📐 Mathematics · Class 8
Proportional Reasoning – 2
Chapter 3 · Ganita Prakash (NCERT) · Complete Study Notes
Ratio & Proportion
Map Scale
Pie Charts
Inverse Proportion
Direct Proportion
Map Scale
Pie Charts
Inverse Proportion
Direct Proportion
📋 Contents
3.1 Proportionality — A Quick Recap
When two or more quantities change by the same factor, they share a proportional relationship. We use ratio notation to show this.
Real-Life Example — Idli Batter!
A classic recipe uses 2 cups rice : 1 cup urad dal (ratio 2 : 1).
Viswanath uses 6 : 3 and Puneet uses 4 : 2.
Are these proportional? Yes! Both simplify to 2 : 1, so the idlis taste the same.
A classic recipe uses 2 cups rice : 1 cup urad dal (ratio 2 : 1).
Viswanath uses 6 : 3 and Puneet uses 4 : 2.
Are these proportional? Yes! Both simplify to 2 : 1, so the idlis taste the same.
✅ Cross-Multiplication Method
To check if two ratios a : b and c : d are proportional:
a × d = b × c or equivalently a/c = b/d
Example: Is 6 : 3 proportional to 4 : 2?
6 × 2 = 12
3 × 4 = 12
✓ Both products are equal → Ratios ARE proportional!
3 × 4 = 12
✓ Both products are equal → Ratios ARE proportional!
Key Rule
Two ratios are proportional if and only if their cross-products are equal.
Two ratios are proportional if and only if their cross-products are equal.
3.2 Ratios in Maps (Representative Fraction)
Maps use a special ratio called a Representative Fraction (RF) to relate distances on the map to real distances on the ground.
RF = Distance on Map : Actual Distance on Ground
Important: This is geographical distance, NOT road distance!
Road distance can be longer due to turns and curves.
Road distance can be longer due to turns and curves.
🔢 How to Use Map Scale
- Note the RF on the map (e.g., RF = 1 : 60,00,000)
- Measure the distance between two cities on the map using a ruler (in cm)
- Multiply the measured distance by the second part of the RF
- Convert to appropriate units (cm → km)
Unit Conversion Tip
RF 1 : 60,00,000 means 1 cm on map = 60,00,000 cm on ground = 60 km on ground.
(Divide by 1,00,000 to convert cm → km)
RF 1 : 60,00,000 means 1 cm on map = 60,00,000 cm on ground = 60 km on ground.
(Divide by 1,00,000 to convert cm → km)
📝 Worked Example
Map RF = 1 : 60,00,000
Measured distance (Bengaluru–Chennai) on map = ~1.5 cm
Actual distance = 1.5 × 60,00,000 cm = 90,00,000 cm
= 90 km (geographical distance)
Measured distance (Bengaluru–Chennai) on map = ~1.5 cm
Actual distance = 1.5 × 60,00,000 cm = 90,00,000 cm
= 90 km (geographical distance)
🌍
Fun Fact about Maps
Different maps can have different scales for the same region. But if you calculate correctly, you should get approximately the same geographical distance — because only the scale ratio differs, not the actual land!
Different maps can have different scales for the same region. But if you calculate correctly, you should get approximately the same geographical distance — because only the scale ratio differs, not the actual land!
3.3 Ratios with More than 2 Terms
Ratios don’t have to be just two terms! We can have 3, 4, or more terms when multiple quantities maintain proportional relationships.
Spice Mix Powder Example
Viswanath’s recipe: Coriander seeds : Red chillies : Toor dal : Fenugreek = 8 : 4 : 2 : 1.
If Puneet has only 2 red chillies (half of 4), he must halve everything else too → 4 : 2 : 1 : 0.5
Viswanath’s recipe: Coriander seeds : Red chillies : Toor dal : Fenugreek = 8 : 4 : 2 : 1.
If Puneet has only 2 red chillies (half of 4), he must halve everything else too → 4 : 2 : 1 : 0.5
📐 General Rule for Multi-Term Proportions
If a : b : c : d :: p : q : r : s then a/p = b/q = c/r = d/s
🎨 Worked Example 1 — Purple Paint
To make purple paint: Red : Blue : White = 2 : 3 : 5. Yasmin has 10 litres of white. Find red and blue needed.
