SSC CGL Maths · Percentage, Profit and Loss

Percentage Word Problems

Elections, exam marks, savings and mixed groups: long stories that reduce to a chain of percentages or a weighted average.

All Percentage, Profit and Loss notes

Why this matters

43 questions in the bank, 29 of them MODERATE, and elections alone account for 15. The stems are long but the arithmetic is short once you see the chain. Reading carefully which total each percentage is taken of is where these marks are won or lost.

Concept 1 of 4: Elections: voters, votes cast, valid votes and the margin

An election question is a chain of shares: the voter list, then the people who voted, then the valid votes, then each candidate's share. With two candidates every valid vote goes to one of them, so the winning margin is the gap between their two shares.

Definition

Valid votes $=$ voters on the list $\times$ share who voted $\times$ share of votes that were valid.
With two candidates, if the winner gets w% of the valid votes, the loser gets $(100 - w)\%$ and the margin is $(2w - 100)\%$ of the valid votes.
Invalid votes given as a number are subtracted from votes cast; given as a percentage, they are a factor.
Fast route: go from the number you are given (a margin, a candidate's votes) to the valid votes first, then divide back up the chain one factor at a time.

Margin and valid votes

\text{margin} = (2w - 100)\% \text{ of valid votes} \qquad \text{valid} = N \times \frac{p}{100} \times \frac{q}{100}

wwinner's share of the valid votes, in per cent
Nvoters on the list
ppercentage of voters who voted
qpercentage of votes cast that were valid

Worked example

In an election between two candidates, 75% of the 40,000 voters on the list voted and 4% of the votes cast were invalid. The winner got 55% of the valid votes. By how many votes did he win?

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 13 Sep 2025, Shift 2 · Q59Moderate

Example 1 · Percentage, Profit and Loss · Percentage Word Problems

In an election,

80\%

of eligible voters cast their votes. Of these votes,

5\%

were declared invalid. If a winning candidate received 11,400 votes, which accounted for

60\%

of the total valid votes, what was the total number of eligible voters enrolled in that election?

Taking the winner's share as the margin

If the winner gets 56% of the valid votes, the margin is

56 - 44 = 12\%

of them, not 56%.

Taking invalid votes out of the whole list

'10% of the votes cast were invalid' is 10% of the people who voted, not of everyone on the list. Apply the turnout factor first, then the invalid share.

Drill 14 more on elections: voters, votes cast, valid votes and the margin

Concept 2 of 4: Pass marks and marks scored

The pass mark is a fixed percentage of the maximum marks. A student who 'fails by 14' is 14 below it; one who scores '20 more than the pass mark' is 20 above it. Either way the pass mark is the bridge to the maximum.

Definition

Pass mark $= \frac{p}{100} \times$ maximum marks.
Score $+$ shortfall $=$ pass mark $=$ score $-$ surplus.
Two students on the same paper: write each one's pass-mark expression and set them equal; solve for the maximum.
Overall percentage over several papers $= \frac{\text{total marks}}{\text{total maximum}} \times 100$ . It equals the simple average of the percentages only when every paper has the same maximum.

Pass mark

\frac{p}{100} \times M = \text{score} + \text{shortfall}

Worked example

The pass mark in a test is 40%. A student scores 150 and fails by 30 marks. Find the maximum marks.

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 15 Sep 2025, Shift 3 · Q58Easy

Example 2 · Percentage, Profit and Loss · Percentage Word Problems

In a competitive exam with 120 questions, there are three sections: General Knowledge (20 questions), Reasoning (50 questions), and English (50 questions). A candidate answered 70% of General Knowledge, 50% of Reasoning, and 30% of English questions correctly. If the minimum passing score is 60%, how many more questions did the candidate need to answer correctly to pass?

Subtracting the shortfall

'Failed by 30 marks' means the pass mark is 30 above the score. Add the shortfall to the score; do not take it away.

Averaging percentages from papers of different sizes

80% on a 50-mark paper and 60% on a 150-mark paper is 40 + 90 = 130 out of 200, which is 65%, not the 70% a simple average gives.

Drill 5 more on pass marks and marks scored

Concept 3 of 4: Income, expenditure and savings

Income splits into expenditure and savings. When income and expenditure change by different percentages, the savings change by a third, often much larger, percentage, because savings are the small piece left over.

