Data Interpretation: Methods and Shortcuts

SSC · Banking · Railways · State exams — method by method

3 Part16 worked examples24 practice questions

In data interpretation (DI) you get a table or a chart and 4–5 questions on it. The maths is light; the test is of time and care in reading. In this chapter we learn the methods that work for every kind of chart: how to read a chart, quick conversion between fractions and percents, moving ahead by estimation, and finding change, ratio, average and growth rate. Each method has the idea, the formula (with the meaning of every letter), the steps, solved questions with a chart, and questions to try yourself; the solution stays hidden, so try first.

Part 2: methods special to pie, bar and line charts, cumulative and missing values, two tables / two charts, caselets, option elimination and a 5-question set in 4 minutes — open Data Interpretation: Chart Methods and Speed Plan in the chapter list.

Type of chart	Open this for more practice
Table	Data Table
Pie chart	Pie Chart
Bar chart	Bar Chart
Line graph	Line Graph
Two charts together	Mixed Graph
“Is it enough?”	Data Sufficiency

How to useOrder of work: learn the method here, then do the hundreds of questions in the chart-type chapters listed above. Their questions are not repeated in this chapter.

Reading and quick arithmetic

The right number, fast

Care in reading, the fraction-percent table, estimation and comparing fractions.

Reading a chart or table without mistakes

Five looks in the first 30 seconds

Stopping reading slips by checking unit, scale, labels and base.

The idea

Most DI questions are lost at reading, not at maths. The setter hides a trap in the chart: different units, a scale that does not start at zero, or a percent of something else. A hurried reader does correct maths on the wrong number. So before any calculation spend 30 seconds on a five-point check.

◭ Diagram

Four looks at a chart: ① title ② unit (thousand tonnes) ③ scale: the axis starts at 40, not 0 (red break mark) ④ labels: which bar is which year

Look at	Ask yourself	If missed
① Title	What, where and when is measured?	working on the wrong thing (imports instead of exports)
② Unit	Thousands or lakhs? ₹ or %?	answer 100 or 1,000 times off
③ Scale	Does the axis start at 0? How much between two ticks?	wrong comparison from bar heights
④ Labels	Which colour or shape is which series? Years or months on the x-axis?	two series swapped
⑤ Base	A percent of what: the total, last year, or one part?	adding percents that have different bases

Steps

Read the title and unit; jot the unit in a corner of your sheet.
Check where the axis starts and the gap between two ticks.
Match labels and legend: colour ↔ series, axis ↔ year / month.
Read the last line of each question: which numbers are needed?
Write the numbers with the unit, then calculate.

Commonest trapWhere the axis has no 0, a bar’s height is not its value. In the chart above the 2021 bar looks 2.5 times the 2019 bar, but output rose only 30%.

Worked questions

Table: milk collected by three dairies. Spring figures are in lakh litres, summer figures in thousand litres.

Dairy	Spring (lakh litres)	Summer (thousand litres)
Aravali	9.2	780
Banas	7.5	810
Chambal	6.4	645

Example 1 · Basic

In which season did Aravali collect more milk, and how much more (in litres)?

Solution, step by step

1The two columns have different units (1 lakh = 1,00,000; 1 thousand = 1,000), so subtracting 9.2 and 780 directly is meaningless.
2Spring = 9.2 lakh = 9,20,000 L; summer = 780 thousand = 7,80,000 L.
3Difference $= 9, 20, 000 - 7, 80, 000 = 1, 40, 000$ L; more in spring.

Answer Spring, by 1,40,000 litres

Bring both to one unit first, then compare. Lakhs also works here: 780 thousand = 7.8 lakh, gap 9.2 − 7.8 = 1.4 lakh.

Example 2 · Basic

What is the total summer collection of the three dairies, in lakh litres?

Solution, step by step

1All three are in one unit (thousand litres), so add directly: $780 + 810 + 645 = 2235$ thousand litres.
2The question wants lakhs; 100 thousand = 1 lakh, so $2235 \div 100 = 22.35$ lakh.

Answer 22.35 lakh litres

Example 3 · Standard

In the rice chart above the 2021 bar looks 2.5 times as tall as the 2019 bar. By what percent did output really rise from 2019 to 2021?

