CSAT Part 3: Quant — Percentages, Ratio and the Speed-Math Toolkit, Exam-Ready Notes

Category: Civil Exams · Series: UPSC CSAT Paper II · Read time: ~10 minutes

Basic numeracy carries roughly a quarter of CSAT marks, and almost every question passes through two gates: percentage and ratio. Speed here is not talent — it is a toolkit of fraction conversions, multiplication tricks and proportion thinking. This note assembles that toolkit, then applies it to the classic CSAT wrappers.

Table of Contents

1. Why These Two Topics Run the Section
2. The Fraction-Percentage Conversion Table (Memorise)
3. Percentage Techniques That Save a Minute Each
4. Ratio: The Scaling Habit
5. Average, Mixture and Alligation
6. Common CSAT Wrappers Solved the Fast Way
7. How Exams Probe This Topic
8. Quick Revision: One-Glance Facts

1. Why These Two Topics Run the Section

• The syllabus line. CSAT’s basic numeracy covers numbers and their relations, orders of magnitude, percentages, ratio and proportion, averages, and basic algebra — but profit-loss, simple interest, data growth and even time-work are all percentages in costume.
• The payoff. A conversion-table user solves “37.5% of 640” in five seconds (3/8 × 640 = 240); a decimal-multiplier user takes forty. Across 25 questions, that is the paper.

2. The Fraction-Percentage Conversion Table (Memorise)

Also: 3/8 = 37.5%, 5/8 = 62.5%, 7/8 = 87.5%, 2/3 = 66.67%, 3/4 = 75%. Reverse direction matters equally: seeing 14.28% should trigger “1/7” instantly.

3. Percentage Techniques That Save a Minute Each

• Successive percentage change: a ±a% then ±b% change = a + b + ab/100 (signs included). Price up 20% then down 10% → 20 − 10 − 2 = +8%.
• The multiplier habit: +25% = ×5/4; −20% = ×4/5. Two changes become two quick multiplications; the “restore original price” question (up 25%, what % fall restores? → 20%) becomes obvious: the multipliers must multiply to 1.
• Percentage-point vs percent: 10% → 12% is 2 points but 20% relative growth — a favourite CSAT trap in population/literacy lines.
• Base discipline: “A is 25% more than B” means A = 1.25B, so B is 20% less than A, not 25% less. Check the base every time.

4. Ratio: The Scaling Habit

• The unit method. A:B = 3:5 with total 64 → 8 parts → part = 8 → A = 24, B = 40. Never write equations; scale.
• Combining ratios. A:B = 2:3, B:C = 4:5 → make B common (12): A:B = 8:12, B:C = 12:15 → A:B:C = 8:12:15.
• Ratios as multipliers in work/pipe questions: capacities and speeds in ratio combine by LCM of the “one unit” times — covered fully in Part 5.

5. Average, Mixture and Alligation

• Averages: new average after adding/removing a value = old average + (deviation of the new value ÷ new count). “Class of 20 averages 40; student scoring 60 joins → 40 + 20/21.”
• Alligation: mix two prices to get a mean price → ratio of quantities = (dearer − mean) : (mean − cheaper). The cross-diagram solves milk-water, profit-mix and interest-mix questions alike.
• Replacement questions (a fraction of the liquid replaced n times): final original fraction = (1 − 1/x)ⁿ — memorise the formula; CSAT uses it directly.

6. Common CSAT Wrappers Solved the Fast Way

1. Population growth: P(1 + r/100)ⁿ, but with r = 6.25% use ×17/16 per year — fraction conversion again.
2. Election/voting: two candidates, one gets x% of valid votes, invalid votes given — always compute the valid base first.
3. Marks and pass percentage: compute total marks, pass mark, then compare each student’s deficit — one line each with fractions.
4. Successive discount: two discounts of 10% and 20% = 28% total, not 30% (use a+b+ab/100 with minus signs).
5. Number series with ratios: geometric-type series (2, 6, 18…) — recognise ×3 rather than testing differences.

7. How Exams Probe This Topic

• Prelims CSAT papers 2018–2025: basic numeracy questions cluster around successive change, ratio-sharing, averages after replacement, and base-switching traps. Solve the last eight papers’ quant sections with the toolkit — if a question takes a full minute, it is telling you which tool you skipped.
• Drill standard: 25 basic-numeracy questions in 25 minutes at 90%+ accuracy before moving to Part 4’s data interpretation.
• Companion exams: CDS, CAPF and state PCS prelims reuse the identical wrappers — one toolkit, many papers.

