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Line Graph

Competitive Mathematics · Data Interpretation

6 Part2 worked examples12 practice questions

Think of a fever chart or a share-price graph — a point for each time and a line joining them. That is a line graph, and it is best at showing how a value changed over time — a rising line is growth, a falling line a drop, and the steeper the line the faster the change. So its questions revolve around change (how much, what percent, in which interval most) and comparison (the gap between two lines).

The formulas for line-graph questions
1

Reading a line graph

Points, axes, unit

Reading a point’s value, totals, averages and the peak.

One point, one value

Check the axes first

The basic rules of a line graph.

Below is a line graph of a village’s population (in hundreds).

05010015020020182019202020212022120140135160180Population
Population of a village (in hundreds)
  • Time (year/month/week) is on the horizontal axis, the value on the vertical axis. If a point has no label, read it off the scale.
  • Trend: a rising line means the value grew, a falling one that it dropped, a flat one that it stayed the same.
  • Total = sum of the points; average = total ÷ number of points; peak = the highest point.
  • With two lines, use the legend (line style and marker) to choose the right line.
Example 1 · Basic
From the graph above, the total percent increase from 2018 to 2022, and the year when the population fell.
Solution, step by step
  1. 1Increase .
  2. 2In 2020 the value went from 140 to 135 — the population fell in 2020
Answer 50%; 2020
Example 2 · Standard
In the graph above, which interval has the largest rise and what is the average yearly rise?
Solution, step by step
  1. 1Changes: ⟹ the largest rise is in 2020–2021 ().
  2. 2Average yearly rise hundred
Answer 2020–2021; 15 hundred

Line graph: Population of a village (in hundreds). Study the graph and answer the questions.

05010015020020162017201820192020110120140150175Population
Population of a village (in hundreds)
BasicTry it yourself first
What is the value of “Population” in 2020 (in hundred)?
BasicTry it yourself first
What is the total of “Population” over all the years (in hundred)?
BasicTry it yourself first
Find the average value of “Population” (in hundred, to one decimal).
BasicTry it yourself first
Find the difference between the values of “Population” in 2019 and 2017 (in hundred).
BasicTry it yourself first
In which year/period is the peak value of “Population”?
BasicTry it yourself first
Find the ratio of “Population” in 2019 to “Population” in 2016.

Line graph: Weekly price of a share (₹). Study the graph and answer the questions.

0100200300400Week 1Week 2Week 3Week 4Week 5Week 6210200240280230230Price
Weekly price of a share (₹)
BasicTry it yourself first
What is the value of “Price” in Week 3 (in ₹)?
BasicTry it yourself first
What is the total of “Price” over all the years (in ₹)?
BasicTry it yourself first
In which year/period is the peak value of “Price”?
BasicTry it yourself first
Find the difference between the values of “Price” in Week 3 and Week 2 (in ₹).
BasicTry it yourself first
Find the average value of “Price” (in ₹, to one decimal).
StandardTry it yourself first
Find the percent increase in “Price” from Week 2 to Week 5. (to one decimal)
2

Change and trend

How much rise, what percent

Change between two points, percent change, the fastest rise or fall and the average change.

Questions on change

The old value is the base

Calculating change on a line graph.

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3

Comparing two lines

Gap, ratio, who leads when

Gaps, ratios, who leads when, and comparing averages in graphs with two or three lines.

The distance between lines

Read both points at the same time

Rules for comparing a two-series graph.

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4

Traps and fast tricks

Slips · tricks

Where marks go in line-graph questions.

The commonest mistakes

Seven traps

The slips that cost marks.

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Fast tricks

Estimating from the shape of the line

Habits for speed.

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5

The practice ladder

Basic → standard → advanced

New line graphs and questions of every type, in three levels.

Level 1 — Basic

Reading points, totals, averages

One-step questions.

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Level 2 — Standard

Change, percent, comparison

Exam-level questions.

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Level 3 — Advanced

Many steps and mixed lines

Questions at the hard end of competitive exams.

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6

Revision — formula cards

Flip each card and test yourself

Read the front, think, then flip.

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