Solution, step by step

1The axis starts at 40, so the visible height = value − 40: $10$ for 2019 and $25$ for 2021. That is the 2.5-times illusion.
2The real values are 50 and 65: rise $= 15$ thousand tonnes.
3Rise % $= \frac{15}{50} \times 100 = 30%$ (the base is the older value, 50).

Answer 30% (not 150%)

If the scale does not start at zero, read the real values first, then find the percent.

Table: households of two zones and the percent that have solar water heaters (each percent is of that zone’s own households).

Zone	Households (thousand)	With heaters (%)
North	40	15%
South	60	25%

Example 4 · Standard

What percent of the households of both zones together have heaters?

Solution, step by step

1The two percents have different bases (40 thousand and 60 thousand), so averaging 15 and 25 to 20% is wrong.
2North: $15%$ of $40 = 6$ thousand; South: $25%$ of $60 = 15$ thousand.
3Total households $= 100$ thousand, with heaters $= 21$ thousand ⟹ $21%$ .

Answer 21%

Before adding or averaging percents ask: is the base the same?

Try yourself

Chart: water used in two hostels (in thousand litres). Note: the scale starts at 30. All questions below use this chart.

◭ Diagram

Water use, June to October (axis starts at 30)

BasicTry it yourself first

How much water did the two hostels use together in August?

BasicTry it yourself first

In June, by what percent was the boys’ hostel use more than the girls’ hostel use?

StandardTry it yourself first

In which month was the gap between the two largest, and how many litres was it?

StandardTry it yourself first

What is the average monthly use of the girls’ hostel?

StandardTry it yourself first

By what percent did the boys’ hostel use rise from June to October?

AdvancedTry it yourself first

Water costs ₹ 12 per thousand litres. What is the girls’ hostel bill for July to September?

The fraction ↔ percent table (1/2 to 1/20)

What you know by heart is fast

Doing percent and ratio work in seconds by knowing the percents of unit fractions.

The idea

DI keeps asking “what percent is this of that?” and “what is the value after a rise of this percent?”. Working out 350 ÷ 2100 × 100 costs 25 seconds, but if you know $\frac{1}{6} = 16 \frac{2}{3} %$ you see 350 and 2100 and say “one-sixth”. So recognise the fraction → know the percent (or the reverse). In the picture below each bar is the percent of $\frac{1}{n}$ ; the colours show which fractions are one family.

◭ Diagram

The percent of

\frac{1}{n}

: inside a family each fraction is half the earlier one (50 → 25 → 12.5 → 6.25)

∑ Formula

\frac{k}{n} = k \times \frac{100}{n} %

Any fraction = numerator × (percent of the unit fraction)

$k$: numerator of the fraction
$n$: denominator of the fraction
$\frac{100}{n} %$: percent of $\frac{1}{n}$ (from the picture)

Fraction	Percent	How	Fraction	Percent	How
$\frac{3}{8}$	37.5%	3 × 12.5	$\frac{2}{3}$	$66 \frac{2}{3} %$	2 × 33⅓
$\frac{5}{8}$	62.5%	5 × 12.5	$\frac{5}{6}$	$83 \frac{1}{3} %$	100 − 16⅔
$\frac{7}{8}$	87.5%	100 − 12.5	$\frac{5}{12}$	$41 \frac{2}{3} %$	5 × 8⅓
$\frac{3}{16}$	18.75%	3 × 6.25	$\frac{4}{9}$	$44 \frac{4}{9} %$	4 × 11⅑
$\frac{3}{5}$	60%	3 × 20	$\frac{3}{7}$	$42 \frac{6}{7} %$	3 × 14²⁄₇

Here: $k$ = numerator, $n$ = denominator; the “How” column multiplies the numerator by the percent of the unit fraction.

Steps

Write the two numbers as a fraction and reduce it (350/2100 = 1/6).
Recognise the percent of the reduced fraction from the table; if the numerator is more than 1, use unit fraction × numerator.
Reverse: turn the percent into a fraction (37.5% = 3/8) and divide and multiply on the number itself.

Spot itSpot it: options with 12.5, 16⅔, 33⅓, 37.5, 62.5, 83⅓, or table numbers that divide neatly by a small denominator, tell you to use the fraction method.

Worked questions

Example 5 · Basic

Output rose from 640 to 720 units. What is the percent rise?

Solution, step by step

1Rise $= 720 - 640 = 80$ ; the base is the old value 640.
2 $\frac{80}{640} = \frac{1}{8}$ (divide both by 80).
3 $\frac{1}{8} = 12.5%$ (from the table).

