Numerical Ability and Data Interpretation for Mains
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Chapter 23: Numerical Ability and Data Interpretation for Mains
The Mains paper tests numerical ability differently from Prelims, and treating the two sections with the same preparation strategy is one of the more common reasons candidates who score comfortably in Prelims arithmetic underperform in the Mains numerical ability and DI section. Where Prelims rewards raw calculation speed on standalone questions, Mains data interpretation sets present a table, bar graph, pie chart, or line graph, and then ask four or five linked questions against the same dataset — meaning a mistake in reading the data correctly propagates through every question in the set. The core skill is not calculation, which is usually simple once you have the right two numbers, but disciplined data extraction: reading the correct row and column, respecting the units printed on the axis or table header, and not confusing "total" with "average" or "percentage share" with "percentage change." This chapter works through five full DI sets in the exact formats AP Police Mains has historically used, with every calculation shown in full and cross-checked, so you can see precisely how an examiner expects the working to be laid out.
1. The DI Reading Discipline — Before You Touch a Single Number
Every DI set should be read in the same fixed order, regardless of format: first read the title (what is being measured), then the units (thousands, crores, percentages), then the categories (rows or bars or slices), and only then the actual numbers. Candidates who skip straight to the numbers routinely answer "in lakhs" when the table says "in thousands," or compute a percentage of the wrong total. This thirty-second discipline, applied before every DI set, prevents the single most common Mains scoring loss: a correct calculation performed on a misread number.
2. DI Set 1 — Table: Number of Students Enrolled in Police Training Academies (2023 vs 2024)
| Academy | 2023 | 2024 |
| A | 420 | 460 |
| B | 380 | 420 |
| C | 510 | 540 |
| D | 290 | 330 |
| E | 460 | 500 |
Q1. What was the total enrolment across all five academies in 2023?
Working: 420+380+510+290+460 = 2,060. Sum row by row, checking the running total after every addition rather than adding all five numbers in one pass — this halves the chance of a slip.
Q2. What was the total enrolment in 2024, and by how much did the combined total grow?
Working: 2024 total = 460+420+540+330+500 = 2,250. Growth = 2,250 - 2,060 = 190. Overall growth% = 190/2,060 × 100 = 9.22% (rounded to two decimals).
Q3. Which academy recorded the highest percentage growth from 2023 to 2024?
Working: Compute each academy's growth% individually — never assume the academy with the largest absolute increase also has the largest percentage increase, since percentage depends on the base too. A: (460-420)/420×100 = 9.52%. B: (420-380)/380×100 = 10.53%. C: (540-510)/510×100 = 5.88%. D: (330-290)/290×100 = 13.79%. E: (500-460)/460×100 = 8.70%. Academy D has the highest percentage growth (13.79%), even though its absolute increase of 40 is the smallest among all five academies. This is exactly the kind of trap Mains DI sets are built around — always compute percentage growth against each row's own base, not by eyeballing the absolute changes.
Q4. What is the ratio of Academy C's 2023 enrolment to Academy D's 2023 enrolment?
Working: 510:290. Divide both by their HCF, which is 10: 51:29. This does not simplify further since 51 = 3×17 and 29 is prime, sharing no common factor.
3. DI Set 2 — Bar Graph: Recruitment Applications Received by District (in numbers)
Bar graph: Krishna 1,200; Guntur 1,500; Visakhapatnam 1,800; Nellore 900; Kurnool 1,100.
Q1. What is the total number of applications received across all five districts?
Working: 1200+1500+1800+900+1100 = 6,500.
Q2. What percentage of the total applications came from Visakhapatnam?
Working: 1,800/6,500 × 100 = 27.69% (rounded to two decimals). Note the denominator is the grand total across all districts, not any single district's figure — a frequent misread in bar-graph percentage-share questions.
Q3. Find the average number of applications received per district.
Working: 6,500/5 = 1,300. This is the simple average across the five bars — do not confuse this with a weighted average, which is not applicable here since each district is one equally-weighted category.
Q4. What is the ratio of applications from Visakhapatnam to applications from Nellore?
Working: 1,800:900 = 2:1. Visakhapatnam received exactly double Nellore's applications — always check for such clean ratios in bar-graph data, since exam-setters frequently build the underlying numbers around round ratios.
4. DI Set 3 — Pie Chart: Annual Police Welfare Budget Allocation (Total Budget: Rs 800 crore)
Pie chart sectors (percentage of total budget): Training 15%, Infrastructure 25%, Welfare 20%, Technology 18%, Administration 22%.
