UPSC CSE IAS · UPSC General Studies & Ethics

CSAT Basic Numeracy & Data Interpretation

Covers basic numeracy at Class X level and data interpretation using charts, graphs, tables and data sufficiency.

Seven concepts for CSAT Paper II numeracy at Class X level: percentages and successive change, profit on cost price, simple versus compound interest, time–speed–distance with unit conversion, averages and alligation, reading charts (especially pie angles), and data sufficiency as a stop-early judgment — not a full solve.

UPSC CSE IAS
Medium level
7 concepts
6 practice questions

1Percentages and successive change

A percentage is a fraction out of 100. Percentage change is always \dfrac{\text{new} - \text{old}}{\text{old}} \times 100 — the earlier value is the base, and swapping the base is the standard distractor in DI and profit–loss stems.

Two successive percentage changes of a\% and b\% do not add. The net change is a + b + \dfrac{ab}{100} with signs kept. Equal up-then-down percentage moves always leave a net decrease — never a wash — of \dfrac{x^{2}}{100}\% when each move has size x\%.

Figure. Successive +20% then −20% on 100 lands at 96, not back at 100 — factors multiply. Values are the worked net-change shape, not a generic percent icon.

How successive change works

Write signed percentsTake each change with its sign — increase positive, decrease negative — before combining.
Apply the net formulaNet % = a + b + \dfrac{ab}{100}. The cross term carries the product of the two signed changes.
Check with factorsMultiply the remaining factors (1 + a/100)(1 + b/100) and convert back to a percent change from 1 — same answer, different arithmetic path.

Percentage–fraction shortcuts
Percent	Fraction	Use
12.5%	1/8	DI mental share
16⅔%	1/6	Pie 60° sector
25%	1/4	Pie 90° sector
33⅓%	1/3	Pie 120° sector
37.5%	3/8	Three-eighths share
62.5%	5/8	Five-eighths share

Equal up-then-down percentage change

The price of a commodity is increased by 20\% and then decreased by 20\%. What is the net percentage change in price?

a = +20, b = −20signed successive changes
ab/100 = (20)(−20)/100−4
net = 20 + (−20) + (−4)−4
Check: 1.20 × 0.800.96 → 4% decrease

Pro tip. Equal up-then-down of x\% always nets -x^{2}/100\% — here 400/100 = 4\% down. Adding +20 and -20 to get zero is the trap the cross term exists to kill.

A price rises 10% and then falls 10%. Relative to the original price, the final price is

unchanged
1% lower
1% higher

Net = 10 + (−10) + (10)(−10)/100 = −1, so 1% lower. Unchanged ignores the cross term; 1% higher flips the sign of ab/100.

2Profit percent sits on cost price

Profit percentage is \dfrac{\text{Profit}}{\text{Cost Price}} \times 100, and selling price from a known profit percent is \text{SP} = \text{CP}\left(1 + \dfrac{\text{Profit\%}}{100}\right). The base is always cost price — dividing by selling price produces a tidy wrong percent that almost always appears among the options.

Loss uses the same base with the opposite difference: loss % = (\text{CP} - \text{SP})/\text{CP} \times 100. Marked-price discounts are a different chapter of arithmetic; when CSAT gives only CP and SP, stay on the cost-price base.

Figure. Profit percent sits on cost price: 20 on a CP of 100 is 20%, even though SP is 120. Using SP as the base is the standard distractor.

How it works

Name CP and SPWrite both in rupees. Profit is SP − CP; loss is CP − SP.
Divide by CPPercent = (difference / CP) × 100. Never divide by SP for profit or loss percent.
Rebuild SP if neededSP = CP × (1 + profit%/100), or CP × (1 − loss%/100).

Profit percent from CP and SP

An article bought for Rs. 400 is sold for Rs. 500. Find the profit percent.

Profit = SP − CP = 500 − 400100
Base = CP = 400divide by 400
Profit % = 100/400 × 10025%
Check: SP = 400 × (1 + 25/100)500

Pro tip. Dividing the same profit by SP gives 100/500 × 100 = 20% — the classic trap option. Whenever both 25% and 20% appear, the question is testing the base, not the subtraction.

Cost price Rs. 240, selling price Rs. 300. Profit percent is

Profit 60 on CP 240 is 25%. Dividing by SP gives 20%. Taking 60 against 200 invents a third base and yields 30%.

3Simple interest versus compound interest

Simple interest is linear in time: \text{SI} = \dfrac{P \times R \times T}{100}, so each year adds the same rupee interest on the original principal. Compound interest grows the amount: A = P\left(1 + \dfrac{R}{100}\right)^{T}, and CI = A - P.