Answer undefined

Example 6 · Basic

What is 37.5% of ₹ 1,840 crore?

Solution, step by step

1 $37.5 = 3 \times 12.5$ and $12.5% = \frac{1}{8}$ , so $37.5% = \frac{3}{8}$ .
2 $1840 \div 8 = 230$ , then $230 \times 3 = 690$ .

Answer ₹ 690 crore

Divide first, then multiply: the numbers stay small.

Example 7 · Standard

350 of 2,100 items were sold. What percent were sold?

Solution, step by step

1 $\frac{350}{2100} = \frac{1}{6}$ (both are multiples of 350).
2From the table $\frac{1}{6} = 16 \frac{2}{3} %$ .

Answer undefined

Example 8 · Standard

A sum is left at 640 after a fall of

11 \frac{1}{9} %

. What was the original sum?

Solution, step by step

1 $11 \frac{1}{9} % = \frac{1}{9}$ , i.e. 1 part out of 9 parts of the original was lost.
2The remaining $9 - 1 = 8$ parts $= 640$ , so 1 part $= 80$ .
3Original $= 9 \times 80 = 720$ .

Answer 720

You cannot get the original by adding the percent back to the reduced sum; think in parts.

Try yourself

BasicTry it yourself first

8 \frac{1}{3} %

of 4,800 = ?

BasicTry it yourself first

A salary rose from ₹ 36,000 to ₹ 40,500. What is the percent rise?

BasicTry it yourself first

What percent is

\frac{7}{16}

StandardTry it yourself first

A sum rose by

14 \frac{2}{7} %

to reach 1,400. What was the original?

StandardTry it yourself first

By how much does

41 \frac{2}{3} %

of 3,720 exceed

\frac{5}{12}

of 3,600?

AdvancedTry it yourself first

Sales rose from 330 thousand to 360 thousand. What is the percent rise?

Estimation: rounding and 10%-1% blocks

Options are far apart, so why be exact?

Getting close to the answer by rounding and by adding up 10%, 5% and 1%.

The idea

DI options are usually 2–3% or more apart. So there is no need to do the full division with 6,023 and 6,987: treat them as 6,000 and 7,000 and the estimate will not cross the gap between options. The second tool is breaking a percent into 10%, 5% and 1% blocks: move the decimal point one place for 10% and two places for 1%.

◭ Diagram

35% = 10% + 10% + 10% + 5%; 5% is half of 10%

∑ Formula

error % = \frac{∣ actual - rounded ∣}{actual} \times 100

Rounding is safe only when this error is smaller than the gap between options

$a c t u a l$: the real number in the question
$ro u n d e d$: the rounded number you use

Move	Example
10% = decimal one place left	10% of 2,480 = 248
5% = half of 10%	5% of 2,480 = 124
1% = decimal two places left	1% of 2,480 = 24.8
Other % = add / subtract	18% = 10% + 5% + 3×1%; 19% = 20% − 1%

Steps

Look at the options: how far apart are the two closest? That is your allowance.
Round each number to 2 significant digits (6,023 → 6,000). Rounding the numerator up and the denominator down makes the fraction bigger (and the reverse).
Calculate with the rounded numbers.
Think of the direction of the error: is the true answer a little above or below the estimate? Pick the nearer option.

CautionWhen two options are very close (say 41% and 42%), do not round: calculate exactly.

Worked questions

Example 9 · Basic

Find 23% of 1,840.

Solution, step by step

1 $23% = 10% + 10% + 1% + 1% + 1%$ (two tens and three ones).
210% $= 184$ ; twice $= 368$ . 1% $= 18.4$ ; thrice $= 55.2$ .
3Add: $368 + 55.2 = 423.2$ .

Answer undefined

Example 10 · Standard

Sales rose from 6,023 to 6,987. About what percent is the rise? (12, 14, 16, 20)

Solution, step by step

1The options are 2–4 apart, so an error of up to 1 is fine.
26,023 → 6,000 and 6,987 → 7,000 (each moved by less than 0.5%).
3Rise $= 1000$ ; $\frac{1000}{6000} = \frac{1}{6} \approx 16.7%$ .
416 is nearest (14 and 20 are far).

Answer 16%

Exact:

\frac{964}{6023} = 16.01%

. The estimate was 0.7 high, but good enough to choose.

