Data Sufficiency — Question: What percentage of the students in a class passed the examination? Statement I: 36 students of the class passed. Statement II: 9 students of the class failed. Which of the following is correct?

Statement II alone is sufficient

Even both statements together are not sufficient

Data Interpretation & Data Sufficiency

Data Interpretation (DI) is the most generous topic in the whole paper. There is nothing to memorise — the answer is already sitting inside the table or the chart. All you have to do is read carefully and calculate a little. Data Sufficiency is its clever twin: it does not want the answer at all, it only asks "is this much information enough?" Together they hand out easy marks to the calm reader and punish the one who rushes. In this chapter we will keep every dataset as a table — because once you can read a table, you can read any bar, line or pie chart, since all of them are only pictures of a table.

What DI Tests — and the Marks Strategy

DI does not test mathematics. It tests three things: whether you can read a label, whether you can estimate instead of grinding out long division, and whether you can stay honest about what the data actually says. Every DI set is a small comprehension passage made of numbers.

🎯 Why this chapter pays

A DI set usually carries 4–5 questions on one dataset. You spend 40 seconds understanding the table once, and then answer five questions from it. No other topic gives you that return. And in the descriptive General Studies papers, statistical analysis with graphs and diagrams is a listed part of the syllabus — so the same skill earns marks twice, in Prelims and in Mains.

The four-step attack on any DI set

Step 1 — Read the head
Read the title, the units and the footnote before a single number. "In lakh" vs "in thousand" vs "in %" decides everything.
Step 2 — Check the scale
Never assume the axis starts at zero. A truncated axis makes a tiny difference look enormous. Read the numbers, not the height of the bar.
Step 3 — Approximate first
Round the numbers and get a rough answer. Then look at the options — often only one option is near your estimate, and you are done.
Step 4 — Calculate only if forced
Do the exact arithmetic only when two options are close. Exact calculation is the last resort, not the first move.

⚠️ The three classic reading traps

(1) Units — the table says "in lakh" and the option is in crore.
(2) Rate vs number — the highest literacy rate is not the highest number of literates.
(3) Absolute vs percentage — the biggest rise in tonnes is often not the biggest rise in per cent. Nearly every wrong DI answer comes from one of these three.

ℹ️ Note

Negative marking applies in the objective paper. So in DI, attempt what you have actually read and leave the one question whose footnote you did not understand. One careless DI guess can eat the mark you earned in the previous question.

Table-Based DI — A Worked Dataset

A table is the purest DI question — nothing is hidden, everything must be computed. Study the dataset below. The figures are illustrative, invented for practice — they are not official data. (For reference, the real all-Bihar literacy rate in Census 2011 was 61.8%.)

Illustrative dataset (NOT official): population and literacy rate of five districts
District	Population (in lakh)	Literacy rate (%)	Literate persons (in lakh)
Patna	58	70	40.6
Gaya	44	60	26.4
Saran	40	60	24.0
Rohtas	30	75	22.5
Nalanda	28	65	18.2
Total	200	65.85 (overall)	131.7

The fourth column was not given — we built it. Literate persons = population × literacy rate. For Patna: 58 × 70 ÷ 100 = 40.6 lakh. Do this once for every row and the whole set becomes trivial. Total literates = 40.6 + 26.4 + 24.0 + 22.5 + 18.2 = 131.7 lakh.

Overall literacy rate of the five districts? Not the average of 70, 60, 60, 75, 65 — that would be wrong. It is total literates ÷ total population = 131.7 ÷ 200 = 65.85%.
Percentage share of Patna in total population? 58 ÷ 200 × 100 = 29%.
Ratio of literates in Gaya to literates in Saran? 26.4 : 24.0 = 264 : 240 = 11 : 10.
Rank by number of literates: Patna (40.6) > Gaya (26.4) > Saran (24.0) > Rohtas (22.5) > Nalanda (18.2).
Share of Patna in total literates? 40.6 ÷ 131.7 × 100 ≈ 30.8%.

