MathsCBSE Class 8 · Proportional Reasoning-2

Pie Charts and Inverse Proportion for Class 8

Updated 17 October 2026 · Fact-checked

In a pie chart each sector angle is proportional to its share: angle = (value / total) x 360 degrees, and 1 per cent is 3.6 degrees. Two quantities are in inverse proportion when one changes by a factor n while the other changes by 1/n, so x1 y1 = x2 y2 (constant product). Exams ask for angles, reading a chart, workers-days-speed problems and combined work using rates.

Each pie angle is its share of 360°, and in inverse proportion halving the workers doubles the days so workers × days stays 80.

Illustration drawn for QuestionBuddy.

Understand Pie Charts and Inverse Proportion

Pie charts. A pie chart shows how a whole is shared among categories. The angle of each slice is proportional to its share, and a full circle is $36 0^{\circ}$ .

*Sector angle* $= \frac{value}{total} \times 36 0^{\circ}$ . In percentage terms, $1% = 3. 6^{\circ}$ , so $25%$ is $9 0^{\circ}$ and $15%$ is $5 4^{\circ}$ .
*Constructing a chart:* divide $360$ in the ratio of the data. For grades with counts $12 : 10 : 8 : 6 : 4$ the HCF is 2, so the ratio is $6 : 5 : 4 : 3 : 2$ and one part is $360 \div 20 = 1 8^{\circ}$ , giving angles $10 8^{\circ}, 9 0^{\circ}, 7 2^{\circ}, 5 4^{\circ}, 3 6^{\circ}$ . Draw a circle and a radius, then measure each angle in turn with a protractor.
*Reading a chart:* if the angle for one sector and its count are known, the total is count $\times \frac{360}{angle}$ , and other counts follow from their angles. Sectors with equal angles have equal numbers.

Direct proportion (rule of three). When $a : b :: c : d$ , $d = \frac{b c}{a}$ ; both quantities change by the same factor.

Inverse proportion. When speed goes up, the time for the same distance goes down. In the chapter's table, cycling is 3 times as fast as walking and takes one third of the time. Two quantities $x$ and $y$ are inversely proportional when, if one changes by a factor $n$ , the other changes by the factor $\frac{1}{n}$ . Then $x \cdot y = k$ , a constant, so for two situations $x_{1} y_{1} = x_{2} y_{2}$ .

20 workers take 4 days to lay a road; 10 workers (half) need double the time, 8 days: $20 \times 4 = 10 \times 8$ .
2 pumps fill a tank in 18 hours; 4 pumps need 9 hours.
Food for 80 students lasts 15 days; for 100 students it lasts $80 \times 15 \div 100 = 12$ days.

Always ask: does more of the first quantity mean less of the second (inverse), or more (direct)? Assumptions matter: all workers or pumps work at the same rate.

Working together. Treat the whole job as 1 unit of work and find the work done in 1 hour (the rate). If Ram finishes a job in 1 hour his rate is 1; if Shyam takes 1.5 hours his rate is $\frac{2}{3}$ per hour. Together the rate is $1 + \frac{2}{3} = \frac{5}{3}$ per hour, so the time is $1 \div \frac{5}{3} = \frac{3}{5}$ hour (36 minutes). Rates add; times do not. A drain that empties has a negative rate.

Map scales and multi-term ratios are on the previous topic page of this chapter.

Pie Charts and Inverse Proportion: formulas to remember

Sector angle: $θ = \frac{value}{total} \times 36 0^{\circ}$; 1 per cent corresponds to 3.6 degrees.
Total from one sector: $total = count \times \frac{36 0 ^{\circ}}{θ}$; For a sector with a known count.
Inverse proportion: $x_{1} y_{1} = x_{2} y_{2} = k$; The product stays constant.
Direct proportion: $\frac{x _{1}}{y _{1}} = \frac{x _{2}}{y _{2}}$; The ratio stays constant.
Work rate: $rate = \frac{1}{time to finish}, t_{together} = \frac{1}{r _{1} + r _{2}}$; Rates add.

