Episode 8 — Aptitude and Reasoning / 8.2 — Profit and Loss
8.2.c Solved Examples -- Profit and Loss
Basic Level (Problems 1-8)
Problem 1: Basic Profit Calculation
A person buys a book for Rs. 250 and sells it for Rs. 310. Find the profit and profit percentage.
Given:
CP = Rs. 250
SP = Rs. 310
Solution:
Profit = SP - CP = 310 - 250 = Rs. 60
Profit % = (Profit / CP) x 100
= (60 / 250) x 100
= 24%
Answer: Profit = Rs. 60, Profit % = 24%
Problem 2: Basic Loss Calculation
A trader purchases goods for Rs. 1,200 and sells them for Rs. 1,020. Find the loss percentage.
Given:
CP = Rs. 1,200
SP = Rs. 1,020
Solution:
Loss = CP - SP = 1200 - 1020 = Rs. 180
Loss % = (Loss / CP) x 100
= (180 / 1200) x 100
= 15%
Answer: Loss % = 15%
Problem 3: Finding SP from Profit%
A shopkeeper buys a chair for Rs. 800 and wants to make a 35% profit. What should be the selling price?
Given:
CP = Rs. 800
Profit % = 35%
Solution:
SP = CP x (100 + Profit%) / 100
= 800 x (100 + 35) / 100
= 800 x 135 / 100
= Rs. 1,080
Answer: SP = Rs. 1,080
Problem 4: Finding CP from SP and Loss%
An article is sold for Rs. 540 at a loss of 10%. Find the cost price.
Given:
SP = Rs. 540
Loss % = 10%
Solution:
CP = SP x 100 / (100 - Loss%)
= 540 x 100 / (100 - 10)
= 540 x 100 / 90
= Rs. 600
Verification: 600 x 90/100 = Rs. 540 (matches SP)
Answer: CP = Rs. 600
Problem 5: Simple Discount Problem
The marked price of a shirt is Rs. 1,500. A discount of 20% is offered. Find the selling price.
Given:
MP = Rs. 1,500
Discount = 20%
Solution:
SP = MP x (100 - Discount%) / 100
= 1500 x (100 - 20) / 100
= 1500 x 80 / 100
= Rs. 1,200
Answer: SP = Rs. 1,200
Problem 6: Markup and Profit
A dealer buys an article for Rs. 500 and marks it at Rs. 700. Find the markup percentage.
Given:
CP = Rs. 500
MP = Rs. 700
Solution:
Markup = MP - CP = 700 - 500 = Rs. 200
Markup % = (Markup / CP) x 100
= (200 / 500) x 100
= 40%
Answer: Markup % = 40%
Problem 7: Overhead Expenses
A person buys a second-hand scooter for Rs. 18,000 and spends Rs. 2,000 on repairs. He then sells it for Rs. 22,000. Find the profit percentage.
Given:
Purchase Price = Rs. 18,000
Repair Cost = Rs. 2,000
SP = Rs. 22,000
Solution:
Effective CP = 18,000 + 2,000 = Rs. 20,000
Profit = SP - CP = 22,000 - 20,000 = Rs. 2,000
Profit % = (2,000 / 20,000) x 100 = 10%
Answer: Profit % = 10%
Problem 8: Finding SP for Desired Profit
A trader buys goods for Rs. 3,500. At what price should he sell to gain 14%?
Given:
CP = Rs. 3,500
Desired Profit = 14%
Solution:
SP = CP x (100 + 14) / 100
= 3500 x 114 / 100
= Rs. 3,990
Answer: He should sell at Rs. 3,990
Medium Level (Problems 9-16)
Problem 9: Markup and Discount Combined
A shopkeeper buys a TV for Rs. 20,000. He marks it 30% above the cost price and then offers a 10% discount. Find his profit percentage.
Given:
CP = Rs. 20,000
Markup = 30%
Discount = 10%
Solution:
Method 1: Step by step
MP = 20,000 x 130/100 = Rs. 26,000
SP = 26,000 x 90/100 = Rs. 23,400
Profit = 23,400 - 20,000 = Rs. 3,400
Profit % = (3,400 / 20,000) x 100 = 17%
Method 2: Combined formula
Net % = M - D - (M x D)/100
= 30 - 10 - (30 x 10)/100
= 30 - 10 - 3
= 17% profit
Answer: Profit % = 17%
Problem 10: Successive Discounts
A store offers successive discounts of 20% and 15% on an item marked at Rs. 2,000. Find the selling price and the equivalent single discount.