⚠️ Rate is not number

Look at Rohtas: it has the highest literacy rate (75%) in the table, yet it is only fourth in the number of literate persons, because its population is small. Examiners love this. Whenever a question says "highest", stop and ask: highest rate or highest number?

⚠️ Exam trap

The average of percentages is not the overall percentage unless the groups are of equal size. Here the simple average of the five rates is (70+60+60+75+65) ÷ 5 = 66%, but the true overall rate is 65.85%. In a real exam the gap is often much larger. Always use total ÷ total.

Bar Graphs & Line Graphs

A bar graph compares separate items side by side — its job is comparison. A line graph joins points across time — its job is trend. Both are only tables in disguise, so the very first thing a good student does is write the numbers down as a table. Then no picture can trick you.

Suppose a line graph shows the production of a crop in one district over five years, rising steadily and then jumping. Written as a table, the same graph reads like this — and now the "steepest rise" is a matter of arithmetic, not eyesight.

The same line graph, written as a table (illustrative figures)
Year	Production ('000 tonnes)	Increase over previous year	Percentage increase
2019	100	—	—
2020	150	+50	50.00%
2021	180	+30	20.00%
2022	240	+60	33.33%
2023	288	+48	20.00%

Largest increase in absolute terms: 2022, with +60 thousand tonnes (240 − 180).
Largest percentage increase: 2020, with 50% (50 ÷ 100 × 100) — even though its absolute rise (+50) was smaller than 2022’s (+60).
Trend: production rose in every single year — the trend is uniformly upward, but the rate of growth is not.
Total rise over the period: 288 − 100 = 188, i.e. 188% growth over 2019 (288 is 2.88 times of 100).

⚠️ Absolute rise vs percentage rise

Two shops. Shop A: ₹100 lakh → ₹140 lakh. Shop B: ₹400 lakh → ₹480 lakh. Shop B rose by ₹80 lakh, twice as much money as A’s ₹40 lakh. But in percentage, A grew 40% and B only 20%. So the bigger absolute rise had the smaller percentage rise — because percentage always divides by the starting value. When the question says "grew fastest", it means percentage; when it says "increased the most", check whether the units are rupees or per cent.

💡 Think of it like this

Think of a ₹40 raise. To a person earning ₹100 it is life-changing (40%); to a person earning ₹4,000 it is nothing (1%). The same rupee amount, two completely different stories. That is the whole difference between absolute and percentage change.

Pie Charts — Degrees, Percentages and Values

A pie chart shows parts of one whole. The whole circle is 360° and also 100%. So the single conversion you must never forget is: 1% = 3.6°, and therefore 1° = 5/18 %. To go from degrees to percentage, divide by 3.6; to go the other way, multiply by 3.6.

A pie chart written as a table: a total budget of ₹7,200 crore (illustrative)
Sector	Central angle	Percentage (angle ÷ 3.6)	Amount (₹ crore)
Education	108°	30%	2,160
Health	72°	20%	1,440
Agriculture	54°	15%	1,080
Social welfare	54°	15%	1,080
Roads	36°	10%	720
Others	36°	10%	720
Total	360°	100%	7,200

Component value from the total: Education = 108 ÷ 360 × 7,200 = ₹2,160 crore. Or use the shortcut: ₹7,200 crore ÷ 360° = ₹20 crore per degree, so 108° × 20 = 2,160.
Total from a component: if you are told only that Health is ₹1,440 crore at 72°, then total = 1,440 × 360 ÷ 72 = ₹7,200 crore.
Comparing two slices: Education : Health = 108 : 72 = 3 : 2. Ratios of amounts equal ratios of angles — you never need the total for a ratio question.
Difference between two slices: Education − Agriculture = (108 − 54)° × ₹20 crore = ₹1,080 crore.

🧠 The degree ladder

Memorise these five and you can eyeball any pie: 10% = 36°, 25% = 90° (a right angle), 33⅓% = 120°, 50% = 180° (a straight line), 75% = 270°. If a slice looks like a perfect right angle, it is a quarter — you already know the answer before calculating.

