Mathematics for SGT and TET Paper 1A (Classes III-VIII) · Chapter 5

Profit and Loss, Discount, GST, Simple and Compound Interest

What to remember

Profit and loss percentages are always on the cost price (CP), discount is always on the marked price (MP), and GST is charged on the price before tax.
Simple interest grows by the same amount every year (SI = P × R × T / 100); compound interest is interest on interest, A = P(1 + R/100)ⁿ, and for the first year the two are equal.
For two years, CI − SI = P(R/100)²; for half-yearly compounding halve the rate and double the number of periods.

Profit and loss

Cost price (CP) is what the shopkeeper pays. Selling price (SP) is what the customer pays.

Profit = SP − CP (when SP is more than CP). Loss = CP − SP (when CP is more than SP).
Profit % = (Profit / CP) × 100. Loss % = (Loss / CP) × 100.
SP = CP × (100 + p)/100 for a profit of p%. SP = CP × (100 − l)/100 for a loss of l%.
CP = SP × 100/(100 + p) for a profit of p%, and CP = SP × 100/(100 − l) for a loss of l%.
Overhead and transport costs are added to the CP.

Worked examples

CP ₹400, SP ₹460: profit ₹60, profit % = 60/400 × 100 = 15%.
CP ₹500, SP ₹450: loss ₹50, loss % = 10%.
CP ₹250 sold at 20% profit: SP = 250 × 1.2 = ₹300.
SP ₹720 with a 20% profit: CP = 720/1.2 = ₹600.
SP ₹540 with a 10% loss: CP = 540/0.9 = ₹600.
A trader buys 20 notebooks for ₹400 and sells each at ₹25: total SP = ₹500, profit = ₹100, profit % = 25%.

Two useful results

If two articles are sold at the same price, one at a gain of x% and the other at a loss of x%, the overall result is always a loss of x²/100 per cent. For x = 20: loss of 4%.
A profit of 25% on cost equals a profit of 20% on the selling price (25 on 125 = 20%).

Discount

Marked price (MP) or list price is the price printed on the article.
Discount = MP − SP. Discount % = (Discount / MP) × 100.
SP = MP × (100 − d)/100. An MP of ₹800 with a 15% discount gives ₹680.
If SP after a 10% discount is ₹990, then MP = 990/0.9 = ₹1100.
Marking above cost: if goods are marked 25% above cost and a 20% discount is given, SP = 1.25 × 0.8 = 1 times CP, so there is no profit and no loss.

Successive discounts: the single equal discount for discounts a% and b% is a + b − ab/100. For 20% and 10%: 20 + 10 − 2 = 28%. For 25% and 20%: 25 + 20 − 5 = 40%. Two discounts of 10% are not 20%; they are 19%.

GST (Goods and Services Tax)

GST is an indirect tax on the supply of goods and services, charged at each stage on the value added, and it replaced many earlier central and state taxes. The GST Council decides the rates; the actual rate for an item should be checked against the latest official rate schedule, and textbook problems usually give the rate.

GST amount = Price before tax × GST rate / 100. Bill amount = Price + GST.
Intra-state supply (within one state): the tax is split into CGST (Central GST) and SGST (State GST), each half of the total rate.
Inter-state supply (between states or into the country): IGST (Integrated GST) is charged, and it equals CGST + SGST.
Working backwards: if the bill including 12% GST is ₹1120, the price before tax is 1120/1.12 = ₹1000.

Worked example: an item costs ₹2000 and GST is 18%. GST = ₹360. Bill = ₹2360. Within a state, CGST = ₹180 and SGST = ₹180.

Simple interest (SI)

Interest is the money paid for the use of a sum of money. The sum lent is the principal (P), the interest rate per year is R%, and the time is T years.

SI = P × R × T / 100.
Amount (A) = P + SI = P(1 + RT/100).
P = SI × 100/(R × T); R = SI × 100/(P × T); T = SI × 100/(P × R).
Time must be in years: 8 months = 8/12 = 2/3 year; 6 months = 1/2 year.
Interest is the same in every year, so the amounts form an arithmetic progression.