Example 11 · Standard

2,483 is about what percent of 6,017? (36, 38, 41, 45)

Solution, step by step

1Numerator $2483 \to 2500$ (up), denominator $6017 \to 6000$ (down): both changes raise the fraction, so the true value is below the estimate.
2 $\frac{2500}{6000} = \frac{5}{12} = 41 \frac{2}{3} % \approx 41.7%$ .
3True value a little below 41.7 ⟹ 41 (45 is far).

Answer 41%

The exact value is 41.27%: the direction guess was right.

Example 12 · Standard

Find 35% of 2,480.

Solution, step by step

1 $35% = 3 \times 10% + 5%$ .
210% $= 248$ ; thrice $= 744$ ; 5% $= 124$ .
3 $744 + 124 = 868$ .

Answer undefined

Try yourself

BasicTry it yourself first

15% of 4,260 = ?

BasicTry it yourself first

About how much is 38% of 5,010? (1,804 / 1,904 / 2,004 / 1,954)

StandardTry it yourself first

Passengers rose from 7,980 to 9,120. About what percent is the rise? (12, 14, 16, 18)

StandardTry it yourself first

3,570 is about what percent of 8,980? (36, 40, 44, 48)

StandardTry it yourself first

About what is 18.2% of 2,410 plus 11.8% of 3,190? (780 / 815 / 850 / 895)

AdvancedTry it yourself first

Four months’ sales are 4,812, 5,196, 5,087 and 4,905. What is the average monthly sale?

Comparing fractions and ratios without dividing

Cross-multiply, benchmark, gap

Deciding “which ratio or percent is highest?” without dividing.

The idea

“In which year (or institute) is the ratio highest?” is the most frequent DI question. Turning every fraction into a decimal takes time, and close values invite slips. For a comparison you do not need the full value, only which is bigger. Four moves do it, and all rest on one idea: put the fractions side by side without dividing.

◭ Diagram

Cross-multiplication: numerator × the other denominator; the side whose numerator gives the bigger product is the bigger fraction

∑ Formula

\frac{a}{b} > \frac{c}{d} ⟺ a \times d > c \times b

valid when

b

and

d

are positive

$a, b$: numerator and denominator of the first fraction
$c, d$: numerator and denominator of the second fraction

Move	When	Example
① Cross-multiply	any two fractions	$\frac{5}{8}$ vs $\frac{7}{11}$ : 55 < 56
② Benchmark ½	many fractions; “who is above a half?”	$\frac{23}{45}$ : $2 \times 23 = 46 > 45$ ⟹ above a half
③ Gap from the whole	fractions near 1	$\frac{72}{80}$ (gap $\frac{8}{80}$ ) vs $\frac{85}{95}$ (gap $\frac{10}{95}$ ): the smaller gap is the bigger fraction
④ Same numerator or denominator	numerators (or denominators) easy to match	$\frac{6}{17}$ vs $\frac{12}{35}$ = $\frac{6}{17.5}$ : smaller denominator ⟹ bigger fraction

Here: $a, c$ = numerators, $b, d$ = denominators; “gap” = denominator − numerator, how far short of the whole.

Steps

Write each as a fraction: numerator = the thing asked about, denominator = what it is compared with.
Pick a move (table above); with five or more fractions run a tournament: pick the bigger of the first two, test it against the third, and so on.
Each comparison is one cross-multiplication; a list of $n$ fractions needs only $n - 1$ comparisons.

SpeedDividing to 3 decimals each time is 4–5 times slower; cross-multiplying is just two products.

Worked questions

Table: seats filled and seats available in four training institutes.

Institute	Seats filled	Seats available
Aarohi	44	56
Bodh	51	66
Chetna	62	79
Disha	39	50

Example 13 · Standard

Which institute has the highest share of seats filled?

Solution, step by step

1Share $= \frac{filled}{available}$ : $\frac{44}{56}$ , $\frac{51}{66}$ , $\frac{62}{79}$ , $\frac{39}{50}$ .
2First two: $44 \times 66 = 2904$ , $51 \times 56 = 2856$ . $2904 > 2856$ ⟹ Aarohi leads.
3Aarohi vs Chetna: $44 \times 79 = 3476$ , $62 \times 56 = 3472$ ⟹ Aarohi leads (by only 4!). Aarohi vs Disha: $44 \times 50 = 2200 > 39 \times 56 = 2184$ .