⚠️ Two pies, two different totals

State X has a budget of ₹7,200 crore and spends 20% on health → ₹1,440 crore. State Y has a budget of ₹4,000 crore and spends 30% on health → ₹1,200 crore. Y’s slice is bigger in percentage but smaller in rupees. A percentage of a smaller whole can be a smaller amount. Whenever two pie charts appear together, hunt for the two totals first — if the totals differ, percentages alone can never be compared as money.

Caselet DI — Turn the Paragraph into a Table

A caselet gives the data as a short paragraph with no table and no chart. It feels hard only because the numbers are hiding inside sentences. The cure is mechanical: read one sentence, write one number into a table. Do not try to solve while reading.

📖 The caselet

In a mock test at a coaching centre in Patna, 500 candidates appeared. 60% of them were male. Of the male candidates, 40% passed; of the female candidates, 25% passed. Find (a) how many passed in all, (b) the overall pass percentage, (c) the ratio of passed males to passed females.

Step one, and the only step that matters: build the table. Males = 60% of 500 = 300, so females = 500 − 300 = 200. Passed males = 40% of 300 = 120. Passed females = 25% of 200 = 50.

The caselet, converted into a table
Category	Appeared	Passed	Failed	Pass %
Male	300	120	180	40%
Female	200	50	150	25%
Total	500	170	330	34%

(a) Total passed = 120 + 50 = 170.
(b) Overall pass percentage = 170 ÷ 500 × 100 = 34% — which lies between 25% and 40%, as it must.
(c) Passed males : passed females = 120 : 50 = 12 : 5.

🧠 Memory trick

Use the sanity check every time: the overall percentage must always sit between the group percentages, and closer to the bigger group. Here 34% sits between 25% and 40%, nearer to 40% because males were more numerous. If your answer falls outside that range, you have made an arithmetic slip — recheck before you tick.

Speed Techniques for the Exam Hall

There is no calculator in the exam hall, and there is no time for long division. DI speed comes from exactly two habits: converting percentages into fractions, and rounding before dividing. Learn the table below by heart — it is the single highest-return memorisation in the whole quantitative section.

Percentage ↔ fraction conversions (memorise)
Fraction	Percentage	Use it when you see
1/2	50%	half of the total
1/3	33.33%	33%, 33⅓%, 120° in a pie
1/4	25%	25%, 90° in a pie
1/5	20%	20%, 72° in a pie
1/6	16.67%	16%, 16⅔%
1/7	14.28%	roughly 14%
1/8	12.5%	12.5%, 45° in a pie
1/9	11.11%	roughly 11%
1/10	10%	10%, 36° in a pie
1/11	9.09%	roughly 9%

Why does this help? Because 16.67% of 4,272 looks terrifying, but 4,272 ÷ 6 = 712 takes five seconds. Percentages are slow; fractions are fast. The moment you see a familiar percentage, switch to its fraction.

Percentage change formula: (New − Old) ÷ Old × 100. The denominator is always the older/base value — this one slip causes more wrong DI answers than any other.
Percentage share: part ÷ whole × 100. Before dividing, check that your "whole" is really the total asked for (all districts? only two districts? only males?).
Approximation discipline: 4,986 ÷ 19.8 ≈ 5,000 ÷ 20 = 250. Round both numbers in the same direction so the errors partly cancel, and never approximate when the two closest options differ by less than about 2%.
Ratio before value: if a question only asks for a ratio or "how many times", cancel the common total and skip the multiplication entirely.

🧠 The 10% ladder

Build any percentage from 10% and 5%. For 35% of 480: 10% = 48, so 30% = 144; 5% = 24; total 168. For 15%: 10% + half of it. This mental ladder replaces almost all multiplication in DI.

Data Sufficiency — Decide, Do Not Solve

In Data Sufficiency you are given a question and two statements. You are not asked for the answer. You are asked only whether the statements give enough information to reach a single, definite answer. This is a test of judgement, not of arithmetic.