How to solve Pie Charts and Inverse Proportion questions

Follow these steps for pie charts and inverse proportion.

1For a pie chart, find the total, then the angle for each category as value divided by total times 360. Check that the angles add to 360.
2For a reading problem, use one known (angle, count) pair to find the total or the count per degree.
3To decide between direct and inverse, ask whether the second quantity rises or falls when the first rises.
4For inverse proportion, write x1 y1 = x2 y2 and solve for the unknown.
5For work problems, find the rate of each worker or machine as 1 divided by its time; add (or subtract) rates and invert to get the time.
6State the assumption that all workers or machines work at the same rate.
7Check the answer: fewer workers should take more days.

Pie Charts and Inverse Proportion in real life

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Inverse proportion

Suppose a hostel kitchen has food for 60 students for 20 days.

If 20 students go home and the rest stay, how long will the food last?

Try it first, then show the solutionSolution

Food = $60 \times 20 = 1200$ student-days.
With 40 students: $1200 \div 40 = 30$ days.

Answer: 30 days

Memory hook: More people, fewer days: the total of person-days is fixed.

Working together

Imagine two taps: tap X fills a bucket in 10 minutes and tap Y fills the same bucket in 15 minutes.

If both are opened together, how long will the empty bucket take to fill?

Try it first, then show the solutionSolution

Rates per minute: X $= \frac{1}{10}$ , Y $= \frac{1}{15}$ .
Together $= \frac{3}{30} + \frac{2}{30} = \frac{5}{30} = \frac{1}{6}$ of the bucket per minute.
Time $= 1 \div \frac{1}{6} = 6$ minutes.

Answer: 6 minutes

Memory hook: Add the rates, not the times.

Tricks to remember Pie Charts and Inverse Proportion

Short memory aids. Say each one aloud once, then use it on a question the same day.

1 per cent is 3.6 degrees
Multiply the percentage by 3.6 to get the sector angle.
Use it when: Quick pie chart angles.
Product constant means inverse
If x times y stays the same, x and y are inversely proportional.
Use it when: Workers and days, speed and time.
Person-days are fixed
For a fixed job, workers x days is a constant number of person-days.
Use it when: Solving workers-days problems in one line.
Add rates, not times
Working together: rate = 1/t1 + 1/t2; time = 1 divided by that.
Use it when: Pump and pipe problems.

Pie Charts and Inverse Proportion: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: Sector angle from percentage · 1 mark · MCQBoard level

The angle at the centre for a sector representing 15% in a pie chart is: (a) 15 degrees (b) 36 degrees (c) 54 degrees (d) 90 degrees

Try it first, then show the solutionSolution

$15% \times 36 0^{\circ} = 0.15 \times 36 0^{\circ} = 5 4^{\circ}$ .

Answer: (c) 54 degrees

Example 2: Angles for a pie chart · 2 marks · Short answerBoard level

In a survey of 240 students, 80 chose cricket, 60 football, 40 kabaddi and 60 other sports. Find the central angle of each sector.

Try it first, then show the solutionSolution

One student corresponds to $\frac{36 0 ^{\circ}}{240} = 1. 5^{\circ}$ . (1 mark)
Cricket $12 0^{\circ}$ , football $9 0^{\circ}$ , kabaddi $6 0^{\circ}$ , other $9 0^{\circ}$ ; total $36 0^{\circ}$ . (1 mark)

Answer: 120, 90, 60 and 90 degrees

Example 3: Reading a pie chart · 3 marks · Short answerHarder board level

In a pie chart of children and their modes of travel, the bus sector has an angle of 72 degrees and 18 children use the bus. How many children took part, and how many are in a sector of 120 degrees?

