Marks, income–expenditure–savings & price–consumption — notes, question types & practice · Percentage

01

Pass marks with two candidates

One candidate scores $x\%$ and fails by $a$ marks. Another scores $y\%$ and gets $b$ marks more than the pass mark.

The gap between their scores covers $(y - x)\%$ of the maximum marks and equals $a + b$ marks.

M = \frac{(a + b) \times 100}{y - x}

Worked: A scores $33\%$ and fails by 45 marks. B scores $45\%$ and gets 15 marks more than the pass mark.

Span of percents $= 45 - 33 = 12$ .
Span of marks $= 45 + 15 = 60$ .
$M = 60 \times \dfrac{100}{12} = 500$ .
Pass mark $= 33\%$ of $500 + 45 = 165 + 45 = 210$ .

Rule: One candidate sits below the pass mark, the other above. The whole distance is $a + b$ .

02

Pass marks with one candidate

"Fails by $a$ " means the score plus $a$ equals the pass mark.

A student scores $30\%$ and fails by 20 marks; the pass mark is 152. Then $30\%$ of $M = 152 - 20 = 132$ , so $M = 440$ .

Tip: Write " $\text{score} + \text{shortfall} = \text{pass}$ " as a line before substituting anything.

03

Income, expenditure and savings

Income $=$ expenditure $+$ savings. Take income as a friendly 100 units and the percents become the money itself.

Item	Before	After
Income	400	$400 \times \dfrac{23}{20} = 460$
Expenditure (80% of income)	320	$320 \times \dfrac{11}{10} = 352$
Savings	80	$460 - 352 = 108$

Savings rose by $\dfrac{28}{80} \times 100 = 35\%$ .

Watch: The savings per cent is measured on the old savings (80), never on income.

04

Price rises and consumption

Same budget, higher price, so the quantity must fall.

\text{quantity chip} = \frac{\text{budget chip}}{\text{price chip}}

Petrol price rises $25\%$ : price chip $\dfrac{5}{4}$ , budget chip 1, so quantity chip $= 1 \div \dfrac{5}{4} = \dfrac{4}{5}$ . Consumption must fall $20\%$ .

Note: A $25\%$ rise pairs with a $20\%$ cut. This is the more/less flip of the previous subtopic.

05

A price cut and the extra quantity

Same money after a price cut buys more.

Price falls $20\%$ : chip $\dfrac{4}{5}$ . Quantity chip $= 1 \div \dfrac{4}{5} = \dfrac{5}{4}$ , a $25\%$ rise.

Worked: Rs 400 spent on rice. After a $20\%$ price cut, the same money buys 5 kg more.

Old price Rs $p$ per kg: old quantity $= \dfrac{400}{p}$ .
New price $= 0.8p$ ; new quantity $= \dfrac{400}{0.8p} = \dfrac{500}{p}$ .
Extra $= \dfrac{100}{p} = 5$ , so $p = 20$ .

Careful: The extra quantity per cent is not the cut per cent. A $20\%$ cut buys $25\%$ more.

06

Checking the direction

Every question in this subtopic is a chain of chips. Before answering, say the direction out loud: are you moving from old to new, or back?

Marks: percents of $M$ move up to marks.
Income: chips move income and expenditure forward, savings are the remainder.
Prices: budget chip over price chip gives the quantity chip.

Example: Income 400 rising $15\%$ is $400 \times \dfrac{23}{20} = 460$ : multiply forward, never add 15% of something else.

07

Question types you will see

Each type: how to recognise it, the method step by step, and one question to try.

Type 1very common2 practice Q

Two candidates and the maximum marks

How to spot it:

One candidate fails by some marks, another crosses the pass mark; maximum marks or the pass mark is asked.

M = \frac{(a+b) \times 100}{y - x}

Method

Note the two percents and the two distances from the pass mark.
Marks span = shortfall + surplus.
Percent span = higher percent minus lower percent.
Maximum marks = marks span x 100 / percent span.

Why it works:

Both candidates' scores are percents of the same maximum, so their gap fixes it.

Try this

In an examination, A scores 33% and fails by 45 marks, while B scores 45% and gets 15 marks more than the pass mark. What are the maximum marks?

Show solution

Marks span $= 45 + 15 = 60$ .
Percent span $= 45 - 33 = 12$ .
$M = 60 \times \dfrac{100}{12} = 500$ .

Answer

500

Type 2common2 practice Q

Income rise and the change in savings

How to spot it:

Income and expenditure both change by percents; the effect on savings is asked.

\text{savings change} = \frac{\text{new savings} - \text{old savings}}{\text{old savings}} \times 100

Method

Assume a friendly income (say 400) and the expenditure percent.
Move income and expenditure forward with chips.
New savings = new income - new expenditure.
Compare with old savings for the per cent change.

Why it works:

Savings are just the remainder, so both chips flow into it.

Try this

A family earns Rs 400 a month and spends 80% of it. Income rises by 15% and expenditure rises by 10%. By what percentage do the savings increase?

Show solution

New income $= 400 \times \dfrac{23}{20} = 460$ ; new spending $= 320 \times \dfrac{11}{10} = 352$ .
Old savings $= 80$ ; new savings $= 460 - 352 = 108$ .
$\dfrac{28}{80} \times 100 = 35$ .

Answer

35%

Type 3very common2 practice Q

Price rise, consumption cut

How to spot it:

A price rises by a percent and the matching cut in consumption (same budget) is asked.

\frac{100x}{100 + x}\%

Method

Note the price rise x.
Compute 100x over 100 + x.
State it as the cut needed.

Why it works:

With a fixed budget, quantity chip is one over the price chip.

