MathsCBSE Class 8 · Chapter 10

Proportional Reasoning-2 Class 8: Map Scales, Multi-term Ratios, Pie Charts and Inverse Proportion

Q: Where is Proportional Reasoning-2 in the Class 8 NCERT book?

It is Chapter 3 of Ganita Prakash Part II (Chapter 10 in our list), the NCERT Grade 8 Mathematics textbook (Reprint 2026-27). It follows Proportional Reasoning-1 from Part I.

Q: Is inverse proportion new in Class 8?

Yes. Proportional Reasoning-1 warned that some problems cannot be solved with the Rule of Three. This chapter names such relationships inverse proportions and shows that the product xy stays constant.

Q: Do I need to draw pie charts accurately?

Yes. Calculate each angle, check that they add to 360°, and use a protractor. Marks are usually given for correct angles, accurate drawing and clear labels.

Q: Why do different maps give slightly different distances?

Measuring on a small map is less precise, and printed maps may round their scales. Each correctly used map should give roughly the same straight-line distance.

Q: How do I know if a work problem is direct or inverse?

For a fixed amount of work, more workers means fewer days: inverse. For a fixed number of workers, more work means more days: direct. Decide which quantity is fixed first.

Updated 15 October 2026 · Fact-checked

Proportional Reasoning-2 is Chapter 3 of Ganita Prakash Part II for Class 8 (Chapter 10 overall). It extends ratios to map scales, ratios with three or more terms, dividing a whole in such a ratio, drawing pie charts with proportional angles, and inverse proportion, where one quantity falls by the same factor that the other rises.

What Proportional Reasoning-2 is about

This chapter takes the ratio ideas of Proportional Reasoning-1 into new situations.

3.1 Proportionality: A Quick Recap: two ratios $a : b$ and $c : d$ are proportional when $a d = b c$ .
3.2 Ratios in Maps: the representative fraction (RF), such as $1 : 60, 00, 000$ , links map distance to real (straight-line) distance; making a classroom plan to a scale such as $1 : 50$ .
3.3 Ratios with More than 2 Terms: spice mixes, paints and concrete with ratios like $8 : 4 : 2 : 1$ or $1 : 1.5 : 3$ ; scaling all terms by the same factor.
3.4 Dividing a Whole in a Given Ratio: each part is $x \times \frac{a}{a + b + c + \dots}$ ; angles of a triangle in a given ratio.
3.5 A Slice of the Pie: drawing pie charts, where each slice's angle is proportional to its share of 360°.
3.6 Inverse Proportions: when one quantity is multiplied by $n$ and the other is divided by $n$ , so $x y = k$ (speed and time, workers and days, pumps and hours); working together.

School exams ask map-scale calculations, multi-term shares, pie-chart angles and constructions, and inverse-proportion word problems, and expect you to decide whether a situation is direct or inverse. This page explains each idea with an example and gives exam-level and harder questions with full solutions.

Key concepts

Recap: testing proportionality

NCERT section 3.1 Proportionality — A Quick Recap

When related quantities change by the same factor, their relationship is proportional. Idli batter made with 6 cups of rice and 3 cups of urad dal ( $6 : 3$ ) tastes like batter made with 4 and 2 ( $4 : 2$ ), because both simplify to $2 : 1$ .

Cross-multiplication test: $a : b :: c : d$ exactly when $a \times d = b \times c$ . For $6 : 3$ and $4 : 2$ : $6 \times 2 = 12 = 3 \times 4$ .

To find a missing term, rearrange: $d = \frac{b \times c}{a}$ (the Rule of Three).

Worked example

Are the batter mixes $9 : 4.5$ and $6 : 3$ (rice : dal) proportional? How much dal goes with 7 cups of rice in the same proportion?

Try it first, then show the solutionSolution

$9 \times 3 = 27$ and $4.5 \times 6 = 27$ : equal, so proportional (both are $2 : 1$ ).
For 7 cups of rice: $2 : 1 :: 7 : d$ .
$d = \frac{1 \times 7}{2} = 3.5$ cups.

Answer: Yes; 3.5 cups of dal.

Ratios in maps

NCERT section 3.2 Ratios in Maps

A map shows a large area on a small sheet, so every length is shrunk by the same factor. The representative fraction (RF) gives that factor as a ratio of map distance to real distance in the same unit.

RF $1 : 25, 00, 000$ means 1 cm on the map stands for 25,00,000 cm on the ground. Since $1 km = 1, 00, 000$ cm, that is 25 km.
A map distance of 7.4 cm then means $7.4 \times 25 = 185$ km.
The answer is the straight-line (geographical) distance, not the road distance, which is usually longer.

