Percentages, Profit and Loss — AAI ATC GK Lesson

Before you start

Learning objectives and outcomes

Objective 1Convert fluently between percentages, fractions and reversed (original-value) calculations.

Objective 2Correctly handle successive percentage changes, especially the up-then-down trap.

Objective 3Apply CP/SP/MP relationships to profit, loss and discount problems without confusing the bases.

Outcome ASolve any percentage or profit/loss word problem in one pass, without needing to re-derive the formula.

Subtopic 1

Percentage fundamentals

A percentage is simply a fraction with denominator 100. Converting between percentages, fractions and decimals fluently is the single most useful skill in this whole topic.

The core conversions

x% of a number N = (x/100) × N
If a number increases by x% to become N, the original number = N × 100/(100+x)
If a number decreases by x% to become N, the original number = N × 100/(100−x)

Working backwards from a final value to an original value is one of the most common traps — students often subtract or add the percentage directly instead of using the correct ratio.

✈ AAI ATC relevance — percentage word problems appear in nearly every aptitude section, often disguised as profit/loss or ratio questions

Memory trick — "Increase divides by more, decrease divides by less"

To reverse an increase, you divide by a number bigger than 100 (100+x). To reverse a decrease, you divide by a number smaller than 100 (100−x). Getting this backwards is the most common percentage mistake.

Easy

If 20% of a number is 50, the number is:

20% of N = 50, so N = 50 × 100/20 = 250.

Moderate

A number is increased by 25% and becomes 100. The original number was:

Using original = final × 100/(100+x) = 100 × 100/125 = 80.

Subtopic 2

Successive percentage changes

When a value is changed by one percentage and then another, the two changes do not simply add together — the second change applies to the already-changed value.

The successive change trap

Net effect of two successive changes of x% and y% = x + y + xy/100
A rise of x% followed by a fall of x% gives a net change of −x²/100 (always a loss)

A price reduced by 20% and then increased by 20% does NOT return to the original price — it ends up lower, because the second 20% is calculated on a smaller base.

Memory trick — "Same percentage up then down is always a net loss"

Whenever a value goes up by x% and then down by the same x% (or vice versa), the net result is always a decrease of x²/100 percent — never a full return to the original value.

Hard

The price of an item is reduced by 20% and then increased by 20%. Compared to the original price, the final price is:

Net change = −20+20+(−20×20)/100 = −4%, so the final price is 4% lower than the original.

Subtopic 3 · highest yield

Profit, loss and discount

Profit and loss problems are percentage problems in disguise, built around cost price (CP), selling price (SP), and marked price (MP).

Core profit/loss relationships

Profit% = (SP−CP)/CP × 100 · Loss% = (CP−SP)/CP × 100
SP = MP − Discount · Discount is always calculated on the marked price, never on the cost price

A common trap: when a trader marks goods above cost price and then offers a discount, the discount percentage is NOT simply the markup minus the profit — both must be calculated through the marked price step by step.

Figure 3.1 — CP, MP and SP on one line A simple horizontal diagram showing Cost Price, then an arrow labelled "mark up" to Marked Price, then an arrow labelled "discount" to Selling Price, with profit or loss measured as the gap between CP and SP. Upload image here

✈ AAI ATC relevance — profit/loss/discount questions, especially the marked-price traps, are the single biggest chunk of this whole topic on the real exam

Memory trick — "Discount always eats into the mark-up, never the cost"

Discount is subtracted from the Marked Price to get the Selling Price. It never touches the Cost Price directly — whether the trader still profits depends on how the mark-up and discount percentages compare.

Moderate

A shopkeeper buys an article for Rs. 400 and sells it for Rs. 460. His profit percentage is:

Profit% = (460−400)/400 × 100 = 60/400 × 100 = 15%.

Hard

A trader marks his goods 40% above the cost price and allows a discount of 20% on the marked price. His actual profit percentage is:

Let CP = 100. Marked price = 140. Selling price after 20% discount = 140 × 0.8 = 112. Profit = 112−100 = 12, so profit% = 12%. (This uses the standard net-effect formula on +40% and −20%.)

Exam strategy

Facts worth having automatic

x% of N(x/100) × N.

Reversing an increaseDivide by 100+x, not subtract x%.

Reversing a decreaseDivide by 100−x, not add x%.

Same % up then downAlways a net loss of x²/100 percent.

Discount ruleAlways calculated on Marked Price, never on Cost Price.

Quick revision

Chapter summary sheet

x% of N(x/100) × N.

Reverse an increaseN × 100/(100+x).

Reverse a decreaseN × 100/(100−x).

Successive % changex + y + xy/100.

Same % up/downNet loss of x²/100 percent, always.

Profit%(SP−CP)/CP × 100.

Loss%(CP−SP)/CP × 100.

Discount baseAlways the Marked Price, never Cost Price.

↓ Free download — Percentages, Profit and Loss question PDF

All 5 questions with a full answer key.

Download the free PDF Join our live classes — extremely affordable

End of chapter

Comprehensive assessment

Full solutions are in the downloadable PDF above.

Multiple choice questions

1. If 20% of a number is 50, find the number.

2. A number is increased by 25% to become 100. Find the original number.

3. A price is reduced by 20% and then increased by 20%. Find the net percentage change.

4. Define profit percentage and loss percentage in terms of CP and SP.

5. A shopkeeper buys an article for Rs. 400 and sells it for Rs. 460. Find the profit percentage.

6. A trader marks goods 40% above cost price and gives a 20% discount. Find the actual profit percentage.

7. Explain why discount is calculated on the marked price, not the cost price.

8. If A's income is 20% more than B's, find what percent less B's income is than A's.

9. A student needs 40% to pass and scores 150, failing by 30 marks. Find the total marks.

10. By selling an item for Rs. 3,540, a shopkeeper gains 18%. Find the cost price.