Percentage

high importance~2 Q in Tier 121 formulas⚡ 15 shortcuts5 subtopics

Overview Learn (5)Formulas (21)Examples (5)Practice (66)

Subtopic 5 of 5·← Population growth, depreciation & elections

Marks, income–expenditure–savings & price–consumption

Pass-marks template: candidate 1 scores $p\%$ and fails by $f$ marks ⇒ pass mark $= \frac{pM}{100} + f$ ; candidate 2 scores $q\%$ and exceeds by $e$ ⇒ pass mark $= \frac{qM}{100} - e$ . Equate and solve for maximum marks $M$ .

Income–expenditure: savings = income − expenditure. If income is I and savings s%, expenditure = (100 − s)% of I.

Price rises, keep expenditure fixed: consumption must fall by $\frac{100r}{100 + r}\%$ . Price falls, same quantity-money: extra quantity bought $= \frac{\text{money} \times r}{100 \times \text{reduced price}}$ ; reduced price $= \frac{r\% \text{ of outlay}}{\text{extra quantity}}$ .

📖 Notes 🧩 4 question types ⚡ Formulas 🎯 13 practice Q

Detailed notes

Pass-marks problems

The pass mark is one fixed number seen from two candidates:

A scores p% and fails by f marks → pass $= \frac{pM}{100} + f$ .
B scores q% and scores e marks more than the pass mark → pass $= \frac{qM}{100} - e$ . Equate the two: $\frac{(q - p)M}{100} = f + e$ → find M, then the pass mark. Example: 33% fails by 45, 45% exceeds by 15 → 12% of M = 60 → M = 500, pass = 210. Signs matter: fail-by pushes the pass mark up from the score, exceeds-by pulls it down. Single-candidate version: if the pass mark is given directly, $\frac{pM}{100} + f = \text{pass}$ → M.

Income–expenditure–savings bookkeeping

savings = income − expenditure. Set income = 100 and all three numbers stay visible. Example: spends 80% of income; income rises 15%, expenditure rises 10%: $100/80/20 \to 115/88/27$ → savings rise $\frac{7}{20} \times 100 = 35\%$ . The percentage change in savings is almost never equal to the change in income — always recompute the new savings. If a person saves s% of income, expenditure is (100 − s)% of it.

Price vs consumption

expenditure = price × quantity, so quantity chip = expenditure chip ÷ price chip.

Price +25%, expenditure unchanged → quantity chip $= 1 \div \frac{5}{4} = \frac{4}{5}$ → cut 20%. (Standard formula: cut $= \frac{100r}{100 + r}\%$ .)
Price +20%, expenditure allowed to rise 8% → chip $= \frac{108}{120} = 0.9$ → cut 10%.
Price +40%, consumption cut 25% → expenditure chip $= 1.4 \times 0.75 = 1.05$ → +5%. Never equate the consumption cut to the price rise — they are related by division, not equality.

Price cut with extra quantity

A price cut of r% on an outlay M saves r% of M; the saving buys the extra quantity at the reduced price. reduced price = (r% of M) ÷ extra quantity → original price = reduced ÷ (1 − r/100). Example: a 20% cut lets a family buy 5 kg more for ₹400 → saved = ₹80 → reduced price = 16/kg → original $= 16 \times \frac{5}{4} = ₹20$ .

Exam traps

Fail-by and exceed-by act in opposite directions (+f, −e); a same-sign slip ruins M.
Savings % change ≠ income % change; compute the new savings from scratch.
Consumption cut ≠ price rise (20% ≠ 25%): divide the chips.
Depreciation/price-cut chains: work in chips, not in added percentages.

Quick revision

$\frac{(q - p)M}{100} = f + e$ for two candidates.
Income = 100 bookkeeping solves every savings question.
Quantity chip = expenditure chip ÷ price chip.
An r% cut saves r% of the outlay → extra quantity at the reduced price.

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: Pass marks with two candidatesvery common2 practice Q

How to spot it:

'A candidate scoring p% fails by f marks; another scoring q% gets e marks more than the pass mark.'

\frac{(q - p)M}{100} = f + e \ \Rightarrow\ M = \frac{100(f + e)}{q - p}

Write pass mark from each candidate: pM/100 + f and qM/100 − e.
Equate → (q − p)% of M = f + e → maximum marks M.
Pass mark = pM/100 + f (verify with the second candidate).

Why: both expressions equal the same pass mark; subtracting them cancels it and leaves M.

Example: A candidate scoring 33% fails by 45 marks, while another scoring 45% gets 15 marks more than the pass marks. Find the maximum marks.

$12\%$ of M $= 45 + 15 = 60$ → M $= 500$ ; pass mark $= 165 + 45 = 210$ .

