📘 Class 8 · Chapter 7 · Ganita Prakash
Proportional Reasoning — 1
Understand Ratios, Proportion, Cross Multiplication, Dividing in a Ratio & Unit Conversions — with real-life examples!
⚖️ Ratios
🔗 Proportion
✖️ Cross Multiplication
🍋 Rule of Three
➗ Dividing in a Ratio
📐 Unit Conversions
🔗 Proportion
✖️ Cross Multiplication
🍋 Rule of Three
➗ Dividing in a Ratio
📐 Unit Conversions
📋 Table of Contents
- Chapter Introduction — Similarity in Change
- 7.2 What is a Ratio?
- 7.3 Ratios in Simplest Form
- Proportion & the ‘::’ Symbol
- 7.4 Problem Solving with Proportional Reasoning
- Trairasika — The Rule of Three (Cross Multiplication)
- 7.5 Sharing in a Given Ratio
- 7.6 Unit Conversions
- Chapter Summary & Quick Revision
- Exam Practice Questions with Solutions
7.1 Observing Similarity in Change
Consider 5 images of a tiger — A, B, C, D, E — all of different sizes. Some look like the original (natural) and some look distorted (stretched or squished).
| Image | Width (mm) | Height (mm) | Width : Height (Simplest) | Looks Like Original? |
|---|---|---|---|---|
| A | 60 | 40 | 3 : 2 | ✅ Yes (Original) |
| B | 40 | 20 | 2 : 1 | ❌ No (Elongated) |
| C | 30 | 20 | 3 : 2 | ✅ Yes |
| D | 90 | 60 | 3 : 2 | ✅ Yes |
| E | 60 | 60 | 1 : 1 | ❌ No (Square/Fat) |
Key Observation
Images A, C, and D look similar because both the width and height changed by the same factor (multiplication). Images B and E look distorted because their width and height changed by different factors.
Images A, C, and D look similar because both the width and height changed by the same factor (multiplication). Images B and E look distorted because their width and height changed by different factors.
Adding/subtracting the same number does NOT preserve similarity. Only multiplying/dividing by the same factor does!
Why is this important?
This is exactly what proportional reasoning is all about — when two quantities change by the same multiplicative factor, they are said to be proportional to each other.
This is exactly what proportional reasoning is all about — when two quantities change by the same multiplicative factor, they are said to be proportional to each other.
7.2 What is a Ratio?
We use the idea of a ratio to express proportional relationships in mathematics.
A ratio a : b means “for every ‘a’ units of the first quantity, there are ‘b’ units of the second quantity.”
- The numbers in a ratio are called its terms.
- In ratio 60 : 40, 60 and 40 are the terms.
- A ratio compares two quantities of the same kind and same units.
- Order matters: 3 : 2 is NOT the same as 2 : 3.
Common Mistake
Never compare ratios using addition or subtraction. Ratios are always compared using multiplication or division. Even if both terms of two ratios differ by the same additive amount, the ratios may not be equal!
Never compare ratios using addition or subtraction. Ratios are always compared using multiplication or division. Even if both terms of two ratios differ by the same additive amount, the ratios may not be equal!
Example: 60:40 and 40:20 differ by 20 each, but 60:40 = 3:2 ≠ 2:1 = 40:20
Also True for Negative Changes!
When we add (or subtract) the same number from the terms of a ratio, the new ratio is generally NOT proportional to the original. Example: Neelima’s age ratio changes from 1:10 to 4:13 after 9 years — adding 9 to both terms does not preserve proportion.
When we add (or subtract) the same number from the terms of a ratio, the new ratio is generally NOT proportional to the original. Example: Neelima’s age ratio changes from 1:10 to 4:13 after 9 years — adding 9 to both terms does not preserve proportion.
7.3 Ratios in Simplest Form
🔢 How to Simplify a Ratio
Divide both terms of the ratio by their HCF (Highest Common Factor).