Try it first, then show the solutionSolution

$7 2^{\circ}$ is $\frac{72}{360} = \frac{1}{5}$ of the circle. (1 mark)
Total children $= 18 \times 5 = 90$ . (1 mark)
A $12 0^{\circ}$ sector is $\frac{120}{360} \times 90 = 30$ children. (1 mark)

Answer: 90 children; 30 children in the 120 degree sector

Example 4: Workers and days · 3 marks · Short answerHarder board level

12 workers can finish a job in 15 days. How many workers are needed to finish it in 9 days, and how many more is that? State the assumption.

Try it first, then show the solutionSolution

The quantities are inversely proportional: $12 \times 15 = x \times 9$ . (1 mark)
$x = \frac{180}{9} = 20$ workers. (1 mark)
That is $20 - 12 = 8$ more workers, assuming all work at the same steady rate. (1 mark)

Answer: 20 workers; 8 more

Example 5: Pipes together · 4 marks · Long answerHarder board level

Pipe A fills a tank in 6 hours and pipe B in 4 hours. A drain C empties the full tank in 12 hours. How long does it take to fill the empty tank if all three are open?

Try it first, then show the solutionSolution

Rates (tanks per hour): A $= \frac{1}{6}$ , B $= \frac{1}{4}$ , C $= - \frac{1}{12}$ . (1 mark)
Net rate $= \frac{2}{12} + \frac{3}{12} - \frac{1}{12}$ . (1 mark)
$= \frac{4}{12} = \frac{1}{3}$ of the tank per hour. (1 mark)
Time $= 1 \div \frac{1}{3} = 3$ hours. (1 mark)

Answer: 3 hours

Where students lose marks in Pie Charts and Inverse Proportion

The slips examiners see most often, and what to write instead.

Mistake: Using direct proportion for workers and days
More workers means fewer days: the relationship is inverse.
Wrong working
"12 workers, 15 days; 20 workers: 15 x 20 / 12 = 25 days."
Right working
$12 \times 15 = 20 \times x$ , so $x = 9$ days.
Do this: Ask: more workers, more or less time?
Usually costs: The whole answer.
Mistake: Angles of a pie chart that do not add to 360
The angles must sum to 360 degrees; a wrong total used in the fraction breaks this.
Wrong working
"Total taken as 200 instead of 240, so the angles are 144, 108, 72, 108 (sum 432)."
Right working
Use the correct total 240 and check the angles add to $36 0^{\circ}$ .
Do this: Add the angles at the end.
Usually costs: 2 marks.
Mistake: Adding times when working together
It is rates that add, not times.
Wrong working
"Together they take 12 + 18 = 30 days."
Right working
Rate $= \frac{1}{12} + \frac{1}{18} = \frac{5}{36}$ , time $= 7.2$ days.
Do this: Convert times to rates first.
Usually costs: The answer.
Mistake: Forgetting to subtract a draining rate
A drain works against the filling pipes.
Wrong working
"Net rate = 1/6 + 1/4 + 1/12"
Right working
Net rate $= \frac{1}{6} + \frac{1}{4} - \frac{1}{12}$ .
Do this: Give filling rates a plus sign and draining rates a minus sign.
Usually costs: 2 marks.

Exam tips

Always check that the sector angles add up to 360 degrees.
State "inverse proportion" and write x1 y1 = x2 y2 before substituting.
In work problems, write the rate of each person as 1 over the time, then combine.
Mention the assumption (same rate, no breaks) when the question asks for it.
Check reasonableness: fewer workers must give more days.

