Quantitative Aptitude — Ratios, Powers, Mensuration, Statistics and Data
None of the mathematics in this part of General Aptitude is beyond Class 10, and that is precisely the risk: an engineering candidate reads the syllabus — ratios, percentages, logarithms, mensuration, elementary statistics, data interpretation — decides there is nothing to prepare, and then loses four or five marks to questions they could all have done. What costs marks here is never the mathematics. It is reading what was asked (a percentage change is not a percentage of, and a rejection rate is not a rejection count), choosing the cheapest route (a ratio question that yields to one division should not become simultaneous equations), and not spending three minutes on a one-mark question because it felt solvable. Treat these five questions as five marks of the same value as any in your discipline paper, budget about a minute each on the 1-mark ones, and check the units in the answer against the units in the question before you click. Every figure in this chapter was recomputed by script, so where your arithmetic disagrees with an explanation, the explanation is the place to look for the reason rather than the error.
The Whole Toolkit, on One Page
- Ratios. A ratio a : b over a total T gives parts of T·a/(a+b) and T·b/(a+b). Two numbers in the ratio 3 : 5 with sum 64 are 24 and 40 — one division, not two equations.
- Powers and logs. A negative exponent is a reciprocal: (0.5)⁻³ = 2³ = 8. A logarithm is a question about an exponent: log₂ 32 asks what power of 2 gives 32, and the answer is 5.
- Counting. Order matters → permutation; order does not → combination. Choosing 3 of 4 distinct letters in order is 4 × 3 × 2 = 24.
- Mensuration. Curved surface of a cylinder 2πrh, total surface 2πr(r + h), volume πr²h. Read which of the three was asked — the question usually says "curved" or "total" and the options include both.
- Statistics. Mean is the total over the count, median the middle value once sorted, mode the most frequent. For a skewed set the three differ, and a question that gives an odd-looking set is usually asking you to notice that.
- Probability. Favourable outcomes over total outcomes, when all outcomes are equally likely. Two dice have 36 outcomes, and the fastest route is almost always to count the favourable ones rather than to reason about them.
Key takeaways
- The mathematics here is Class 10. The marks go to reading the question, so check the denominator, the units and whether a rate or a count was asked before computing.
- A percentage change divides by the earlier value; "what percentage of X" puts X in the denominator. Reversing either is the commonest error in the section.
- Successive percentage changes multiply: +20% then −20% is 1.2 × 0.8 = 0.96, a 4% net fall, and never a return to the start.
- A ratio over a known total is one division, not a pair of simultaneous equations. 3 : 5 of 64 is 24 and 40.
- The plant with the most rejections is rarely the one with the highest rejection rate — a rate needs its base, and every data-interpretation set is built on that gap.
- For a cylinder, decide between 2πrh, 2πr(r + h) and πr²h by rereading the question — the options will contain more than one of them.
- These five questions are worth the same as five from your discipline paper, and are far cheaper to secure. Budget about a minute each on the one-markers and move.
Practice questions (8)
Attempt each one before opening the answer. Every explanation names the tempting wrong option as well as the right one, because that is where marks are lost.
The price of a component is increased by 20% and then reduced by 20%. Compared with the original price, the final price is:
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Answer: B — 4% lower
Multiply the factors rather than adding the percentages: 1.20 × 0.80 = 0.96, so the final price is 96% of the original — a 4% fall. Option A is what adding +20 and −20 suggests, and it is wrong because the 20% reduction is taken on the larger price, so it removes more than the increase added. The general result: a rise of x% followed by a fall of x% always leaves 1 − x²/10 000 of the original, so it is always a net loss, whichever order the two changes come in.Two numbers are in the ratio 3 : 5 and their sum is 64. Their difference is:
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Answer: B — 16
The ratio has 3 + 5 = 8 parts, so one part is 64/8 = 8, and the numbers are 24 and 40. The difference is 16 — which is also 2 parts, since 5 − 3 = 2, and reading it straight off as 2 × 8 skips finding the numbers at all. Option C is the smaller number and option D is 4 parts; both are quantities that appear in the working, which is why they are offered. Setting up 3k + 5k = 64 in one line beats writing two simultaneous equations.The value of (0.5)⁻³ is:
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Answer: C — 8
A negative exponent means the reciprocal, not a negative number: (0.5)⁻³ = 1/(0.5)³ = 1/0.125 = 8. Equivalently, 0.5 = 2⁻¹, so (2⁻¹)⁻³ = 2³ = 8. Options A and D come from reading the minus sign as belonging to the answer, which is the error the question is set to catch; option B is (0.5)³, the reciprocal of the right answer. Converting the decimal to a power of 2 first makes this a one-step question.How many three-letter arrangements can be formed from the letters of GATE, using no letter more than once?