Q1. Verify that the percentages add up correctly, then find the amount allocated to Infrastructure.
Working: 15+25+20+18+22 = 100%, confirmed. Infrastructure amount = 800 × 25/100 = Rs 200 crore.
Q2. What is the central angle (in degrees) representing the Welfare allocation in the pie chart?
Working: A full pie chart spans 360 degrees, and each percentage point equals 360/100 = 3.6 degrees. Welfare angle = 20 × 3.6 = 72 degrees. Memorise the "3.6 degrees per percentage point" conversion factor; it turns every pie-chart-to-angle question into a single multiplication.
Q3. Find the combined amount allocated to Technology and Administration together.
Working: Combined percentage: Technology 18% + Administration 22% = 40%. Amount = 800 × 40/100 = Rs 320 crore. Adding the percentages before multiplying by the total (rather than computing each amount separately and then adding) saves one step.
Q4. By what percentage does the Infrastructure allocation exceed the Training allocation?
Working: Infrastructure is 25%, Training is 15%, both of the same total budget, so comparing the percentages directly is valid. Percentage excess = (25-15)/15 × 100 = 10/15 × 100 = 66.67% (rounded to two decimals). Note this is not simply "10 percentage points" — the question asks by what percentage one exceeds the other, which requires dividing by the smaller base value (Training's 15%), not subtracting the two percentages directly.
5. DI Set 4 — Ratio-Based DI: Male and Female Recruits Over Three Years
| Year | Male Recruits | Female Recruits |
| 2021 | 1,400 | 600 |
| 2022 | 1,650 | 750 |
| 2023 | 1,800 | 900 |
Q1. What was the male-to-female ratio of recruits in 2021?
Working: 1,400:600. Divide both by their HCF, 200: 7:3.
Q2. In which year was the percentage of female recruits (out of that year's total) the highest?
Working: 2021: total 2,000, female% = 600/2,000×100 = 30%. 2022: total 2,400, female% = 750/2,400×100 = 31.25%. 2023: total 2,700, female% = 900/2,700×100 = 33.33% (rounded). The female percentage rises every year even though the male numbers also keep rising — this is only visible by computing each year's percentage independently, not by comparing absolute female counts, which would misleadingly suggest 2023 is simply "more of everything."
Q3. What is the combined male-to-female ratio across all three years put together?
Working: Total male = 1,400+1,650+1,800 = 4,850. Total female = 600+750+900 = 2,250. Ratio = 4,850:2,250. Divide both by their HCF, 50: 97:45. This combined ratio is not simply an average of the three yearly ratios (7:3, 22:10, 2:1) — it must be computed from the summed totals, because averaging ratios directly, without weighting by the actual base numbers in each year, gives a mathematically invalid result.
Q4. By what percentage did total recruitment (male plus female) increase from 2021 to 2023?
Working: 2021 total = 2,000. 2023 total = 2,700. Increase = 700. Percentage increase = 700/2,000 × 100 = 35%.
6. DI Set 5 — Line Graph: Reported Crimes in a District Over Five Years (in thousands)
Line graph values: 2020: 45, 2021: 52, 2022: 48, 2023: 60, 2024: 54.
Q1. What was the percentage change in reported crimes from 2020 to 2021?
Working: Change = 52-45 = 7. Percentage change = 7/45 × 100 = 15.56% (rounded to two decimals), an increase.
Q2. In which year-to-year interval was the percentage decrease the largest?
Working: Compute every consecutive interval, not just the ones that look large in absolute terms. 2021→2022: (48-52)/52×100 = -7.69%. 2023→2024: (54-60)/60×100 = -10%. The 2023→2024 interval shows the larger percentage decrease (10% versus 7.69%), even though the absolute drop in 2021→2022 (4 thousand) is smaller than in 2023→2024 (6 thousand) — checking both intervals independently, rather than assuming the visually steeper line segment on the graph is automatically the larger percentage change, is essential, since the percentage always depends on the starting value as well as the drop.
Q3. What is the average number of reported crimes per year over the five-year period?
Working: Total = 45+52+48+60+54 = 259. Average = 259/5 = 51.8 thousand.
Q4. What is the overall percentage change in reported crimes from 2020 to 2024?