Simplified Ratio = (a ÷ HCF) : (b ÷ HCF)
Example 1: Simplify 60 : 40
HCF of 60 and 40 = 20
60 ÷ 20 = 3, 40 ÷ 20 = 2
Simplest form = 3 : 2Example 2: Simplify 72 : 96
HCF of 72 and 96 = 24
72 ÷ 24 = 3, 96 ÷ 24 = 4
Simplest form = 3 : 4
HCF of 60 and 40 = 20
60 ÷ 20 = 3, 40 ÷ 20 = 2
Simplest form = 3 : 2Example 2: Simplify 72 : 96
HCF of 72 and 96 = 24
72 ÷ 24 = 3, 96 ÷ 24 = 4
Simplest form = 3 : 4
Example 3: Simplify 90 : 60
HCF of 90 and 60 = 30
90 ÷ 30 = 3, 60 ÷ 30 = 2
Simplest form = 3 : 2 (same as Image A → proportional!)
🔗 When Are Two Ratios Proportional?
Definition of Proportion
Two ratios a : b and c : d are said to be in proportion (or proportional) if they are equal in their simplest forms.
Two ratios a : b and c : d are said to be in proportion (or proportional) if they are equal in their simplest forms.
We write this as: a : b :: c : d (read as “a is to b as c is to d”)
Example: 60 : 40 :: 30 : 20 :: 90 : 60 (all simplify to 3 : 2)
| Ratio | HCF | Simplest Form | Proportional to A (3:2)? |
|---|---|---|---|
| 60 : 40 (Image A) | 20 | 3 : 2 | ✅ Yes (Original) |
| 30 : 20 (Image C) | 10 | 3 : 2 | ✅ Yes |
| 90 : 60 (Image D) | 30 | 3 : 2 | ✅ Yes |
| 40 : 20 (Image B) | 20 | 2 : 1 | ❌ No |
| 60 : 60 (Image E) | 60 | 1 : 1 | ❌ No |
Proportion — Understanding the ‘::’ Symbol
a : b :: c : d means “a is to b as c is to d”
→ Both ratios are equal in their simplest form
→ Both ratios are equal in their simplest form
📐 Checking Proportionality — Two Methods
Method 1: Simplest Form
Reduce both ratios to their simplest forms using HCF. If both simplest forms are equal, the ratios are proportional.
Reduce both ratios to their simplest forms using HCF. If both simplest forms are equal, the ratios are proportional.
Example: 4:7 and 12:21
12:21 ÷ 3 = 4:7 ✅
Method 2: Same Factor
Check if each term in the second ratio is obtained by multiplying each term of the first ratio by the same factor.
Check if each term in the second ratio is obtained by multiplying each term of the first ratio by the same factor.
Example: 60:40 → ×(3/2) → 90:60 ✅
🔍 Finding Missing Terms in Proportional Ratios
Example: Find the missing terms in ratios proportional to 14 : 21.
14 : 21 → simplest form = 2 : 3Case 1: __ : 42
21 × 2 = 42, so multiply first term by 2 also.
14 × 2 = 28 → 28 : 42
21 × 2 = 42, so multiply first term by 2 also.
14 × 2 = 28 → 28 : 42
Case 2: 6 : __
14y = 6 → y = 6/14 = 3/7
21 × (3/7) = 9 → 6 : 9
Case 3: 2 : __
14 ÷ 7 = 2 (HCF of 14,21 is 7)
21 ÷ 7 = 3 → 2 : 3 (simplest form itself!)
7.4 Problem Solving with Proportional Reasoning
🍋 Example: Lemonade (Same Sweetness)
Kesang’s Lemonade Problem
For 6 glasses of lemonade, she uses 10 spoons of sugar. How much sugar for 18 glasses?
For 6 glasses of lemonade, she uses 10 spoons of sugar. How much sugar for 18 glasses?
Set up: 6 : 10 :: 18 : ?
Factor of change: 18 ÷ 6 = 3
Sugar = 10 × 3 = 30 spoons
🧱 Example: Wall Strength (Cement to Length)
Nitin & Hari’s Wall Problem
Nitin: 60 ft wall, 3 bags cement → 60:3 = 20:1
Hari: 40 ft wall, 2 bags cement → 40:2 = 20:1
Both ratios are the same → Walls are equally strong. Nitin was wrong to worry!
Nitin: 60 ft wall, 3 bags cement → 60:3 = 20:1
Hari: 40 ft wall, 2 bags cement → 40:2 = 20:1
Both ratios are the same → Walls are equally strong. Nitin was wrong to worry!
