Chapter 7 of 14

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West Bengal (WBBSE)Class 8 Mathematics★ 12-15 marks · CBSE Class 8 Maths (Ganita Prakash Chapter 7)

Proportional Reasoning

"'Proportional Reasoning — 1' covers ratios and proportion, dividing a quantity in a given ratio, rates such as population density and cost per square foot, and unit conversions. Learn the cross-product test, why adding to a ratio destroys it, when to round up, and why 4 times faster means a quarter of the time. CBSE Class 8 Maths (Ganita Prakash) Chapter 7."

⏱ 22 min readMedium difficulty✓ Free · NCERT-aligned

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By the end of this chapter you'll be able to…

1Test whether two ratios are in proportion using the cross-product rule
2Generate equivalent ratios by multiplying both parts by the same number, and explain why adding does not work
3Divide a quantity in a given ratio by finding the total number of parts
4Handle mixtures where one component is added, recognising that the other component is unchanged
5Compute and compare rates — population density, cost per unit area, bricks per foot
6Convert between units before setting up a proportion, including acres to square feet and litres to millilitres
7Recognise inverse proportion, where being n times faster means taking one nth of the time
8Decide when to round up rather than to the nearest whole number, for indivisible units like buses

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Why this chapter matters

Proportional reasoning is the single most used piece of school mathematics in adult life — recipes, medicine doses, map scales, currency, fuel economy, population density, land prices. This chapter's real lesson is that comparing raw totals is usually meaningless: Delhi has more people than Mumbai yet is far less crowded, because what matters is people per square kilometre. Reducing to a per-unit rate is the move that makes two different-sized things comparable.

Before you start — revise these

A 5-minute refresher here will save you 30 minutes of confusion below.

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Fractions — simplifying, and multiplying by a reciprocal to divide

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HCF, for reducing a ratio to its simplest form

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Unit conversions between metric units

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Forming and solving a simple linear equation in one unknown

Proportional Reasoning — Class 8 Mathematics (Ganita Prakash)

What the book actually covers (2026-27) The book titles this chapter "Proportional Reasoning — 1", and it has six sections: Observing Similarity in Change, Ratios, Ratios in their Simplest Form, Problem Solving with Proportional Reasoning, Sharing, but Not Equally! and Unit Conversions. Percentages, profit and loss, discount and simple interest are not in this chapter. They belong to Fractions in Disguise (Part 2, Chapter 1 — the eighth chapter of the course). That material is still Class 8 and is still on this page below, but study it with that chapter, not this one. The four things this chapter is really examined on: the cross-product test for proportion, dividing a quantity in a given ratio (add the parts first), rates and unit conversions — especially 1 acre = 43,560 sq ft — and inverse proportion, where being four times faster means taking a quarter of the time.

"Almost every real-world problem — recipes, currency conversion, taxes, salary hikes, election polls — is at heart a problem of proportions."

1. About the Chapter

'Proportional Reasoning' is one of the most practically useful chapters in your school career. It teaches:

Ratios — comparing two quantities
Proportions — equality of two ratios
Direct proportion (one quantity grows as another grows)
Inverse proportion (one quantity grows as another shrinks)
Unitary method — find the value of one unit, then scale
Percentages and their applications
Profit/loss/discount in commercial math
Simple interest in banking

Key Idea

Proportional reasoning is the mathematics of comparing rates and scaling. Whether you're doubling a recipe, calculating tax, or planning a trip, you're using proportions.

2. Ratios — Comparing Quantities

Definition

A ratio compares two quantities of the same kind by division.

The ratio of a to b is a : b (read "a is to b") or a/b.

Simplification

Ratios should be in simplest form (HCF = 1).

6 : 9 → 2 : 3 (divide both by 3)
25 : 75 → 1 : 3

Equivalent Ratios

Multiply both terms by the same non-zero number.

1 : 2 = 3 : 6 = 5 : 10 = 100 : 200

Comparing Ratios

To compare 3:4 and 5:7, convert to common denominator (LCM):

3:4 = 21:28
5:7 = 20:28
21:28 > 20:28, so 3:4 > 5:7

3. Proportion

Definition

A proportion is an equality of two ratios.