Making a plan: to draw a classroom at $1 : 50$ , every real length is divided by 50: an 8 m (800 cm) wall becomes 16 cm.

Different maps, same answer: a larger RF number means a smaller map; the real distance worked out from any correct map should be about the same.

Worked example

On a map with RF $1 : 25, 00, 000$ , two towns are 7.4 cm apart. Find the real straight-line distance.

Try it first, then show the solutionSolution

1 cm represents 25,00,000 cm $= 25$ km.
7.4 cm represents $7.4 \times 25$ km.
$= 185$ km.

Answer: 185 km (straight-line).

Ratios with more than two terms

NCERT section 3.3 Ratios with More than 2 Terms

A recipe or mixture can have three or more ingredients in a fixed ratio, such as coriander : chillies : toor dal : methi $= 8 : 4 : 2 : 1$ . To keep the same taste or strength, every term must change by the same factor:

$a : b : c : d :: p : q : r : s$ means $\frac{p}{a} = \frac{q}{b} = \frac{r}{c} = \frac{s}{d}$ .

Method: find the factor from the ingredient you know, then multiply all the others by it. If paint is mixed red : blue : white $= 3 : 4 : 5$ and you have 12 L of blue, the factor is $12 \div 4 = 3$ , so red $= 9$ L, white $= 15$ L, total 36 L.

Concrete for strong structures is mixed cement : sand : gravel $= 1 : 1.5 : 3$ ; ratios may contain decimals. With 3 bags of cement: $3 : 4.5 : 9$ , a total of 16.5 bags.

Worked example

A concrete mix is cement : sand : gravel $= 1 : 2 : 4$ . If 3 bags of cement are used, how many bags of sand and gravel are needed, and how many bags of mixture result?

Try it first, then show the solutionSolution

The factor is 3 (cement goes from 1 to 3).
Sand $= 2 \times 3 = 6$ bags; gravel $= 4 \times 3 = 12$ bags.
Total $= 3 + 6 + 12 = 21$ bags.

Answer: 6 bags sand, 12 bags gravel; 21 bags in all.

Dividing a whole in a ratio of several terms

NCERT section 3.4 Dividing a Whole in a Given Ratio

To divide a quantity $x$ in the ratio $a : b : c : \dots$ :
1. Add the terms: $a + b + c + \dots$ equal parts.
2. Each part is $\frac{x}{a + b + c + \dots}$ .
3. The shares are $x \times \frac{a}{a + b + c + \dots}$ , $x \times \frac{b}{a + b + c + \dots}$ , and so on.

Example: ₹7,200 in the ratio $2 : 3 : 4$ : 9 parts of ₹800, so ₹1,600, ₹2,400 and ₹3,200.

Angles of a triangle in a ratio: the angles add to 180°, so angles in the ratio $2 : 3 : 7$ are $180° \times \frac{2}{12} = 30°$ , $45°$ and $105°$ .

Sides of a triangle in a ratio: the ratio fixes the shape but not the size, so triangles with sides in the ratio $3 : 4 : 5$ need not be congruent. Some ratios are impossible: $1 : 3 : 5$ fails because $1 + 3 < 5$ , so the two shorter sides cannot meet.

Worked example

The angles of a triangle are in the ratio $2 : 3 : 7$ . Find them and say what kind of triangle it is.

Try it first, then show the solutionSolution

Total parts $= 2 + 3 + 7 = 12$ ; one part $= 180° \div 12 = 15°$ .
Angles: $30°$ , $45°$ , $105°$ .
One angle is more than 90°, so it is an obtuse-angled triangle.

Answer: $30°, 45°, 105°$ : obtuse-angled.

Pie charts

NCERT section 3.5 A Slice of the Pie

A pie chart shows how a whole is split into parts, as slices of a circle. Each slice's angle is proportional to its share, and all the angles add to 360°.

Steps to draw one:
1. List the values and simplify their ratio by dividing by the HCF.
2. Angle of a slice $= \frac{value}{total} \times 360°$ (or percentage $\times 3.6°$ ).
3. Check the angles add to 360°.
4. Draw a circle and a radius; measure the first angle from it with a protractor, then each next angle from the last radius.
5. Label each slice (and colour it).

Example: 60 students choose cricket 24, football 15, badminton 12, kabaddi 9. The ratio simplifies to $8 : 5 : 4 : 3$ (20 parts of 18°), giving 144°, 90°, 72° and 54°.

Pie charts are best for comparing parts of one whole; they do not show changes over time.

Worked example

Find the pie-chart angles for 60 students: cricket 24, football 15, badminton 12, kabaddi 9.

