Episode 8 — Aptitude and Reasoning / 8.2 — Profit and Loss
8.2.b Tips, Tricks, and Shortcuts -- Profit and Loss
1. The Fraction Method (Fastest for Mental Math)
Instead of working with percentages, convert common profit/loss percentages to fractions. This dramatically speeds up calculations.
1.1 Common Percentage-to-Fraction Table
Profit/Loss % | Fraction | Multiplier for SP (Profit) | Multiplier for SP (Loss)
---------------|------------|------------------------------|-------------------------
5% | 1/20 | 21/20 | 19/20
10% | 1/10 | 11/10 | 9/10
12.5% | 1/8 | 9/8 | 7/8
15% | 3/20 | 23/20 | 17/20
16.67% | 1/6 | 7/6 | 5/6
20% | 1/5 | 6/5 | 4/5
25% | 1/4 | 5/4 | 3/4
30% | 3/10 | 13/10 | 7/10
33.33% | 1/3 | 4/3 | 2/3
40% | 2/5 | 7/5 | 3/5
50% | 1/2 | 3/2 | 1/2
60% | 3/5 | 8/5 | 2/5
66.67% | 2/3 | 5/3 | 1/3
75% | 3/4 | 7/4 | 1/4
100% | 1 | 2 | 0
1.2 How to Use It
To find SP at 25% profit when CP = Rs. 360:
SP = CP x 5/4 = 360 x 5/4 = Rs. 450
To find SP at 20% loss when CP = Rs. 600:
SP = CP x 4/5 = 600 x 4/5 = Rs. 480
To find CP when SP = Rs. 780 at 30% profit:
CP = SP x 10/13 = 780 x 10/13 = Rs. 600
2. Quick Profit/Loss Percentage Shortcuts
2.1 Direct Percentage Calculation
Shortcut: Profit % = ((SP - CP) / CP) x 100
Trick: Think of it as "how much extra per hundred of CP?"
If CP = 250 and SP = 300:
Extra = 50 on 250
Per 100: 50/250 x 100 = 20%
Quick mental math: 50 is 1/5 of 250 = 20%
2.2 When SP and CP are Close
If SP is slightly more than CP:
Profit = SP - CP
Approximate Profit % = (Profit / CP) x 100
Tip: Round CP to nearest convenient number for quick estimation.
Example: CP = 498, SP = 572
Profit = 74
Approx: 74/500 x 100 = 14.8%
Exact: 74/498 x 100 = 14.86%
3. The CP-SP Swap Trick
3.1 When CP and SP are Interchanged
Scenario: If the CP and SP of two transactions are swapped,
the profit % and loss % are NOT equal.
Example:
Transaction 1: CP = 400, SP = 500 --> Profit % = 25%
Transaction 2: CP = 500, SP = 400 --> Loss % = 20%
Rule: The profit percentage is always LARGER than the loss
percentage because profit is calculated on a smaller base (CP)
and loss on a larger base.
3.2 Useful Identity
If selling at x% profit gives SP = A,
and selling at x% loss gives SP = B,
then:
CP = (A + B) / 2 x 100 / 100 ... (only when percentages are equal)
Actually more precisely:
A = CP x (100 + x)/100
B = CP x (100 - x)/100
A + B = CP x 200/100 = 2 x CP
Therefore: CP = (A + B) / 2
Example:
Selling at 10% profit gives SP = Rs. 550.
Selling at 10% loss gives SP = Rs. 450.
CP = (550 + 450) / 2 = Rs. 500
Verify: 500 x 110/100 = 550 (correct)
500 x 90/100 = 450 (correct)
4. Buy X Get Y Free -- Quick Calculation
"Buy X Get Y Free" means customer gets (X + Y) items for the price of X.
Effective Discount = Y/(X+Y) x 100
Common Offers:
Buy 1 Get 1 Free --> 50% discount
Buy 2 Get 1 Free --> 33.33% discount
Buy 3 Get 1 Free --> 25% discount
Buy 4 Get 1 Free --> 20% discount
Buy 5 Get 1 Free --> 16.67% discount
Buy 9 Get 1 Free --> 10% discount
Quick Check: To find profit when shopkeeper gives "Buy X Get Y Free"
with a markup of M%:
Total items given = X + Y
Revenue = X items x MP
Cost = (X + Y) items x CP
Profit % = ((X x MP) / ((X + Y) x CP) - 1) x 100
5. Markup-Discount Combined Formula
This is one of the most useful shortcuts for competitive exams.
When a shopkeeper marks up by M% and gives a discount of D%:
Net Profit/Loss % = M - D - (M x D)/100
If result is positive --> Profit
If result is negative --> Loss
Quick Reference Table
Markup % | Discount % | Net Effect
----------|-------------|-------------
20% | 10% | +8% profit
25% | 20% | 0% (break-even)
30% | 20% | +4% profit
40% | 20% | +12% profit
50% | 30% | +5% profit
30% | 30% | -9% loss
20% | 25% | -10% loss
40% | 25% | +5% profit
50% | 20% | +20% profit
25% | 10% | +12.5% profit
Break-Even Condition
For no profit no loss: M - D - (M x D)/100 = 0
This gives: M = 100D / (100 - D)
Or: D = 100M / (100 + M)
Example: If discount is 20%, what markup gives break-even?
M = 100 x 20 / (100 - 20) = 2000/80 = 25%
6. Two Articles, Same SP, Equal Profit% and Loss%
GOLDEN RULE: Always results in a net LOSS.
Loss % = x^2 / 100 where x is the equal profit/loss percentage.
