Comparing Quantities — Questions and Answers
Welcome to HSLC Guru. This page gives complete, step-by-step solutions to Exercise 8.1 to 8.5 of Chapter 8 Comparing Quantities of ASSEB Class 8 General Mathematics, along with every formula on profit and loss, discount, overhead expenses, simple and compound interest, and GST.
Summary
This chapter compares quantities using percentage. Profit and loss always depend on the cost price (CP): if the selling price (SP) is more than CP there is profit, if less there is loss. Profit% $= \frac{SP-CP}{CP}\times 100\%$ and Loss% $= \frac{CP-SP}{CP}\times 100\%$.
Discount is always calculated on the Marked Price (MP): Discount $= MP – SP$. Small extra costs after buying, such as transport, labour and tax, are called overhead expenses, and Actual CP $= $ CP $+$ Overhead Expenses.
In compound interest the interest is added to the principal after each period to form a new principal; the amount for $n$ periods is $A = P\left(1+\frac{r}{100}\right)^{n}$ and CI $= A – P$. For the first year, simple interest and compound interest are equal. Finally the chapter introduces Goods and Services Tax (GST) — an intra-state sale attracts CGST + SGST while an inter-state sale attracts IGST.
Summary: This ASSEB Class 8 General Mathematics Chapter 8 (Comparing Quantities) solution covers profit and loss, discount, overhead expenses, simple and compound interest, and Goods and Services Tax (GST). Every question of Exercise 8.1 to 8.5 is solved step by step with the formulas SP−CP, discount = MP−SP, and A = P(1 + r/100)ⁿ, making it a complete worked guide for Assam Board Class 8 students.
Key Formulae
Profit $= SP – CP$ (when $SP>CP$), Loss $= CP – SP$ (when $CP>SP$).
$$SP = \frac{100+p}{100}\times CP, \qquad CP = \frac{100}{100+p}\times SP$$
$$SP = \frac{100-l}{100}\times CP, \qquad CP = \frac{100}{100-l}\times SP$$
Discount $= MP – SP$, Rate of discount $= \frac{MP-SP}{MP}\times 100\%$.
Simple Interest $SI = \frac{P\times R\times T}{100}$, Amount $A = P\left(1+\frac{r}{100}\right)^{n}$, Compound Interest $CI = A – P$.
For two years $CI_2 – SI_2 = P\left(\dfrac{r}{100}\right)^{2}$.
Textbook Questions and Answers
Exercise 8.1 (Profit and Loss)
1. A watch was bought for 250 and sold for 260. Find the profit and profit per cent.
Answer: Here $CP = 250$, $SP = 260$; since $SP>CP$ there is profit. Profit $= SP – CP = 260 – 250 = 10$. Profit% $= \frac{10}{250}\times 100\% = 4\%$. So profit is 10 and profit per cent is 4%.
2. Buying a pen for 60, at what price should it be sold to get a profit of 15%?
Answer: $CP = 60$, $p = 15$. $SP = \frac{100+p}{100}\times CP = \frac{115}{100}\times 60 = 69$. So the pen should be sold for 69.
3. Ramen sold a mobile for 13,500 at a loss of 20%. Find the CP of the mobile.
Answer: $SP = 13500$, $l = 20$. $CP = \frac{100}{100-l}\times SP = \frac{100}{80}\times 13500 = 16875$. So the cost price of the mobile is 16,875.
4. If the SP of 10 pens is equal to the CP of 8 pens, calculate the loss or gain percentage.
Answer: Let the CP of one pen be $x$. SP of 10 pens $= $ CP of 8 pens $= 8x$; so SP of one pen $= \frac{8x}{10} = \frac{4x}{5} = 0.8x$. Since $0.8x < x$, there is loss. Loss $= x - 0.8x = 0.2x$; Loss% $= \frac{0.2x}{x}\times 100\% = 20\%$.