Working: Change = 54-45 = 9. Percentage change = 9/45 × 100 = 20%. Note that this overall five-year change is not the sum or average of the four yearly percentage changes computed individually — it must always be calculated directly from the first and last values of the period being asked about.
7. DI Set 6 — Two-Category Table: Equipment Procurement (Vehicles and Weapons, Rs lakh) Over Three Years
| Year | Vehicles | Weapons | Total |
| 2022 | 80 | 120 | 200 |
| 2023 | 95 | 140 | 235 |
| 2024 | 110 | 150 | 260 |
Q1. In which year did vehicles account for the highest percentage share of that year's total procurement?
Working: 2022: 80/200×100 = 40.00%. 2023: 95/235×100 = 40.43%. 2024: 110/260×100 = 42.31%. The vehicle share rises every year; 2024 has the highest share at 42.31%. As with the earlier DI sets, this requires computing each year's share against that year's own total, not comparing the raw 80/95/110 figures directly, since the totals also change each year.
Q2. What is the combined vehicles-to-weapons ratio across all three years put together?
Working: Total vehicles = 80+95+110 = 285. Total weapons = 120+140+150 = 410. Ratio = 285:410. Divide both by their HCF, 5: 57:82. Once again, this combined ratio comes from the summed totals, not from averaging the three yearly ratios (2:3, 19:28, 11:15).
Q3. By what percentage did weapons procurement increase from 2022 to 2024?
Working: Increase = 150-120 = 30. Percentage increase = 30/120 × 100 = 25%.
Q4. What was the total procurement spend (vehicles plus weapons) across all three years?
Working: 200+235+260 = 695 lakh. Alternatively, total vehicles (285) plus total weapons (410) = 695 — the same figure reached by summing the yearly totals or the category totals, which is a useful self-check: both routes to the grand total must agree, and if they don't, one of your earlier row or column sums has an error.
8. DI Set 7 — Derived-Ratio DI: Population and Sanctioned Police Personnel by District
| District | Population (lakh) | Sanctioned Personnel |
| Krishna | 45 | 900 |
| Guntur | 52 | 1,040 |
| Nellore | 30 | 540 |
| Kurnool | 40 | 720 |
This kind of DI set is common in Mains because it forces you to construct a derived ratio (personnel per lakh of population) that is not printed directly in the table — you have to compute it before you can compare districts meaningfully.
Q1. Find the number of sanctioned personnel per lakh of population for Krishna district.
Working: Personnel per lakh population = Sanctioned personnel ÷ Population (in lakh) = 900/45 = 20. Krishna has 20 sanctioned personnel per lakh of population.
Q2. Which two districts have identical personnel-per-lakh-population ratios, and what is that ratio?
Working: Krishna: 900/45 = 20. Guntur: 1,040/52 = 20. Nellore: 540/30 = 18. Kurnool: 720/40 = 18. Krishna and Guntur share a ratio of 20 personnel per lakh; Nellore and Kurnool share a ratio of 18 personnel per lakh. Computing all four ratios individually, rather than assuming districts with larger populations automatically have proportionally larger personnel counts, is what reveals this pairing.
Q3. If Nellore's population grows to 36 lakh next year with personnel unchanged, what will its new personnel-per-lakh ratio be?
Working: New ratio = 540/36 = 15. Because personnel stayed fixed while population rose, the per-lakh availability of police personnel actually falls — from 18 to 15 — even though the raw personnel count hasn't changed. This is a good example of why per-capita or per-lakh figures, not raw totals, are what should drive resourcing questions in real police administration, not merely as an exam trick.
Q4. What is the combined population of all four districts, and the combined sanctioned personnel?
Working: Combined population = 45+52+30+40 = 167 lakh. Combined personnel = 900+1,040+540+720 = 3,200. Combined personnel-per-lakh ratio = 3,200/167 = 19.16 (rounded to two decimals) — again, this combined figure must come from the summed totals, not from averaging the four individual ratios of 20, 20, 18, and 18 (which would incorrectly give 19.0, ignoring that the districts have different population weights).
9. Recurring DI Traps to Watch For
Trap one — confusing "of total" with "of category." When a question asks "what percentage of the total budget," the denominator is always the grand total, never a single sector's value, unless the question explicitly says "as a percentage of [that specific sector]."
Trap two — averaging ratios or percentages directly. As shown in DI Set 4, Q3, a combined ratio across multiple years must be computed from the summed raw totals, not by averaging the individual yearly ratios. The same applies to combined percentages: you cannot average 30%, 31.25%, and 33.33% to get the three-year combined female percentage; you must sum the raw female and total counts first.