8. Quick Revision: One-Glance Facts

• The table. 1/8 = 12.5% through 1/15 = 6.67%; both directions automatic.
• Successive change. a + b + ab/100.
• Base rule. 25% more ≠ 25% less back; use multipliers that multiply to 1.
• Unit method for all ratios; alligation for all mixtures.
• Replacement. (1 − 1/x)ⁿ survives.
• Drill bar. 25 questions / 25 minutes / 90% accuracy.

Conclusion. CSAT quant speed is a memory product, not a maths talent. Burn the conversion table in until fractions and percentages are the same thing in your head, run every mixture through alligation and every ratio through the unit method — then the 25-odd basic numeracy marks become the most reliable block on your path past 66.67.

Twelve Worked Items (The Speed-Toolkit-Tested)

1. 63.5% of 640 = 406.4 (5/8 × 640 = 400 + 1% × 640 = 6.4).
2. A’s-salary is 25% more than B’s; B’s-is-what-%-less-than-A’s? — 20% (the multiplier-inversion — never 25).
3. Three-successive-discounts-of-10% = a-single-discount-of 27.1%.
4. If 6-men-do-a-job-in-12-days, 8-men-take 9 days (the inverse-proportion).
5. A-mixture-of-milk-and-water (3:1) — half-the-mixture-removed-and-replaced-with-water: the-new-ratio = 3:5 (the replacement-formula (1−1/2)-applied-to-the-milk-only).
6. The-average-of-5-consecutive-even-numbers-is-24 → the-largest = 28.
7. A-train-150-m-crosses-a-pole-in-9-s → speed = 60 km/h.
8. Two-trains-100-m-and-150-m-cross-each-other (opposite, 54-and-36 km/h) in 10 s (250 ÷ 25 m/s).
9. ₹12,000-at-10%-CI-for-2-years = ₹14,520 (1.21×).
10. A-buys-at-₹400, marks-60%-up, gives-20%-discount → profit-% = 28%.
11. If-x:y = 3:4 and y:z = 2:3 → x:z = 1:2 (the y-made-common-at-4-and-2’s-LCM-4: 3:4-and-4:6).
12. Population-grows-5%-then-4% → the-net-growth = 9.2% (a+b+ab/100).
    Twelve-items-covering-the-whole-toolkit: the-inversions (2, 11), the-successive-operations (3, 10, 12), the-LCM-and-replacement-methods (4, 5), and the S-D-family (7, 8) — attempt-in-15-minutes; every-item-over-75-seconds names-the-tool-to-redrill.

The Estimation-Decision-Tree (When-to-Compute-Exactly)

At-each-question, three-branches: the-options-10%-apart → estimate-to-the-nearest-clean-number (the 63.5%-of-640-becomes-5/8-plus); the-options-2-3%-apart → compute-exactly (and-check-the-units-first — the exact-questions-are-usually-unit-traps); the-options-in-different-magnitudes → order-of-magnitude-only (the 0.047%-vs-4.7%-vs-47%-question-answered-in-15-seconds). The-tree’s-discipline: the decision-precedes-the-computation — the candidates-who-estimate-everything-eventually-lose-the-tight-option-questions, and-the-ones-who-compute-everything-lose-the-paper’s-clock.

The Percentage-Word-Problem Translations (The Sentence-to-Equation Bank)

Every percentage word problem is one of five sentence-patterns, and translating it correctly is half the solution. Pattern one — the comparison: “A’s income is 25% more than B’s” means A equals 1.25 times B; the reverse question (“B is what percent less than A”) uses the inversion multiplier — twenty percent, never twenty-five. Pattern two — the successive operation: “increased by 20%, then decreased by 10%” composes through the formula a plus b plus ab upon hundred — eight percent here; the composition’s order-independence (the same result either way) surprises candidates and appears as a trap regularly. Pattern three — the consumption case: “price rises 25%, expenditure constant” — consumption falls by twenty percent; the multiplier’s product-to-one logic (five-fourths times four-fifths) solves every variant without equations. Pattern four — the population or depreciation: repeated equal changes compound — the 5%-then-4% growth is 9.2 percent, and the 10%-annual-depreciation over three years is 1 minus 0.9 cubed, twenty-seven point one percent — the compounding’s distinction from the simple addition is the recurring discriminator. Pattern five — the base-shift: “x% of y versus y% of x” are equal (the commutative insight); “30% of a number equals 45” reverses to the number equals 150 — the base’s identification before any computation. Five patterns, five translations — drill each with two examples until the sentence’s pattern-name arrives with the sentence itself, and the word problem converts to arithmetic before any hesitation.