The standard five-option format
Option	Meaning
(a) Statement I alone is sufficient	I answers it; II does not.
(b) Statement II alone is sufficient	II answers it; I does not.
(c) Both together are sufficient	Neither alone works, but the two combined do.
(d) Either alone is sufficient	I works by itself and II also works by itself.
(e) Neither is sufficient	Even both together leave the answer undecided — more data is needed.

Read the question and note exactly what one quantity is being asked.
Test Statement I alone, covering Statement II with your finger. Sufficient or not?
Now forget I completely and test Statement II alone. This is where students fail — they carry information from I into II.
Only if both fail alone, combine them and test again.
Match your finding to the option letters. Do not compute the final value.

ℹ️ Worked example 1

Question: What is Ram’s average monthly income? Statement I: Ram’s annual income is ₹6,00,000. Statement II: Ram’s income is 25% more than Shyam’s income. — I alone: annual ÷ 12 gives a unique figure, so sufficient. II alone: Shyam’s income is unknown, so no number can be found — not sufficient. Answer: (a) Statement I alone is sufficient. Notice we never actually divided 6,00,000 by 12; knowing that we could was enough.

ℹ️ Worked example 2

Question: What percentage of students in a class passed the examination? Statement I: 36 students passed. Statement II: 9 students failed. — I alone: 36 out of how many? Total unknown — not sufficient. II alone: 9 failed out of how many? Total unknown — not sufficient. Together: total = 36 + 9 = 45, so pass percentage = 36 ÷ 45 × 100 = 80%, a single definite answer — sufficient. Answer: (c) both statements together are sufficient, but neither alone.

⚠️ The golden rule

Never solve fully — only decide sufficiency. Three traps follow from breaking it:
(1) you waste 90 seconds computing a number nobody asked for;
(2) you unconsciously use a figure from Statement I while testing Statement II;
(3) you accept a statement that yields two possible answers (for example "x² = 49" gives x = 7 or −7) — two answers means not sufficient. Sufficiency demands a unique answer, and it does not care what that answer is.

Key takeaways

Read the title, units and footnote first; never assume an axis starts at zero; approximate before you calculate.
Every bar, line, pie or caselet is a table in disguise — write the numbers as a table and the set solves itself.
Highest rate ≠ highest number (a small district with 75% literacy can have fewer literates than a big one with 60%).
The largest absolute rise can have the smallest percentage rise — percentage always divides by the older/base value: (New − Old) ÷ Old × 100.
Pie chart: 1% = 3.6°; 25% = 90°, 50% = 180°. Value of a slice = angle ÷ 360 × total. With two pies of different totals, a bigger percentage can still be a smaller amount.
Learn the fraction table (1/2=50%, 1/3≈33.33%, 1/4=25%, 1/5=20%, 1/6≈16.67%, 1/7≈14.28%, 1/8=12.5%, 1/9≈11.11%, 1/11≈9.09%) — fractions are fast, percentages are slow.
Data Sufficiency: test each statement alone, then together; the golden rule is do not solve — only decide sufficiency, and two possible answers means not sufficient.

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.