Try it first, then show the solutionSolution

Ratio $24 : 15 : 12 : 9 = 8 : 5 : 4 : 3$ (HCF 3), a total of 20 parts.
One part $= 360° \div 20 = 18°$ .
Angles: $8 \times 18 = 144°$ , $90°$ , $72°$ , $54°$ ; sum $= 360°$ .

Answer: 144°, 90°, 72°, 54°.

Inverse proportion

NCERT section 3.6 Inverse Proportions

Some quantities change by the same factor but in opposite directions. To cover 90 km, walking at 5 km/h takes 18 hours, cycling at 15 km/h takes 6 hours: the speed is 3 times, the time is $\frac{1}{3}$ .

Definition: $x$ and $y$ are in inverse proportion if $x y = k$ for a constant $k$ . Then

$x_{1} y_{1} = x_{2} y_{2}$ , or equivalently $\frac{x _{1}}{x _{2}} = \frac{y _{2}}{y _{1}}$ ;
if $x$ is multiplied by $n$ , $y$ is divided by $n$ .

Examples: speed and time for a fixed distance; number of workers and days for a fixed job; number of pumps and hours to fill a tank; number of people and days a stock of food lasts.

Check a table: multiply each pair. If all products are equal, it is inverse proportion; if all quotients are equal, it is direct proportion.

Contrast with direct proportion ( $\frac{y}{x} = k$ ): cost and quantity, distance and petrol used. Not every relationship is either; your age and height are neither.

Worked example

12 workers take 15 days to build a wall. How long would 18 workers take, working at the same rate?

Try it first, then show the solutionSolution

More workers means fewer days: inverse proportion, so the product is constant.
$12 \times 15 = 180$ worker-days.
$18 \times d = 180$ , so $d = 10$ days.

Answer: 10 days.

Working together

NCERT section 3.6 Inverse Proportions

When people (or pipes) work at different rates, think about the work done in 1 unit of time.

If A finishes a job in 6 hours, A does $\frac{1}{6}$ of it per hour; if B finishes in 3 hours, B does $\frac{1}{3}$ per hour.
Together they do $\frac{1}{6} + \frac{1}{3} = \frac{1}{2}$ per hour.
Work and time are directly proportional at a fixed rate, so the whole job takes $1 \div \frac{1}{2} = 2$ hours.

A pipe that empties a tank counts as negative work: with a third pipe emptying the tank in 4 hours, the net rate is $\frac{1}{6} + \frac{1}{3} - \frac{1}{4} = \frac{1}{4}$ , so 4 hours.

The combined time is always less than the faster worker's time alone.

Worked example

Ravi can paint a fence in 4 hours and Sunil in 6 hours. How long will they take together?

Try it first, then show the solutionSolution

Ravi does $\frac{1}{4}$ of the fence per hour; Sunil $\frac{1}{6}$ .
Together: $\frac{1}{4} + \frac{1}{6} = \frac{5}{12}$ per hour.
Time $= 1 \div \frac{5}{12} = \frac{12}{5} = 2.4$ hours (2 hours 24 minutes).

Answer: 2.4 hours (2 h 24 min).

Maths in real life: Proportional Reasoning-2

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.

Map scale

Suppose a road atlas has scale $1 : 10, 00, 000$ and two villages are 3.5 cm apart on it.

What is the straight-line distance between them?

Try it first, then show the solutionSolution

1 cm represents 10 km.
$3.5 \times 10 = 35$ km.

Answer: 35 km (the road may be longer).

Memory hook: Knock off five zeros to read km per cm.

Inverse proportion

Suppose a camp has food packed for 30 campers for 8 days, but 40 campers come.

How many days will the food last?

Try it first, then show the solutionSolution

More campers, fewer days: $30 \times 8 = 40 \times d$ .
$d = 240 \div 40 = 6$ days.

Answer: 6 days

Memory hook: Share the same food among more people and it runs out sooner.

Multi-term ratio

Suppose you make masala tea with tea decoction : milk : water $= 1 : 3 : 4$ and want 400 mL in all.

How much of each do you use?

Try it first, then show the solutionSolution

8 parts; one part $= 50$ mL.
Decoction 50 mL, milk 150 mL, water 200 mL.

Answer: 50 mL, 150 mL, 200 mL.

Memory hook: Add the terms to count the cups, then fill each.

Working together

Suppose two friends paint a wall: one alone takes 3 hours, the other 6 hours.

How long do they take together?

Try it first, then show the solutionSolution

Rates $\frac{1}{3} + \frac{1}{6} = \frac{1}{2}$ wall per hour.
Time $= 2$ hours.

