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CAT (Common Admission Test) · Data Interpretation & Logical Puzzles

Bar and Line Graphs

Reading trends, comparisons and growth from bar charts and line graphs.

Bar and line graphs are the fast marks in CAT DILR — if the axis is read before any arithmetic starts. Six ideas cover the set: scale, bar comparison, stacked segments, slope, percentage change, and reading two series at the same x.

  • CAT (Common Admission Test)
  • Medium level
  • 6 concepts
  • 17 practice questions

1Read the axis scale before any bar

Every graph question starts on the axes, not on the tallest bar. Note the y-axis unit and the value of one gridline, then estimate each height against those gridlines rather than guessing from how the ink looks. A bar that looks 'about half way' is useless until you know whether the top of the chart is 50 or 500.

Watch for a truncated axis that does not start at zero — the same absolute gap then looks dramatic. Mark the gridline value once and read every bar and every point relative to it; that single note is what keeps a rushed set from inventing numbers.

Figure. Both bars are built from the same 40-unit grid. A is three gridlines (120); B is five (200). The dashed rules are the gridlines — read height by counting them, then multiply.

How to read a height

  1. Name the unitRead the axis caption once — Rs crore, thousand units, index points — and keep that unit on every answer.
  2. Fix the gridlineCount what one horizontal gridline is worth before estimating any bar.
  3. Count, then multiplyA bar that reaches n gridlines is n times the gridline value; fractions of a gridline are fine.

Two bars off the same grid

Each horizontal gridline on a bar chart is worth 40 units. Bar A reaches 3 gridlines above the baseline; Bar B reaches 5. What is B − A?

  • Gridline value40
  • Bar A = 3 × 40120
  • Bar B = 5 × 40200
  • B − A80

Pro tip. If the difference is asked, subtract after converting both bars — never subtract '2 gridlines' and then forget to multiply by the gridline value.

A bar reaches 3.5 gridlines above a baseline. Each gridline is Rs 20 crore. The bar's value is
  1. Rs 3.5 crore
  2. Rs 20 crore
  3. Rs 70 crore
  4. Rs 23.5 crore

3.5 × 20 = 70. The trap options keep the 3.5, keep the 20, or add them — none of those is a height reading.

2Grouped bars compare side by side

A simple bar chart compares quantities category by category: taller means more, once the scale is fixed. Grouped bars place two or more series in each category so the comparison is local — Product A against Product B in the same quarter — without stacking them into a total.

Questions usually ask how many categories one series beats the other, or the gap in a named category. Read the pair at the same x-position; do not compare A's Q1 bar to B's Q3 bar.

Figure. Each quarter is a cluster. A beats B in Q2 and Q3 only — two quarters — because the comparison never leaves its own group.

How to compare groups

  1. Identify the seriesName which bar in each cluster is A and which is B before comparing heights.
  2. Stay inside the clusterFor 'in how many quarters did A beat B', decide A vs B once per group, never across groups.
  3. Match the wording'How many times' divides; 'how much more' subtracts. The chart is the same; the operation is in the stem.

A versus B across three quarters

Grouped bars show Product A as 30, 45, 60 and Product B as 50, 40, 30 across Q1, Q2, Q3. In how many quarters did A outsell B?

  • Q1: A = 30, B = 50B wins
  • Q2: A = 45, B = 40A wins
  • Q3: A = 60, B = 30A wins
  • Quarters with A > B2

Pro tip. When the question is a count of wins, skip the totals and the grand averages — each cluster is its own comparison.

In the same A/B chart (A: 30, 45, 60; B: 50, 40, 30), the largest A − B gap in any one quarter is
  1. 10
  2. 15
  3. 20
  4. 30

Q1: 30 − 50 = −20; Q2: 45 − 40 = 5; Q3: 60 − 30 = 30. The largest gap is 30 in Q3. Option 20 is the size of B's lead in Q1, which is a gap the other way.

3A stacked segment is top minus bottom

In a stacked bar the full height is the total; each component is a segment of that height. The segment's value is the length between its two boundaries — upper gridline minus lower gridline — never the number printed at the top of the segment alone.

Reading a stacked segment as its top gridline value overstates that component whenever anything sits below it. The total is free once the top of the stack is marked; an individual share still needs the subtraction.

Figure. A fills 0→30; B fills 30→50. B is the segment length 20, not the top reading 50. The dashed rules mark the two boundaries the subtraction needs.

How to read a stack

  1. Find the two boundariesFor the segment you want, note where it starts and where it ends on the value axis.
  2. SubtractSegment value = upper boundary − lower boundary.
  3. Use the top for totalsThe top of the whole bar is the total when you need a share of the year, not a second addition of every segment.