☕ Filter Coffee — Real Life Ratio
Regular: 15 mL decoction : 35 mL milk → ratio 15:35 = 3:7
| Coffee (mL) | Milk (mL) | Ratio | Simplest | Type |
|---|---|---|---|---|
| 15 | 35 | 15:35 | 3:7 | 🟡 Regular |
| 20 | 30 | 20:30 | 2:3 | 💪 Stronger (more coffee per unit milk) |
| 10 | 40 | 10:40 | 1:4 | ☁️ Lighter (less coffee per unit milk) |
| 300 | 600 | 300:600 | 1:2 | 💪 Stronger |
| 150 | 500 | 150:500 | 3:10 | ☁️ Lighter |
| 24 | 56 | 24:56 | 3:7 | 🟡 Regular |
Why is 20:30 stronger than 15:35?
In 20:30, for every 2 parts coffee there are 3 parts milk. In regular (15:35 = 3:7), for every 3 parts coffee there are 7 parts milk. More coffee per unit of milk = STRONGER taste!
In 20:30, for every 2 parts coffee there are 3 parts milk. In regular (15:35 = 3:7), for every 3 parts coffee there are 7 parts milk. More coffee per unit of milk = STRONGER taste!
Trairasika — The Rule of Three (Cross Multiplication)
🎯 What is the Rule of Three?
This is a method to find the fourth (unknown) quantity when three quantities are known and all four are proportionally linked. This was described by Āryabhaṭa (199 CE) in ancient India.
Āryabhaṭa’s Rule of Three
Three given quantities:
• Pramāṇa (Measure) = a
• Phala (Fruit) = b
• Ichchhā (Requisition) = c
Three given quantities:
• Pramāṇa (Measure) = a
• Phala (Fruit) = b
• Ichchhā (Requisition) = c
To find the Ichchhāphala (Yield) = d:
d = (phala × ichchhā) ÷ pramāṇa = (b × c) ÷ a
✖️ Cross Multiplication — The Algebraic Proof
When a : b :: c : d, we can prove mathematically that:
If a : b :: c : d, then ad = bc
(Cross Multiply — multiply the outer and inner terms)
And: d = (b × c) ÷ a
(Cross Multiply — multiply the outer and inner terms)
And: d = (b × c) ÷ a
Proof: Since both ratios are proportional with the same factor f:
c = f × a → f = c/a and d = f × b → f = d/b
Therefore c/a = d/b → bc = ad ✓
🚗 Example: Distance Travelled (Car)
Problem: Car travels 90 km in 150 minutes.
How far in 4 hours (= 240 minutes)?Set up: 150 : 90 :: 240 : x
How far in 4 hours (= 240 minutes)?Set up: 150 : 90 :: 240 : x
Cross multiply: 150 × x = 240 × 90
x = (240 × 90) / 150
x = 21600 / 150
x = 144 km
⚠️ NOTE: Always use SAME units! Convert 4 hours → 240 minutes first.
🍵 Example: Tea Price Comparison
Himachal: 200g = ₹200 → ratio = 200:200 = 1:1 (1g = ₹1)
Meghalaya: 1kg = ₹800 → 1000:800 = 5:4 (1g = ₹0.80)To compare, find cost per kg:
Himachal: ₹200 per 200g → ₹1000 per kg
Meghalaya: ₹800 per kg
Meghalaya: 1kg = ₹800 → 1000:800 = 5:4 (1g = ₹0.80)To compare, find cost per kg:
Himachal: ₹200 per 200g → ₹1000 per kg
Meghalaya: ₹800 per kg
Since ₹1000 > ₹800, Himachal Pradesh tea is MORE EXPENSIVE!
When Can’t We Use Rule of Three? (Inverse Proportion)
Puneeth’s father takes 2 hours at 50 km/h. How long at 75 km/h?
This CANNOT be modelled as 50:2 :: 75:__ because when speed increases, time decreases! This is an inverse proportion situation — NOT covered by the standard Rule of Three.
Puneeth’s father takes 2 hours at 50 km/h. How long at 75 km/h?
This CANNOT be modelled as 50:2 :: 75:__ because when speed increases, time decreases! This is an inverse proportion situation — NOT covered by the standard Rule of Three.
7.6 Unit Conversions
Always make sure both ratios use the same units before comparing or solving proportion problems!