Answer: 2 hours.

Memory hook: Together is faster than the faster one alone.

Topics in this chapter

Proportional Reasoning-2 formula sheet

Proportion test: $a : b :: c : d ⟺ a d = b c$
Map scale: $real distance = map distance \times RF number$
Unit conversion for maps: $1 km = 1, 00, 000 cm$
Multi-term proportion: $a : b : c :: p : q : r ⟺ \frac{p}{a} = \frac{q}{b} = \frac{r}{c}$
Dividing in a ratio: $x \times \frac{a}{a + b + c + \dots}$
Pie chart angle: $\frac{value}{total} \times 36 0^{\circ}$
Inverse proportion: $x y = k, x_{1} y_{1} = x_{2} y_{2}$
Work rate: $rate = \frac{1}{time for whole job}; together: add rates$

How the school exam asks Proportional Reasoning-2

The question patterns to expect, and the method that works for each.

Question type	How it is asked	Method
Map scales	Find real distances from a map, or map lengths for a plan at a given scale.	Convert the RF into cm-to-km, then multiply or divide; remember it gives straight-line distance.
Multi-term ratios	Recipes, paints, concrete, coins in a ratio; find the other quantities from one.	Find the factor from the known term and multiply every term by it.
Dividing in a ratio	Share money, time or angles in a ratio of three or more terms.	Add the terms, find one part, multiply; check the shares add back.
Pie charts	Calculate angles and draw a pie chart; read values from a pie chart.	Angle $= \frac{value}{total} \times 360°$ ; check the total is 360°; draw with a protractor.
Direct or inverse?	Classify pairs of quantities or tables.	Ask whether doubling one doubles or halves the other; test tables with products and quotients.
Inverse proportion and work	Workers and days, pumps and hours, food and people, people working together.	Use $x_{1} y_{1} = x_{2} y_{2}$ or add work rates.

Tricks to remember Proportional Reasoning-2

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

1 cm on the map, RF cm on the ground
Knock five zeros off the RF number to get kilometres per centimetre.
Use it when: Map scales.
One factor for all
In a multi-term ratio, the same multiplier touches every term.
Use it when: Scaling recipes and mixtures.
Pie = parts of 360
Each slice is its fraction of 360°; the slices must close the circle.
Use it when: Pie-chart angles.
Direct divides, inverse multiplies
Direct: $\frac{y}{x}$ stays the same. Inverse: $x \times y$ stays the same.
Use it when: Classifying tables.
Add rates, not times
Working together, add the fractions of the job done per hour, never the hours.
Use it when: Work problems.

Proportional Reasoning-2: exam-level solved questions

Ordered from exam level to HOTS. Cover the solution, write your own, then compare step by step.

Example 1: Map distance · 1 mark · MCQExam level

On a map of scale $1 : 50, 000$ , two temples are 4 cm apart. The real distance is
(a) 0.2 km
(b) 2 km
(c) 20 km
(d) 200 km

Try it first, then show the solutionSolution

$4 \times 50, 000 = 2, 00, 000$ cm.
$= 2$ km.

Answer: (b) 2 km

Example 2: Workers and days · 1 mark · Assertion-reasonExam level

Assertion (A): For a fixed job, the number of workers and the number of days are in inverse proportion.
Reason (R): When two quantities are in inverse proportion, their product 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.

Try it first, then show the solutionSolution

Workers × days = total worker-days, fixed for the job, so the product is constant.
That is exactly what R describes, so R explains A.

Answer: (a)

Example 3: Sharing in three parts · 2 marks · Very short answerExam level

Divide ₹8,400 among three children in the ratio $3 : 4 : 5$ .

Try it first, then show the solutionSolution

12 parts; one part $= ₹700$ .
Shares: ₹2,100, ₹2,800, ₹3,500.

Answer: ₹2,100, ₹2,800 and ₹3,500

Example 4: Pie-chart angles · 3 marks · Short answerHarder exam level

45 students each picked the sport they like most: cricket 18, football 12, badminton 9, kabaddi 6. Find the angle for each slice of a pie chart.

Try it first, then show the solutionSolution

Each student is $360° \div 45 = 8°$ .
Cricket $144°$ , football $96°$ , badminton $72°$ , kabaddi $48°$ .
Sum $= 360°$ .

Answer: 144°, 96°, 72°, 48°

Example 5: Concrete mix · 2 marks · Very short answerHarder exam level

Concrete is mixed cement : sand : gravel $= 1 : 2 : 4$ . How much of each is needed for 56 bags of concrete?