Type 2: Income–expenditure–savings changevery common2 practice Q

How to spot it:

'He spends 80% of his income; income rises a% and expenditure rises b% — % change in savings?'

\text{savings} = \text{income} - \text{expenditure}, \ \text{track with income} = 100

Set income = 100; write expenditure and savings.
Apply the two % changes to income and expenditure separately.
New savings = new income − new expenditure; % change on the old savings.

Why: savings is a leftover, not a base — its % change must be computed, never guessed.

Example: A person spends 80% of his income. His income increases by 15% and his expenditure increases by 10%. The percentage increase in his savings is:

Income 100/80/20 → 115/88/27 → savings rise $\frac{7}{20} \times 100 = 35\%$ .

Type 3: Price rise vs consumption (expenditure target)very common2 practice Q

How to spot it:

'The price rises by r% — by what % must consumption be cut so the expenditure stays the same?' (Or the expenditure is allowed to move by s%.)

\text{cut}\% = \frac{100r}{100 + r}\%, \qquad \text{quantity chip} = \frac{\text{expenditure chip}}{\text{price chip}}

Write chips: price $\frac{100 + r}{100}$ , expenditure $\frac{100 + s}{100}$ (s = 0 for 'same').
Quantity chip = expenditure chip ÷ price chip.
1 − quantity chip is the cut.

Why: expenditure = price × quantity, so quantity adjusts by the ratio of the other two chips.

Example: The price of petrol rises by 25%. By what per cent must a family reduce its consumption so that the expenditure on petrol remains the same?

Quantity chip $= 1 \div \frac{5}{4} = \frac{4}{5}$ → cut $= 20\%$ ( $= \frac{100 \times 25}{125}$ ).

Type 4: Price cut → extra quantity for the same moneycommon2 practice Q

How to spot it:

'A reduction of r% in the price enables a family to buy k kg more for ₹M — find the (original/reduced) price.'

\text{reduced price} = \frac{r\% \text{ of } M}{k}, \qquad \text{original} = \frac{\text{reduced}}{1 - r/100}

The r% cut saves r% of M — that saving is spent on the extra k units.
Reduced price = saving ÷ k.
Original price = reduced price ÷ (1 − r/100).

Why: the outlay is unchanged, so the money freed by the cut is exactly what buys the extra quantity.

Example: A reduction of 20% in the price of cooking oil enables a family to buy 5 kg more for ₹400. The original price per kg is:

Saving = 20% of 400 = ₹80 → reduced price $= \frac{80}{5} = ₹16$ → original $= 16 \times \frac{5}{4} = ₹20$ .

Formulas

Pass marks equation

\frac{pM}{100} + f = \frac{qM}{100} - e

Savings identity

\text{savings} = \text{income} - \text{expenditure}

Price up r%, expenditure fixed

\text{reduce consumption by } \frac{100r}{100+r}\%

Price cut & extra quantity

\text{reduced price} = \frac{r\% \text{ of outlay}}{\text{extra quantity}},\quad \text{old price} = \frac{\text{reduced price}}{1 - r/100}

Shortcut tricks

⚡ Marks gap = percentage gap

In two-candidate mark problems the difference of the two percentages times M equals the total mark gap (fail-by + exceed-by).

Example: A candidate scoring 33% fails by 45 marks; another scoring 45% gets 15 marks more than the pass mark. Find the maximum marks.

$(45 - 33)\%$ of M = 45 + 15 = 60 ⇒ $M = 500$ .

⚡ Income = 100 bookkeeping

Set income to 100, express savings and expenditure, then apply every change and read the answer off.

Example: A man saves 20% of his income. His income rises 15% and expenditure rises 10%. % increase in savings?

Income 100 → save 20, spend 80. New: income 115, spend 88 ⇒ save 27. Increase = 7/20 = 35%.

⚡ Price cut: r% of money buys the extra

A r% price cut saves r% of the outlay, and that saved money buys the extra quantity at the REDUCED price.

Example: A 20% cut in sugar price lets a buyer get 5 kg more for ₹400. Find the original price.

Saved money = 20% of 400 = ₹80 = price of extra 5 kg ⇒ reduced price ₹16/kg ⇒ original price = ₹ $16 \times \frac{100}{80}$ = ₹20 per kg.

Where students lose marks

Using the fail-by and exceed-by marks with the same sign (they act in opposite directions: +f and −e on the two sides).
In savings problems, raising expenditure by 10% as 'reduce savings by 10%'.
For the price-cut question, dividing the full outlay (not r% of it) by the extra quantity.
Consumption reduction for a 25% price rise answered as 25% instead of 20%.

Practice sets — 13 questions

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

Topic test · 13 questions

Suggested time 9 min · wrong answers go to your mistake notebook automatically.