = 5 years
900 = (P × 6 × 5)/100
900 = 30P/100
P = (900 × 100)/30 = Rs 3,000
Answer: Rs 3,000
Q10. A man borrows Rs 10,000 at 8% per annum. After 3 years, he pays back Rs 6,000 cash and a TV set for the balance. Find the value of the TV set.
P = Rs 10,000, R = 8%, T = 3 years
SI = (10,000 × 8 × 3)/100 = Rs 2,400
Amount = 10,000 + 2,400 = Rs 12,400
He pays Rs 6,000 cash.
Balance = 12,400 - 6,000 = Rs 6,400
Answer: Value of TV set = Rs 6,400
Exercise 4.4 — Compound Interest
Q1. Find the compound interest on Rs 10,000 at 10% per annum for 2 years.
P = Rs 10,000, R = 10%, n = 2
A = P(1 + R/100)^n
= 10,000 × (1 + 10/100)^2
= 10,000 × (11/10)^2
= 10,000 × 121/100
= Rs 12,100
CI = A - P = 12,100 - 10,000 = Rs 2,100
Answer: CI = Rs 2,100
Q2. Find the compound interest on Rs 8,000 at 5% per annum for 3 years.
P = Rs 8,000, R = 5%, n = 3
A = 8,000 × (1 + 5/100)^3
= 8,000 × (21/20)^3
= 8,000 × 9,261/8,000
= Rs 9,261
CI = 9,261 - 8,000 = Rs 1,261
Answer: CI = Rs 1,261
Q3. Find the difference between compound interest and simple interest on Rs 5,000 at 10% per annum for 2 years.
Simple Interest:
SI = (5,000 × 10 × 2)/100 = Rs 1,000
Compound Interest:
A = 5,000 × (1 + 10/100)^2
= 5,000 × (11/10)^2
= 5,000 × 121/100
= Rs 6,050
CI = 6,050 - 5,000 = Rs 1,050
Difference = CI - SI = 1,050 - 1,000 = Rs 50
Answer: Difference = Rs 50
Q4. A sum of Rs 15,000 is invested at 8% per annum compounded annually. Find the amount after 2 years.
P = Rs 15,000, R = 8%, n = 2
A = 15,000 × (1 + 8/100)^2
= 15,000 × (108/100)^2
= 15,000 × 11,664/10,000
= Rs 17,496
Answer: Rs 17,496
Q5. Find the compound interest on Rs 20,000 at 4% per annum for 2 years compounded half-yearly.
P = Rs 20,000, R = 4% per annum
Half-yearly rate = 4/2 = 2%
Number of half-years = 2 × 2 = 4
A = 20,000 × (1 + 2/100)^4
= 20,000 × (102/100)^4
= 20,000 × (1.02)^4
= 20,000 × 1.08243216
= Rs 21,648.64 (approximately)
CI = 21,648.64 - 20,000 = Rs 1,648.64
Answer: CI = Rs 1,648.64 (approximately)
Q6. The compound interest on Rs 10,000 at 10% per annum for 1 year compounded half-yearly is:
P = Rs 10,000, R = 10% per annum
Half-yearly rate = 10/2 = 5%
Number of half-years = 1 × 2 = 2
A = 10,000 × (1 + 5/100)^2
= 10,000 × (21/20)^2
= 10,000 × 441/400
= Rs 11,025
CI = 11,025 - 10,000 = Rs 1,025
Answer: CI = Rs 1,025
Q7. A sum amounts to Rs 5,670 at 8% per annum compound interest in 2 years. Find the principal.
A = Rs 5,670, R = 8%, n = 2
A = P(1 + R/100)^n
5,670 = P × (1 + 8/100)^2
5,670 = P × (108/100)^2
5,670 = P × (27/25)^2
5,670 = P × 729/625
P = 5,670 × 625/729
P = 35,43,750/729
P = Rs 4,861.11 (approximately)
Answer: Principal = Rs 4,861.11 (approximately)
Q8. The population of a town increases by 5% every year. If the present population is 40,000, what will it be after 2 years?
P = 40,000, R = 5%, n = 2
A = P(1 + R/100)^n
= 40,000 × (1 + 5/100)^2
= 40,000 × (21/20)^2
= 40,000 × 441/400
= 40,000 × 1.1025
= 44,100
Answer: 44,100
Q9. The value of a car depreciates by 15% per year. If its present value is Rs 4,00,000, find its value after 2 years.
For depreciation: A = P(1 - R/100)^n
P = Rs 4,00,000, R = 15%, n = 2
A = 4,00,000 × (1 - 15/100)^2
= 4,00,000 × (85/100)^2
= 4,00,000 × (17/20)^2
= 4,00,000 × 289/400
= 2,89,000
Answer: Rs 2,89,000
Q10. Find the compound interest on Rs 25,000 at 12% per annum for 1.5 years.