Answer Aarohi

Aarohi is 78.57% and Chetna 78.48%: miss the second decimal and the answer flips.

Example 14 · Standard

Which is bigger:

\frac{72}{80}

\frac{85}{95}

Solution, step by step

1Both are near 1, so look at the “gap from the whole”: $1 - \frac{72}{80} = \frac{8}{80} = 0.100$ and $1 - \frac{85}{95} = \frac{10}{95} \approx 0.105$ .
2The smaller the gap, the bigger the fraction: $\frac{72}{80}$ .

Answer $\frac{72}{80}$ (90%) vs $\frac{85}{95}$ (89.5%)

Example 15 · Standard

How many of

\frac{17}{35}, \frac{23}{45}, \frac{31}{59}

are bigger than

\frac{1}{2}

Solution, step by step

1 $\frac{a}{b} > \frac{1}{2}$ exactly when $2 a > b$ .
2 $2 \times 17 = 34 < 35$ ✗; $2 \times 23 = 46 > 45$ ✓; $2 \times 31 = 62 > 59$ ✓.

Answer undefined

The test is one doubling and one comparison: very fast.

Example 16 · Advanced

Which is bigger,

\frac{6}{17}

\frac{12}{35}

Solution, step by step

112 = 2 × 6, so to make the numerator 6 divide both by 2: $\frac{12}{35} = \frac{6}{17.5}$ .
2Numerators are equal and $17 < 17.5$ ⟹ $\frac{6}{17}$ is bigger.

Answer undefined

Try yourself

Table: parts made and workers in five factories. “Output per worker” = parts ÷ workers.

Factory	Parts made	Workers
F1	372	12
F2	551	17
F3	464	15
F4	595	19
F5	413	13

StandardTry it yourself first

Which factory has the highest output per worker?

StandardTry it yourself first

Which factory has the lowest output per worker?

StandardTry it yourself first

Between F4 and F5, which has the higher output per worker?

StandardTry it yourself first

Which factory is second when output per worker is arranged from highest to lowest?

AdvancedTry it yourself first

How many factories have output per worker above 31.3?

AdvancedTry it yourself first

How many more parts must F3 make at least to reach F1’s output per worker?

Change, ratio, average and growth

Four ways to compare values

Percent change, ratio, percent-to-units, average and growth rate.

Percent change and percentage points

The base is always the older value

Percent rise and fall, the difference between percent and percentage points, and reverse questions.

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Ratios from a chart and comparing ratios

Think in equal blocks

The ratio of two values, ratios of sums, and ratios from percents with different bases.

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Turning a percent into units

When one amount is equal in two cases

Treating discount, profit or commission percents as parts of a base to find ratios across two cases.

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Average and weighted average

Total ÷ count, but only if groups are equal

Average from a chart, average rise, and the weighted average of groups of different sizes.

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Growth rate and compound growth rate (CAGR)

The yearly %, and “on average per year”

Year-on-year growth, the year of highest growth, estimating CAGR, and projecting forward.

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Summary, mixed practice, revision

All methods in one place

A table of all methods, two mixed question sets and revision.

Summary of all methods

Pick the method as you read

Every method on one page: when to use it and the key formula.

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Mixed practice 1: a table

Five methods, one table

Questions from different methods on one table.

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Mixed practice 2: a bar chart

Reading + calculating

Questions from different methods on one bar chart.

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Revision: the commonest slips

Read once before the exam

One slip per method that costs marks.

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Data Interpretation: Methods and Shortcuts

Reading and quick arithmetic

Reading a chart or table without mistakes

The idea

Steps

Worked questions

Try yourself

The fraction ↔ percent table (1/2 to 1/20)

The idea

Steps

Worked questions

Try yourself

Estimation: rounding and 10%-1% blocks

The idea

Steps

Worked questions

Try yourself

Comparing fractions and ratios without dividing

The idea

Steps

Worked questions

Try yourself

Change, ratio, average and growth

Percent change and percentage points

Ratios from a chart and comparing ratios

Turning a percent into units

Average and weighted average

Growth rate and compound growth rate (CAGR)

Summary, mixed practice, revision

Summary of all methods

Mixed practice 1: a table

Mixed practice 2: a bar chart

Revision: the commonest slips

Free All Access Passevery mind map · Hindi + English · 12 months

How is it going?