Practice questions on Pie Charts and Inverse Proportion

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · Assertion-reasonBoard level
Assertion (A): If 8 workers finish a job in 6 days, then 12 workers finish it in 4 days.
Reason (R): The number of workers and the days for a fixed job are inversely proportional, so the product 8 x 6 = 12 x 4 = 48 stays constant.
(a) Both A and R are true and R is the correct explanation of A.
(b) Both A and R are true but R is not the correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true.
Hint
Check the product.
Answer
(a). $8 \times 6 = 48 = 12 \times 4$ , so the days fall in the same ratio as the workers rise.
Q2 · 2 marks · Very short answerBoard level
A hostel has food for 60 students for 20 days. If 20 students leave, for how many days will the food last?
Hint
Person-days are fixed.
Answer
Total $= 60 \times 20 = 1200$ student-days. With 40 students: $1200 \div 40 = 30$ days.
Q3 · 3 marks · Short answerHarder board level
A train takes 7.5 hours for a journey at 64 km/h. How long will it take at 80 km/h? Find the distance as well.
Hint
Speed and time are inversely proportional for a fixed distance.
Answer
Distance $= 64 \times 7.5 = 480$ km. Time at 80 km/h $= 480 \div 80 = 6$ hours. Check: $64 \times 7.5 = 80 \times 6$ .
Q4 · 4 marks · Long answerHOTS / competency
A can paint a wall in 12 days and B in 18 days. A starts alone and works for 4 days, then B joins. How many more days are needed to finish the wall, and what is the total time from the start?
Hint
Find the work done by A in 4 days, then use the combined rate.
Answer
A's rate $= \frac{1}{12}$ , so in 4 days A completes $\frac{4}{12} = \frac{1}{3}$ of the wall, leaving $\frac{2}{3}$ . B's rate $= \frac{1}{18}$ ; together $\frac{1}{12} + \frac{1}{18} = \frac{3}{36} + \frac{2}{36} = \frac{5}{36}$ per day. Time for the rest $= \frac{2}{3} \div \frac{5}{36} = \frac{2}{3} \times \frac{36}{5} = 4.8$ days. Total $= 4 + 4.8 = 8.8$ days.
Q5 · 4 marks · Case-basedHOTS / competency
Case study: 600 students were asked how they travel to school. 35% walk, 25% take a bus, 20% cycle and the rest use a car.
(i) Find the central angle of each sector of the pie chart. (1 mark)
(ii) Find the number of students in each category. (1 mark)
(iii) Next term 30 students switch from walking to cycling. Find the new angles for walking and for cycling. (2 marks)
Hint
1 per cent is 3.6 degrees. After the switch, 180 walk and 150 cycle out of 600.
Answer
(i) Walk $35% \times 360 = 12 6^{\circ}$ ; bus $25% \to 9 0^{\circ}$ ; cycle $20% \to 7 2^{\circ}$ ; car $20% \to 7 2^{\circ}$ . Total 360. (ii) Walk 210, bus 150, cycle 120, car 120. (iii) Walking becomes $210 - 30 = 180$ students: $\frac{180}{600} \times 36 0^{\circ} = 10 8^{\circ}$ . Cycling becomes $120 + 30 = 150$ students: $\frac{150}{600} \times 36 0^{\circ} = 9 0^{\circ}$ .

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.How many degrees are in a full pie chart?Show answer
360.
2.What is the angle of a sector representing 25%?Show answer
90 degrees.
3.What stays constant in inverse proportion?Show answer
The product of the two quantities.
4.If workers double, what happens to the days for the same job?Show answer
They are halved.
5.How do you combine work rates?Show answer
Add the rates (per day or per hour); the time is 1 divided by the total rate.
6.What is the angle for 1 per cent?Show answer
3.6 degrees.

Pie Charts and Inverse Proportion: frequently asked questions

How do I decide between direct and inverse proportion?

If one quantity rises when the other rises, it is direct; if it falls, it is inverse. Check with a halving or doubling.

What if the pie chart percentages do not add to 100?

Then data is missing or wrong; the angles must sum to 360 degrees, so check the total.

Do workers always work at the same rate?

Not in real life; the chapter asks you to state this assumption in such problems.

Can a pie chart have a sector of more than 180 degrees?

Yes, if one category is more than half of the whole.