If a:b = c:d, then a, b, c, d are in proportion.

This is often written: a/b = c/d or a : b :: c : d (read "a is to b as c is to d").

The Cross-Product Rule

If a/b = c/d, then ad = bc (cross-multiplication).

This is the most useful rule for solving proportion problems.

Example

Solve: x/4 = 12/16

Cross multiply: 16x = 48
x = 3 ✓

4. Direct Proportion

Definition

Two quantities x and y are in direct proportion if y/x = constant, i.e., as x increases, y increases (and vice versa).

We write: y ∝ x (y is proportional to x) → y = kx where k is a constant.

Examples

Distance traveled is directly proportional to time (at constant speed)
Cost is directly proportional to quantity
Salary is directly proportional to hours worked (at fixed hourly rate)

Solving

Method 1 — Unitary:

Find the value of one unit
Multiply by required number of units

Example: If 5 pens cost ₹25, find cost of 7 pens.

Cost of 1 pen = ₹25/5 = ₹5
Cost of 7 pens = ₹35

Method 2 — Proportion:

Set up: x₁/y₁ = x₂/y₂
Cross multiply

Same example:

5/25 = 7/y → 5y = 175 → y = ₹35

5. Inverse Proportion

Definition

Two quantities x and y are in inverse proportion if xy = constant, i.e., as x increases, y decreases.

We write: y ∝ 1/x → y = k/x.

Examples

Time and speed (for a fixed distance): more speed = less time
Number of workers and time (for a fixed job): more workers = less time per worker
Number of people and food rations (for a fixed amount): more people = less per person

Solving

Example: 8 workers complete a job in 12 days. How many days will 6 workers take?

xy = constant
8 × 12 = 6 × y → y = 96/6 = 16 days

Dividing a quantity in a given ratio is the single most-tested skill in this chapter.

The method

To divide a total T in the ratio a : b:

Add the parts — the total splits into (a + b) equal shares.
One share = T ÷ (a + b).
The two amounts are a/(a+b) × T and b/(a+b) × T.

₹4,500 in the ratio 2 : 3. Total parts 5, one share ₹900, so the amounts are ₹1,800 and ₹2,700. Always check both ways: they add back to ₹4,500 ✓ and 1800 : 2700 = 2 : 3 ✓

The most common error in the whole chapter. In acid : water = 1 : 5, the acid is one part out of six, so one sixth of the solution — not one fifth. In 240 mL, the acid is 240 ÷ 6 = 40 mL. Add the parts before you do anything else.

When one component is added later

Adding to a mixture changes only that component; the other is untouched and the total simply grows.

Blue : yellow = 3 : 5 in 40 mL gives 15 mL and 25 mL. Add 20 mL of yellow → blue still 15 mL, yellow now 45 mL, so the new ratio is 15 : 45 = 1 : 3.

Watch the units of "one bucket". If a bucket of paint is 3 : 5 red to yellow and a whole extra bucket of yellow is added, the red stays at 3/8 of a bucket while the yellow rises to 5/8 + 1 = 13/8. The new ratio is 3 : 13 — not 3 : 6, because a whole bucket is 8 parts, not 1.

5B. Rates and Unit Conversions

A rate compares quantities in different units — km per hour, people per sq km, bricks per foot. Reducing to a per-unit figure is what makes differently-sized things comparable.

Why raw totals mislead

City	Population	Area	Density
Delhi	30 million	1,484 sq km	≈ 20,216 per sq km
Mumbai	20 million	550 sq km	≈ 36,364 per sq km

Delhi has more people but Mumbai is far more crowded — two-thirds of the population on barely a third of the land. Crowding is a rate, not a total.

A mental check that avoids long division: Delhi's area is about 2.7× Mumbai's while its population is only 1.5× as large. Since 1.5 < 2.7, Delhi must be less dense.

Convert units BEFORE setting up a proportion

Conversion	Value
1 acre	43,560 sq ft
1 litre	1000 mL
1 tonne	1000 kg
1 year	52 weeks

Mixing a per-acre rate with an area in square feet gives an answer wrong by a factor of tens of thousands.

