Profit and loss compares what a seller pays with what the seller gets.
This lesson explains the terms, gives each formula with its reason, and solves three examples. It also lists the traps that cost marks.
- Base: percent is always on the cost price.
- Gain: SP − CP. Loss: CP − SP.
- Multiplier: gain g% is × (100 + g)/100.
- Discounts: x + y − xy/100.
- Trap: equal gain and loss at one price is a loss.
What the Terms Mean
What are cost price and selling price?
Cost price (CP) is what the seller pays for the item. Extra costs such as transport are added to it.Selling price (SP) is what the seller receives. If SP is above CP, the seller gains. If SP is below CP, the seller loses.
| Term | Meaning | Formula |
|---|---|---|
| Gain | SP is more than CP | SP − CP |
| Loss | CP is more than SP | CP − SP |
| Marked price | Price on the tag | Fixed by the shop |
| Discount | Cut from the marked price | Marked price − sale price |
Which base is used for the percentage?
Gain percent and loss percent always use the cost price as base.
Why: you compare the money gained or lost with the money first spent.
Check 1
Gain percent is worked out on the selling price.
Core Formulas and Why They Work
How do you find gain or loss percent?
Gain% = (SP − CP)/CP × 100. Loss% = (CP − SP)/CP × 100.
Why it works: it is just the profit as a percentage of what you spent.
How do you find the selling price?
With a gain of g%: SP = CP × (100 + g)/100. With a loss of l%: SP = CP × (100 − l)/100.
Why it works: treat CP as 100 parts. A gain adds g parts. A loss removes l parts.
How do you find the cost price?
CP = SP × 100/(100 + g) for a gain, and SP × 100/(100 − l) for a loss.
Why it works: this just reverses the step above.
How do two discounts combine?
Single discount = x + y − xy/100.
Why it works: the second discount works on the lower price. So the two do not simply add.
What if two items sell at the same price, one at x% gain and one at x% loss?
There is always a loss of x²/100 percent.
Why it works: the item sold at a gain cost less. The item sold at a loss cost more. So the total cost beats the total selling price.
What if a dealer buys n items for the price of m?
Gain% = (n − m)/m × 100.
Why it works: he pays for m items but sells n items at the usual price.
Check 2
Selling two items at the same price, one at 10% gain and one at 10% loss, gives no overall result.
Three Solved Examples
Easy: an item costs Rs. 1408 and sells for Rs. 1760. What is the gain percent?
Gain = 1760 − 1408 = Rs. 352.
Gain% = 352/1408 × 100.
352 is one quarter of 1408. So the gain is 25%.
Medium: an item is sold for Rs. 288 at a loss of 64%. What is the cost price?
A 64% loss means SP is 36% of CP.
So CP = 288 × 100/36 = 28800/36.
The cost price is Rs. 800. Check: 36% of 800 is 288.
Exam level: what single discount equals discounts of 25% and 20%?
After 25% off, 75% of the price is left. After 20% off that, 80% of 75% is left.
0.75 × 0.80 = 0.60. So 60% remains.
The single discount is 40%, not 45%. The formula agrees: 25 + 20 − (25 × 20)/100 = 40.
Step by step: a shopkeeper marks an item 25% above cost of Rs 800. He gives a 10% discount. Find his gain percent.
Step 1: marked price = 800 × 1.25 = Rs 1000.
Step 2: selling price = 1000 × 0.90 = Rs 900.
Step 3: gain = 900 − 800 = Rs 100.
Step 4: gain% = 100/800 × 100 = 12.5%.
Shortcut check: 1.25 × 0.90 = 1.125. That is a 12.5% rise on cost.
Check 3
A dealer buys 10 articles for the price of 8 and sells at the usual price. His gain is 25%.
Common Traps and Shortcuts
Which traps cost marks?
Using SP as the base. Always divide by CP.
Adding discounts. 25% and 20% is not 45%.
Thinking equal gain and loss at the same SP cancel out. They never do.
Forgetting overhead costs. Transport and repair costs raise the CP.
Which shortcuts save time?
Use multipliers. A gain of 20% is × 1.2. A loss of 20% is × 0.8.
A loss of 20% means SP is 80% of CP. Say it that way and the sum is one step.
To find CP from SP, divide by the multiplier. For a 25% gain, divide by 1.25.
Why is the gain percent not the same as the markup minus the discount?
A markup of 25% and a discount of 10% do not give 15%.
The two work on different bases. The markup is on cost. The discount is on the marked price.
Always multiply the two multipliers instead of subtracting.
What patterns should you expect in exams?
Questions fall into a few patterns. Direct gain or loss, hidden cost price, successive discounts, and false weights or free items.
For each, first write the cost price as 100. Then move through each step with multipliers.
Keep gain, loss and discount apart in your mind. Gain and loss use cost price. Discount uses marked price.
If a question gives a final price, work backwards by dividing by the multiplier.
Check 4
A loss of 20% means the selling price is 80% of the cost price.
MCQ on Profit and Loss
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