Bar Graphs: Formulas, Shortcuts and Solved Examples

What is a bar graph?

A bar graph shows data as bars. Each bar stands for one category. The height of the bar shows its value.

Bars have equal width and equal gaps. So only the height carries information.

Sources: NCERT Class 8 Mathematics, Data Handling.

Which parts of the graph must you check first?

Read the title to know what is measured. Read the unit, such as crore, lakh or thousand. Read the scale on the vertical axis.

If there are two or more colours, read the legend. It tells you which bar belongs to which year or group.

Many wrong answers come from skipping this step.

How do you find a difference or a total?

Difference = larger value − smaller value. Total = sum of all the bars.

Why it works: each bar is a plain number. So you add or subtract the numbers you read, in the same unit.

How do you find an average from a graph?

Average = total of the bars ÷ number of bars.

Why it works: the average is the height every bar would have if all were levelled out to the same size.

How do you write one bar as a percentage of another?

Percentage = (bar A ÷ bar B) × 100.

Why it works: percent means per hundred. You compare A with B, and B is the base.

The base is the value after the word “of”.

How do you find the share of one bar in the total?

Share % = (one bar ÷ total) × 100.

Why it works: the total is the whole. A share is one part of that whole, scaled to 100.

How do you find a percentage change between two years?

% change = (new − old) ÷ old × 100.

Why it works: you measure the change against where you started. So the old value is always the base.

A negative answer means a fall. A positive answer means a rise.

How do you write a ratio from two bars?

Ratio = bar A : bar B. Divide both by their common factor.

Why it works: a ratio compares sizes. Dividing both sides by the same number does not change the comparison.

Easy: using the shop graph, how much more are the sales of shop B than shop D?

Step 1: read shop B. It is 65.

Step 2: read shop D. It is 25.

Step 3: subtract. 65 − 25 = 40 lakh.

Medium: shop D sales are what percentage of shop B sales? Round to the nearest whole number.

Step 1: the base is shop B, since it comes after “of”. So the base is 65.

Step 2: percentage = 25 ÷ 65 × 100 = 38.46.

Step 3: round it. The answer is about 38%.

Exam level, step by step: using the double bar graph, find the percentage fall in shop B between 2022 and 2023.

Step 1: read the 2022 bar of B. It is 65. This is the old value.

Step 2: read the 2023 bar of B. It is 52. This is the new value.

Step 3: change = 52 − 65 = −13.

Step 4: % change = −13 ÷ 65 × 100 = −20%. So the fall is 20%.

Exam level: by what percentage did the total of all four shops rise from 2022 to 2023?

Step 1: the 2022 total is 40 + 65 + 50 + 25 = 180.

Step 2: the 2023 total is 50 + 52 + 60 + 30 = 192.

Step 3: the rise is 192 − 180 = 12.

Step 4: 12 ÷ 180 × 100 = 6.67. So the total rose by about 6.7%.

Which traps cost marks?

Reading the wrong year or the wrong colour. Check the legend each time.

Missing the unit. Rs 65 lakh and Rs 65 crore are very different.

Using the new value as the base for a percentage change.

Ignoring a scale that does not start at zero. Then the bar heights are not in proportion.

Which shortcuts save time?

Compare bars by eye first. For “highest” or “lowest” questions, you need no calculation.

Write down every value you need before you start. Then do the sums in one go.

For “approximately” questions, round the numbers before you divide.

Estimate first. If shop D is about a third of shop B, then 38% is sensible and 83% is not.

How do exam questions use bar graphs?

Most questions are of five kinds. Highest or lowest, difference, total or average, percentage, and ratio.

The graph is the same for the whole set. So read all the values once and keep them in a small table.

Then each question takes only one or two lines of working.