5. A cycle bought for 5,000 is sold at a profit of 12%. Find the SP of the cycle.
Answer: $CP = 5000$, $p = 12$. $SP = \frac{100+12}{100}\times 5000 = \frac{112}{100}\times 5000 = 5600$. So the selling price of the cycle is 5600.
6. Kamal bought a water filter for 4500 and sold it at 4230. Calculate the loss per cent.
Answer: $CP = 4500$, $SP = 4230$; Loss $= 4500 – 4230 = 270$. Loss% $= \frac{270}{4500}\times 100\% = 6\%$.
7. A shopkeeper sold a watch for 785 at a loss of 5%. What is the cost price of the watch?
Answer: $SP = 785$, $l = 5$. $CP = \frac{100}{100-5}\times 785 = \frac{100}{95}\times 785 = \frac{78500}{95} \approx 826.32$. So the cost price of the watch is about 826.32.
8. If the selling price of 10 items is equal to the cost price of 11 items of the same type, find the profit or loss percentage.
Answer: Let the CP of one item be $x$. SP of 10 items $= $ CP of 11 items $= 11x$; so SP of one item $= \frac{11x}{10} = 1.1x$. Since $1.1x > x$, there is profit. Profit $= 1.1x – x = 0.1x$; Profit% $= \frac{0.1x}{x}\times 100\% = 10\%$.
9. A man bought two cars for 99,000 each. He sold one at a profit of 10% and the other at a loss of 10%. Calculate the profit or loss percentage in the whole transaction.
Answer: First car: $SP = \frac{110}{100}\times 99000 = 108900$. Second car: $SP = \frac{90}{100}\times 99000 = 89100$. Total CP $= 2\times 99000 = 198000$; total SP $= 108900 + 89100 = 198000$. Since total SP $= $ total CP, there is neither profit nor loss (0%).
Exercise 8.2 (Discount and Overhead Expenses)
1. The marked price of a radio is 2055. If it is sold at 3% discount, find the selling price.
Answer: Discount $= 3\%$ of $2055 = \frac{3}{100}\times 2055 = 61.65$. $SP = MP – $ Discount $= 2055 – 61.65 = 1993.35$.
2. Suman bought a Mathematics book for 190 at a discount of 10%. What was the marked price?
Answer: $SP = 190$, discount $= 10\%$; so $SP = MP\times\frac{90}{100}$. $MP = 190\times\frac{100}{90} = \frac{19000}{90} \approx 211.11$. So the marked price is about 211.11.
3. Ramen bought an article for 630 whose marked price was 700. What is the percentage of discount he got?
Answer: Discount $= MP – SP = 700 – 630 = 70$. Discount% $= \frac{70}{700}\times 100\% = 10\%$.
4. The marked price of a sofa set is 30,000. On the occasion of new year the shopkeeper sold it at 25,000. What is the percentage of discount?
Answer: Discount $= 30000 – 25000 = 5000$. Discount% $= \frac{5000}{30000}\times 100\% \approx 16.67\%$, i.e. $16\tfrac{2}{3}\%$.
5. After a discount of 10%, a fan is sold at 1260. What was the marked price of the fan?
Answer: $SP = 1260$, discount $= 10\%$; $MP = 1260\times\frac{100}{90} = 1400$. So the marked price of the fan is 1400.
6. The marked price of a watch is 1150. On account of Puja, what rate of discount must be given so that it is sold at 1000?
Answer: Discount $= 1150 – 1000 = 150$. Discount% $= \frac{150}{1150}\times 100\% \approx 13.04\%$.
7. A cloth seller advertises a 10% discount and sells accordingly. A customer buys a pair of suit for 6050, a shirt for 575 and a saree for 875. Calculate the total discount she got.
Answer: Total marked price $= 6050 + 575 + 875 = 7500$. Total discount $= 10\%$ of $7500 = \frac{10}{100}\times 7500 = 750$. So the customer got a total discount of 750.
8. What is the rate of retail discount on a book priced at 200 and sold at 175?
Answer: Discount $= 200 – 175 = 25$. Rate of retail discount $= \frac{25}{200}\times 100\% = 12.5\%$.