For T > 1 at a positive rate, compound interest exceeds simple interest because each year's interest itself earns interest. For small integer T, compute A directly rather than expanding the binomial.

Figure. Schematic growth: simple interest is a straight climb; compound interest bends upward as interest earns interest. Axes are qualitative — use the two-year ledger for exact rupees.

How to choose the formula

Read the wording"Simple interest" or "interest per year on the principal" → SI formula. "Compounded annually" or "amount after T years" → CI amount formula.
Compute SI or ASI = PRT/100. For CI, evaluate P(1 + R/100)^{T} first; do not add SI and a patchwork correction unless the question asks for the difference.
Extract CI if askedCI = A − P. The difference CI − SI for two years is P(R/100)^{2}, a useful check.

SI and CI for two years

Find the simple interest and the compound interest (annual compounding) on Rs. 1000 at 10% per annum for 2 years.

SI = (1000 × 10 × 2)/100200
A = 1000 × (1.10)² = 1000 × 1.211210
CI = A − P = 1210 − 1000210
Check: CI − SI = 1000 × (0.10)²10

Pro tip. For two years the CI−SI gap is exactly P(R/100)^{2}. If your CI and SI differ by something else, one of the two calculations slipped.

On the same principal, rate and time greater than one year, which statement is correct?

Simple interest is always greater than compound interest
Compound interest is always greater than simple interest
They are equal whenever the rate is 10%

Compounding credits interest on interest, so CI > SI for T > 1 at positive rate. Equality holds only for T = 1 (or R = 0), not for a special rate like 10%.

4Distance, speed and the 5/18 conversion

The core relation is \text{Distance} = \text{Speed} \times \text{Time}. Trains, boats and pipes in CSAT are the same relation with a named length or rate — crossing a pole means the train travels its own length; crossing a platform means train length plus platform length.

Unit mixing is the silent killer: 1\ \text{km/h} = \dfrac{5}{18}\ \text{m/s} and 1\ \text{m/s} = \dfrac{18}{5}\ \text{km/h}. Convert as soon as metres meet hours, or seconds meet km/h. For equal distances at speeds x and y, average speed is the harmonic mean \dfrac{2xy}{x+y}, not (x+y)/2.

Figure. Crossing a pole: the train travels its own length L. Platform crossing adds platform length — the figure is the pole case only. Convert km/h to m/s with ×5/18 when lengths are in metres.

How a crossing problem works

Name the distancePole or man → train length only. Platform or bridge → train length + platform (or bridge) length.
Speed = distance / timeKeep SI units consistent — metres with seconds gives m/s.
Convert if askedm/s → km/h multiply by 18/5; km/h → m/s multiply by 5/18.

Train crossing a pole

A train 150 m long crosses a pole in 9 seconds. Find its speed in km/h.

Distance to cross a pole150 m (own length)
Speed = 150/950/3 m/s
(50/3) × (18/5)60 km/h
Check: 60 × (5/18) × 9150 m

Pro tip. Crossing a pole uses only the train's length; crossing a platform adds the platform. Applying the 18/5 factor immediately prevents leaving the answer in m/s when the options are in km/h.

A 180 m train crosses a man standing on the platform in 12 s. Its speed is

54 km/h
15 km/h
45 km/h

Distance = 180 m, speed = 180/12 = 15 m/s → 15 × (18/5) = 54 km/h. Leaving the answer as 15 reports m/s as if it were km/h. 45 km/h is 12.5 m/s converted, a slip in the division 180/12.

5Averages and alligation

The ordinary average is \dfrac{\text{sum of observations}}{\text{number of observations}}. When CSAT asks for a combined average of two groups, weight each group's average by its size — never average the two averages unless the groups are equal.

Alligation finds the mixing ratio of two ingredients at different prices or concentrations that produce a known mean. On the alligation line, the ratio of quantities is (dear − mean) : (mean − cheap) — the gaps swap sides. It is the same arithmetic as a weighted average solved for the weights.

Figure. Alligation cross for mixing 20 and 50 to mean 32: opposite differences give 18 parts cheap to 12 parts dear (3:2). Schematic layout — recompute from the three scalars in the ledger.

How alligation sets the ratio

Name the three valuesCheap price (or %), dear price (or %), and the desired mean.
Take the gapsGap above = dear − mean; gap below = mean − cheap.
Swap onto quantitiesQuantity of cheap : quantity of dear = (dear − mean) : (mean − cheap).