📏 Length
1 metre = 3.281 feet
1 km = 1000 m
1 metre = 3.281 feet
1 km = 1000 m
📦 Area
1 sq metre = 10.764 sq feet
1 acre = 43,560 sq feet
1 hectare = 10,000 sq m
1 hectare = 2.471 acres
1 sq metre = 10.764 sq feet
1 acre = 43,560 sq feet
1 hectare = 10,000 sq m
1 hectare = 2.471 acres
🥤 Volume
1 mL = 1 cubic cm (cc)
1 litre = 1,000 mL
1 litre = 1,000 cc
1 mL = 1 cubic cm (cc)
1 litre = 1,000 mL
1 litre = 1,000 cc
🌡️ Temperature
°F = (9/5 × °C) + 32
°C = (5/9) × (°F − 32)
0°C = 32°F, 25°C = 77°F
°F = (9/5 × °C) + 32
°C = (5/9) × (°F − 32)
0°C = 32°F, 25°C = 77°F
Always Use Same Units!
In Example 9 (Car), 150 minutes and “4 hours” are different units. You must convert 4 hours = 240 minutes BEFORE writing the proportion. Never mix units in a ratio!
In Example 9 (Car), 150 minutes and “4 hours” are different units. You must convert 4 hours = 240 minutes BEFORE writing the proportion. Never mix units in a ratio!
🌾 Example: Manure for Farming
Apply 10 tonnes manure per 1 acre.
Field: 200 ft × 500 ft = 100,000 sq ftConvert: 1 acre = 43,560 sq ft
Field area = 100,000 / 43,560 ≈ 2.296 acres
Field: 200 ft × 500 ft = 100,000 sq ftConvert: 1 acre = 43,560 sq ft
Field area = 100,000 / 43,560 ≈ 2.296 acres
Manure needed: 1 acre : 10 tonnes :: 2.296 acres : ?
x = 10 × 2.296 = ≈ 22.96 tonnes of manure
Chapter Summary — Quick Revision
Ratio
a : b = “for every a units of quantity 1, there are b units of quantity 2”
Terms are ‘a’ and ‘b’
a : b = “for every a units of quantity 1, there are b units of quantity 2”
Terms are ‘a’ and ‘b’
Simplest Form
Divide both terms by their HCF.
a : b in simplest = (a÷HCF) : (b÷HCF)
Divide both terms by their HCF.
a : b in simplest = (a÷HCF) : (b÷HCF)
Proportion
a : b :: c : d if they have the same simplest form. Written using ‘::’ symbol.
a : b :: c : d if they have the same simplest form. Written using ‘::’ symbol.
Cross Multiplication
If a:b :: c:d, then ad = bc.
Unknown d = bc ÷ a (Rule of Three)
If a:b :: c:d, then ad = bc.
Unknown d = bc ÷ a (Rule of Three)
Dividing in a Ratio
Divide x in ratio m:n:
Part 1 = m × x/(m+n)
Part 2 = n × x/(m+n)
Divide x in ratio m:n:
Part 1 = m × x/(m+n)
Part 2 = n × x/(m+n)
Key Warning
Adding/subtracting same number from ratio terms does NOT preserve proportion. Only multiplication works!
Adding/subtracting same number from ratio terms does NOT preserve proportion. Only multiplication works!
🗂️ All Key Formulas at a Glance
| Concept | Formula / Rule | Example |
|---|---|---|
| Simplest Ratio | a:b → (a÷HCF):(b÷HCF) | 60:40 = 3:2 |
| Check Proportion | a:b :: c:d if ad = bc | 3:4 :: 6:8 since 3×8=4×6=24 ✓ |
| Find 4th term | d = bc/a | 6:10::18:? → d=18×10/6=30 |
| Divide x in m:n | Part1 = mx/(m+n), Part2 = nx/(m+n) | Divide 42 in 4:3 → 24 and 18 |
| Proportional change | Multiply both terms by same factor | 3:2 × 5 = 15:10 |
🏺
Ancient Indian Mathematics — Trairasika!
The Rule of Three (Trairasika) was used by Āryabhaṭa (199 CE), Brahmagupta (628 CE), and Bhaskaracharya in his famous work Lilavati (1150 CE). It was one of the most important practical tools in ancient commerce, astronomy, and everyday life!