Trap three — "percentage increase" versus "how many times." If a value grows from 40 to 100, that is a 150% increase (60/40×100), not a "2.5 times" increase read carelessly as "150% of the original" (100 is indeed 2.5 times 40, and a 150% increase over 40 does correctly land at 100 — but candidates frequently misapply this by computing 100/40 = 2.5 and calling it "a 2.5% increase" or conversely computing the percentage increase correctly but then mislabelling it as a multiplicative factor in a later linked question). Always be explicit about which of the two you are being asked for, and convert between them only via: final value = original × (1 + percentage increase/100).
Trap four — degree-percentage conversion errors in pie charts. Always use 3.6 degrees per percentage point (since 360/100 = 3.6). A common error is dividing by 100 instead of multiplying, or forgetting that the chart's "100%" corresponds to 360 degrees rather than some other reference.
Trap five — rounding too early. When a DI question requires several intermediate steps, always carry at least two decimal places through each intermediate calculation and round only the final answer. Rounding an intermediate ratio like 900/45 to "20" is safe because it divides evenly, but rounding an intermediate figure like 540/36 = 15 exactly is fine too — the danger arises specifically with recurring decimals such as 3,200/167 = 19.1616..., where rounding to 19 at an intermediate step and then continuing further calculations from that rounded 19 can shift a final answer enough to make you pick the wrong MCQ option, especially when the four options are closely spaced (as in Q4 of DI Set 7 above, where 19.00, 19.16, 18.50, and 20.00 are all plausible-looking choices).
10. A Faster Way to Handle Multi-Part DI Sets Under Time Pressure
Mains DI sets typically bundle four or five sub-questions to one dataset, and the efficient approach is to extract every number the set is likely to need — row totals, column totals, and the grand total — in one pass immediately after reading the data, before looking at any individual sub-question. Write these totals in the margin. This "pre-compute the totals" habit means that when you then read Q1 through Q5, at least two or three of them can be answered by a single division or subtraction using numbers you have already calculated, rather than re-deriving a total from scratch for each sub-question separately.
11. Numerical Ability Beyond DI — Mixed Standalone Questions in Mains
Alongside DI sets, the Mains numerical ability portion also includes standalone questions similar in flavour to Prelims arithmetic but typically requiring two or three linked steps rather than one, and often phrased as short word problems tied to policing or administrative contexts. The approach is the same layered technique from Chapter 21 — convert percentages to fractions, use the LCM method for time-and-work, use capital-time products for partnership — but Mains rewards showing clean, labelled working more than Prelims does, since some Mains numerical sections are evaluated with partial credit for correct method even when the final answer has a small arithmetic slip. Always write out what each number in your working represents (e.g., "Personnel per lakh = 900/45") rather than leaving a bare string of numbers, since a clearly labelled step also makes it much easier for you to catch a unit or base error before submitting.
Worked example — a typical Mains-style word problem. A police department received a budget increase from Rs 450 crore to Rs 540 crore. In the same period, the number of sanctioned posts increased from 3,000 to 3,300. Did the average budget per sanctioned post increase or decrease, and by what percentage?
Fast method: Budget per post, before: 450/3,000 = 0.15 crore per post = Rs 15 lakh per post. Budget per post, after: 540/3,300 = 0.1636 crore per post = Rs 16.36 lakh per post (rounded to two decimals). Percentage change = (16.36-15)/15 × 100 = 1.36/15 × 100 = 9.09% (rounded to two decimals), an increase. This question layers two DI-style ratios (budget-to-post) computed at two points in time and then asks for the percentage change between them — exactly the kind of multi-step construction Mains numerical ability favours over single-step Prelims-style questions.
12. Time Allocation for DI Sets in the Mains Numerical Section
A full DI set with four linked sub-questions should take no more than six to seven minutes once you have practised the pre-computation habit from Section 10 — roughly ninety seconds to read and extract totals, and about ninety seconds per sub-question thereafter. If a single sub-question within a set is taking longer than two minutes on its own, the more efficient move is almost always to mark it for review and proceed to the next sub-question in the same set, since the other three or four questions are usually answerable from the same pre-computed totals regardless of whether you have cracked the difficult one. Losing five minutes trying to force one hard sub-question, when three easier ones from the same set remain unattempted, is a poor trade under Mains time constraints — treat each DI set as a small basket of individually gradable questions, not a single monolithic problem that must be solved top to bottom in order.