Given:
MP = Rs. 2,000
Discount 1 = 20%
Discount 2 = 15%
Solution:
After 1st discount: SP1 = 2000 x 80/100 = Rs. 1,600
After 2nd discount: SP = 1600 x 85/100 = Rs. 1,360
Total discount = 2000 - 1360 = Rs. 640
Equivalent single discount = (640 / 2000) x 100 = 32%
Verification using formula:
Equivalent discount = 20 + 15 - (20 x 15)/100
= 35 - 3 = 32%
Answer: SP = Rs. 1,360; Equivalent single discount = 32%
Problem 11: Two Articles with Same SP
Two watches are sold for Rs. 1,980 each. On one there is a gain of 10% and on the other a loss of 10%. Find the overall gain or loss percentage.
Given:
SP of each = Rs. 1,980
Profit on 1st = 10%
Loss on 2nd = 10%
Solution:
Since profit% = loss% = 10%, use the shortcut:
Net Loss % = x^2 / 100 = (10)^2 / 100 = 100/100 = 1%
Verification:
CP of 1st watch = 1980 x 100/110 = Rs. 1,800
CP of 2nd watch = 1980 x 100/90 = Rs. 2,200
Total CP = 1800 + 2200 = Rs. 4,000
Total SP = 1980 + 1980 = Rs. 3,960
Loss = 4000 - 3960 = Rs. 40
Loss % = (40 / 4000) x 100 = 1%
Answer: Overall Loss = 1%
Problem 12: Buy X Get Y Free with Markup
A shopkeeper marks his goods 50% above cost price and offers "Buy 2 Get 1 Free." Find his profit or loss percentage.
Given:
Markup = 50%
Offer = Buy 2, Get 1 Free
Solution:
Let CP of each item = Rs. 100
MP of each item = Rs. 150
Customer buys 2, gets 1 free --> receives 3 items
Total CP for shopkeeper = 3 x 100 = Rs. 300
Revenue = 2 x 150 = Rs. 300
Profit = 300 - 300 = 0
Profit % = 0%
Answer: Neither profit nor loss (Break-even)
Problem 13: Successive Transactions
A buys an article for Rs. 2,000 and sells it to B at 20% profit. B sells it to C at 10% profit. C sells it to D at a loss of 25%. What does D pay?
Given:
A's CP = Rs. 2,000
A to B: 20% profit
B to C: 10% profit
C to D: 25% loss
Solution:
B's CP = A's SP = 2000 x 120/100 = Rs. 2,400
C's CP = B's SP = 2400 x 110/100 = Rs. 2,640
D's CP = C's SP = 2640 x 75/100 = Rs. 1,980
Combined multiplier: 120/100 x 110/100 x 75/100
= 1.2 x 1.1 x 0.75
= 0.99
D pays = 2000 x 0.99 = Rs. 1,980
Answer: D pays Rs. 1,980
Problem 14: Dishonest Dealer
A shopkeeper claims to sell rice at cost price but uses a false weight of 800g instead of 1 kg. Find his profit percentage.
Given:
Claimed weight = 1000g (1 kg)
Actual weight = 800g
Solution:
Error = 1000 - 800 = 200g
Profit % = (Error / False Weight) x 100
= (200 / 800) x 100
= 25%
Explanation: The shopkeeper charges for 1000g but gives only 800g.
His cost is for 800g worth of rice, but he receives payment for 1000g.
Answer: Profit % = 25%
Problem 15: Buying and Selling Rates
A man buys 15 pens for Rs. 300 and sells 12 pens for Rs. 300. Find his profit percentage.
Given:
Buys 15 pens for Rs. 300
Sells 12 pens for Rs. 300
Solution:
CP per pen = 300/15 = Rs. 20
SP per pen = 300/12 = Rs. 25
Profit per pen = 25 - 20 = Rs. 5
Profit % = (5/20) x 100 = 25%
Shortcut (same amount, different quantities):
Since buying quantity (15) > selling quantity (12):
Profit % = ((15 - 12) / 12) x 100 = (3/12) x 100 = 25%
Answer: Profit % = 25%
Problem 16: Finding MP When Profit% and Discount% are Given
A shopkeeper wants to earn 26% profit after giving a 10% discount. By what percentage should he mark the goods above the cost price?