9. Buying a table for 2750 from a furniture shop, a customer gets a discount of $8\tfrac{1}{3}\%$. What was the price of the table fixed by the shopkeeper?
Answer: Rate of discount $= 8\tfrac{1}{3}\% = \frac{25}{3}\%$. $SP = MP\left(1-\frac{25/3}{100}\right) = MP\times\frac{275}{300} = MP\times\frac{11}{12}$. So $MP = 2750\times\frac{12}{11} = 3000$. The shopkeeper had fixed the marked price at 3000.
10. A shopkeeper gave a discount of 30% on a shirt with marked price 600. He further gave a discount of 20% on that shirt. At what price did he sell the shirt and what is the total percentage of discount?
Answer: After 30% discount, $SP_1 = 600\times\frac{70}{100} = 420$. On this 420, a further 20% discount gives $SP_2 = 420\times\frac{80}{100} = 336$. Total discount $= 600 – 336 = 264$; total discount% $= \frac{264}{600}\times 100\% = 44\%$. So the shirt was sold for 336 and the total discount is 44%.
11. Kamal bought a car for 4,00,000 and spent 10,000 on repairing. He sold it to Suresh at 10% profit, and Suresh sold it to Deepak at 5% profit. At what price did Deepak buy the car?
Answer: Kamal’s actual CP $= 400000 + 10000 = 410000$. Kamal $\to$ Suresh: $SP = 410000\times\frac{110}{100} = 451000$. Suresh $\to$ Deepak: $SP = 451000\times\frac{105}{100} = 473550$. So Deepak bought the car for 4,73,550.
12. A shopkeeper bought a radio from a man for 800. He spent 200 on repairing and sold it to another person for 1300. What was his percentage of profit?
Answer: Actual CP $= 800 + 200 = 1000$. $SP = 1300$; Profit $= 1300 – 1000 = 300$. Profit% $= \frac{300}{1000}\times 100\% = 30\%$.
13. Migam bought an iron for 1200. He spent 40 on transportation. At what price should he sell it to get a profit of 25%?
Answer: Actual CP $= 1200 + 40 = 1240$. $SP = \frac{100+25}{100}\times 1240 = \frac{125}{100}\times 1240 = 1550$. So he should sell the iron for 1550.
Exercise 8.3 (Compound Interest — Amount and CI)
Find the amount and the compound interest of the following (Q1 to Q6), using $A = P\left(1+\frac{r}{100}\right)^{n}$ and $CI = A – P$.
1. 300 for 2 years at 3% per annum.
Answer: $A = 300\left(1+\frac{3}{100}\right)^{2} = 300\times(1.03)^2 = 300\times 1.0609 = 318.27$. $CI = 318.27 – 300 = 18.27$.
2. 4,000 for 3 years at 2% per annum.
Answer: $A = 4000\times(1.02)^3 = 4000\times 1.061208 = 4244.83$. $CI = 4244.83 – 4000 = 244.83$.
3. 10,000 for 2 years at 4% per annum.
Answer: $A = 10000\times(1.04)^2 = 10000\times 1.0816 = 10816$. $CI = 10816 – 10000 = 816$.
4. 7,000 for 3 years at 3% per annum.
Answer: $A = 7000\times(1.03)^3 = 7000\times 1.092727 = 7649.09$. $CI = 7649.09 – 7000 = 649.09$.
5. 1,500 for 2 years at 10% per annum.
Answer: $A = 1500\times(1.10)^2 = 1500\times 1.21 = 1815$. $CI = 1815 – 1500 = 315$.
6. 900 for 3 years at 5% per annum.
Answer: $A = 900\times(1.05)^3 = 900\times 1.157625 = 1041.86$. $CI = 1041.86 – 900 = 141.86$.
7. Find the compound interest on 2000 for $1\tfrac{1}{2}$ years at 4% per annum compounded half-yearly.
Answer: Half-yearly rate $= \frac{4}{2} = 2\%$, number of half-years in $1\tfrac{1}{2}$ years $n = 3$. $A = 2000\times(1.02)^3 = 2000\times 1.061208 = 2122.42$. $CI = 2122.42 – 2000 = 122.42$.