Worked examples

P = ₹5000, R = 8%, T = 3: SI = 5000 × 8 × 3/100 = ₹1200. Amount = ₹6200.
P = ₹6000, R = 5%, T = 2: SI = ₹600; amount = ₹6600.
₹2000 earns ₹300 in 3 years, so R = 300 × 100/(2000 × 3) = 5%.
SI ₹360 on ₹1500 at 8%: T = 360 × 100/(1500 × 8) = 3 years.
A sum doubles in 10 years at SI: the interest equals the principal, so R = 100/10 = 10%.
₹1500 at 6% for 8 months: 1500 × 6 × (2/3)/100 = ₹60.

Compound interest (CI)

In compound interest, the interest of each period is added to the principal, and the next interest is earned on the larger sum.

A = P(1 + R/100)ⁿ, where n is the number of years. CI = A − P.
Half-yearly compounding: rate = R/2, number of periods = 2n. Quarterly: rate = R/4, periods = 4n.
Different rates in successive years: A = P(1 + R₁/100)(1 + R₂/100).
For the first year, CI = SI.
For two years: CI − SI = P(R/100)².

Worked examples

₹10000 at 10% for 2 years: A = 10000 × 1.1 × 1.1 = ₹12100. CI = ₹2100. SI = ₹2000. Difference = ₹100 = 10000 × (0.1)².
₹8000 at 10% a year, compounded half-yearly for 1 year: rate 5% for 2 periods, A = 8000 × 1.05 × 1.05 = ₹8820. CI = ₹820.
₹4000 at 5% for 2 years: A = 4000 × 1.1025 = ₹4410.
A sum becomes 1.21 times in 2 years at CI: (1 + r)² = 1.21, so r = 10%.
Difference between CI and SI on ₹7500 at 6% for 2 years: 7500 × 36/10000 = ₹27.

Classroom angle: use a table to show year-by-year growth. In SI the "interest" column stays the same; in CI it grows. Children can see why CI is larger after the first year.

Comparison table

Feature	Simple interest	Compound interest
Interest earned on	Original principal only	Principal plus earlier interest
Yearly interest	Constant	Increases every year
Formula	P R T / 100	P(1 + R/100)ⁿ − P
Growth pattern	Arithmetic (equal steps)	Geometric (equal ratio)
After 1 year	Same as CI	Same as SI

Table of bases

Item	Percentage is on
Profit or loss	Cost price
Discount	Marked price
GST	Price before tax
Interest	Principal

Exam traps

Profit % is on CP, not on SP; discount % is on MP, not on SP.
Two successive discounts of 10% are 19%, not 20%.
Selling two articles at the same price with equal gain and loss percentages is always a loss.
In SI, time must be in years: 8 months is 2/3 of a year, not 8.
In half-yearly compounding, the rate is halved and the periods are doubled; do not halve only one of them.
CI and SI are equal for one year, not for two.
GST is added to the price; it is not a part of the marked price unless stated as inclusive.
Inter-state supply carries IGST; intra-state supply carries CGST plus SGST.

One-liners

Profit % = Profit/CP × 100.
SP = CP × (100 + p)/100.
Discount is on MP.
Single discount for 20% and 10% is 28%.
Marking 25% up and giving 20% off means no profit and no loss.
Equal-percentage gain and loss at the same SP gives a loss of x²/100 %.
SI = PRT/100.
A = P(1 + R/100)ⁿ.
Two-year CI − SI = P(R/100)².
Half-yearly means half the rate and double the periods.
CGST + SGST = IGST.
₹10000 at 10% for two years amounts to ₹12100 under CI.