Mixing two concentrations

In what ratio must a 20% spirit solution be mixed with a 40% spirit solution to get a 25% spirit solution?

dear − mean = 40 − 2515
mean − cheap = 25 − 205
cheap : dear = 15 : 53 : 1
Check: (3×20 + 1×40)/425%

Pro tip. The larger gap sits on the cheaper ingredient — you need more of the solution that is closer to the mean. Writing 1 : 3 reverses which bottle dominates.

Solutions at 10% and 30% are mixed to get 18%. The ratio of 10% solution to 30% solution is

3 : 2
2 : 3
1 : 1

Gaps: 30 − 18 = 12 and 18 − 10 = 8, so cheap : dear = 12 : 8 = 3 : 2. Writing 2 : 3 swaps the bottles; 1 : 1 would mean a 20% mean, not 18%.

6Reading tables, bars and pie charts

Before any calculation, read the title, the axis labels and the units. The load-bearing check is scale and units — treating thousands as units, or reading a bar against the wrong axis — before the arithmetic.

A pie chart uses 360^\circ for the whole: category angle = \dfrac{\text{value}}{\text{total}} \times 360^\circ, and category percentage = \dfrac{\text{angle}}{360^\circ} \times 100. The conversion 1\% = 3.6^\circ makes 90^\circ = 25\%, 120^\circ = 33\tfrac{1}{3}\% and 180^\circ = 50\% instant.

Figure. Bar stand-in for pie shares: 25–50–25 maps to 90°–180°–90° of a full circle. Read titles, units and axes before arithmetic — the vocabulary has no wedge, so angles live in the angle table.

How to attack a DI set

Read titles and unitsWrite down what each axis or column measures before touching a question.
Park shared totalsSum the relevant series once. Every later percent or ratio is one division off that total.
Convert pie anglesAngle/360° is the fraction; multiply by the stated total for the rupee (or unit) value.

Pie angles that should be instant
Central angle	Fraction	Percentage
36°	1/10	10%
45°	1/8	12.5%
60°	1/6	16⅔%
90°	1/4	25%
120°	1/3	33⅓%
180°	1/2	50%

Value from a pie sector

A family's monthly budget of Rs. 36000 is shown as a pie chart. The Food sector has central angle 120^\circ. Find the amount spent on food.

Fraction = 120°/360°1/3
(1/3) × 3600012000
Check: 120°/3.6°33⅓%
33⅓% of 3600012000

Pro tip. A 120^\circ sector is always one-third. Keep 90^\circ, 120^\circ and 180^\circ as ready fractions so pie arithmetic collapses before a calculator comes out.

A pie chart totals Rs. 48000. A sector of 90° represents

Rs. 12000
Rs. 16000
Rs. 24000

90° is one-fourth of 360°, so (1/4)×48000 = 12000. Taking 90/270 (dropping a zero wrongly) gives 16000; treating 90° as half gives 24000.

7Data sufficiency is enough-or-not

Data sufficiency asks whether the given statements are enough to answer the question — not what the numerical answer is. You may stop as soon as you know the value is determined; computing that value is optional and often a time sink.

Evaluate each statement alone first. Only if each is insufficient do you combine them. The standard five-way choice (statement I alone, II alone, either, both together, neither) rewards this order; jumping to a combined solve hides which statement was already enough.

Figure. Data-sufficiency decision spine: test each statement alone, then together. The verdict is enough-or-not — not the numerical answer. Map each leaf to the five standard option wordings in the table.

Sufficiency order

Restate the askName the exact quantity or yes/no the question wants — "find x" and "is x > 10?" need different information.
Test statement I aloneAsk whether I determines a unique answer. Do not peek at II.
Test statement II aloneSame test, ignoring I. Mark each as sufficient or not.
Combine only if neededIf both alone fail, use I and II together. If that still fails, the data are insufficient.

Standard sufficiency verdicts
Verdict	Meaning
I alone	I sufficient; II not needed
II alone	II sufficient; I not needed
Either alone	Each statement alone is sufficient
Both together	Neither alone; both combined suffice
Neither	Even together, answer not determined

What is the value of x? (I) 2x + 4 = 10. (II) x² = 9. Which sufficiency verdict fits?

I alone is sufficient, but II alone is not
II alone is sufficient, but I alone is not
Both statements together are needed, and even then x is not unique

I gives 2x = 6 so x = 3 uniquely. II gives x = 3 or x = −3, so II alone is not sufficient. The trap is treating II as enough because 9 is a perfect square, or assuming both are needed because II looks relevant.