In a pie chart, a component that forms 25% of the total will be represented by a central angle of:
1. 60°
2. 72°
3. 90°
4. 120°
Show answer
Answer: C — 90°
The whole circle is 360° = 100%, so 1% = 3.6°. Therefore 25% × 3.6 = 90°. (Equivalently, one quarter of 360° is 90°.)
A pie chart shows how a total budget of ₹7,200 crore is divided. The sector for education has a central angle of 108°. The amount spent on education is:
1. ₹1,440 crore
2. ₹1,800 crore
3. ₹2,160 crore
4. ₹2,400 crore
Show answer
Answer: C — ₹2,160 crore
Amount = angle ÷ 360 × total = 108 ÷ 360 × 7,200 = ₹2,160 crore. Shortcut: 7,200 ÷ 360 = ₹20 crore per degree, so 108 × 20 = 2,160. (108° is also 30%, and 30% of 7,200 = 2,160.)
The production of a crop in a district (in thousand tonnes) was: 2020 = 150, 2021 = 180, 2022 = 240, 2023 = 288.
Consider the following statements:
1. The largest increase in absolute terms took place in 2022 over 2021.
2. The largest percentage increase over the previous year also took place in 2022.
Which is/are correct?
1. Only 1
2. Only 2
3. Both 1 and 2
4. Neither 1 nor 2
Show answer
Answer: C — Both 1 and 2
Absolute increases: 2021 = +30, 2022 = +60, 2023 = +48 — largest is 2022. Percentage increases: 2021 = 30/150 = 20%, 2022 = 60/180 = 33.33%, 2023 = 48/240 = 20% — largest is again 2022. So both statements are correct.
Sales of two shops rose as follows: Shop A from ₹100 lakh to ₹140 lakh, and Shop B from ₹400 lakh to ₹480 lakh.
Consider the following statements:
1. Shop B recorded the larger increase in rupees.
2. Shop A recorded the higher percentage increase.
Which is/are correct?
1. Only 1
2. Only 2
3. Both 1 and 2
4. Neither 1 nor 2
Show answer
Answer: C — Both 1 and 2
Shop B rose by ₹80 lakh and Shop A by only ₹40 lakh, so statement 1 is correct. In percentage, A grew 40/100 = 40% while B grew 80/400 = 20%, so statement 2 is also correct. This is the classic trap: the larger absolute rise here has the smaller percentage rise.
In a test, 400 candidates appeared, of whom 55% were girls. If 40% of the girls and 50% of the boys passed, how many candidates passed in all?
1. 172
2. 178
3. 182
4. 190
Show answer
Answer: B — 178
Girls = 55% of 400 = 220; boys = 400 − 220 = 180. Passed girls = 40% of 220 = 88; passed boys = 50% of 180 = 90. Total passed = 88 + 90 = 178 (an overall pass rate of 44.5%).
Data Sufficiency — Question: What percentage of the students in a class passed the examination?
Statement I: 36 students of the class passed.
Statement II: 9 students of the class failed.
Which of the following is correct?
1. Statement I alone is sufficient
2. Statement II alone is sufficient
3. Both statements together are sufficient, but neither alone
4. Even both statements together are not sufficient
Show answer
Answer: C — Both statements together are sufficient, but neither alone
A percentage needs a total. Statement I gives only the number who passed, and Statement II only the number who failed — neither alone gives the class strength. Together, total = 36 + 9 = 45, so the pass percentage = 36 ÷ 45 × 100 = 80%, a unique answer. Hence both together are sufficient but neither alone.
State X has an annual budget of ₹8,000 crore and allots 15% of it to health. State Y has an annual budget of ₹5,000 crore and allots 20% of it to health.
Consider the following statements:
1. State Y allots a higher percentage of its budget to health than State X.
2. State Y spends more money on health than State X.
Which is/are correct?
1. Only 1
2. Only 2
3. Both 1 and 2
4. Neither 1 nor 2
Show answer
Answer: A — Only 1
Statement 1 is correct: 20% is greater than 15%. Statement 2 is wrong: X spends 15% of 8,000 = ₹1,200 crore while Y spends 20% of 5,000 = ₹1,000 crore, so X actually spends more. This is the two-pie-chart trap — when the totals differ, a bigger percentage can still be a smaller amount.
District P has a population of 40 lakh with a literacy rate of 60%, while District Q has a population of 25 lakh with a literacy rate of 80%. Which district has more literate persons, and by how many?
1. P, by 4 lakh
2. Q, by 4 lakh
3. P, by 5 lakh
4. Q, by 15 lakh
Show answer
Answer: A — P, by 4 lakh
Literates in P = 60% of 40 lakh = 24 lakh. Literates in Q = 80% of 25 lakh = 20 lakh. So P has 24 − 20 = 4 lakh more literate persons, even though Q has the higher literacy rate. Highest rate is not the same as highest number.