Common values:
x = 10% --> Net Loss = 1%
x = 15% --> Net Loss = 2.25%
x = 20% --> Net Loss = 4%
x = 25% --> Net Loss = 6.25%
x = 30% --> Net Loss = 9%
x = 50% --> Net Loss = 25%
This is a very frequently asked question. Memorize the formula.
7. Successive Transactions Shortcut
7.1 Successive Profit/Loss
If an article passes through multiple hands, each adding their profit:
For two successive profits of a% and b%:
Net Profit % = a + b + (a x b)/100
For profit of a% followed by loss of b%:
Net effect % = a - b - (a x b)/100
(Positive = profit, Negative = loss)
For two successive losses of a% and b%:
Net Loss % = a + b - (a x b)/100
(Note: Both are losses, so the formula subtracts the product term)
Actually, the universal formula is:
Use multipliers: (1 + a/100)(1 + b/100) where a,b are signed
(positive for profit, negative for loss)
Net % change = ((1 + a/100)(1 + b/100) - 1) x 100
Example:
A sells to B at 20% profit. B sells to C at 10% loss.
If A's CP = Rs. 500, find C's CP (i.e., B's SP).
Method 1: Step by step
B's CP = 500 x 120/100 = Rs. 600
C's CP = 600 x 90/100 = Rs. 540
Method 2: Combined multiplier
Combined = 120/100 x 90/100 = 108/100
C's CP = 500 x 108/100 = Rs. 540
Net effect = 8% profit on A's CP
8. Dishonest Dealer Shortcuts
8.1 Quick Formula
Profit % = (Error / True Value - Error) x 100
= (Error / Value Given) x 100
Where:
Error = True Weight - False Weight
Value Given = False Weight (what customer actually gets)
8.2 Common False Weight Problems
Sells 1 kg as: | False Weight | Profit %
----------------|----------------|----------
950g | 950g | 5.26%
900g | 900g | 11.11%
850g | 850g | 17.65%
800g | 800g | 25%
750g | 750g | 33.33%
700g | 700g | 42.86%
Memorize at least the 900g (11.11%) and 800g (25%) cases.
8.3 Dishonest Dealer with Profit/Loss Claim
Total Profit % = ((Claimed Weight / Actual Weight) x (100 +/- P%) / 100 - 1) x 100
Where P% is the profit (+) or loss (-) the dealer claims.
9. Common Exam Patterns to Recognize Instantly
Pattern 1: "Sells at Cost Price but Uses False Weights"
Direct formula: Profit% = (Error / False Weight) x 100
Pattern 2: "Marks Up by M% and Gives D% Discount"
Direct formula: Net% = M - D - (MD/100)
Pattern 3: "Two Articles Sold at Same Price, One at x% Profit, Other at x% Loss"
Direct formula: Net Loss% = x^2/100
Pattern 4: "Buys at X for Rs. A and Sells at Y for Rs. B"
CP per item = A/X
SP per item = B/Y
Profit/Loss% = ((B/Y - A/X) / (A/X)) x 100
= ((BX - AY) / AY) x 100
Pattern 5: "If SP is Increased/Decreased by Rs. N, Profit Changes by P%"
CP = N x 100 / P
(Because changing SP by N changes profit by N, and P% of CP = N)
Pattern 6: "Sells 2/3 at 30% Profit, Rest at 10% Loss -- Overall Profit?"
Overall Profit% = (2/3) x 30 + (1/3) x (-10)
= 20 - 3.33 = 16.67%
(Weighted average of individual profit/loss percentages)
Pattern 7: "At What Price Should He Sell to Make X% Profit?"
Required SP = CP x (100 + X) / 100
If CP is not given, derive it from the existing SP and its profit/loss.
10. Percentage Change and Ratio Tricks
10.1 If Profit is Given as a Fraction of SP
If Profit = (1/n) of SP, then:
Profit % = 100 / (n - 1)
Example: Profit = 1/4 of SP
Profit % = 100 / (4 - 1) = 33.33%
Derivation:
Profit = SP/n
CP = SP - SP/n = SP(n-1)/n
Profit% = (SP/n) / (SP(n-1)/n) x 100 = 100/(n-1)
10.2 If Loss is Given as a Fraction of SP
If Loss = (1/n) of SP, then:
Loss % = 100 / (n + 1)
Example: Loss = 1/5 of SP
Loss % = 100 / (5 + 1) = 16.67%
Derivation:
Loss = SP/n
CP = SP + SP/n = SP(n+1)/n
Loss% = (SP/n) / (SP(n+1)/n) x 100 = 100/(n+1)
10.3 If Profit/Loss is Given as a Fraction of CP
If Profit = (1/n) of CP, then Profit % = 100/n (straightforward)
If Loss = (1/n) of CP, then Loss % = 100/n (straightforward)
11. Speed Tips for Exams
1. ALWAYS identify what is given -- CP, SP, or MP -- before starting.
2. When a problem mentions "marked price" and "discount," work from
MP backwards to SP, then compare with CP.
3. For chain problems (A sells to B, B sells to C), multiply all
the multipliers together and apply once.
4. When stuck, assume CP = Rs. 100. This simplifies all percentage
calculations and you can scale up later.
5. In "Buy X Get Y Free" problems, always think about total cost
vs. total revenue for the correct number of items.
6. Watch for the trap: "x% profit on SP" is different from
"x% profit on CP." Unless specified, profit% is always on CP.
7. Negative answers in net effect formula mean LOSS, not a
calculation error.
8. If a problem seems to have insufficient data, check if you
can use ratios instead of absolute values.