Try this

The price of petrol rises by 25%. By what percentage must consumption be reduced to keep the expenditure unchanged?

Show solution

Price chip $= \dfrac{5}{4}$ .
Quantity chip $= 1 \div \dfrac{5}{4} = \dfrac{4}{5}$ .
Fall $= 1 - \dfrac{4}{5} = \dfrac{1}{5} = 20\%$ .

Answer

20%

Type 4common2 practice Q

Price cut and the extra quantity

How to spot it:

A price falls by a percent; the same money now buys extra kilograms, and the price is asked.

\text{extra}\% = \frac{100c}{100 - c}\%

Method

Write the quantity chip as one over the price chip.
Extra per cent = quantity chip minus 1.
Use the extra kilograms to find the old quantity.
Price = money divided by old quantity.

Why it works:

A fixed budget divided by a smaller price gives a bigger quantity, and the surplus pins the old quantity.

Try this

A household spends Rs 400 on rice every month. After a 20% fall in the price of rice, the same money buys 5 kg more. What was the original price per kg?

Show solution

Quantity chip $= \dfrac{5}{4}$ , so extra $= \dfrac{1}{4}$ of old quantity $= 5$ kg.
Old quantity $= 20$ kg.
Price $= \dfrac{400}{20} = 20$ .

Answer

Rs 20 per kg

Type 5common

One candidate and the pass mark

How to spot it:

A single candidate's percent, the shortfall, and the pass mark are given; maximum marks is asked.

M = \frac{(\text{pass} - a) \times 100}{x}

Method

Turn the shortfall into the candidate's marks: pass minus shortfall.
That equals x% of the maximum.
Divide and multiply by 100 for M.

Why it works:

One candidate's score is one percent slice of the maximum, so one equation is enough.

Try this

A student scores 30% in an examination and fails by 20 marks. If the pass mark is 152, what are the maximum marks?

Show solution

Student's marks $= 152 - 20 = 132$ .
$\dfrac{30}{100}M = 132$ .
$M = 132 \times \dfrac{100}{30} = 440$ .

Answer

440

08

Formula sheet

Maximum marks (two candidates)

M = \frac{(a+b) \times 100}{y - x}

a = shortfall below pass, b = marks above pass, x and y are the two percents.

Pass mark from failing

\text{pass} = \frac{x}{100}M + a

a is the shortfall.

Consumption cut after a price rise

\frac{100x}{100 + x}\%

Budget unchanged, price up x%.

Extra quantity after a price cut

\frac{100c}{100 - c}\%

Budget unchanged, price down c%.

09

Shortcuts that save time

⚡ Span of marks over span of percents

Two candidates give one clean fraction: marks apart over percents apart. Multiply by 100 for maximum marks.

Example

In an examination, A scores 33% and fails by 45 marks. B scores 45% and gets 15 marks more than the pass mark. Find the maximum marks.

Show solution

Marks apart $= 45 + 15 = 60$ .
Percents apart $= 45 - 33 = 12$ .
$M = 60 \times \dfrac{100}{12} = 500$ .

Answer

500

⚡ Price rise to consumption cut is a flip

Reuse the more/less flip: price up $x\%$ with the same budget means consumption down $\dfrac{100x}{100+x}\%$ .

Example

The price of petrol rises by 25%. By what per cent must a family cut its consumption to keep its spending the same?

Show solution

Cut $= \dfrac{100 \times 25}{125}$ .
$= 20$ .

Answer

20%

⚡ Extra kilograms reveal the price

After a price cut, extra kilograms equal money times the difference of the chips, divided by the new price. Solve for the price.

Example

Rs 400 is spent on rice. After a 20% fall in price, the same money buys 5 kg more. What was the original price per kg?

Show solution

Quantity chip $= \dfrac{5}{4}$ , so extra $= \dfrac{1}{4}$ of old quantity.
Old quantity $= 20$ kg, so extra $= 5$ kg matches.
Price $= \dfrac{400}{20} = 20$ per kg.

Answer

Rs 20 per kg

10

Mistakes to avoid

Where most students lose marks on this subtopic.

Mistake 01

Dividing by $(y - x)$ but forgetting to add $b$ to $a$ .

The marks span is the whole distance, shortfall plus surplus: $a + b$ .

Mistake 02

Writing pass mark $= x\%$ of $M - a$ for a failing candidate.

Failing means below pass: pass $= \dfrac{x}{100}M + a$ .

Mistake 03

Measuring the savings change on the new savings.

Per cent change uses the old value as the base.

Mistake 04

Cutting consumption by the same per cent as the price rise.

Price up $25\%$ needs only a $20\%$ cut: $\dfrac{100x}{100+x}$ .

Mistake 05

Calling the extra quantity per cent equal to the price cut.

A $20\%$ cut buys $\dfrac{100 \times 20}{80} = 25\%$ more.

Mistake 06

Taking income as 100 but expenditure as a real rupee figure.

Keep every number in the same units, assumed or real.

11

Quick revision

Read this the night before the exam.

Two candidates: $M = (a+b) \times 100 \div (y-x)$ .
Pass $= x\%$ of $M + a$ when a candidate fails by $a$ .
Income $=$ expenditure $+$ savings; assume income 100.
Savings change is measured on the old savings.
Price up $x\%$ , same budget: consumption down $\dfrac{100x}{100+x}\%$ .
Price down $c\%$ : quantity up $\dfrac{100c}{100-c}\%$ .

12

Practice: 13 questions

Sets of 10, mixed across the question types above. Every answer has a step-by-step explanation.

Topic test · 13 questions

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