Exercise 8.4 (Compound Interest using the formula)
Find the amount and the compound interest of the following (Q1 to Q6) using the formula.
1. 300 for 2 years at 3% per annum.
Answer: $A = 300\left(1+\frac{3}{100}\right)^{2} = 300\times 1.0609 = 318.27$; $CI = 318.27 – 300 = 18.27$.
2. 4,000 for 3 years at 2% per annum.
Answer: $A = 4000\left(1+\frac{2}{100}\right)^{3} = 4000\times 1.061208 = 4244.83$; $CI = 244.83$.
3. 10,000 for 2 years at 4% per annum.
Answer: $A = 10000\left(1+\frac{4}{100}\right)^{2} = 10000\times 1.0816 = 10816$; $CI = 816$.
4. 7,000 for 3 years at 3% per annum.
Answer: $A = 7000\left(1+\frac{3}{100}\right)^{3} = 7000\times 1.092727 = 7649.09$; $CI = 649.09$.
5. 1,500 for 2 years at 10% per annum.
Answer: $A = 1500\left(1+\frac{10}{100}\right)^{2} = 1500\times 1.21 = 1815$; $CI = 315$.
6. 900 for 3 years at 5% per annum.
Answer: $A = 900\left(1+\frac{5}{100}\right)^{3} = 900\times 1.157625 = 1041.86$; $CI = 141.86$.
7. Find the compound interest on 1000 for 9 months at 4% per annum compounded quarterly.
Answer: Quarterly rate $= \frac{4}{4} = 1\%$, number of quarters in 9 months $n = 3$. $A = 1000\times(1.01)^3 = 1000\times 1.030301 = 1030.30$; $CI = 1030.30 – 1000 = 30.30$.
8. Find the compound interest on 2000 for $1\tfrac{1}{2}$ years at 4% per annum compounded half-yearly.
Answer: Half-yearly rate $= 2\%$, $n = 3$. $A = 2000\times(1.02)^3 = 2122.42$; $CI = 2122.42 – 2000 = 122.42$.
9. What principal will amount to 4500 in 2 years at 4% per annum compounded annually?
Answer: $A = P\left(1+\frac{4}{100}\right)^{2}$ ⇒ $4500 = P\times(1.04)^2 = P\times 1.0816$. $P = \frac{4500}{1.0816} \approx 4160.50$. So the principal is about 4160.50.
10. At what rate of compound interest will 576 amount to 625 in 2 years?
Answer: $\left(1+\frac{r}{100}\right)^{2} = \frac{625}{576} = \left(\frac{25}{24}\right)^{2}$. Taking the square root of both sides, $1+\frac{r}{100} = \frac{25}{24}$ ⇒ $\frac{r}{100} = \frac{25}{24}-1 = \frac{1}{24}$ ⇒ $r = \frac{100}{24} = \frac{25}{6} = 4\tfrac{1}{6}\% \approx 4.17\%$.
11. At what rate of compound interest will 64 amount to 125 in 3 years?
Answer: $\left(1+\frac{r}{100}\right)^{3} = \frac{125}{64} = \left(\frac{5}{4}\right)^{3}$. Taking the cube root of both sides, $1+\frac{r}{100} = \frac{5}{4}$ ⇒ $\frac{r}{100} = \frac{1}{4}$ ⇒ $r = 25\%$.
12. Find the difference of compound interest and simple interest on a sum of 500 at the rate of 10% per annum for 2 years.
Answer: $SI = \frac{500\times 10\times 2}{100} = 100$. For CI: $A = 500\times(1.10)^2 = 605$ ⇒ $CI = 605 – 500 = 105$. Difference $= CI – SI = 105 – 100 = 5$. (By formula $CI_2 – SI_2 = P\left(\frac{r}{100}\right)^2 = 500\times\left(\frac{10}{100}\right)^2 = 5$.)
