ExamShortcut
🔍Log in

Percentage

🔒 Log in to track
high importance~2 Q in Tier 121 formulas⚡ 15 shortcuts5 subtopics
OverviewLearn (5)Formulas (21)Examples (5)Practice (66)
Subtopic 2 of 5·← Percentage meaning & conversions·'x% more than' ↔ 'x% less than' & chains →

Percentage increase / decrease & successive change

🔒 Log in to track

Percentage change =final−initialinitial×100= \frac{\text{final} - \text{initial}}{\text{initial}} \times 100 (initial = the base being changed).

Two successive changes of a%a\% and b%b\% combine to a+b+ab100a + b + \frac{ab}{100} (keep minus signs for decreases).

+10% then −10% is not zero: net =−102100=−1%= -\frac{10^2}{100} = -1\% (a fall of a²/100 %).

To restore an original value after a multiplier, divide back: if F=I×120100F = I \times \frac{120}{100} then I=F×100120I = F \times \frac{100}{120}.

Model every change as a fraction multiplier: +20% → ×6/5, −25% → ×3/4. Chain the multipliers.

📖 Notes🧩 5 question types⚡ Formulas🎯 14 practice Q

Detailed notes

Percentage change and its base

% change=final−initialinitial×100\%\ \text{change} = \frac{\text{final} - \text{initial}}{\text{initial}} \times 100 The denominator is the value before the change. ₹250 → ₹200 is a fall of 50250=20%\frac{50}{250} = 20\%; going back from ₹200 to ₹250 is a rise of 50200=25%\frac{50}{200} = 25\%. Same ₹50, different base, different percentage.

The multiplier ("chip") idea — the master shortcut

Every change is a chip: +20%→×65+20\% \to \times\frac{6}{5}, −25%→×34-25\% \to \times\frac{3}{4}, +8%→×2725+8\% \to \times\frac{27}{25}. Forward: multiply the chips. Backward: divide by them. Example: +20%+20\% then −10%-10\%: 65×910=2725\frac{6}{5} \times \frac{9}{10} = \frac{27}{25} → net +8%+8\%. Example: two −10%-10\% steps leave 8100 → before: 8100×109×109=100008100 \times \frac{10}{9} \times \frac{10}{9} = 10000.

Two successive changes: a+b+ab100a + b + \frac{ab}{100}

Keep the signs: +30%+30\% then −20%-20\%: 30−20−600100=+4%30 - 20 - \frac{600}{100} = +4\%. Both increases: +15%+15\% then +12%+12\%: 15+12+1.8=+28.8%15 + 12 + 1.8 = +28.8\%. Works only for exactly two changes; for three or more, chain the chips.

The round-trip rule: +x%+x\% and −x%-x\% never cancel

Same size up then down (either order): net =−x2100%= -\frac{x^2}{100}\%. +20%+20\% then −20%-20\%: net −4%-4\%. A fall of 25% then a rise of 25%: net −6.25%-6.25\%. The percentage always lands a little below the start — remember this to kill "no change" options at sight.

Numerator–denominator changes

If the numerator changes by a%a\% and the denominator by b%b\%, the fraction's chip is 100+a100+b\frac{100 + a}{100 + b} (a ratio of chips): numerator +50%+50\%, denominator −25%-25\%: 150100÷75100=2\frac{150}{100} \div \frac{75}{100} = 2 → the fraction doubles (+100%+100\%).

Restoring the original value

To undo +25%+25\%, divide by 54\frac{5}{4} — do not apply −25%-25\%. A salary of ₹50,000 after a 25% rise was 50000×45=₹40,00050000 \times \frac{4}{5} = ₹40{,}000 before. Applying −25%-25\% gives 37,500 — the classic wrong answer. Same for cuts: after a 12.5%12.5\% cut a price is ₹3,500 → original =3500÷78=₹4,000= 3500 \div \frac{7}{8} = ₹4{,}000.

How the questions are dressed up

The same five ideas arrive as prices, salaries, output of a factory, exam scores, or the value of a share — the chips do not care about the story. Two changes in the wording ("and then", "subsequently", "after some months") always means successive; a single change with two numbers (from A to B) is a plain c1. The options are built from classic slips: wrong base in the change %, subtracting instead of dividing to undo, and "no change" for the round trip.

Quick revision

  • % change: base is the value before the change.
  • Chips: multiply forward, divide backward.
  • Two changes: a+b+ab100a + b + \frac{ab}{100} with signs.
  • +x%+x\% and −x%-x\% in any order → net −x2100%-\frac{x^2}{100}\%.
  • Undo a change with the reciprocal chip, never the opposite percentage.

Types of questions asked

Every way this subtopic shows up in exams — how to recognise it, the formula or logic to use, and a solved example.

Type 1: Single percentage increase or decreasevery common2 practice Q
How to spot it:

A value moves from A to B and the % change is asked — price, population, marks, anything.

% change=final−initialinitial×100\%\ \text{change} = \frac{\text{final} - \text{initial}}{\text{initial}} \times 100
  1. Compute the absolute change (final − initial).
  2. Divide by the INITIAL value — the one before the change.
  3. Multiply by 100; sign tells the direction.

Why: a percentage change always measures the movement against where the value started.

Example: The salary of an employee rises from ₹18,000 to ₹20,700. Find the percentage increase.

Increase =2700= 2700 on base 18000 → 270018000×100=15%\frac{2700}{18000} \times 100 = 15\%.