Product B in a two-segment stack

A stacked bar for 2023 shows Product A from 0 to 30 and Product B stacked from 30 to 50. What is Product B's value?

  • Lower boundary of B30
  • Upper boundary of B50
  • B = 50 − 3020
  • Product B's value20

Pro tip. The wrong answer 50 is sitting on the page as the top of the bar. A stacked segment's value is top boundary minus bottom boundary, never just the top number.

A stacked bar has Online from 0 to 20, Retail from 20 to 55, and Wholesale from 55 to 100. Retail's value is
  1. 55
  2. 35
  3. 20
  4. 100

Retail runs from 20 to 55, so 55 − 20 = 35. 55 is the upper boundary (the trap); 20 is Online; 100 is the total.

4Slope carries the rate of change

A line graph connects values over an ordered category — usually time — and its power is the slope of each segment. A steeper segment means faster change; the sign of the slope shows increase or decrease. Comparing slopes answers 'largest growth' questions before exact arithmetic when the options allow a visual shortlist.

Slope of a segment is \Delta y / \Delta x: the change in the plotted value over the change in the category. Equal horizontal steps (year to year) make the comparison collapse to which rise is larger — still check percentage change when the question asks for rate relative to the starting level.

Figure. Same horizontal steps; the second leg drops much farther on the screen because \Delta y is four times larger. Steepness is the rate — here, the raw rise — when \Delta x is shared.

How to use slope

  1. Shortlist by eyeFor 'maximum increase', pick the steepest rising segments before computing every year.
  2. Check the signA falling segment is a decrease — skip it when the question asks for growth.
  3. Confirm with numbersWhen two slopes look close, compute \Delta y (and the percentage if asked) only for the shortlist.

Which segment rose faster?

A line graph of output reads 20 in Year 1, 30 in Year 2 and 70 in Year 3. The horizontal gap between consecutive years is the same. Which year-to-year rise is steeper, and by what factor in \Delta y?

  • Year 1 → 2: \Delta y30 − 20 = 10
  • Year 2 → 3: \Delta y70 − 30 = 40
  • Same \Delta x, so slope ratio40 / 10 = 4
  • Steeper segmentYear 2 → 3, four times the rise

Pro tip. Equal year gaps mean you can compare rises as a proxy for slope. Unequal category spacing — a missing year, a quarterly then annual mix — breaks that shortcut.

On a line graph with equal year gaps, values go 40 → 50 → 45 → 60. The steepest rising segment is
  1. 40 → 50
  2. 50 → 45
  3. 45 → 60
  4. 40 → 60 overall

Rises: +10, then a fall of 5, then +15. The steepest rise is 45 → 60. The overall 40 → 60 chord is not a single segment on the graph.

5Growth is a percentage of the earlier value

Percentage change between two points is (\text{later} - \text{earlier}) / \text{earlier} \times 100. That is the growth-rate formula that turns any two readings on a line — or any two bars in a time series — into a comparable rate.

For 'highest growth', compare those percentages, not the raw increases. A jump of 15 on a base of 25 is 60%; a jump of 6 on a base of 40 is only 15%. A big jump on a small base wins, and the graph's steepest-looking ink is only a shortlist until the division is done.

Figure. The 2021→2022 step is the largest raw rise (+15) and the largest percentage (+60%). When raw rise and percentage disagree, the percentage is the growth ranking — the quick check carries a series built for that split.

How to rank growth

  1. Pair consecutive pointsFor each year asked, take that year's value and the previous year's value.
  2. Divide by the earlierCompute (\text{later} - \text{earlier}) / \text{earlier} \times 100 for each pair.
  3. Rank the percentagesThe largest percentage is the answer — even when another year has a larger raw rise.

Largest year-on-year increase

A line graph shows profit (Rs crore): 2020: 20, 2021: 25, 2022: 40, 2023: 46. In which year was the percentage increase over the previous year the highest?

  • 2021 over 2020: (25 - 20) / 2025%
  • 2022 over 2021: (40 - 25) / 2560%
  • 2023 over 2022: (46 - 40) / 4015%
  • Highest percentage increase2022 (60%)

Pro tip. Raw increases here are 5, 15 and 6 — so 2022 wins on both measures. When they disagree, the percentage is what 'highest growth' means; keep the base in the denominator.