10 tonnes of manure per acre, plot 200 ft × 500 ft. Area = 100,000 sq ft = 100,000 ÷ 43,560 ≈ 2.3 acres → about 23 tonnes.

Inverse proportion

When one quantity rises as the other falls, divide instead of multiplying.

A tractor 4 times faster than oxen takes one quarter of the time: 6 hours per acre becomes 1.5 hours. Over 20 acres that is 120 hours against 30.

Rounding for indivisible units

204 people at 54 per bus needs 204 ÷ 54 = 3.78 buses. You cannot hire 0.78 of a bus, so round up to 4 — then say that 4 × 54 = 216 seats leaves 12 empty.

Always round up for buses, boxes and tickets, even when ordinary rounding would go down.

6. Percentages — A Power Concept

Not in Chapter 7. Percentages, profit and loss, discount and simple interest belong to Fractions in Disguise (Part 2, Chapter 1 — the eighth chapter of the course). Everything from here to the end of the page is kept for reference and revision, but study it alongside that chapter.

What is a Percentage?

A percentage is a rate per hundred. The symbol % means "per 100".

25% means 25 per 100 = 25/100 = 0.25
60% means 60/100 = 0.6

Converting

Fraction to Percentage: multiply by 100.

3/5 → (3/5) × 100% = 60%

Percentage to Fraction: divide by 100.

75% → 75/100 = 3/4

Decimal to Percentage: multiply by 100.

0.4 → 40%

Percentage to Decimal: divide by 100.

35% → 0.35

Finding Percentage of a Number

x% of N = (x/100) × N

20% of 500 = (20/100) × 500 = 100
15% of 80 = (15/100) × 80 = 12

Percentage Increase / Decrease

% increase = (Increase / Original) × 100 % decrease = (Decrease / Original) × 100

Example: A salary increases from ₹40,000 to ₹46,000. Find % increase.

Increase = 6000
% increase = (6000/40000) × 100 = 15%

7. Profit and Loss

Definitions

CP (Cost Price): the price at which an item is bought
SP (Selling Price): the price at which an item is sold
Profit = SP − CP (when SP > CP)
Loss = CP − SP (when SP < CP)

Profit/Loss Percentage

Profit % = (Profit / CP) × 100 Loss % = (Loss / CP) × 100

(ALWAYS calculate percentage on CP!)

Examples

Example 1: An item bought for ₹400 is sold for ₹500. Find profit %.

Profit = 500 − 400 = ₹100
Profit % = (100/400) × 100 = 25%

Example 2: A book bought for ₹250 is sold at 20% loss. Find SP.

Loss = 20% of 250 = ₹50
SP = 250 − 50 = ₹200

Discount

Discount is a reduction from the marked price (MP):

SP = MP − Discount
Discount % = (Discount / MP) × 100

Example: A shirt's MP is ₹800, discount 25%. Find SP.

Discount = 25% of 800 = ₹200
SP = 800 − 200 = ₹600

8. Simple Interest

Definition

Simple Interest (SI) is the interest charged on a fixed principal at a fixed rate, calculated only on the original principal.

Formula

SI = (P × R × T) / 100

where:

P = Principal (original amount)
R = Rate per annum (% per year)
T = Time (in years)

Amount

A = P + SI

Examples

Example 1: ₹10,000 deposited at 6% per annum for 3 years. Find SI and Amount.

SI = (10000 × 6 × 3) / 100 = ₹1,800
A = 10000 + 1800 = ₹11,800

Example 2: An amount becomes ₹6,000 from ₹5,000 in 2 years. Find the rate.

SI = 6000 − 5000 = ₹1,000
R = (SI × 100) / (P × T) = (1000 × 100) / (5000 × 2) = 10%

9. Common Word Problems

Type 1: Unitary Method

"If A objects cost B rupees, how much do C objects cost?"

Cost per object = B/A
Total for C = (B/A) × C

Type 2: Direct Proportion

"x is to y as p is to q" → cross multiply

Type 3: Inverse Proportion

"M workers do a job in N days; how many days for X workers?"