Given:
Desired Profit = 26%
Discount = 10%
Find: Markup %
Solution:
Using the combined formula:
Net % = M - D - (M x D)/100
26 = M - 10 - (M x 10)/100
26 = M - 10 - M/10
36 = M - M/10
36 = M(1 - 1/10)
36 = M x 9/10
M = 36 x 10/9
M = 40%
Verification:
Let CP = Rs. 100
MP = 100 x 140/100 = Rs. 140
SP = 140 x 90/100 = Rs. 126
Profit = 126 - 100 = Rs. 26 = 26% (correct)
Answer: Markup = 40% above CP
Advanced Level (Problems 17-25)
Problem 17: Dishonest Dealer with False Claim
A dealer claims to sell goods at 10% loss but uses a weight of 750g instead of 1 kg. What is his actual profit or loss percentage?
Given:
Claimed: 10% loss
True weight = 1000g
False weight = 750g
Solution:
Effective multiplier = (True Weight / False Weight) x (100 - Loss%) / 100
= (1000/750) x (90/100)
= (4/3) x (9/10)
= 36/30
= 1.2
Actual Profit % = (1.2 - 1) x 100 = 20% profit
Explanation:
He charges for 1 kg at 10% below cost.
If CP of 1 kg = Rs. 100, he charges Rs. 90 per kg.
But he gives only 750g, whose cost = Rs. 75.
Profit = 90 - 75 = Rs. 15
Profit % = (15/75) x 100 = 20%
Answer: Actual Profit = 20%
Problem 18: Partnership Profit Sharing
A, B, and C start a business. A invests Rs. 30,000 for 12 months, B invests Rs. 45,000 for 8 months, and C invests Rs. 50,000 for 6 months. If the total profit is Rs. 90,000, find each partner's share.
Given:
A: Rs. 30,000 for 12 months
B: Rs. 45,000 for 8 months
C: Rs. 50,000 for 6 months
Total profit = Rs. 90,000
Solution:
A's capital-time = 30,000 x 12 = 3,60,000
B's capital-time = 45,000 x 8 = 3,60,000
C's capital-time = 50,000 x 6 = 3,00,000
Ratio = 360 : 360 : 300 = 6 : 6 : 5
Sum of ratios = 6 + 6 + 5 = 17
A's share = 90,000 x 6/17 = Rs. 31,764.71 (approx Rs. 31,765)
B's share = 90,000 x 6/17 = Rs. 31,764.71 (approx Rs. 31,765)
C's share = 90,000 x 5/17 = Rs. 26,470.59 (approx Rs. 26,471)
Check: 31,765 + 31,765 + 26,471 = Rs. 90,001 (rounding difference)
Answer: A = Rs. 31,765, B = Rs. 31,765, C = Rs. 26,471 (approximately)
Problem 19: Conditional Profit Problem
A man sold an article at a 12% profit. Had he sold it for Rs. 60 more, he would have gained 18%. Find the cost price.
Given:
Actual profit = 12%
If SP increased by Rs. 60, profit = 18%
Solution:
Let CP = C
Actual SP = C x 112/100
Increased SP = C x 118/100
Difference = C x 118/100 - C x 112/100 = 60
C x (118 - 112)/100 = 60
C x 6/100 = 60
C = 60 x 100/6
C = Rs. 1,000
Verification:
Actual SP = 1000 x 112/100 = Rs. 1,120
Increased SP = 1120 + 60 = Rs. 1,180
Check: 1000 x 118/100 = Rs. 1,180 (correct)
Answer: CP = Rs. 1,000
Problem 20: Profit on Selling Some and Loss on Rest
A man bought 100 oranges. He sold 60 at 20% profit and the remaining at 10% loss. Find his overall profit or loss percentage.
Given:
Total oranges = 100
60 sold at 20% profit
40 sold at 10% loss
Solution:
Let CP of each orange = Rs. 1
Total CP = Rs. 100
Revenue from 60 oranges at 20% profit = 60 x 1.2 = Rs. 72
Revenue from 40 oranges at 10% loss = 40 x 0.9 = Rs. 36
Total SP = 72 + 36 = Rs. 108
Profit = 108 - 100 = Rs. 8
Profit % = 8%
Shortcut (weighted average):
Overall % = (60/100) x 20 + (40/100) x (-10)
= 12 + (-4)
= 8% profit
Answer: Overall Profit = 8%
Problem 21: Three Successive Discounts
An item has a marked price of Rs. 5,000. Three successive discounts of 10%, 15%, and 20% are offered. Find the final selling price and the equivalent single discount.