Try it first, then show the solutionSolution

7 parts; one part $= 8$ bags.
Cement 8, sand 16, gravel 32 bags.

Answer: 8, 16 and 32 bags

Example 6: Inverse table · 3 marks · Short answerHarder exam level

$x$ and $y$ are inversely proportional, with $y = 30$ when $x = 12$ . Complete: $x = 18$ , $y = ?$ ; $y = 9$ , $x = ?$ . Write the rule connecting them.

Try it first, then show the solutionSolution

$k = 12 \times 30 = 360$ , so $x y = 360$ .
$x = 18$ : $y = 20$ .
$y = 9$ : $x = 40$ .

Answer: $y = 20$ , $x = 40$ ; $x y = 360$ .

Example 7: Food provisions · 5 marks · Long answerHarder exam level

A hostel has food for 120 students for 30 days. After 10 days, 30 more students join. How many more days will the remaining food last? Show your reasoning.

Try it first, then show the solutionSolution

After 10 days, food remains for 120 students for 20 more days.
That is $120 \times 20 = 2400$ student-days of food.
Now there are $120 + 30 = 150$ students.
More students, fewer days: inverse proportion, so $150 \times d = 2400$ .
$d = 16$ days.

Answer: 16 more days.

Example 8: Two fill, one empties · 4 marks · Case-basedHOTS / competency

Pipe A fills a tank in 8 hours and pipe B in 12 hours. Pipe C empties the full tank in 6 hours. If all three are opened together on an empty tank, how long will it take to fill?

Try it first, then show the solutionSolution

Rates per hour: A $\frac{1}{8}$ , B $\frac{1}{12}$ , C $- \frac{1}{6}$ .
Net rate $= \frac{3}{24} + \frac{2}{24} - \frac{4}{24} = \frac{1}{24}$ per hour.
Time $= 1 \div \frac{1}{24} = 24$ hours.
Without C it would take $1 \div \frac{5}{24} = 4.8$ hours, so C makes it 5 times as long.

Answer: 24 hours.

Example 9: Triangles from side ratios · 3 marks · Short answerHOTS / competency

(a) A triangle has sides in the ratio $3 : 4 : 5$ and perimeter 36 cm. Find the sides and show it is right-angled. (b) Explain why no triangle has sides in the ratio $1 : 3 : 5$ .

Try it first, then show the solutionSolution

(a) 12 parts of 3 cm: sides 9, 12, 15 cm.
$9^{2} + 1 2^{2} = 81 + 144 = 225 = 1 5^{2}$ , so right-angled.
(b) Sides $k, 3 k, 5 k$ : $k + 3 k = 4 k < 5 k$ , so the two shorter sides cannot reach each other.

Answer: 9, 12, 15 cm (right-angled); $1 : 3 : 5$ is impossible.

Example 10: Two maps · 3 marks · Short answerHOTS / competency

Two towns are 6 cm apart on a map of scale $1 : 20, 00, 000$ . How far apart would they be on a map of scale $1 : 50, 00, 000$ ?

Try it first, then show the solutionSolution

Real distance $= 6 \times 20 = 120$ km.
On the second map 1 cm $= 50$ km.
Map distance $= 120 \div 50 = 2.4$ cm.

Answer: 2.4 cm

Case-study practice (Section E style)

Read the situation, then answer every part. In a school exam set on the board pattern this is a 4-mark question split into 1 + 1 + 2 marks.

A family's monthly budget

Suppose a family spends its monthly income of ₹36,000 as follows: food ₹9,000, rent ₹12,000, education ₹6,000, transport ₹3,000, and savings ₹6,000. They want to show this in a pie chart.

(i) · 1 mark
What angle represents ₹1,000?
Answer
Answer: 10°
$360° \div 36 = 10°$ per ₹1,000.
(ii) · 1 mark
Find the angle for each item.
Answer
Answer: Food 90°, rent 120°, education 60°, transport 30°, savings 60°
Multiply each amount (in thousands) by 10°; the angles add to 360°.
(iii) · 2 marks
Next year rent rises to ₹15,000 while income stays the same and transport is unchanged; the rise comes out of savings. Find the new angles for rent and savings, and the fraction of income saved.
Answer
Answer: Rent 150°, savings 30°; $\frac{1}{12}$
Rent $= 15 \times 10 = 150°$ ; savings fall to ₹3,000, so 30° (1 mark). Saved fraction $= \frac{3000}{36000} = \frac{1}{12}$ (1 mark).

Where the marks are

How a typical answer is split into step marks. Show every step: a right final answer alone rarely earns full marks.