P = Rs 25,000, R = 12%, n = 1.5 years
For 1.5 years: A = P(1 + R/100)^1 × (1 + (R × 0.5)/100)
= 25,000 × (112/100) × (106/100)
= 25,000 × 1.12 × 1.06
= 25,000 × 1.1872
= Rs 29,680
CI = 29,680 - 25,000 = Rs 4,680
Answer: CI = Rs 4,680
Q11. Compare the simple interest and compound interest on Rs 10,000 at 5% per annum for 3 years.
Simple Interest:
SI = (10,000 × 5 × 3)/100 = Rs 1,500
Compound Interest:
A = 10,000 × (1 + 5/100)^3
= 10,000 × (21/20)^3
= 10,000 × 9,261/8,000
= Rs 11,576.25
CI = 11,576.25 - 10,000 = Rs 1,576.25
Difference = CI - SI = 1,576.25 - 1,500 = Rs 76.25
Answer: SI = Rs 1,500, CI = Rs 1,576.25, Difference = Rs 76.25
Q12. A bank offers 8% compound interest compounded annually. If Rs 20,000 is deposited, what is the difference between CI for the 2nd year and the 1st year?
CI for 1st year:
A_1 = 20,000 × 108/100 = Rs 21,600
CI_1 = 21,600 - 20,000 = Rs 1,600
CI for 2nd year:
A_2 = 21,600 × 108/100 = Rs 23,328
CI for 2nd year = 23,328 - 21,600 = Rs 1,728
Difference = 1,728 - 1,600 = Rs 128
Answer: Difference = Rs 128
Q13. The compound interest on a sum at 10% per annum for 2 years is Rs 1,260. Find the sum.
CI = Rs 1,260, R = 10%, n = 2
CI = P[(1 + R/100)^n - 1]
1,260 = P[(1 + 10/100)^2 - 1]
1,260 = P[(11/10)^2 - 1]
1,260 = P[121/100 - 1]
1,260 = P[21/100]
P = 1,260 × 100/21
P = 1,26,000/21
P = Rs 6,000
Answer: Principal = Rs 6,000
Q14. Find the amount on Rs 50,000 at 8% per annum compounded annually for 3 years.
P = Rs 50,000, R = 8%, n = 3
A = 50,000 × (1 + 8/100)^3
= 50,000 × (108/100)^3
= 50,000 × (27/25)^3
= 50,000 × 19683/15625
= Rs 62,985.6
Answer: Rs 62,985.60 (approximately)
Q15. A sum of money amounts to Rs 6,655 at 10% per annum compound interest in 3 years. Find the sum.
A = Rs 6,655, R = 10%, n = 3
A = P(1 + R/100)^n
6,655 = P × (11/10)^3
6,655 = P × 1331/1000
P = 6,655 × 1000/1331
P = 6,655,000/1331
P = Rs 5,000
Answer: Principal = Rs 5,000
Exercise 4.5 — Word Problems and Applications
Q1. A shopkeeper gives a discount of 15% on a shirt marked at Rs 800 and still makes a profit of 10%. Find the cost price.
Step 1: Find SP.
SP = MP × (1 - 15/100) = 800 × 85/100 = Rs 680
Step 2: Find CP.
CP = SP × 100/(100 + Profit%)
= 680 × 100/110
= Rs 618.18
Answer: CP = Rs 618.18 (approximately)
Q2. A shopkeeper buys an article for Rs 1,200 and marks it at 40% above cost. He then gives a discount of 25%. Find the profit percentage.
CP = Rs 1,200
MP = 1,200 × (1 + 40/100) = 1,200 × 1.4 = Rs 1,680
SP = 1,680 × (1 - 25/100) = 1,680 × 0.75 = Rs 1,260
Profit = SP - CP = 1,260 - 1,200 = Rs 60
Profit % = (60/1,200) × 100 = 5%
Answer: 5% profit
Q3. A man deposited Rs 20,000 in a bank at 8% per annum compound interest. How much will he receive after 2 years? What is the interest earned?
P = Rs 20,000, R = 8%, n = 2
A = 20,000 × (1 + 8/100)^2
= 20,000 × (108/100)^2
= 20,000 × 11664/10000
= Rs 23,328
CI = 23,328 - 20,000 = Rs 3,328
Answer: Amount = Rs 23,328, Interest = Rs 3,328
Q4. The price of a TV is Rs 25,000. A sales tax of 8% is added. Find the total amount to be paid.
Sales tax = 8% of 25,000 = (8/100) × 25,000 = Rs 2,000
Total amount = 25,000 + 2,000 = Rs 27,000
Answer: Rs 27,000
Q5. A person borrows Rs 15,000 at 6% per annum simple interest and lends it at 8% per annum simple interest. Find his gain after 3 years.