Practice questions

An article bought for ₹400 is sold for ₹460. What is the profit percentage?
1. 15%
2. 12%
3. 13.04%
4. 10%
Answer
A. 15%
Profit = 60; 60/400 × 100 = 15%.
An article bought for ₹500 is sold for ₹450. What is the loss percentage?
1. 5%
2. 10%
3. 12%
4. 11.11%
Answer
B. 10%
Loss = 50; 50/500 × 100 = 10%.
A trader buys an article for ₹250 and sells it at a 20% profit. What is the selling price?
1. ₹320
2. ₹310
3. ₹300
4. ₹270
Answer
C. ₹300
SP = 250 × 120/100 = ₹300.
The marked price of a shirt is ₹800 and a 15% discount is given. What is the selling price?
1. ₹700
2. ₹660
3. ₹720
4. ₹680
Answer
D. ₹680
Discount = 120, so SP = 800 − 120 = ₹680.
An article is sold for ₹720 at a 20% profit. What is its cost price?
1. ₹600
2. ₹640
3. ₹650
4. ₹576
Answer
A. ₹600
CP = 720 × 100/120 = ₹600.
An article is sold for ₹540 at a 10% loss. What is its cost price?
1. ₹594
2. ₹600
3. ₹560
4. ₹486
Answer
B. ₹600
CP = 540 × 100/90 = ₹600.
What single discount is equal to two successive discounts of 20% and 10%?
1. 30%
2. 26%
3. 28%
4. 25%
Answer
C. 28%
20 + 10 − (20 × 10)/100 = 28%.
A shopkeeper marks goods 25% above cost price and allows a 20% discount. What is the result?
1. 20% profit
2. 5% profit
3. 5% loss
4. No profit and no loss
Answer
D. No profit and no loss
SP = 1.25 × 0.80 = 1.00 times CP.
An article sold for ₹1120 gives a profit of 12%. What was the cost price?
1. ₹980
2. ₹1008
3. ₹1000
4. ₹1050
Answer
C. ₹1000
CP = 1120/1.12 = ₹1000.
Two articles are each sold for ₹1200, one at a 20% gain and the other at a 20% loss. What is the overall result?
1. A gain of 4%
2. A loss of 2%
3. No gain, no loss
4. A loss of 4%
Answer
D. A loss of 4%
Equal percentage gain and loss at the same SP always gives a loss of x²/100 = 400/100 = 4%.
The profit or loss percentage is calculated on which quantity?
1. Marked price
2. Cost price
3. Selling price
4. Discount
Answer
B. Cost price
Profit % and loss % are always taken on the cost price unless stated otherwise.
The discount percentage is calculated on which quantity?
1. Marked price
2. Cost price
3. Selling price
4. Tax amount
Answer
A. Marked price
Discount = MP − SP, and its percentage is taken on the marked price.
What is the full form of GST?
1. Goods and Services Tax
2. General Sales Tax
3. Goods and Sales Turnover
4. Government Service Tariff
Answer
A. Goods and Services Tax
GST stands for Goods and Services Tax.
Which component of GST is charged on the supply of goods between two states?
1. VAT
2. SGST
3. IGST
4. CGST
Answer
C. IGST
Inter-state supply carries Integrated GST (IGST).
On a supply within a single state, GST is divided into
1. IGST and CGST
2. CGST and SGST
3. SGST and IGST
4. CGST only
Answer
B. CGST and SGST
Intra-state supply carries equal CGST and SGST.
An item priced at ₹2000 carries 18% GST. What is the total bill?
1. ₹2400
2. ₹2320
3. ₹2180
4. ₹2360
Answer
D. ₹2360
GST = 360, so the bill is 2000 + 360 = ₹2360.
An item priced at ₹2000 carries 18% GST on a sale within a state. What is the CGST amount?
1. ₹180
2. ₹320
3. ₹360
4. ₹90
Answer
A. ₹180
Total GST = 360, split equally: CGST = SGST = ₹180.
A bill including 12% GST comes to ₹1120. What was the price before tax?
1. ₹985
2. ₹1008
3. ₹1012
4. ₹1000
Answer
D. ₹1000
Price = 1120/1.12 = ₹1000.
What is the simple interest on ₹5000 at 8% per annum for 3 years?
1. ₹1500
2. ₹1600
3. ₹1200
4. ₹1000
Answer
C. ₹1200
SI = 5000 × 8 × 3/100 = ₹1200.
What is the amount when ₹6000 is lent at 5% simple interest for 2 years?
1. ₹7000
2. ₹6600
3. ₹6615
4. ₹6300
Answer
B. ₹6600
SI = 600, so A = 6000 + 600 = ₹6600.
A sum of ₹2000 earns simple interest of ₹300 in 3 years. What is the rate per annum?
1. 5%
2. 4%
3. 7.5%
4. 6%
Answer
A. 5%
R = 300 × 100/(2000 × 3) = 5%.
In how many years does ₹1500 earn simple interest of ₹360 at 8% per annum?
1. 4 years
2. 3 years
3. 2 years
4. 5 years
Answer
B. 3 years
T = 360 × 100/(1500 × 8) = 3.
A sum doubles itself in 10 years at simple interest. What is the rate per annum?
1. 20%
2. 5%
3. 10%
4. 12.5%
Answer
C. 10%
Interest = principal in 10 years, so R = 100/10 = 10%.
What is the amount on ₹10000 at 10% per annum compound interest for 2 years?
1. ₹12200
2. ₹12000
3. ₹11000
4. ₹12100
Answer
D. ₹12100
A = 10000 × 1.1 × 1.1 = ₹12100.
What is the compound interest on ₹10000 at 10% per annum for 2 years?
1. ₹2100
2. ₹2200
3. ₹1900
4. ₹2000
Answer
A. ₹2100
A = 12100, so CI = 12100 − 10000 = ₹2100.
What is the difference between compound interest and simple interest on ₹10000 at 10% for 2 years?