On a map with RF $1 : 75, 000$ , a lake is 6 cm long. Find its real length in km. (2 marks)

Step	Marks
Real length $= 6 \times 75, 000 = 4, 50, 000$ cm	1
$4, 50, 000 \div 1, 00, 000 = 4.5$ km	1

16 workers can lay a road in 9 days. How many days will 24 workers take? State the type of proportion. (3 marks)

Step	Marks
More workers, fewer days: inverse proportion	1
$16 \times 9 = 24 \times d$	1
$d = 144 \div 24 = 6$ days	1

72 students were asked how they come to school: walk 18, cycle 24, bus 20, car 10. Calculate the angles and describe how to draw the pie chart. (5 marks)

Step	Marks
Each student is $360° \div 72 = 5°$	1
Walk $18 \times 5 = 90°$ ; cycle $24 \times 5 = 120°$	1
Bus $20 \times 5 = 100°$ ; car $10 \times 5 = 50°$	1
Check: $90 + 120 + 100 + 50 = 360°$	1
Draw a circle and a radius; measure 90° for walk, then 120°, 100° and 50° in turn from each new radius; label each slice	1

Where students lose marks in Proportional Reasoning-2

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

Mistake: Forgetting to convert cm to km on maps
The RF number is used directly as kilometres.
Wrong working
RF $1 : 25, 00, 000$ , 7.4 cm: distance $= 7.4 \times 25, 00, 000 = 1, 85, 00, 000$ km.
Right working
$7.4 \times 25, 00, 000$ cm $= 1, 85, 00, 000$ cm $= 185$ km.
Do this: Divide by 1,00,000 to change cm to km.
Usually costs: Whole answer.
Mistake: Using direct proportion for an inverse situation
Every 'three numbers' problem is set up the same way.
Wrong working
12 workers, 15 days; 18 workers: $12 : 15 :: 18 : d$ , $d = 22.5$ days.
Right working
More workers, fewer days: $12 \times 15 = 18 \times d$ , $d = 10$ days.
Do this: Decide direct or inverse before writing anything.
Usually costs: Whole answer.
Mistake: Pie angles that do not add to 360°
Rounding or a missed category.
Wrong working
Angles 144°, 90°, 72° and the last slice drawn by eye.
Right working
Compute every angle: 144°, 90°, 72°, 54°; check the sum is 360°.
Do this: Always add the angles before drawing.
Usually costs: 1-2 marks.
Mistake: Adding times in a work problem
Rates and times are confused.
Wrong working
A takes 4 h, B takes 6 h, so together 10 h (or 5 h on average).
Right working
Rates $\frac{1}{4} + \frac{1}{6} = \frac{5}{12}$ per hour, so 2.4 h.
Do this: Convert each time to a rate, add the rates, then invert.
Usually costs: Whole answer.
Mistake: Scaling only one term of a multi-term ratio
Only the given ingredient is changed.
Wrong working
Spice mix $6 : 3 : 2 : 1$ with 9 chillies: $6 : 9 : 2 : 1$ .
Right working
Factor $9 \div 3 = 3$ : $18 : 9 : 6 : 3$ .
Do this: Multiply every term by the same factor.
Usually costs: 1-2 marks.

NCERT exercise guide: Proportional Reasoning-2

What each exercise in the NCERT chapter tests, how to approach it, and a question of the same kind solved step by step. Questions here are written in our own words.

Figure it Out (ratios with more terms)

What it tests: Sharing session time in a four-term ratio, finding other quantities from one known term, the value of coins in a ratio, and whether triangles with sides in a ratio are congruent or even possible.

How to approach it: Find one part from the total or from a known term; multiply each term; check triangle side ratios with the triangle inequality.

Like the coins question in Section 3.4

A box has 120 coins of ₹5, ₹2 and ₹1 in the ratio $1 : 2 : 3$ . How much money is in the box?

Show the solutionSolution

Parts $= 1 + 2 + 3 = 6$ ; one part $= 20$ coins.
Coins: 20 of ₹5, 40 of ₹2, 60 of ₹1.
Money $= 100 + 80 + 60 = ₹240$ .

Answer: ₹240

Figure it Out (pie charts)

What it tests: Drawing pie charts for survey data given as counts or as percentages, and for data collected in class.

How to approach it: Angle $= \frac{value}{total} \times 360°$ , or percentage $\times 3.6°$ ; check the sum and draw.

Like the percentage pie-chart question in Section 3.5

Find the pie-chart angles for: rice 40%, wheat 25%, millets 20%, pulses 15%.

Show the solutionSolution

Each 1% is $3.6°$ .
Rice $144°$ , wheat $90°$ , millets $72°$ , pulses $54°$ .
Check: $144 + 90 + 72 + 54 = 360°$ .