M × N = X × ? → ? = M × N / X

Type 4: Profit/Loss with Discount

"MP is X, discount Y%, profit Z%; find CP."

SP = X(1 − Y/100)
CP = SP / (1 + Z/100)

Type 5: Simple Interest

Direct formula application.

10. Worked Examples

Example 1: Simplify Ratio

Simplify 36 : 60.

HCF(36, 60) = 12
36/12 : 60/12 = 3 : 5

Example 2: Direct Proportion

If 12 books cost ₹360, find cost of 18 books.

12 books → ₹360
1 book → ₹30
18 books → ₹540

Example 3: Inverse Proportion

20 workers complete a job in 15 days. How many days will 25 workers take?

20 × 15 = 25 × ?
300 = 25 × ?
? = 12 days

Example 4: Percentage Increase

A population grew from 50,000 to 55,000. Find % increase.

Increase = 5,000
% increase = (5000/50000) × 100 = 10%

Example 5: Profit %

SP = ₹920, CP = ₹800. Find profit %.

Profit = 120
Profit % = (120/800) × 100 = 15%

Example 6: Discount and Profit

MP = ₹1000, discount = 20%, profit = 10%. Find CP.

Discount = 200, SP = 800
CP = SP / (1 + 10/100) = 800/1.1 ≈ ₹727.27

Example 7: Simple Interest

P = ₹15,000, R = 8% p.a., T = 5 years. Find SI and A.

SI = (15000 × 8 × 5) / 100 = ₹6,000
A = ₹21,000

Example 8: Find Time

P = ₹4000 becomes ₹4960 at R = 6% per annum. Find T.

SI = 4960 − 4000 = ₹960
T = (SI × 100) / (P × R) = (960 × 100) / (4000 × 6) = 4 years

11. Common Mistakes

Confusing direct and inverse proportion
- Direct: more of one → more of other (y = kx)
- Inverse: more of one → less of other (xy = constant)
Calculating % on SP instead of CP
- Profit % and Loss % are ALWAYS on CP
- Discount % is on MP
Forgetting to convert percentage
- 5% means 5/100 = 0.05 (not 5)
Misreading 'per annum'
- 'p.a.' means PER YEAR. For 6 months, use T = 0.5 years.
Wrong formula for SI
- SI = PRT/100 (not PRT)
Adding percentages on different bases
- 20% increase followed by 20% decrease is NOT 0% change
- Original 100 → 120 (after +20%) → 96 (after −20%) = 4% net decrease

12. Real-World Applications

Shopping

Discount calculations
Bulk buying (unitary method)
Comparing prices

Salary and Taxes

Income tax (percentage of salary)
GST (added to MRP)
Annual increments (percentage increase)

Banking

Simple interest on savings
Personal loans (rate per annum)
Fixed deposits

Cooking

Scaling recipes (direct proportion)
Converting units (cups to grams)

Travel

Speed-distance-time (direct/inverse)
Currency conversion

Election Polling

Percentage of votes
Margin calculations

Cricket Statistics

Strike rate (runs per 100 balls)
Run rate (runs per over)
Required run rate

13. Tips for Mastery

For Ratios

Always reduce to lowest terms
Use LCM for comparison

For Proportions

Cross-multiplication is your friend
Set up: "First ratio = Second ratio"

For Direct/Inverse

Read the problem CAREFULLY
Ask: "If first increases, does second increase (direct) or decrease (inverse)?"

For Percentages

1% = 1/100 = 0.01
10% = 1/10 = 0.1
50% = 1/2 = 0.5
25% = 1/4 = 0.25

For Profit/Loss

ALWAYS calculate on CP
For discount, calculate on MP

For Simple Interest

Memorise SI = PRT/100
Time in YEARS

14. Conclusion

'Proportional Reasoning' is the most practical chapter in your math course. The skills you learn here will be used:

Every time you shop
Every time you check a salary
Every time you compare prices
Every time you read a news report with percentages

Master ratios, proportions, percentages, profit/loss, and simple interest. These are not just exam topics — they are life skills.