Profits (Rs crore): 2020: 20, 2021: 35, 2022: 50, 2023: 80. The year with the highest percentage increase over the previous year is
  1. 2021
  2. 2022
  3. 2023
  4. 2022 and 2023 tied

Percentages: (35−20)/20 = 75%; (50−35)/35 ≈ 42.9%; (80−50)/50 = 60%. Raw rises are 15, 15 and 30 — so 2023 wins on raw increase and loses on percentage. The answer is 2021.

6Read both series at the same x

Line-plus-bar combinations and dual-line charts ask for a ratio or a difference between the two series in a given period. Align both readings at the same x-value — the same year, the same quarter — then divide or subtract. Mixing one series' 2021 point with the other's 2022 point is the standard cross-period error.

The ratio of two series at a point is (\text{value of series 1}) / (\text{value of series 2}). Averages over a period, when asked, are (\text{sum of values}) / (\text{number of periods}) for the series named in the stem — compute each series on its own timeline before combining.

Figure. Each year is one vertical slice: read Sales and Costs together, then subtract or divide. Crossing Year 2's Sales with Year 3's Costs invents a pair the chart never drew.

How to combine two series

  1. Fix the periodFind the named year or category on the shared horizontal axis.
  2. Read bothTake the bar height and the line point — or both line points — at that same x.
  3. Apply the operationRatio divides; difference subtracts; a share of the combined total divides one by the sum of both.

Sales against costs in one year

In 2022 a chart shows Sales = 80 and Costs = 50 (same units). Find the Sales : Costs ratio and Sales − Costs.

  • Sales at 202280
  • Costs at 202250
  • Sales : Costs = 80 : 508 : 5
  • Sales − Costs30

Pro tip. Both numbers come from the same vertical slice of the chart. If a later question asks for 2023, re-read both series — do not reuse the 2022 pair.

Sales and Costs over three years are Sales: 60, 80, 100 and Costs: 40, 50, 55. The year when Sales − Costs is largest is
  1. Year 1 (difference 20)
  2. Year 2 (difference 30)
  3. Year 3 (difference 45)
  4. Year 3 (difference 55)

Differences: 60−40=20, 80−50=30, 100−55=45. Year 3 is largest. 55 is the Costs reading, not the gap.

Notes

  • Bar Graphs - Read the Axis Scale: Note the y-axis units and gridline value; estimate bar heights against gridlines rather than guessing. Grouped/stacked bars need you to distinguish each segment.
  • Line Graphs - Trend and Slope: A steeper slope means faster change; the sign of the slope shows increase or decrease. Compare slopes to answer 'largest growth' questions without exact values when options allow.
  • Stacked Bars: The total is the full bar height; an individual component is the segment length, read as the difference between two gridline levels.
  • Combining Two Series: Line-plus-bar combos often ask for the ratio or difference between the two series in a given period - read both at the same x-value.
  • Common trap: Reading a stacked segment as its top gridline value instead of its length (top minus bottom) overstates that component.

Formulas

  • Segment value in a stacked bar = (upper boundary) - (lower boundary).
  • Percentage change between two points = \dfrac{\text{later}-\text{earlier}}{\text{earlier}}\times 100.
  • Average over a period = \dfrac{\text{sum of values}}{\text{number of periods}}.
  • Slope of a line segment = \dfrac{\Delta y}{\Delta x} indicates rate of change.
  • Ratio of two series at a point = \dfrac{\text{value of series 1}}{\text{value of series 2}}.

Exam traps & shortcuts

  • Mark the gridline value once and read every bar/point relative to it for speed.
  • For 'maximum increase', compare slopes/steepness visually before computing exact differences.
  • In stacked bars, read a component as the length between its two boundaries, not the top value.
  • Align both series at the same x-value when a question mixes a bar and a line.

Reference tables

The five relations that turn a bar or line reading into an answer. Segment length is the one that fights the ink on the page.

Graph reading identities
QuantityFormula
Stacked segmentupper boundary − lower boundary
Percentage change( later - earlier ) / earlier \times 100
Average over a periodsum of values / number of periods
Slope of a segment\Delta y / \Delta x
Ratio of two series at xseries 1 at x / series 2 at x

Recap

Read only this the night before a DILR graph set.

Scale first
Name the unit and one gridline value before estimating any height. Truncated axes exaggerate gaps.
Stacked trap
A segment is top minus bottom. The top number alone overstates every component that is not on the baseline.
Slope then percent
Shortlist the steepest rising segments by eye; rank 'highest growth' by percentage of the earlier value, not by raw rise.
Same x
For two series, align both readings at the same category. Ratio divides; difference subtracts.

Practise Bar and Line Graphs

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