Answer: 144°, 90°, 72°, 54°.

Figure it Out (inverse proportion tables)

What it tests: Deciding which tables show inverse proportion and filling missing values when $x$ and $y$ are inversely proportional.

How to approach it: Multiply each pair; equal products mean inverse proportion; use $k$ to fill gaps.

Like the fill-in-the-table question in Section 3.6

$x$ and $y$ are inversely proportional. When $x = 14$ , $y = 30$ . Find $y$ when $x = 21$ and $x$ when $y = 12$ .

Show the solutionSolution

$k = 14 \times 30 = 420$ .
$y = 420 \div 21 = 20$ .
$x = 420 \div 12 = 35$ .

Answer: $y = 20$ ; $x = 35$ .

Figure it Out (direct and inverse problems)

What it tests: Classifying pairs of quantities, cost of pencils, how long a water supply lasts for more families, sleep hours from charts, and reading a transport pie chart.

How to approach it: Decide direct or inverse, then use the matching rule; state assumptions such as equal use per family.

Like the water-supply question in Section 3.6

A tank holds enough water for 40 people for 6 days. If 48 people use it at the same rate per person, how long will it last? What assumption is needed?

Show the solutionSolution

More people, fewer days: inverse proportion.
$40 \times 6 = 48 \times d$ , so $d = 5$ days.
Assumption: every person uses the same amount of water each day.

Answer: 5 days, assuming equal daily use per person.

Practice questions on Proportional Reasoning-2

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 · MCQExam level
Which pair is in inverse proportion?
(a) Number of bus tickets and the total fare
(b) Number of friends sharing a fixed bill and each person's share
(c) Hours worked and wages at a fixed hourly rate
(d) Weight of rice and its cost
Hint
More friends, smaller share each.
Answer
(b)
Q2 · 1 mark · MCQExam level
On a map of scale $1 : 40, 00, 000$ , 1 cm represents
(a) 4 km
(b) 40 km
(c) 400 km
(d) 4,000 km
Hint
Divide by 1,00,000.
Answer
(b) 40 km
Q3 · 2 marks · Very short answerExam level
The angles of a triangle are in the ratio $1 : 4 : 7$ . Find them.
Hint
180° in 12 parts.
Answer
15°, 60°, 105°
Q4 · 1 mark · Very short answerExam level
15 notebooks cost ₹600. Find the cost of 12 notebooks. Is this direct or inverse proportion?
Hint
Unit price.
Answer
₹480; direct proportion.
Q5 · 2 marks · Very short answerHarder exam level
8 identical taps fill a tank in 27 minutes. How long will 6 such taps take?
Hint
Fewer taps, more time.
Answer
$8 \times 27 = 6 \times t$ , so $t = 36$ minutes.
Q6 · 2 marks · Very short answerHarder exam level
A spice mix uses coriander : chillies : toor dal : methi $= 6 : 3 : 2 : 1$ (spoons). If 9 spoons of chillies are used, find the other amounts.
Hint
Factor 3.
Answer
18 coriander, 6 toor dal, 3 methi: $18 : 9 : 6 : 3$ .
Q7 · 3 marks · Short answerHarder exam level
150 coins of ₹10, ₹5 and ₹2 are in the ratio $1 : 2 : 3$ . How much money is there?
Hint
Count each type first.
Answer
25, 50 and 75 coins: $250 + 250 + 150 = ₹650$ .
Q8 · 2 marks · Very short answerHarder exam level
A survey shows how students reach school: bus 35%, walk 30%, cycle 20%, car 15%. Find the pie-chart angles.
Hint
1% = 3.6°.
Answer
126°, 108°, 72°, 54°
Q9 · 2 marks · Very short answerHarder exam level
A classroom is 9 m long and 6.5 m wide. Find its size on a plan drawn to the scale $1 : 50$ .
Hint
Convert to cm and divide by 50.
Answer
900 cm ÷ 50 = 18 cm; 650 cm ÷ 50 = 13 cm. So 18 cm by 13 cm.
Q10 · 5 marks · Long answerHarder exam level
15 workers can finish a road in 12 days. After 4 days, 5 more workers join. (a) How much work (in worker-days) is left? (b) How many more days are needed? (c) How many days does the whole job take?
Hint
Total is $15 \times 12$ worker-days.
Answer
(a) Total 180 worker-days; done $15 \times 4 = 60$ ; left 120. (b) $120 \div 20 = 6$ days. (c) $4 + 6 = 10$ days.
Q11 · 3 marks · Short answerHOTS / competency
A can do a job in 10 days and B in 15 days. They work together for 2 days, then B leaves. How long does A take to finish the rest?
Hint
Add rates for the first 2 days.
Answer
Together they do $\frac{1}{10} + \frac{1}{15} = \frac{1}{6}$ per day, so $\frac{1}{3}$ in 2 days. A does the remaining $\frac{2}{3}$ at $\frac{1}{10}$ per day: $\frac{20}{3} = 6 \frac{2}{3}$ days.
Q12 · 3 marks · Short answerHOTS / competency
Decide whether each table shows direct proportion, inverse proportion or neither: (i) $x$ : 4, 6, 9; $y$ : 27, 18, 12. (ii) $x$ : 2, 5, 8; $y$ : 7, 17.5, 28. (iii) $x$ : 1, 2, 3; $y$ : 5, 7, 9.
Hint
Test products and quotients.
Answer
(i) Products all 108: inverse. (ii) $\frac{y}{x} = 3.5$ each time: direct. (iii) Neither: $\frac{y}{x}$ is 5, 3.5, 3 and $x y$ is 5, 14, 27.