Two Part 2 chapters carry these ideas forward. Fractions in Disguise (Part 2, Chapter 1) develops percentage into profit, loss, discount and compound interest. Proportional Reasoning-2 (Part 2, Chapter 3) extends ratio to three or more terms, adds map scales and pie charts, and introduces inverse proportion. The foundation built here carries you through both.

Key formulas & results

Everything you need to memorise, in one card. Screenshot this for revision.

Test for proportion

a : b :: c : d  ⟺  ad = bc

Cross-multiplying is one line and needs no simplifying — the safest test in an exam.

Equivalent ratios

a : b = ka : kb

MULTIPLY both parts by the same number. Adding the same number changes the ratio.

Dividing in a ratio

parts of T in a : b are a/(a+b) × T and b/(a+b) × T

The denominator is the SUM of the parts, not one of them.

Rate (per unit)

rate = quantity ÷ number of units

Population density, cost per sq ft, bricks per foot. This is what makes different sizes comparable.

Inverse proportion

n times faster ⟹ 1/n of the time

Speed and time move in opposite directions — divide, do not multiply.

Key conversions

1 acre = 43,560 sq ft;  1 L = 1000 mL;  1 tonne = 1000 kg

Convert BEFORE setting up any proportion.

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Common mistakes & fixes

These are the exact errors that cost students marks in board exams. Read them once, save yourself the trouble.

WATCH OUT

✗ Adding the same number to both parts of a ratio instead of multiplying

✓ Scale by multiplying both parts by the same factor. This is also why enlarging a rectangle means multiplying both dimensions, never adding to them.

WATCH OUT

✗ Treating a : b as meaning the first part is a fraction a/b of the total

✓ Always add the parts first. In a : b the fractions of the total are a/(a+b) and b/(a+b).

WATCH OUT

✗ Setting up a proportion before converting units

✓ Convert first: 1 acre = 43,560 sq ft, 1 L = 1000 mL, 1 tonne = 1000 kg. Then the proportion is safe.

WATCH OUT

✗ Adjusting both components when only one is added to a mixture

✓ Recompute the actual quantities, then form the new ratio. Adding a whole bucket of yellow to 3 : 5 gives 3 : 13, not 3 : 6.

WATCH OUT

✗ Rounding to the nearest whole number when the units are indivisible

✓ For buses, boxes or tickets, always round UP. Then state how much spare capacity is left.

WATCH OUT

✗ Multiplying instead of dividing for inverse proportion

✓ Ask which way the quantity moves. Faster must mean LESS time, so divide.

WATCH OUT

✗ Comparing raw totals instead of rates

✓ Compute a per-unit figure — people per sq km, cost per sq ft — before comparing.

NCERT exercises (with solutions)

Every NCERT exercise from this chapter — what it covers and how many questions to expect.

Figure it Out — Ratios and Proportion

Questions

Figure it Out — Proportional Reasoning in Context

Questions

Figure it Out — Sharing in a Given Ratio

Questions

Figure it Out — Ratios, Rates and Unit Conversions

Questions

Practice problems

Work through this chapter's problems as a readiness check — reveal each solution, mark yourself honestly, and get your gap report at the end.

Readiness check

Are you exam-ready for Proportional Reasoning?

8 problems from this chapter. Try each one, reveal the worked solution, mark yourself honestly — get your gap report at the end.

8 questions~6 min worth ~15 marks in West Bengal (WBBSE) exams

5-minute revision

The whole chapter, distilled. Read this the night before the exam.

•Two ratios are in proportion when ad = bc. Cross-multiplying is faster and safer than simplifying.
•A ratio is unchanged by multiplying both parts by the same number, and destroyed by adding to them.
•To divide T in the ratio a : b, split into (a + b) shares of T/(a+b) each.
•In a : b, the first quantity is a/(a+b) of the total — the denominator is the SUM.
•Reducing to a unit rate (per person, per foot, per acre) is usually the quickest route and gives a built-in check.
•Population density = population ÷ area. Mumbai ≈ 36,364 per sq km beats Delhi ≈ 20,216, despite Delhi's larger population.
•Convert units before setting up a proportion. 1 acre = 43,560 sq ft is the one to memorise for this chapter.
•Inverse proportion: n times faster means 1/n of the time. Speed up, time down.
•For indivisible units, round UP and then state the spare capacity.
•Similar rectangles are exactly those with the same width : height ratio — the ratio fixes the shape, not the size.
•The Lilavati (Bhaskara II, c. 1150 CE) poses proportion problems in verse; the saffron problem gives 52.5 palas.