Definition

Savings $=$ income $-$ expenditure, before and after the change.
Percentage change in savings $= \frac{\text{new savings} - \text{old savings}}{\text{old savings}} \times 100$ .
Fast route: take income as 100. If expenditure is E and savings are S $= 100 - E$ , a rise of i% in income and e% in expenditure changes savings by $s = \frac{100\,i - E\,e}{S}$ per cent.
If income rises and expenditure stays the same, the whole rise goes into savings.
'Spending is 40% more than savings' means spending $= 1.4 \times$ savings, so income $= 2.4 \times$ savings.

Savings and their change

\text{income} = \text{expenditure} + \text{savings} \qquad s = \frac{100\,i - E\,e}{S}

i, e, spercentage changes in income, expenditure and savings
E, Sexpenditure and savings when income is 100

Worked example

Ritu earns ₹50,000 a month and spends ₹40,000. Her income rises by 10% and her spending by 5%. By what percentage do her savings rise?

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 24 Sep 2025, Shift 3 · Q58Moderate

Example 3 · Percentage, Profit and Loss · Percentage Word Problems

A man spends

75\%

of his income. Due to inflation, his expenses increase by

20\%

, and his income increases by

10\%

. What is the percentage change in his savings?

Averaging the two changes

Income up 20% and spending up 12% does not make savings rise by anything in between. Savings are what is left, so work out old and new savings and compare those.

Drill 12 more on income, expenditure and savings

Concept 4 of 4: Combined percentage of several groups

When a whole is made of groups (men and women, two incomes, budget heads), the percentage for the whole is a weighted average: each group's percentage counts in proportion to the group's size.

Definition

Overall per cent $= \frac{A \cdot a + B \cdot b}{A + B}$ , where A and B are the group sizes and a and b their percentages.
The overall figure always lies between the group figures, nearer the bigger group.
Alligation: the group sizes are in the ratio of the opposite gaps, $A : B = (b - \text{overall}) : (\text{overall} - a)$ .
For a budget in a ratio (2 : 4 : 4), give each head its ratio number, apply its own change, and add up.

Weighted percentage and alligation

\text{overall} = \frac{A\,a + B\,b}{A + B} \qquad A : B = (b - \text{overall}) : (\text{overall} - a)

Worked example

A class has 40 boys and 60 girls. 30% of the boys and 50% of the girls wear glasses. What percentage of the class wears glasses?

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 12 Sep 2025, Shift 3 · Q54Easy

Example 4 · Percentage, Profit and Loss · Percentage Word Problems

The expenditure of a household on food, clothing, and entertainment is in the ratio

2 : 4 : 4

. In the coming year, the cost of food is expected to fall by

10\%

, clothing to rise by

5\%

, and entertainment to fall by

15\%

. What is the percent change (increase or decrease) in the total household expenditure?

Taking a plain average of the rates

30% of 40 boys and 50% of 60 girls is 42% of the class, not 40%. Weight each percentage by the size of its group.

Drill 8 more on combined percentage of several groups

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (4)

Elections: voters, votes cast, valid votes and the margin
Margin and valid votes
$\text{margin} = (2w - 100)\% \text{ of valid votes} \qquad \text{valid} = N \times \frac{p}{100} \times \frac{q}{100}$
Pass marks and marks scored
Pass mark
$\frac{p}{100} \times M = \text{score} + \text{shortfall}$
Income, expenditure and savings
Savings and their change
$\text{income} = \text{expenditure} + \text{savings} \qquad s = \frac{100\,i - E\,e}{S}$
Combined percentage of several groups
Weighted percentage and alligation
$\text{overall} = \frac{A\,a + B\,b}{A + B} \qquad A : B = (b - \text{overall}) : (\text{overall} - a)$

Watch out for (6)

Taking the winner's share as the margin
→ Elections: voters, votes cast, valid votes and the margin
Taking invalid votes out of the whole list
→ Elections: voters, votes cast, valid votes and the margin
Subtracting the shortfall
→ Pass marks and marks scored
Averaging percentages from papers of different sizes
→ Pass marks and marks scored
Averaging the two changes
→ Income, expenditure and savings
Taking a plain average of the rates
→ Combined percentage of several groups

Test yourself on Percentage, Profit and Loss

20 past SSC CGL questions from this chapter, timed at 12 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Take the Percentage, Profit and Loss chapter test Drill all 43 questions