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

Proportional Reasoning

'Proportional Reasoning' teaches ratios, proportions, percentages, profit/loss, and simple interest — the practical math that powers daily life, business, and finance. Master direct and inverse proportions, the unitary method, percentage conversions, and the formulas that underlie shopping, salaries, taxes, and banking. 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…

1Simplify and compare ratios
2Solve direct and inverse proportion problems
3Apply unitary method
4Compute percentages, percentage increases/decreases
5Solve profit/loss/discount problems
6Calculate simple interest

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Why this chapter matters

Most practical chapter — ratios, percentages, profit/loss, and simple interest power daily life, shopping, banking, salary, taxes. Foundation for financial literacy.

Before you start — revise these

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

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We Distribute Yet Things Multiply

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Class 7 Ratios and Percentages

Proportional Reasoning — Class 8 Mathematics (Ganita Prakash)

"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

6. Percentages — A Power Concept

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.

In Chapter 10 (Proportional Reasoning II), you'll extend these ideas to compound interest, ratios in motion, and more complex applications. The foundation built here will carry you through.

Key formulas & results

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

Cross-multiplication

If a/b = c/d, then ad = bc

Universal for proportions

Direct proportion

y = kx (y/x = constant)

Inverse proportion

y = k/x (xy = constant)

Percentage

x% = x/100

% of a number

x% of N = (x/100) × N

% increase

(Increase / Original) × 100

Profit %

(Profit / CP) × 100

ALWAYS on CP

Loss %

(Loss / CP) × 100

ALWAYS on CP

Discount

SP = MP − Discount

Discount % is on MP

Simple Interest

SI = PRT / 100

P=Principal, R=rate p.a., T=years

Amount

A = P + SI

⚠️

Common mistakes & fixes

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

WATCH OUT

✗ Confusing direct vs inverse proportion

✓ Direct: y increases as x increases (y = kx). Inverse: y decreases as x increases (xy = constant).

WATCH OUT

✗ Calculating Profit/Loss % on SP

✓ ALWAYS calculate on CP. If you do it on SP, your answer will be wrong.

WATCH OUT

✗ Confusing discount and profit percent

✓ Discount % is on MP (marked price). Profit/Loss % is on CP (cost price).

WATCH OUT

✗ Forgetting /100 in percent

✓ 5% = 5/100, not 5. So 5% of 200 = 10, not 1000.

WATCH OUT

✗ Time in months for SI

✓ SI formula needs T in YEARS. For 6 months, use T = 6/12 = 0.5.

NCERT exercises (with solutions)

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

Try These - Ratio and Proportion

Questions

Try These - Direct/Inverse

Questions

Try These - Percentages

Questions

Ex 7.1

Exercise 7.1

Ratios, proportions, unitary method

Questions

Ex 7.2

Exercise 7.2

Percentages and percentage changes

Questions

Ex 7.3

Exercise 7.3

Profit, loss, discount

Practice problems

Try each one yourself before tapping "Show solution". Active recall > rereading.

Q1EASY· Ratio

Simplify 48 : 84 to its lowest form.

▶ Show solution

✦ Answer: HCF(48, 84) = 12. Divide both: 48/12 = 4, 84/12 = 7. Simplified ratio = 4 : 7.

Q2EASY· Percentage

What is 30% of 250?

▶ Show solution

✦ Answer: 30% of 250 = (30/100) × 250 = 75.

Q3MEDIUM· Direct proportion

If 8 metres of cloth cost ₹560, find the cost of 12 metres.

▶ Show solution

Step 1 — Unitary method.
  8 metres → ₹560
  1 metre → ₹560 / 8 = ₹70
  12 metres → ₹70 × 12 = ₹840

Step 2 — Verify with proportion.
  8/560 = 12/y → 8y = 6720 → y = ₹840 ✓

✦ Answer: 12 metres cost ₹840.

Q4MEDIUM· Profit

A shopkeeper bought a watch for ₹1,200 and sold it for ₹1,500. Find the profit percentage.

▶ Show solution

Step 1 — Identify values.
  CP = ₹1,200
  SP = ₹1,500

Step 2 — Calculate profit.
  Profit = SP − CP = 1500 − 1200 = ₹300

Step 3 — Calculate profit %.
  Profit % = (Profit / CP) × 100
           = (300 / 1200) × 100
           = 25%

Step 4 — Verify.
  25% of 1200 = 300 ✓
  CP + Profit = 1200 + 300 = 1500 = SP ✓

✦ Answer: Profit percentage = 25%.