Type 2: Two successive changes (net change)very common3 practice Q
How to spot it:

'Increased by a% and then decreased by b% — net change?' Often dressed as price, value or output.

a+b+ab100  (+ increase, − decrease)a + b + \frac{ab}{100} \ \ (+\text{ increase},\ -\text{ decrease})
  1. Apply a+b+ab100a + b + \frac{ab}{100} with the signs of the two changes.
  2. Or multiply the chips (100+a100×100+b100\frac{100+a}{100} \times \frac{100+b}{100}) and read the net.
  3. The chip method is safe for any number of changes.

Why: the second change acts on the changed value, and the ab100\frac{ab}{100} term is exactly that correction.

Example: A number is increased by 15% and then decreased by 12%. The net change is:

15−12−15×12100=3−1.8=+3.2%15 - 12 - \frac{15 \times 12}{100} = 3 - 1.8 = +3.2\%. Chips: 2320×2225=506500=1.012\frac{23}{20} \times \frac{22}{25} = \frac{506}{500} = 1.012.

Type 3: Same % up and down (round trip)very common2 practice Q
How to spot it:

'The price rises x% and then falls x%' (or falls then rises) — a guaranteed 'no change' trap among the options.

net=−x2100%\text{net} = -\frac{x^2}{100}\%
  1. Write the net change as −x2100%-\frac{x^2}{100}\% — order does not matter.
  2. Or verify with chips: 120100×80100=2425=0.96\frac{120}{100} \times \frac{80}{100} = \frac{24}{25} = 0.96 → −4%-4\%.
  3. The final value is always BELOW the start (for any x > 0).

Why: the second change works on a bigger or smaller base, so the absolute swings differ.

Example: The price of a commodity first rises by 20% and then falls by 20%. The net change in price is:

Net =−202100=−4%= -\frac{20^2}{100} = -4\%, a 4%4\% decrease. (Check: 100→120→96100 \to 120 \to 96.)

Type 4: Restore the original value (work backwards)very common3 practice Q
How to spot it:

The value AFTER the change(s) is given and the original is asked: 'after a 20% increase the population is 8,400…'.

original=final÷(chip)=final×100100+a\text{original} = \text{final} \div \text{(chip)} = \text{final} \times \frac{100}{100 + a}
  1. Write the chip of each change.
  2. Divide the final value by the chips (or multiply by their reciprocals).
  3. Never 'apply the opposite percentage' — that uses the wrong base.

Why: forward = multiply by chips, so backward = divide by the same chips.

Example: After successive decreases of 10% and 20%, a number becomes 720. Find the number.

N×910×45=720N \times \frac{9}{10} \times \frac{4}{5} = 720 → N×0.72=720N \times 0.72 = 720 → N=1000N = 1000.

Type 5: Fraction changes: numerator and denominator movecommon2 practice Q
How to spot it:

'The numerator is increased by 20% and the denominator decreased by 10% — the fraction becomes?'

new factor=100+a100+b ⇒ % change=100+a100+b−1\text{new factor} = \frac{100 + a}{100 + b} \ \Rightarrow\ \%\ \text{change} = \frac{100 + a}{100 + b} - 1
  1. Write the chip of the numerator and of the denominator.
  2. Divide the numerator chip by the denominator chip — that is the fraction's new factor.
  3. Factor − 1 is the percentage change of the fraction.

Why: the fraction scales by (numerator chip)/(denominator chip), since denominator changes divide.

Example: The numerator of a fraction is increased by 50% and the denominator is decreased by 25%. The fraction becomes:

Chips: 32\frac{3}{2} and 34\frac{3}{4} → factor =32÷34=2= \frac{3}{2} \div \frac{3}{4} = 2 → the fraction is doubled (+100%+100\%).

Formulas

Percentage change
% change=final−initialinitial×100\%\ \text{change} = \frac{\text{final} - \text{initial}}{\text{initial}} \times 100
Successive change
a+b+ab100a + b + \frac{ab}{100}

use signs: + for increase, − for decrease

Same % up and down
+x% then −x%⇒net=−x2100%+x\% \text{ then } -x\% \Rightarrow \text{net} = -\frac{x^2}{100}\%
Multiplier form
final=initial×100+a100×100−b100\text{final} = \text{initial} \times \frac{100 + a}{100} \times \frac{100 - b}{100}
Original value back
initial=final×100100+a\text{initial} = \text{final} \times \frac{100}{100 + a}

Shortcut tricks

⚡ Multiplier chaining

Replace every % by its fraction multiplier and multiply across all steps.

Example: A number is increased by 20% and the result is decreased by 10%. Net change?

65×910=5450=2725\frac{6}{5} \times \frac{9}{10} = \frac{54}{50} = \frac{27}{25} ⇒ net +225=+8%+\frac{2}{25} = +8\%.

⚡ a + b + ab/100 in one line

Add, then adjust by the product term. Fastest when one change is small.

Example: Successive changes: +30% and −20%.

30−20−600100=10−6=4%30 - 20 - \frac{600}{100} = 10 - 6 = 4\% increase.

⚡ Reverse to the original

To undo +25%, divide by 5/4 (not multiply by 75/100).

Example: After a 25% increase a salary is ₹50,000. What was it before?

Salary after increase = ₹50,000. Before = ₹50000×10012550000 \times \frac{100}{125}, i.e. ₹40,000.

Where students lose marks

  • Adding successive percentages without the ab100\frac{ab}{100} term.

  • Assuming +x% followed by −x% is a net zero.

  • Using the new (increased) value as the base when computing a percentage decrease.

  • Undoing +25% by applying −25% (the right move is ÷1.25).

Practice sets — 14 questions

Sets of 10, mixed across the question types above. Each answer comes with a step-by-step explanation.

Topic test · 10 questions

Suggested time 5 min · wrong answers go to your mistake notebook automatically.