Notes

Percentages and ratios: a percentage is a fraction out of 100; percentage change = \dfrac{\text{new} - \text{old}}{\text{old}} \times 100. Most DI and profit-loss problems reduce to percentage manipulation.
Time-Speed-Distance and Time-Work: the core relations \text{Distance} = \text{Speed} \times \text{Time} and 'work done = rate \times time' cover trains, boats-streams and pipes-cisterns problems.
Averages and mixtures: the average = \dfrac{\text{sum of observations}}{\text{number of observations}}; alligation quickly finds mixing ratios of two ingredients at different prices/concentrations.
Data Interpretation: read titles, axes and units of tables, bar/line/pie charts before calculating; a pie chart uses 360^\circ for the whole, so each category's angle = \dfrac{\text{value}}{\text{total}} \times 360^\circ.
Data Sufficiency: decide whether the given statements are ENOUGH to answer - you need not compute the final value, only judge sufficiency of each statement alone and together.

Formulas

Simple Interest = \dfrac{P \times R \times T}{100}; Compound Interest amount A = P\left(1 + \dfrac{R}{100}\right)^{T}.
Profit percentage = \dfrac{\text{Profit}}{\text{Cost Price}} \times 100; Selling Price = \text{CP}\left(1 + \dfrac{\text{Profit\%}}{100}\right).
Speed conversion: 1\ \text{km/h} = \dfrac{5}{18}\ \text{m/s}; and 1\ \text{m/s} = \dfrac{18}{5}\ \text{km/h}.
Average speed for equal distances at speeds x and y = \dfrac{2xy}{x+y} (harmonic mean).
Pie chart: category angle = \dfrac{\text{category value}}{\text{total}} \times 360^\circ; category percentage = \dfrac{\text{angle}}{360^\circ} \times 100.

Exam traps & shortcuts

Percentage-to-fraction shortcuts: 12.5\% = \tfrac{1}{8}, 16.67\% = \tfrac{1}{6}, 25\% = \tfrac{1}{4}, 33.33\% = \tfrac{1}{3} - speeds up mental DI calculations.
For two successive percentage changes of a\% and b\%, net change = a + b + \dfrac{ab}{100} (use with sign).
In data sufficiency, evaluate each statement independently first; avoid combining unless each alone is insufficient.
Use the \tfrac{5}{18} factor immediately whenever a problem mixes km/h with seconds/metres to avoid unit errors.

Reference tables

Night-before relations from the concepts above. Reconstruct the idea from the concept, then use this sheet only to check the formula.

CSAT numeracy formula sheet
Idea	Relation	Watch
Percentage change	(new − old)/old × 100	Base is the earlier value
Successive a%, b%	a + b + ab/100	Keep signs; equal ±x nets −x²/100%
Profit %	(SP − CP)/CP × 100	Base is CP, never SP
Simple interest	SI = PRT/100	Linear in T
Compound amount	A = P(1 + R/100)^T	CI = A − P
Speed conversion	1 km/h = 5/18 m/s	Convert before mixing units
Equal-distance average	2xy/(x + y)	Harmonic mean, not (x+y)/2
Pie sector	θ = (value/total) × 360°	1% = 3.6°

Recap

Read only this the night before.

Percent change: Always divide by the old value. Successive net = a + b + ab/100.
±x then ∓x: Equal up-then-down of x% nets −x²/100% — never zero.
Shortcuts: 12.5% = 1/8, 16⅔% = 1/6, 25% = 1/4, 33⅓% = 1/3.
Profit base: Profit % and loss % divide by CP. SP = CP(1 + profit%/100).
SI vs CI: SI = PRT/100; A = P(1+R/100)^T. For T > 1, CI > SI.
5/18: km/h → m/s ×5/18; m/s → km/h ×18/5. Pole cross = own length.
Alligation: cheap : dear = (dear − mean) : (mean − cheap).
Pie: Angle/360° is the fraction. 90° = 25%, 120° = 1/3, 180° = 1/2.
Sufficiency: Judge each statement alone first; compute the value only if you must.

Practise CSAT Basic Numeracy & Data Interpretation

Reading is free and needs no account. Practice, mocks and progress live in the app.

6 exam-style questions on this topic, with explanations
A 5-question practice set that ends the chapter
Timed mocks scored with the real marking scheme
Readiness tracked per topic, kept on your device

Continue with Google — freeNo card, no trial. Works offline once installed.