Interest paid (at 6%): SI_out = (15,000 × 6 × 3)/100 = Rs 2,700
Interest earned (at 8%): SI_in = (15,000 × 8 × 3)/100 = Rs 3,600
Gain = 3,600 - 2,700 = Rs 900
Answer: Gain = Rs 900
Q6. In a factory, the production of goods increases by 10% every year. If the present production is 50,000 units, find the production after 3 years.
P = 50,000, R = 10%, n = 3
A = 50,000 × (1 + 10/100)^3
= 50,000 × (11/10)^3
= 50,000 × 1331/1000
= 66,550
Answer: 66,550 units
Q7. The population of a village is 20,000. It increases by 10% in the first year and decreases by 5% in the second year. Find the population after 2 years.
After 1st year: 20,000 × 110/100 = 22,000
After 2nd year: 22,000 × 95/100 = 20,900
Answer: 20,900
Q8. A cloth merchant buys cloth at Rs 50 per metre and sells it at Rs 60 per metre. If he also gives a 5% discount on the selling price, find his profit percentage.
CP = Rs 50 per metre
SP before discount = Rs 60 per metre
Discount = 5% of 60 = Rs 3
Actual SP = 60 - 3 = Rs 57 per metre
Profit = 57 - 50 = Rs 7
Profit % = (7/50) × 100 = 14%
Answer: 14% profit
Q9. A sum amounts to Rs 7,920 at 10% per annum compound interest in 2 years. Find the sum and the compound interest.
A = Rs 7,920, R = 10%, n = 2
A = P(1 + R/100)^n
7,920 = P × (11/10)^2
7,920 = P × 121/100
P = 7,920 × 100/121
P = Rs 6,545.45
CI = 7,920 - 6,545.45 = Rs 1,374.55
Answer: P = Rs 6,545.45, CI = Rs 1,374.55
Q10. A shopkeeper sells a shirt at Rs 720 after giving a discount of 10%. If he makes a profit of 20%, find the marked price and the cost price.
Step 1: Find CP from profit.
SP = Rs 720, Profit % = 20%
CP = 720 × 100/120 = Rs 600
Step 2: Find MP from discount.
SP = MP × (1 - 10/100)
720 = MP × 90/100
MP = 720 × 100/90 = Rs 800
Answer: CP = Rs 600, MP = Rs 800
Common Mistakes
1. Dividing by Selling Price instead of Cost Price for profit/loss percentage — Profit % = (Profit / CP) × 100, NOT (Profit / SP) × 100. Similarly for loss. Always divide by the cost price.
2. Confusing simple and compound interest formulas — SI = PRT/100 uses the original principal every year. CI = A - P where A = P(1 + R/100)^n recalculates on the accumulated amount. They give the same result only for 1 year.
3. Wrong time period in interest calculations — If the rate is per annum but time is in months, convert: T = months/12. For days: T = days/365. For half-yearly compounding: rate = R/2, periods = 2n.
4. Successive discounts are NOT additive — A 10% discount followed by a 20% discount is NOT 30%. The second discount applies to the already-reduced price. Example: Rs 100 with 10% off = Rs 90, then 20% off Rs 90 = Rs 72 (total discount is 28%, not 30%).
5. Percentage increase/decrease reversal error — If a price increases by 20%, the new price is 120% of the original. To return to the original, you need a 16.67% decrease (NOT 20%). The percentage change is always relative to the original base value.
6. Forgetting to add overhead costs to CP — When a problem mentions transport, repair, or other costs, these must be ADDED to the purchase price to get the true cost price. CP = purchase price + all additional costs.
7. Confusing Amount and Interest — SI/PRT formula gives the INTEREST, not the Amount. Amount = Principal + Interest. Similarly, CI formula A = P(1+R/100)^n gives the AMOUNT. CI = A - P.
8. Depreciation uses subtraction — For depreciation (value decreasing), the formula is A = P(1 - R/100)^n, using subtraction, NOT addition. Many students write A = P(1 + R/100)^n for depreciation, which is wrong.
Exam Checklist
- [ ] I can calculate percentage, percentage increase, and percentage decrease
- [ ] I can find the original value when a percentage increase or decrease is given
- [ ] I can calculate profit, loss, profit %, and loss % correctly
- [ ] I can find CP given SP and profit/loss percentage
- [ ] I can calculate discount, discount %, and find MP or SP
- [ ] I can handle successive discounts correctly
- [ ] I can calculate simple interest using SI = PRT/100
- [ ] I can find any one of P, R, T given the other values and SI
- [ ] I can calculate compound interest using A = P(1 + R/100)^n
- [ ] I can find the difference between CI and SI
- [ ] I can handle half-yearly and quarterly compounding (adjust R and n)
- [ ] I can solve word problems involving percentage, profit/loss, and interest
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Frequently Asked Questions
Percentage = (part/whole) times 100. For example, if a student scores 450 out of 500, percentage = (450/500) times 100 = 90%. To find a percentage of a number: x% of y = (x/100) times y.