Given:
MP = Rs. 5,000
Discounts: 10%, 15%, 20% (successive)
Solution:
After 10% discount: 5000 x 90/100 = Rs. 4,500
After 15% discount: 4500 x 85/100 = Rs. 3,825
After 20% discount: 3825 x 80/100 = Rs. 3,060
Total discount = 5000 - 3060 = Rs. 1,940
Equivalent single discount = (1940/5000) x 100 = 38.8%
Using multipliers: 0.9 x 0.85 x 0.80 = 0.612
SP = 5000 x 0.612 = Rs. 3,060
Equivalent discount = (1 - 0.612) x 100 = 38.8%
Answer: SP = Rs. 3,060; Equivalent single discount = 38.8%
Problem 22: Cost Price Redistribution
A person buys two articles for Rs. 5,000 combined. He sells one at 20% profit and the other at 30% loss. If the overall loss is 2%, find the CP of each article.
Given:
Combined CP = Rs. 5,000
Article 1: sold at 20% profit
Article 2: sold at 30% loss
Overall: 2% loss
Solution:
Let CP of Article 1 = x
Then CP of Article 2 = 5000 - x
SP of Article 1 = x x 120/100 = 1.2x
SP of Article 2 = (5000 - x) x 70/100 = 0.7(5000 - x)
Total SP = 5000 x 98/100 = Rs. 4,900 (2% loss on 5000)
So: 1.2x + 0.7(5000 - x) = 4900
1.2x + 3500 - 0.7x = 4900
0.5x = 1400
x = Rs. 2,800
CP of Article 1 = Rs. 2,800
CP of Article 2 = Rs. 2,200
Verification:
SP1 = 2800 x 1.2 = 3360
SP2 = 2200 x 0.7 = 1540
Total SP = 3360 + 1540 = 4900
Loss = 5000 - 4900 = 100
Loss% = (100/5000) x 100 = 2% (correct)
Answer: CP of Article 1 = Rs. 2,800, CP of Article 2 = Rs. 2,200
Problem 23: Dishonest Dealer with Profit on Purchase and Sale
A dealer cheats both while buying and selling. While buying, he uses weights 20% more than the true weight (gets more goods). While selling, he uses weights 20% less than the true weight (gives less goods). Find his total profit percentage if he claims to sell at cost price.
Given:
While buying: uses weights 20% more (gets 1200g for every 1000g he pays for)
While selling: uses weights 20% less (gives 800g for every 1000g he charges for)
Sells at cost price
Solution:
Let the true cost of 1000g = Rs. 100
While buying 1000g:
He actually receives 1200g (20% more)
He pays for 1000g = Rs. 100
Effective CP per gram = 100/1200
While selling 1000g:
He charges for 1000g = Rs. 100
He actually gives 800g (20% less)
Revenue per gram = 100/800
Now compare for the same amount of goods (say 1 gram):
CP per gram = 100/1200 = 1/12
SP per gram = 100/800 = 1/8
Profit per gram = 1/8 - 1/12 = (3 - 2)/24 = 1/24
Profit % = (Profit / CP) x 100
= (1/24) / (1/12) x 100
= (1/24) x (12/1) x 100
= 50%
Alternative: Effective multiplier = (1200/1000) x (1000/800)
= 1.2 x 1.25 = 1.5
Profit % = (1.5 - 1) x 100 = 50%
Answer: Total Profit = 50%
Problem 24: Selling Price to Achieve Target After Initial Loss
A shopkeeper sold an article at 20% loss. If he had sold it for Rs. 200 more, he would have earned 5% profit. At what price should he sell it to make 25% profit?
Given:
Actual: 20% loss
If SP increased by Rs. 200: 5% profit
Find: SP for 25% profit
Solution:
Step 1: Find CP
Let CP = C
SP at 20% loss = C x 80/100 = 0.8C
SP at 5% profit = C x 105/100 = 1.05C
Difference = 1.05C - 0.8C = 0.25C = Rs. 200
C = 200 / 0.25 = Rs. 800
Step 2: Find required SP for 25% profit
SP = 800 x 125/100 = Rs. 1,000
Verification:
SP at 20% loss = 800 x 0.8 = Rs. 640
640 + 200 = Rs. 840
840/800 = 1.05 = 5% profit (correct)
Answer: He should sell at Rs. 1,000 for 25% profit
Problem 25: Complex Profit-Loss with Multiple Conditions
A merchant has 2,000 kg of rice. He sells part at 8% profit and the rest at 18% profit. If the total profit is 14%, how many kilograms did he sell at 18% profit?