13. Find the sum on which the difference of compound interest and simple interest will be 1 after 2 years at the rate of 4% per annum.
Answer: $CI_2 – SI_2 = P\left(\frac{r}{100}\right)^{2}$ ⇒ $1 = P\times\left(\frac{4}{100}\right)^{2} = P\times\frac{16}{10000}$. $P = \frac{10000}{16} = 625$. So the sum is 625.
Exercise 8.5 (Goods and Services Tax — GST)
Remember — a sale within the state (intra-state) attracts CGST + SGST, while a sale outside the state (inter-state) attracts only IGST. Tax is always calculated on the selling price (the price after discount).
1. The transaction between a company in Delhi and one in Jaipur is as follows: MRP 60,000, Discount 20%, GST 28%. Find the amount of discount, SP, CGST, SGST, IGST and the bill amount.
Answer: Delhi to Jaipur is an inter-state sale, so only IGST applies. Discount $= 20\%$ of $60000 = 12000$; SP $= 60000 – 12000 = 48000$. $CGST = 0$, $SGST = 0$; $IGST = 28\%$ of $48000 = \frac{28}{100}\times 48000 = 13440$. Bill amount $= 48000 + 13440 = 61440$.
2. The transaction between a distribution centre in Guwahati and one in Dhubri is as follows: MRP 90,000, Discount 30%, SGST 9%, CGST 9%. Find SP, SGST, CGST, IGST and the bill amount.
Answer: Both Guwahati and Dhubri are within Assam, so this is an intra-state sale and CGST + SGST apply. Discount $= 30\%$ of $90000 = 27000$; SP $= 90000 – 27000 = 63000$. $CGST = 9\%$ of $63000 = 5670$; $SGST = 9\%$ of $63000 = 5670$; $IGST = 0$. Bill amount $= 63000 + 5670 + 5670 = 74340$.
3. Fill in the blanks of the following bill and find the bill amount.
Answer: For each row, Total $= $ Number $\times$ MRP; Amount of discount $= $ Discount% of Total; SP $= $ Total $-$ Discount; $CGST = SGST = 2.5\%$ of SP. The completed table is—
| Item | Number | MRP | Total | Discount | Amount of discount | SP | CGST 2.5% | SGST 2.5% |
|---|---|---|---|---|---|---|---|---|
| A | 12 | 50 | 600 | 10% | 60 | 540 | 13.50 | 13.50 |
| B | 30 | 60 | 1800 | 15% | 270 | 1530 | 38.25 | 38.25 |
| C | 10 | 35 | 350 | 12% | 42 | 308 | 7.70 | 7.70 |
| D | 6 | 15 | 90 | 10% | 9 | 81 | 2.03 | 2.03 |
Bill amount of each item $= $ SP $+ CGST + SGST$: A $= 540+13.50+13.50 = 567$; B $= 1530+38.25+38.25 = 1606.50$; C $= 308+7.70+7.70 = 323.40$; D $= 81+2.03+2.03 = 85.06$. Total bill amount $= 567 + 1606.50 + 323.40 + 85.06 = 2581.96$.
Additional Questions and Answers
Multiple Choice Questions (MCQ)
1. If $CP = 200$ and $SP = 250$, the profit per cent is: (a) 10% (b) 20% (c) 25% (d) 50%
Answer: (c) 25% — Profit $= 50$, Profit% $= \frac{50}{200}\times 100\% = 25\%$.
2. Discount is always calculated on the: (a) Selling Price (b) Marked Price (c) Cost Price (d) Profit
Answer: (b) Marked Price.
3. The amount for $n$ years under compound interest is: (a) $P+\frac{PRT}{100}$ (b) $P\left(1+\frac{r}{100}\right)^{n}$ (c) $\frac{PRT}{100}$ (d) $P\left(1-\frac{r}{100}\right)^{n}$
Answer: (b) $P\left(1+\frac{r}{100}\right)^{n}$.
4. For the first year, the relation between simple and compound interest is: (a) $SI>CI$ (b) $SI
Answer: (c) $SI = CI$ (the principal is the same in the first year).