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.When are $a : b$ and $c : d$ proportional?Show answer
When $a d = b c$ .
2.What is a representative fraction (RF)?Show answer
The ratio of a distance on a map to the corresponding real distance, in the same unit.
3.What does RF $1 : 60, 00, 000$ mean in km?Show answer
1 cm on the map represents 60 km.
4.Does a map give road distance?Show answer
No, it gives the straight-line (geographical) distance.
5.How do you share $x$ in the ratio $a : b : c$ ?Show answer
Shares are $x \times \frac{a}{a + b + c}$ , $x \times \frac{b}{a + b + c}$ , $x \times \frac{c}{a + b + c}$ .
6.What do all the angles in a pie chart add up to?Show answer
360°.
7.Define inverse proportion.Show answer
$x$ and $y$ are inversely proportional if $x y = k$ , a constant.
8.Give two examples of inverse proportion.Show answer
Speed and time for a fixed distance; number of workers and days for a fixed job.
9.How do you find the time for two people working together?Show answer
Add their rates (fractions of the job per hour) and take the reciprocal.
10.Why can't a triangle have sides in the ratio $1 : 3 : 5$ ?Show answer
The two shorter sides add to less than the longest, so they cannot meet.

Best order to study it

Revise proportion and cross-multiplication (3.1).
Learn map scales: convert RF to cm-to-km and practise distances and plans (3.2).
Work with ratios of three or four terms: recipes, paints, concrete (3.3).
Divide wholes in multi-term ratios, including triangle angles (3.4); do the first Figure it Out set.
Calculate pie-chart angles and draw pie charts (3.5); do the second set.
Learn inverse proportion, $x y = k$ , and test tables (3.6).
Finish with workers, pumps, food and working-together problems in the last set.

Quick revision checklist

The night before the exam: can you do each of these without looking?

Can you test two ratios by cross-multiplication?
Can you convert a map distance to kilometres?
Can you draw a classroom plan to a scale of 1 : 50?
Can you scale a four-term recipe from one known amount?
Can you divide money in the ratio $2 : 3 : 4$ ?
Can you find triangle angles in a ratio?
Can you calculate and draw a pie chart?
Can you decide whether a table shows direct or inverse proportion?
Can you solve a workers-and-days problem?
Can you find the time taken by two people working together?

Proportional Reasoning-2: frequently asked questions

Where is Proportional Reasoning-2 in the Class 8 NCERT book?

It is Chapter 3 of Ganita Prakash Part II (Chapter 10 in our list), the NCERT Grade 8 Mathematics textbook (Reprint 2026-27). It follows Proportional Reasoning-1 from Part I.

Is inverse proportion new in Class 8?

Yes. Proportional Reasoning-1 warned that some problems cannot be solved with the Rule of Three. This chapter names such relationships inverse proportions and shows that the product xy stays constant.

Do I need to draw pie charts accurately?

Yes. Calculate each angle, check that they add to 360°, and use a protractor. Marks are usually given for correct angles, accurate drawing and clear labels.

Why do different maps give slightly different distances?

Measuring on a small map is less precise, and printed maps may round their scales. Each correctly used map should give roughly the same straight-line distance.

How do I know if a work problem is direct or inverse?

For a fixed amount of work, more workers means fewer days: inverse. For a fixed number of workers, more work means more days: direct. Decide which quantity is fixed first.

Based on: NCERT Ganita Prakash, Mathematics Textbook for Grade 8, Part II (Reprint 2026-27); NCERT chapter PDF: Proportional Reasoning-2