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Answer: C — 24
GATE has four distinct letters and an arrangement is ordered, so this is a permutation: 4 choices for the first position, 3 for the second, 2 for the third — 4 × 3 × 2 = 24, which is ⁴P₃. Option A is ⁴C₃ = 4, the number of ways to choose three letters while ignoring order, and that is the distinction the question tests. Option D is 4³ = 64, which allows repetition. Always settle two questions before computing: does order matter, and is repetition allowed?A closed cylindrical tank has radius 7 m and height 10 m. Its curved surface area, taking π = 22/7, is:
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Answer: B — 440 m²
Curved surface area is 2πrh = 2 × (22/7) × 7 × 10 = 440 m²; the 7 cancels the denominator, which is why the radius was chosen as 7. Option D is the volume, πr²h = 1540, a number with the wrong dimensions that reads plausibly. Option C is the total surface area, 2πr(r + h) = 2 × (22/7) × 7 × 17 = 748, and it is there because the question says the tank is closed — a detail that matters for the total but not for the curved surface. Reread which of the three the question asked before choosing.For the data set 5, 8, 8, 11, 13, the value of (mean − median) is:
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Answer: B — 1
The mean is (5 + 8 + 8 + 11 + 13)/5 = 45/5 = 9. The set is already sorted and has five values, so the median is the third, 8. So mean − median = 9 − 8 = 1. The mode is also 8, since 8 occurs twice. Option A is what you would get if the set were symmetric; this one is pulled right by the 13, which is why the mean sits above the median — and that relationship, mean above median for a right-skewed set, is itself examinable.Set. The table gives the number of units produced and the number rejected at four plants in one quarter.
Units produced and rejected, by plant Plant Produced Rejected P 4,000 120 Q 2,500 100 R 6,000 150 S 1,500 75 Which plant has the highest rejection rate?
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Answer: D — S
A rate needs its base, so divide rather than compare the rejection column. P: 120/4000 = 3%. Q: 100/2500 = 4%. R: 150/6000 = 2.5%. S: 75/1500 = 5% — the highest, from the smallest plant with the fewest rejections in absolute terms. R has the most rejections (150) and the lowest rate, which is the whole point of the set: the rejection column alone ranks the plants in almost the opposite order to the rate. Any table giving a count beside a total is asking you to divide.Set. The table gives the number of units produced and the number rejected at four plants in one quarter.
Units produced and rejected, by plant Plant Produced Rejected P 4,000 120 Q 2,500 100 R 6,000 150 S 1,500 75 Taking the four plants together, the overall rejection rate is closest to:
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Answer: B — 3.0%
Add the columns, then divide once: rejected 120 + 100 + 150 + 75 = 445; produced 4000 + 2500 + 6000 + 1500 = 14 000. So the overall rate is 445/14 000 ≈ 3.18%, closest to 3.0%. The trap is option C, 3.6% — the average of the four rates (3 + 4 + 2.5 + 5)/4 = 3.625, which weights the 1,500-unit plant the same as the 6,000-unit one. A combined rate is a ratio of the totals, never the average of the rates, unless the bases happen to be equal, and this question exists to make that concrete.