Percentage for RRB NTPC: methods, shortcuts and practice

"Per cent" means "per hundred". That one idea sits under half the arithmetic in RRB NTPC. Profit and loss is percentage on cost price. Interest is percentage on a principal. A data interpretation table asks for percentage change. If percentage is slow for you, every one of those topics is slow too.

The good news is that percentage questions come in a small number of shapes. Learn the shapes, and the reason each shortcut works, and you can answer most of them in under 45 seconds.

Percentages are fractions

A percentage is a fraction with denominator 100, so 25% = 25/100 = 1/4. Thinking in fractions turns multiplication into short division: 25% of 360 is 360 ÷ 4 = 90.

Two more habits help. Break a percentage into easy pieces: 15% of 640 = 10% (64) + 5% (32) = 96. Swap the numbers when it helps: x% of y = y% of x, so 16% of 25 = 25% of 16 = 4.

Percentage change is always on the original

Percentage change = (change ÷ original value) × 100.

If a price goes from ₹250 to ₹300, the change is 50 and the original is 250, so the rise is 20%. Going back from ₹300 to ₹250 is a fall of 50 on 300, which is 16 2/3%. Same ₹50, different base, different percentage.

Successive changes: a + b + ab/100

For a change of a% followed by b% (use a minus sign for a fall), the net change is a + b + ab/100 per cent.

Why it works: the multiplier is (1 + a/100)(1 + b/100) = 1 + a/100 + b/100 + ab/10,000. Subtract 1 and multiply by 100. The ab/100 term is the "change on the change", which is why a 20% rise and a 20% fall do not cancel.

"More than" and "less than"

If A is r% more than B, then B is less than A by r/(100 + r) × 100 per cent.
If A is r% less than B, then B is more than A by r/(100 − r) × 100 per cent.

Why: the gap between A and B is the same amount both ways, but the base changes. Take B = 100 and A = 125. The gap of 25 is 25% of B but only 20% of A.

Price and consumption

Expenditure = price × consumption. If the price rises by r% and spending must stay the same, consumption must fall by r/(100 + r) × 100 per cent. If the price falls by r%, consumption can rise by r/(100 − r) × 100 per cent.

Why: this is the "more than, less than" rule again. For the product to stay fixed, consumption must shrink by the same factor that price grows.

Growth and depreciation

A population growing at r% a year for n years becomes P × (1 + r/100)ⁿ. A machine losing value at r% a year becomes P × (1 − r/100)ⁿ. This is the compound interest formula in another setting, so see compound interest for the fast year-by-year method.

Worked questions at NTPC level

Q1. A price rises by 20% and then falls by 20%. Find the net change.

20 − 20 + (20 × −20)/100 = −4. A 4% decrease.
Check: 100 → 120 → 96.

Q2. A's salary is 25% more than B's. By what percentage is B's salary less than A's?

Let B = 100, so A = 125. B is 25 less than A: 25 ÷ 125 × 100 = 20%.

Q3. The price of sugar rises by 25%. By what percentage must a family cut consumption to keep spending the same?

25 ÷ 125 × 100 = 20%.
Check: price 100 → 125, consumption 100 → 80; spending 100 × 100 = 125 × 80 = 10,000.

Q4. A town's population is 20,000 and grows by 10% each year. What will it be after 2 years?

20,000 × 1.1 = 22,000; 22,000 × 1.1 = 24,200.

Q5. A candidate scoring 30% fails by 15 marks. Another scoring 40% gets 35 marks more than the pass mark. Find the maximum marks and the pass mark.

The gap between the two candidates is 40% − 30% = 10% of the maximum, and in marks it is 15 + 35 = 50.
So 10% = 50, and the maximum is 500. Pass mark: 30% of 500 + 15 = 165.
Check: 40% of 500 = 200, and 200 − 35 = 165.

Q6. In an election between two candidates, 10% of voters did not vote. The winner got 60% of the votes cast and won by 1,800 votes. How many voters were on the list?

The loser got 40% of the votes cast, so the margin is 20% of votes cast. 20% = 1,800, so votes cast = 9,000.
Votes cast are 90% of the list: 9,000 ÷ 0.9 = 10,000.

Common mistakes

• Wrong base. Percentage change is on the original value, not the new one.
• Assuming +x% and −x% cancel. They always leave a loss of x²/100 per cent.
• Adding percentages of different bases. 10% of A plus 10% of B is not 20% of anything unless A = B.
• Forgetting non-voters or invalid votes. In election questions, check whether the percentage is of votes cast or of the whole list.
• Answering a percentage when the question asks for a number, or the other way round.

Practice set

1. Find 35% of 360.
2. 45 is what percentage of 180?
3. A number is increased by 10% and then decreased by 10%. Find the net change.
4. If the price of an item falls by 20%, by what percentage can a buyer increase consumption for the same spending?
5. A's income is 20% less than B's. By what percentage is B's income more than A's?
6. When a number is increased by 20%, it becomes 150. Find the number.
7. A town of 50,000 people shrinks by 4% a year. What will its population be after 2 years?
8. To pass, a student needs 35% of the maximum marks. The student scores 212 and fails by 33 marks. Find the maximum marks.

Answers:

1. 126. 10% = 36, so 30% = 108 and 5% = 18; 108 + 18 = 126.
2. 25%. 45/180 = 1/4.
3. 1% decrease. 10 − 10 + (10 × −10)/100 = −1.
4. 25%. 20 ÷ 80 × 100.
5. 25%. Let B = 100, so A = 80; B is 20 more, and 20 ÷ 80 × 100 = 25.
6. 125. 150 ÷ 1.2.
7. 46,080. 50,000 × 0.96 = 48,000; 48,000 × 0.96 = 46,080.
8. 700. Pass mark = 212 + 33 = 245, which is 35%; 245 ÷ 0.35 = 700.

What to do next

• Learn the fraction–percentage table above; test yourself both ways, fraction to percentage and back.
• Solve 20 percentage questions daily for a week, timed at 45 seconds each.
• Write the successive-change and "more than, less than" rules in your formula notebook, with the reason beside each.
• Move on to profit and loss, which is percentage applied to buying and selling, and ratio and proportion.