The Rule of Three (Trairasika) was used by Āryabhaṭa (199 CE), Brahmagupta (628 CE), and Bhaskaracharya in his famous work Lilavati (1150 CE). It was one of the most important practical tools in ancient commerce, astronomy, and everyday life!
Exam Practice Questions with Solutions
Q1. Are the ratios 3 : 4 and 72 : 96 proportional?
Ans: Simplify 72:96. HCF of 72 and 96 = 24. 72÷24 = 3, 96÷24 = 4. So 72:96 = 3:4. Yes, both ratios are proportional. ✅
Q2. Divide ₹4,500 into two parts in the ratio 2 : 3.
Ans: Total parts = 2+3 = 5. Each group = 4500÷5 = ₹900. First part = 2×900 = ₹1,800. Second part = 3×900 = ₹2,700. Check: 1800+2700 = 4500 ✓
Q3. The Earth travels 940 million km around the Sun in a year (365 days). How far in a week (7 days)?
Ans: Set up: 365 : 940 million :: 7 : x. Cross multiply: x = (940 million × 7) / 365 = 6580/365 ≈ 18.03 million km per week.
Q4. In a science lab, acid and water are mixed in ratio 1 : 5 to make 240 mL solution. How much acid and water?
Ans: Total parts = 1+5 = 6. Each group = 240÷6 = 40 mL. Acid = 1×40 = 40 mL. Water = 5×40 = 200 mL. Check: 40+200=240 ✓
Q5. Blue and yellow paints in ratio 3:5 make green. To make 40 mL of green, how much of each?
Ans: Total parts = 3+5 = 8. Each group = 40÷8 = 5 mL. Blue = 3×5 = 15 mL. Yellow = 5×5 = 25 mL.
Q6. A car travels 90 km in 150 minutes. What distance in 4 hours at same speed?
Ans: 4 hours = 240 minutes. Set up: 150:90 :: 240:x. Cross multiply: x = (240×90)/150 = 21600/150 = 144 km.
Q7. Which is proportional? Circle the true ones: (i) 4:7 :: 12:21 (ii) 8:3 :: 24:6 (iii) 21:6 :: 35:10 (iv) 24:8 :: 9:3
Ans: (i) 4:7 = 4:7, 12:21 = 4:7 ✅ TRUE. (ii) 8:3, 24:6=4:1 ❌ FALSE. (iii) 21:6=7:2, 35:10=7:2 ✅ TRUE. (iv) 24:8=3:1, 9:3=3:1 ✅ TRUE.
Q8. Prashanti invested ₹75,000 and Bhuvan invested ₹25,000. Profit = ₹4,000. Find each person’s share.
Ans: Ratio = 75000:25000 = 3:1. Total parts = 4. Group size = 4000÷4 = ₹1000. Prashanti = 3×1000 = ₹3,000. Bhuvan = ₹1,000.
Q9. A tap takes 15 seconds to fill a 500 mL mug. How long to fill a 10-litre (10,000 mL) bucket?
Ans: Set up: 500 mL : 15 sec :: 10000 mL : x. x = (10000×15)/500 = 150000/500 = 300 seconds = 5 minutes.
Q10. A ₹10 coin has copper:nickel = 3:1 and weighs 7.74 grams. Copper costs ₹906/kg, nickel ₹1341/kg. Find the cost of metals in one coin.
Ans: Copper = (3/4)×7.74 = 5.805 g. Nickel = (1/4)×7.74 = 1.935 g. Cost of copper = 5.805×(906/1000) = ₹5.26. Cost of nickel = 1.935×(1341/1000) = ₹2.59. Total = ≈ ₹7.85.
Exam Tips for This Chapter
1. Always convert units before forming a proportion.
2. When asked to divide in a ratio, always add the ratio terms first to get total parts.
3. Use cross multiplication (ad = bc) to verify or find unknown terms.
4. Remember: Adding the same number to ratio terms does NOT preserve proportion!
5. To check if ratios are proportional, reduce to simplest form or cross multiply.
1. Always convert units before forming a proportion.
2. When asked to divide in a ratio, always add the ratio terms first to get total parts.
3. Use cross multiplication (ad = bc) to verify or find unknown terms.
4. Remember: Adding the same number to ratio terms does NOT preserve proportion!
5. To check if ratios are proportional, reduce to simplest form or cross multiply.
Advertisement
Leave a Reply