Given:
Total rice = 2,000 kg
Part sold at 8% profit, rest at 18% profit
Overall profit = 14%
Solution:
Method 1: Alligation (Weighted Average Method)
8% profit .............. 18% profit
\ /
14% (mean)
/ \
18 - 14 = 4 .... 14 - 8 = 6
Ratio = 4 : 6 = 2 : 3
Sold at 8% = 2000 x 2/5 = 800 kg
Sold at 18% = 2000 x 3/5 = 1,200 kg
Method 2: Equation
Let quantity sold at 18% = x kg
Then quantity at 8% = (2000 - x) kg
Let CP per kg = Rs. 1
Total CP = 2000
Total Profit = 14% of 2000 = 280
Profit from 1st part = 0.08 x (2000 - x)
Profit from 2nd part = 0.18 x x
0.08(2000 - x) + 0.18x = 280
160 - 0.08x + 0.18x = 280
0.10x = 120
x = 1,200 kg
Answer: 1,200 kg sold at 18% profit
Problem 26 (Bonus): Multiple Operations on Same Article
A trader buys an article and marks it 80% above CP. He gives a 25% discount on the marked price but cheats on the weight by using 900g instead of 1 kg. What is his overall profit percentage?
Given:
Markup = 80%
Discount = 25%
True weight = 1000g, False weight = 900g
Solution:
Step 1: Effect of markup and discount
Net % from markup-discount = M - D - (M x D)/100
= 80 - 25 - (80 x 25)/100
= 80 - 25 - 20
= 35% profit on honest weight
Step 2: Effect of false weights
The honest selling gives 35% profit.
With false weights, he gives only 900g for 1000g price.
Total multiplier = (100 + 35)/100 x (1000/900)
= 1.35 x 10/9
= 1.35 x 1.1111
= 1.5
Step 3: Overall profit
Overall Profit % = (1.5 - 1) x 100 = 50%
Verification with numbers:
Let CP of 1 kg = Rs. 100
MP of 1 kg = Rs. 180 (80% markup)
SP of 1 kg = 180 x 75/100 = Rs. 135 (25% discount)
But he gives only 900g whose cost = Rs. 90
Profit = 135 - 90 = Rs. 45
Profit % = (45/90) x 100 = 50%
Answer: Overall Profit = 50%
Problem 27 (Bonus): Break-Even Analysis
A manufacturer has a fixed cost of Rs. 50,000. Each unit costs Rs. 120 to produce and sells for Rs. 200. (a) How many units must he sell to break even? (b) What is the profit if he sells 1,000 units?
Given:
Fixed cost = Rs. 50,000
Variable cost per unit = Rs. 120
SP per unit = Rs. 200
Solution:
(a) Break-Even Point:
Contribution per unit = SP - Variable Cost = 200 - 120 = Rs. 80
Break-Even Quantity = Fixed Cost / Contribution per unit
= 50,000 / 80
= 625 units
(b) Profit at 1,000 units:
Total Revenue = 1000 x 200 = Rs. 2,00,000
Total Cost = 50,000 + (1000 x 120) = 50,000 + 1,20,000 = Rs. 1,70,000
Profit = 2,00,000 - 1,70,000 = Rs. 30,000
Alternatively:
Profit = (Units sold - Break-even units) x Contribution per unit
= (1000 - 625) x 80
= 375 x 80 = Rs. 30,000
Answer: (a) 625 units, (b) Rs. 30,000
Summary of Problem Types Covered
Problem | Type | Level
---------|--------------------------------|----------
1 | Basic profit | Basic
2 | Basic loss | Basic
3 | SP from profit% | Basic
4 | CP from SP and loss% | Basic
5 | Simple discount | Basic
6 | Markup | Basic
7 | Overhead expenses | Basic
8 | SP for desired profit | Basic
9 | Markup + discount combined | Medium
10 | Successive discounts | Medium
11 | Two articles same SP | Medium
12 | Buy X Get Y free | Medium
13 | Successive transactions | Medium
14 | Dishonest dealer (basic) | Medium
15 | Buy/sell at different rates | Medium
16 | Finding markup from conditions | Medium
17 | Dishonest dealer + false claim | Advanced
18 | Partnership profit sharing | Advanced
19 | Conditional profit | Advanced
20 | Mixed profit and loss | Advanced
21 | Three successive discounts | Advanced
22 | CP redistribution | Advanced
23 | Double cheating dealer | Advanced
24 | Target selling price | Advanced
25 | Alligation in profit/loss | Advanced
26 | Markup + discount + false wt | Advanced
27 | Break-even analysis | Advanced