Section 5, arithmetic

Percentages

Percentages rarely appear as a standalone CAT question now, but they sit inside profit-loss, interest, mixtures and most word problems. Master the fraction method here and every later chapter gets faster.

How CAT asks it

Recent papers wrap percentages in a story: spending and saving, elections, exam marks, price and consumption, population change. The maths is light; the trap is always the base, “percent of what?”.

Concepts you must know

  • x% of y = y% of x. Example: 8% of 50 = 50% of 8 = 4.
  • Percentage change = (new − old) ÷ old × 100.
  • If A is r% more than B, then B is r ÷ (100 + r) × 100% less than A. If A is r% less than B, then B is r ÷ (100 − r) × 100% more than A.
  • Product rule: if price × quantity stays fixed and price rises by a fraction 1/n, quantity must fall by 1/(n + 1).
  • Percentage points vs percent: a rate moving from 10% to 12% rose by 2 percentage points, which is a 20% increase.
  • Growth over n periods: P × (1 + r/100)ⁿ. Decline: P × (1 − r/100)ⁿ.
  • Two successive changes of a% and b% (use negative values for decreases):
Net change %=a+b+a×b100\text{Net change \%} = a + b + \frac{a \times b}{100}

Thought process

  1. Pick a base: assume the starting value is 100, or an LCM of the denominators.
  2. Convert each percentage into a fraction or multiplying factor.
  3. Ask “percent of what?” for every number. When the base changes (“% of the remainder”), update it.
  4. Chain the factors, then convert back to the quantity asked.

Patterns CAT repeats

Numbered so your error log can point at them: “Pattern 2”.

  1. 1Successive changes: price up then down, salary hikes and cuts, population over years. Multiply the factors.
  2. 2Chained comparisons: “A is 20% more than B, which is 25% less than C”. Set C = 100 and walk down the chain.
  3. 3Product constancy: expenditure = price × quantity, revenue = price × units, area = length × breadth. Fix the product, change one factor, find the other.
  4. 4Spending from the remainder: x% on rent, then y% of the rest on food. Multiply the “remaining” fractions.
  5. 5Elections: registered voters → votes cast → valid votes → winner’s share and margin. Each stage has its own base.
  6. 6Pass marks: “scored 178 and failed by 22” means the pass mark is 200.
  7. 7Mixed groups: different shares of boys and girls pass; the overall share is a weighted average (Section 10).
  8. 8Overlapping groups: passed A, passed B, passed both. Total = A + B − both + neither.

Worked examples

Attempt each one before opening its solution, then compare methods.

  1. Example 1, successive changes

    A phone’s price is raised 25%, cut 20% in a sale, then raised 10%. What is the net change from the original price?

    Show solution

    Factors 5/4 × 4/5 × 11/10 = 11/10. Answer: +10%.

  2. Example 2, product constancy

    Petrol prices rise 25%. By what percent must a family cut consumption to keep its spending unchanged?

    Show solution

    Price × 5/4, so quantity × 4/5, a fall of 1/5. Answer: 20%.

  3. Example 3, base flip

    Ravi earns 25% more than Anil. By what percent is Anil’s salary less than Ravi’s?

    Show solution

    Anil = 100, Ravi = 125. The gap of 25 out of 125 is 1/5. Answer: 20%.

  4. Example 4, spending from the remainder

    Meera spends 30% of her income on rent, 25% of the remainder on food and 20% of what is then left on travel. She saves ₹16,800. Find her income.

    Show solution

    Share left after each step: 0.7 × 0.75 × 0.8 = 0.42 of income. 0.42 × income = 16,800. Answer: ₹40,000.

  5. Example 5, elections

    In a two-candidate election, 80% of registered voters voted and 5% of the votes cast were invalid. The winner got 55% of valid votes and won by 1,520 votes. How many voters were registered?

    Show solution

    Valid votes = 0.95 × 0.8 × V = 0.76V. Margin = (55 − 45)% of valid votes = 0.076V = 1,520. Answer: 20,000.

  6. Example 6, pass marks

    Passing needs 40% of the maximum marks. Arjun scored 178 and failed by 22 marks. What are the maximum marks?

    Show solution

    Pass mark = 178 + 22 = 200 = 40% of the maximum. Answer: 500.

  7. Example 7, mixed groups

    60% of a school’s students are boys. 20% of the boys and 35% of the girls joined the NCC. What percent of students did not join?

    Show solution

    Take 100 students. Joined: 12 boys + 14 girls = 26. Answer: 74%.

  8. Example 8, population chain

    A town grows 10%, shrinks 10%, then grows 20% over three years, ending at 23,760. What was the initial population?

    Show solution

    Net factor 1.1 × 0.9 × 1.2 = 1.188. 23,760 ÷ 1.188 = 20,000. Answer: 20,000.

Traps

  • Averaging percentages from groups of different sizes; weight them by size instead.
  • Treating “+20% then −20%” as no change. It is −4%.
  • Confusing percentage points with percent.
  • Forgetting that “% of the remainder” uses a new base.

Practice set

Timed, about 2.5 minutes per question. Log every miss in the error log.

  1. 1

    A number is increased by 20% and then decreased by 20%. What is the net percentage change?

    Show answer

    −4%: 20 − 20 − 400/100.

  2. 2

    A 20% fall in the price of sugar lets a person buy 2 kg more for ₹400. Find the original price per kg.

    Show answer

    ₹50: the 20% saving is ₹80, which buys 2 kg at the new ₹40/kg; the original was 40 ÷ 0.8.

  3. 3

    A’s income is 20% less than B’s. By what percent is B’s income more than A’s?

    Show answer

    25%: 20/80.

  4. 4

    A rectangle’s length rises 20% and its breadth falls 10%. What is the change in area?

    Show answer

    +8%: 1.2 × 0.9 = 1.08.

  5. 5

    65% of students passed English, 55% passed Maths and 30% passed both. What percent failed both?

    Show answer

    10%: passed at least one = 65 + 55 − 30 = 90.

  6. 6

    A shop’s unit sales fall 10% while the price per unit rises 15%. What is the change in revenue?

    Show answer

    +3.5%: 0.9 × 1.15 = 1.035.

  7. 7

    In a two-candidate contest with no invalid votes, the loser got 35% and lost by 15,000 votes. How many votes were cast?

    Show answer

    50,000: the 30% margin is 15,000.

  8. 8

    Rahul spends 40% of his income on the household and 20% of the rest on rent, and saves ₹14,400. What is his income?

    Show answer

    ₹30,000: 0.6 × 0.8 = 0.48 of income = 14,400.

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