Key Facts at a Glance
- Read title, units, and categories before touching any number in a DI set — misread units cause more lost marks than calculation errors.
- Percentage growth must always be computed against each row's own base value, never assumed from the size of the absolute change.
- A combined ratio or combined percentage across multiple periods must be computed from summed raw totals, never by averaging the individual period ratios or percentages.
- Pie chart angle = percentage × 3.6 degrees; keep this conversion factor memorised as a single fact.
- "By what percentage does A exceed B" always divides by B (the smaller/reference value), not by A, and is not the same as the simple percentage-point difference.
- Pre-compute row totals, column totals, and the grand total immediately after reading the dataset, before reading the individual sub-questions.
- Distinguish "percentage change" (relative, divided by the original) from "percentage point change" (absolute difference between two percentages) in every question that mentions rates or shares.
- When a bar or table has visibly round relationships (like a clean 2:1 ratio), verify it explicitly — exam-setters build datasets around clean underlying ratios deliberately.
Practice MCQs (based on the DI sets above)
- Referring to DI Set 1, what was Academy B's percentage growth from 2023 to 2024?
- (a) 9.52%
- (b) 10.53%
- (c) 11.11%
- (d) 13.79%
- Referring to DI Set 1, what is the combined 2024 enrolment of academies A and E?
- (a) 880
- (b) 920
- (c) 960
- (d) 1000
- Referring to DI Set 2, what percentage of total applications came from Kurnool?
- (a) 15.38%
- (b) 16.92%
- (c) 18.46%
- (d) 20.00%
- Referring to DI Set 2, what is the ratio of applications from Guntur to applications from Kurnool?
- (a) 15:11
- (b) 3:2
- (c) 4:3
- (d) 5:4
- Referring to DI Set 3, what is the central angle for the Administration sector?
- (a) 72 degrees
- (b) 79.2 degrees
- (c) 64.8 degrees
- (d) 90 degrees
- Referring to DI Set 3, what amount is allocated to Training?
- (a) Rs 100 crore
- (b) Rs 110 crore
- (c) Rs 120 crore
- (d) Rs 130 crore
- Referring to DI Set 4, what was the male-to-female ratio of recruits in 2023?
- (a) 2:1
- (b) 9:4
- (c) 5:2
- (d) 3:1
- Referring to DI Set 4, what is the combined male-to-female ratio across all three years?
- (a) 97:45
- (b) 95:43
- (c) 99:47
- (d) 93:41
- Referring to DI Set 5, what was the percentage decrease in reported crimes from 2023 to 2024?
- (a) 8%
- (b) 9%
- (c) 10%
- (d) 12%
- Referring to DI Set 5, what is the average number of reported crimes per year across the five years?
- (a) 50.8 thousand
- (b) 51.8 thousand
- (c) 52.8 thousand
- (d) 53.8 thousand
- If a value rises from 40 to 100, what is the percentage increase?
- (a) 60%
- (b) 100%
- (c) 150%
- (d) 250%
- In a pie chart, a sector occupying 108 degrees represents what percentage of the total?
- (a) 25%
- (b) 28%
- (c) 30%
- (d) 32%
- Referring to DI Set 1, what is the ratio of Academy C's 2023 figure to Academy D's 2023 figure, in lowest terms?
- (a) 51:29
- (b) 50:28
- (c) 52:30
- (d) 17:10
- By what percentage does 25 exceed 15?
- (a) 40%
- (b) 50%
- (c) 60%
- (d) 66.67%
- Referring to DI Set 2, what is the average number of applications received per district?
- (a) 1,200
- (b) 1,250
- (c) 1,300
- (d) 1,350
- Referring to DI Set 6, in which year did vehicles account for the highest percentage of total procurement?
- (a) 2022
- (b) 2023
- (c) 2024
- (d) Equal in all three years
- Referring to DI Set 6, what is the combined vehicles-to-weapons ratio across all three years, in lowest terms?
- (a) 57:82
- (b) 55:80
- (c) 59:84
- (d) 28:41
- Referring to DI Set 7, what is Guntur's sanctioned personnel per lakh of population?
- (a) 18
- (b) 19
- (c) 20
- (d) 21
- Referring to DI Set 7, what is the combined personnel-per-lakh ratio for all four districts put together?
- (a) 18.50
- (b) 19.00
- (c) 19.16
- (d) 20.00