Q5HARD· Application

Rohan deposited ₹25,000 in a bank that offers 6.5% simple interest per annum. (a) Calculate the interest after 4 years. (b) Find the total amount in his account. (c) After how many years will the amount become ₹35,400?

▶ Show solution

Part (a) — Calculate SI for 4 years.
  SI = (P × R × T) / 100
     = (25000 × 6.5 × 4) / 100
     = 6,50,000 / 100
     = ₹6,500

Part (b) — Amount after 4 years.
  A = P + SI = 25000 + 6500 = ₹31,500

Part (c) — Find time T for A = ₹35,400.
  SI needed = 35400 − 25000 = ₹10,400
  Apply SI formula: 10400 = (25000 × 6.5 × T) / 100
  10400 = 1625T
  T = 10400 / 1625 = 6.4 years

Step — Verify Part (c).
  SI in 6.4 years = (25000 × 6.5 × 6.4)/100 = 10,400 ✓
  Amount = 25000 + 10400 = 35,400 ✓

✦ Answer: (a) SI = ₹6,500. (b) Amount after 4 years = ₹31,500. (c) Amount becomes ₹35,400 after 6.4 years.

5-minute revision

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

•Ratio a:b should be in lowest terms
•Proportion: a/b = c/d ⟺ ad = bc
•Direct proportion: y = kx; y/x = constant
•Inverse proportion: xy = constant
•Unitary method: find value of 1, then scale
•Percentage: x% = x/100
•x% of N = (x/100) × N
•% increase = (Increase/Original) × 100
•Profit/Loss % always on CP
•Discount % on MP
•SI = PRT/100 (T in years)
•Amount = Principal + SI

CBSE 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

Prep strategy

Memorise key formulas (SI = PRT/100, profit % = profit/CP × 100)
Practise 5 direct + 5 inverse proportion problems
Master percentage conversions (fraction ↔ decimal ↔ %)
Solve 10 mixed profit/loss/discount problems
Practice 8 simple interest scenarios

Where this shows up in the real world

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

Shopping

Compare unit prices, calculate discounts (e.g., 'Buy 2 Get 1 Free'), apply GST (~18% on most goods).

Banking

Savings accounts (~4% SI), fixed deposits (~7% SI/CI), personal loans (~12% per annum).

Salary and tax

Income tax slabs use percentages; HRA/DA are percentages of basic salary; annual increments are percentage hikes.

News and statistics

Polling shows 45% vote share; inflation 6% per year; population growth 1.2% — all proportional reasoning.

Recipe and cooking

Doubling/halving recipes uses direct proportion. 'Salt: 1 tsp per 4 servings' is a ratio.

Exam strategy

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

Memorise SI formula and profit/loss formulas COLD
Identify problem type — direct, inverse, percentage, profit-loss, or SI
Always state the formula at the start of your answer
Show units throughout (rupees, years, percent)
Verify answer by plugging back
Read 'per annum' as 'per year' for SI questions

Going beyond the textbook

For olympiad aspirants and curious learners — topics that build on this chapter.

Compound interest (Chapter 10)
Mixture and alligation problems
Chain rule (combining direct and inverse)
Population growth modelling
Ratio and partnership problems

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.

Convention has evolved this way because CP represents the merchant's INVESTMENT. The profit percentage tells the merchant how much they earned RELATIVE to what they spent. It's more meaningful than profit/SP. International accounting standards also use 'Return on Investment' (similar to profit/CP). When you see 'Profit margin' (a different metric), it's calculated on SP — but for school problems, profit % is always on CP.

SIMPLE interest is calculated only on the ORIGINAL principal — every year you get the same amount. COMPOUND interest is calculated on the principal PLUS accumulated interest — so each year you earn 'interest on interest'. Compound interest grows much faster. You'll study compound interest in Chapter 10 (Proportional Reasoning II). For now, SI is the foundation.

Yes, in a sense. DIRECT: as x grows, y grows (proportionally). INVERSE: as x grows, y shrinks (proportionally). Mathematically: direct → y = kx; inverse → y = k/x. Direct keeps the RATIO y/x constant. Inverse keeps the PRODUCT xy constant. Many real-world relationships are one or the other — but some are NEITHER (e.g., quadratic relationships).

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