West Bengal (WBBSE) marks blueprint

Where the marks come from in this chapter — so you can plan your prep.

Typical chapter weightage: 12-15 marks per chapter (highest weightage)

Question type	Marks each	Typical count	What it tests
MCQ / Very Short	1	3	Ratios; percentages; SI formula
Short Answer	2-3	3	Direct/inverse problems; percentage conversion; profit/loss
Long Answer	5	1-2	Multi-step word problems; financial scenarios

Where this shows up in the real world

This chapter isn't just an exam topic — it lives in the world around you.

Cooking and baking

Cooking and baking — scaling a recipe up or down keeps every ingredient in the same ratio

Medicine dosage by body weight

Medicine dosage by body weight, where the dose is a rate in mg per kg

Map scales and architectural plans

Map scales and architectural plans, which are similar figures at a fixed ratio

Currency conversion

Currency conversion, fuel economy in km per litre, and unit pricing in shops

Population density and land valuation

Population density and land valuation, both of which are rates per unit area

Alloy composition

Alloy composition — the cupro-nickel in a ₹10 coin is mixed 3 : 1 by mass

Exam strategy

Battle-tested tips from teachers and toppers for this chapter.

Four question types cover almost everything. (1) Is it a proportion? — cross-multiply, one line. (2) Divide in a ratio — always write the total number of parts first, then one part, then each share, and finish by checking the parts add back to the total. (3) Rate and unit-conversion word problems — convert units BEFORE forming the proportion, and prefer the unit-rate route (bricks per foot, rupees per square foot) since it gives an independent check on your cross-multiplication. (4) Comparison questions ('which city is more crowded?') — the marks are for computing a per-unit rate and saying why raw totals will not do. Watch for two traps that appear every year: rounding up for indivisible units, and inverse proportion where faster means dividing.

Where else this chapter is tested

CBSE board isn't the only one — other exams test this chapter too.

CBSE Class 8 School ExamVery High

Class 8 OlympiadHigh

NTSE Mental AbilityVery High

Class 9 StatisticsHigh

CTET / TET (teaching exams)Very High

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Questions students ask

The real ones — pulled from the Q&A community and tutor sessions.

Because a ratio is a fraction, and adding changes its value: 4 : 9 becomes 6 : 11 if you add 2 to each, and 6/11 ≈ 0.545 while 4/9 ≈ 0.444. Multiplying both parts by the same number keeps the fraction identical, which is why scaling always means multiplying.

No — it is one SIXTH. The ratio 1 : 5 means one part to five parts, so there are six parts altogether. In acid : water = 1 : 5 with 240 mL of solution, the acid is 40 mL, which is 240 ÷ 6.

Whenever the units cannot be split — buses, boxes, tickets, cartons. 204 people at 54 per bus gives 3.78, and three buses would strand 42 people, so the answer is 4. Then say how many seats are spare.

It means one quarter of the time, because speed and time are inversely proportional. If oxen take 6 hours per acre, a tractor 4 times faster takes 6 ÷ 4 = 1.5 hours. Multiplying by 4 is the standard error here.

Because crowding is people per unit area, not total people. Delhi has 30 million on 1,484 sq km (about 20,216 per sq km) while Mumbai has 20 million on just 550 sq km (about 36,364 per sq km). Mumbai has two-thirds the population on barely a third of the land.

1 acre = 43,560 sq ft. It matters because several questions give a rate per acre but an area in square feet — the manure question and the land-cost question both hinge on converting first. Skipping the conversion makes the answer wrong by a factor of tens of thousands.

Practice more, ask doubts, find a tutor

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