1. ₹1000
2. ₹200
3. ₹100
4. ₹10
Answer
C. ₹100
Difference = P(R/100)² = 10000 × 0.01 = ₹100.
What is the compound interest on ₹8000 for 1 year at 10% per annum, compounded half-yearly?
1. ₹840
2. ₹820
3. ₹800
4. ₹900
Answer
B. ₹820
Rate 5% for 2 periods: A = 8000 × 1.05 × 1.05 = 8820, so CI = ₹820.
When interest is compounded half-yearly, the rate and number of periods used in the formula are
1. half the rate and the same periods
2. double the rate and half the periods
3. the same rate and double the periods
4. half the rate and double the periods
Answer
D. half the rate and double the periods
Rate R/2 per half-year and 2n half-year periods.
For the first year, compound interest and simple interest on the same sum at the same rate are
1. simple interest is greater
2. double each other
3. compound interest is greater
4. equal
Answer
D. equal
Interest is earned only on the original principal in the first year.
A sum becomes 1.21 times itself in two years at annual compound interest. What is the rate?
1. 10%
2. 20%
3. 11%
4. 21%
Answer
A. 10%
(1 + r)² = 1.21 gives 1 + r = 1.1, so r = 10%.
What is the amount on ₹4000 at 5% per annum compound interest for 2 years?
1. ₹4200
2. ₹4410
3. ₹4400
4. ₹4420
Answer
B. ₹4410
A = 4000 × 1.05 × 1.05 = 4000 × 1.1025 = ₹4410.
A profit of 25% on the cost price is what percentage of the selling price?
1. 30%
2. 25%
3. 20%
4. 15%
Answer
C. 20%
Take CP = 100, SP = 125; profit 25 on 125 is 20%.
Consider these statements. 1. Profit percentage is calculated on the cost price. 2. Discount percentage is calculated on the selling price. Which is/are correct?
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
A. 1 only
Discount is on the marked price, so 2 is wrong.
Consider these statements. 1. Simple interest increases by an equal amount every year. 2. Compound interest earns interest on previously earned interest. Which is/are correct?
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
C. Both 1 and 2
Both describe the two systems correctly.
Consider these statements. 1. For a supply within a state, IGST is charged. 2. For a supply between two states, CGST plus SGST is charged. Which is/are correct?
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
D. Neither 1 nor 2
The two are the wrong way round: within a state it is CGST plus SGST; between states it is IGST.
Consider these statements. 1. Two successive discounts of 10% each equal a single discount of 20%. 2. Two successive discounts of a% and b% equal a + b − ab/100 per cent. Which is/are correct?
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
B. 2 only
Two 10% discounts equal 10 + 10 − 1 = 19%, so 1 is wrong.
Match: (1) Simple interest (2) Compound interest (3) Discount (4) Central GST with (a) PRT/100 (b) P(1 + R/100)ⁿ − P (c) MP − SP (d) Tax shared equally with the state in an intra-state sale. Choose the correct matching.
1. 1-c, 2-b, 3-a, 4-d
2. 1-a, 2-b, 3-c, 4-d
3. 1-b, 2-a, 3-c, 4-d
4. 1-a, 2-b, 3-d, 4-c
Answer
B. 1-a, 2-b, 3-c, 4-d
SI = PRT/100; CI = P(1 + R/100)ⁿ − P; Discount = MP − SP; CGST is half of the GST charged within a state.
A trader buys 20 notebooks for ₹400 and sells all of them at ₹25 each. What is the profit percentage?
1. 30%
2. 15%
3. 20%
4. 25%
Answer
D. 25%
SP = 20 × 25 = 500; profit = 100; 100/400 × 100 = 25%.
A watch is sold for ₹990 after a 10% discount on its marked price. What is the marked price?
1. ₹1100
2. ₹1000
3. ₹1089
4. ₹1090
Answer
A. ₹1100
SP = 0.9 × MP, so MP = 990/0.9 = ₹1100.
What single discount is equal to two successive discounts of 25% and 20%?
1. 42%
2. 35%
3. 40%
4. 45%
Answer
C. 40%
25 + 20 − (25 × 20)/100 = 40%.
In the formula SI = PRT/100, the time T must be expressed in
1. days
2. years
3. weeks
4. months
Answer
B. years
R is the rate per year, so T is in years.
What is the simple interest on ₹1500 at 6% per annum for 8 months?
1. ₹720
2. ₹90
3. ₹60
4. ₹45
Answer
C. ₹60
T = 8/12 = 2/3 year; SI = 1500 × 6 × (2/3)/100 = ₹60.
An article costing ₹250 is sold at a loss of 8%. What is its selling price?
1. ₹220
2. ₹240
3. ₹270
4. ₹230
Answer
D. ₹230
SP = 250 × 92/100 = ₹230.
What is the amount after 4 years on ₹2500 at 4% per annum simple interest?
1. ₹2900
2. ₹2800
3. ₹2916
4. ₹2650
Answer
A. ₹2900
SI = 2500 × 4 × 4/100 = 400; A = ₹2900.
What is the difference between compound interest and simple interest on ₹7500 at 6% per annum for 2 years?
1. ₹27
2. ₹36
3. ₹54
4. ₹18
Answer
A. ₹27
Difference = P(R/100)² = 7500 × 36/10000 = ₹27.

