Bar & Pie Charts
Quantitative Aptitude – Tamil Nadu Patwari / Lekhpal
(Competitive‑exam style – IIT‑JEE, JEE‑Mains, UPSC, etc.)
1. Overview
| Chart | Use | Key Features |
|---|---|---|
| Bar chart | Compare discrete quantities or time‑series data. | Horizontal or vertical bars; axis values may be integers, fractions, or percentages. |
| Pie chart | Visualise parts of a whole. | Sum of all wedges = 100 % of the circle. |
Both charts are frequently asked in competitive exams. Mastering them requires:
- Understanding how to read the data correctly.
- Translating the graph into algebraic relations.
- Using simple arithmetic, ratios, and geometry (for pie charts).
2. Bar Charts
2.1 Reading a Bar Chart
- Identify the axis:
- X‑axis → categories (e.g., years, items).
- Y‑axis → numeric value (units, rupees, %, etc.).
- Read the scale:
- Determine the interval (e.g., 0, 5, 10, …).
- Note if the scale is non‑linear (rare in exams).
- Measure the height (or length) of each bar.
- Match the bar to its category.
2.2 Common Problem Types
| Type | Typical Question |
|---|---|
| Area comparison | “Which bar has the largest area?” |
| Missing value | “If the bar for 2019 has height 14 units, what is the height of 2020?” |
| Percentage change | “From 2018 to 2019, the value increased by 20 %.” |
| Ratio | “The ratio of the heights of bar A to bar B is 3:4.” |
2.3 Quick Computation Tricks
| Situation | Trick |
|---|---|
| Missing height | Let the missing height be (x). Use given ratio or percentage relation to set up a simple equation. |
| Percent change | (\displaystyle \text{New} = \text{Old}\times\Bigl(1 \pm \frac{\text{percent}}{100}\Bigr)). |
| Comparison of two bars | If the scale is linear, compare the numbers directly; if you only have the difference between two bars, use that difference. |
2.4 Sample Bar‑Chart Problem
Problem
A bar chart shows the number of students enrolled in 4 subjects:
- Maths: 80, Science: 64, English: 40, Hindi: 24.
If the total number of students is 200, find the missing subject count.
Solution
Sum of given counts = (80+64+40+24 = 208).
Since the total should be 200, there is a mistake: either the chart is mis‑drawn or the total is wrong.
Trick: Always verify totals before proceeding.
3. Pie Charts
3.1 Basic Geometry
- Area of a full circle
[ A_{\text{full}} = \pi r^{2} ] - Area of a sector with central angle (\theta) (in degrees)
[ A_{\text{sector}} = \frac{\theta}{360},A_{\text{full}} ] - Area of a wedge (pie slice) is exactly the sector area.
3.2 Translating a Pie Chart to Numbers
- Identify the whole: Total = 100 % → area of full circle.
- Read each slice’s percentage: e.g., “25 % of the chart is water.”
- Compute slice area:
[ A_{\text{slice}} = \frac{\text{percentage}}{100},A_{\text{full}} ] - If radius is given, you can find the actual area numerically:
[ A_{\text{slice}} = \frac{\text{percentage}}{100},\pi r^{2} ] - If the radius is not given, you can still work with relative ratios or proportional lengths.
3.3 Common Pie‑Chart Problem Types
| Type | Typical Question |
|---|---|
| Missing percentage | “One slice covers 24 % of the chart. What is the remaining percentage?” |
| Area comparison | “Which slice has the larger area?” |
| Radius relation | “If the radius of the circle is 7 cm, what is the area of a 30 % slice?” |
| Chord length | “The chord opposite a 60 ° slice has length (l). Find (l) when (r = 8) cm.” |
| Sector area vs. whole | “Sector area = 50 % of total area. Find the radius if the sector’s area is 80 cm².” |
3.4 Derivation – Sector Area Formula
[
A_{\text{sector}} = \frac{1}{2} r^{2}\theta_{\text{rad}}
]
where (\theta_{\text{rad}}) is the angle in radians.
Since (\theta_{\text{rad}} = \theta_{\text{deg}}\times\frac{\pi}{180}),
[ A