5. Which taxes apply to an intra-state sale? (a) only IGST (b) CGST + SGST (c) only CGST (d) none
Answer: (b) CGST + SGST.
6. If $MP = 500$ and $SP = 400$, the rate of discount is: (a) 5% (b) 10% (c) 20% (d) 25%
Answer: (c) 20% — Discount $= 100$, rate $= \frac{100}{500}\times 100\% = 20\%$.
7. The correct formula for simple interest is: (a) $\frac{P\times R\times T}{100}$ (b) $\frac{100}{PRT}$ (c) $P\times R\times T$ (d) $\frac{PR}{100T}$
Answer: (a) $\frac{P\times R\times T}{100}$.
8. Two articles of the same cost price are sold, one at 10% profit and one at 10% loss. The overall result is: (a) 1% profit (b) 1% loss (c) no profit no loss (d) 10% loss
Answer: (c) no profit no loss (total SP $= $ total CP).
9. When interest is compounded half-yearly, the number of periods $n$ in $1\tfrac{1}{2}$ years is: (a) 1 (b) 2 (c) 3 (d) 6
Answer: (c) 3.
10. GST is a type of: (a) Direct tax (b) Indirect tax (c) Income tax (d) Property tax
Answer: (b) Indirect tax.
Fill in the Blanks
1. Discount $= $ ________ $- SP$.
Answer: Marked Price ($MP$).
2. Actual CP $= $ CP $+$ ________.
Answer: Overhead expenses.
3. Selling an article at a price lower than its actual price is called ________.
Answer: Discount.
4. Under compound interest, the amount for $n$ years is $A = $ ________.
Answer: $P\left(1+\frac{r}{100}\right)^{n}$.
5. A sale within the state is called a ________ sale.
Answer: intra-state.
True or False
1. Discount is calculated on the selling price.
Answer: False — discount is calculated on the marked price.
2. For the first year, simple interest and compound interest are equal.
Answer: True.
3. IGST is charged on an inter-state sale.
Answer: True.
4. Profit or loss always depends on the selling price.
Answer: False — profit or loss always depends on the cost price.
5. Successive discounts are always calculated on the selling price after each discount.
Answer: True.
Short Answer Questions
1. Write the formulae for profit per cent and loss per cent.
Answer: Profit% $= \frac{SP-CP}{CP}\times 100\%$ (when $SP>CP$) and Loss% $= \frac{CP-SP}{CP}\times 100\%$ (when $CP>SP$).
2. What is the main difference between simple interest and compound interest?
Answer: In simple interest the interest is calculated each year only on the original principal, which stays the same. In compound interest, the interest of each period is added to the principal to form a new principal, and the next interest is calculated on this increased principal. Hence, from the second year onwards, compound interest is greater than simple interest.
3. What is GST? Write its types.
Answer: GST (Goods and Services Tax) is an indirect tax that has combined many indirect taxes such as sales tax, service tax and value added tax. It has two main types — intra-state (CGST + SGST/UTGST) and inter-state (IGST).
4. What do you mean by overhead expenses? Give an example.
Answer: The sum of the small extra expenses incurred after buying an article, such as transport, labour, repairing and tax, is called overhead expenses. Actual CP $= $ CP $+$ Overhead Expenses.
Key Terms
| Term | Meaning |
|---|---|
| Profit | The gain when selling price is greater than cost price |
| Loss | The deficit when cost price is greater than selling price |
| Cost Price (CP) | The price at which an article is bought |
| Selling Price (SP) | The price at which an article is sold |
| Discount | A reduction given on the marked price |
| Marked Price (MP) | The price printed or marked on an article |
| Overhead Expenses | Extra costs incurred after buying an article |
| Simple Interest | Interest calculated only on the principal |
| Compound Interest | Interest calculated on principal together with accumulated interest |
| Principal | The money borrowed or deposited |
| Amount | The sum of the principal and the interest |
| GST | Goods and Services Tax, an indirect tax |