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Chapter quiz · 10 questions

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On Simple Interest and Compound Interest, from the live question bank. 🔒 Answers and solutions are shown to registered students (free).
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1.The compound interest on Rs.5,000 for 3 years at 10% p.a. will amount to
2.The compound interest on Rs.4000 for 4 years at 10% per annum will be
3.The compound interest on a certain sum of money for 2 years at 5% is Rs.328, then the sum is
4.The amount received at 8% per annum compound interest after 2 yrs is Rs72,900. What was the principal (in Rs)?
5.The amount received at 10% per annum compound interest after 3 yrs is Rs.10,648. What was the principal (in Rs)?
6.The sum for 2 years gives a compound interest of Rs.3225 at 15% rate. Then sum is
7.Rs.260200 is divided between Ram and Shyam so that the amount that Ram receives in 3 years is the same as that Shyam receives in 6 years. If the interest is compounded annually at the rate of 4% per annum then Ram's share is
8.What is the rate of interest (in %) if simple interest earned on a certain sum for the 3 years is Rs6,000 and compound interest earned for 2 years is Rs4,160?
9.What is the rate of interest if simple interest earned on a certain sum for the 3rd year is Rs1,750 and compound interest earned for 2 years is Rs3622.5?
10.What is the rate of interest (in%) if simple interest earned on a certain sum for the 3rd year is Rs.2,000 and compound interest earned in 2 years is Rs.4160?

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