White corresponds to 5 parts → 10 litres
So 1 part = 10 ÷ 5 = 2 litres
Red = 2 × 2 = 4 litres
Blue = 3 × 2 = 6 litres
Total purple paint = 4 + 6 + 10 = 20 litres
So 1 part = 10 ÷ 5 = 2 litres
Red = 2 × 2 = 4 litres
Blue = 3 × 2 = 6 litres
Total purple paint = 4 + 6 + 10 = 20 litres
🏗️ Worked Example 2 — Cement Concrete
Concrete mix ratio: Cement : Sand : Gravel = 1 : 1.5 : 3. With 3 bags of cement:
Multiply all terms by 3:
Cement : Sand : Gravel = 3 : 4.5 : 9
Total = 3 + 4.5 + 9 = 16.5 bags of concrete
Cement : Sand : Gravel = 3 : 4.5 : 9
Total = 3 + 4.5 + 9 = 16.5 bags of concrete
Remember
The ratio 8:4:2:1 and 4:2:1:0.5 are proportional. We write this as: 8:4:2:1 :: 4:2:1:0.5
The ratio 8:4:2:1 and 4:2:1:0.5 are proportional. We write this as: 8:4:2:1 :: 4:2:1:0.5
3.4 Dividing a Whole in a Given Ratio
When a total quantity must be divided in a given ratio, use this formula:
Each part = Total × (that term) ÷ (sum of all terms)
📌 General Formula
If quantity x is divided in ratio a : b : c : …:
1st Partx × a ÷ (a+b+c+…)
2nd Partx × b ÷ (a+b+c+…)
3rd Partx × c ÷ (a+b+c+…)
…and so onSame pattern for each remaining term
🏗️ Worked Example 3 — Concrete (110 units)
Ratio is 1 : 1.5 : 3. Need 110 units of concrete total.
Sum of ratio = 1 + 1.5 + 3 = 5.5
Multiplier = 110 ÷ 5.5 = 20
Cement = 1 × 20 = 20 units
Sand = 1.5 × 20 = 30 units
Gravel = 3 × 20 = 60 units
Check: 20 + 30 + 60 = 110 ✓
Multiplier = 110 ÷ 5.5 = 20
Cement = 1 × 20 = 20 units
Sand = 1.5 × 20 = 30 units
Gravel = 3 × 20 = 60 units
Check: 20 + 30 + 60 = 110 ✓
🎨 Worked Example 4 — Purple Paint (50 ml)
Red : Blue : White = 2 : 3 : 5. Need 50 ml total.
Sum = 2 + 3 + 5 = 10
Red paint = 50 × 2/10 = 10 ml
Blue paint = 50 × 3/10 = 15 ml
White paint = 50 × 5/10 = 25 ml
Red paint = 50 × 2/10 = 10 ml
Blue paint = 50 × 3/10 = 15 ml
White paint = 50 × 5/10 = 25 ml
📐 Worked Example 5 — Triangle Angles (Ratio 1 : 3 : 5)
Sum of angles in a triangle = 180°. Divide 180° in ratio 1 : 3 : 5.
Sum = 1 + 3 + 5 = 9
∠A = 180° × 1/9 = 20°
∠B = 180° × 3/9 = 60°
∠C = 180° × 5/9 = 100°
Check: 20 + 60 + 100 = 180° ✓
∠A = 180° × 1/9 = 20°
∠B = 180° × 3/9 = 60°
∠C = 180° × 5/9 = 100°
Check: 20 + 60 + 100 = 180° ✓
Quick Steps to Divide x in Ratio a:b:c
(1) Add all ratio terms to get the sum S.
(2) Each part = x × (its term) ÷ S.
(3) Always verify: all parts must add up to x.
(1) Add all ratio terms to get the sum S.
(2) Each part = x × (its term) ÷ S.
(3) Always verify: all parts must add up to x.
3.5 A Slice of the Pie (Pie Charts)
A pie chart is a circle divided into slices, where each slice’s angle is proportional to the data it represents. A full circle = 360°.
Angle of each slice = (Frequency ÷ Total) × 360°
📊 Worked Example — Grade Distribution
| Grade | Students | Ratio | Angle |
|---|---|---|---|
| A | 12 | 6 | 6 × 18° = 108° |
| B | 10 | 5 | 5 × 18° = 90° |
| C | 8 | 4 | 4 × 18° = 72° |
| D | 6 | 3 | 3 × 18° = 54° |
| E | 4 | 2 | 2 × 18° = 36° |
| Total | 40 | 20 | 360° |
How to Simplify the Ratio First
HCF of 12, 10, 8, 6, 4 is 2. Divide all by 2 → 6 : 5 : 4 : 3 : 2.
Sum = 20. Multiplier = 360 ÷ 20 = 18°.
HCF of 12, 10, 8, 6, 4 is 2. Divide all by 2 → 6 : 5 : 4 : 3 : 2.
Sum = 20. Multiplier = 360 ÷ 20 = 18°.
🖊️ Steps to Draw a Pie Chart
- Calculate the angle for each category using the formula above.
- Draw a circle and mark a radius (starting line).
- Using a protractor, measure and draw the angle for the first slice.
- From the new radius, measure the next angle for the second slice.
- Continue until all slices are drawn and the circle is complete (total = 360°).
- Colour each slice and label it with the category name.
Common Mistake
Always measure angles from the most recently drawn radius, NOT from the original starting line each time. Otherwise all your slices will overlap!
Always measure angles from the most recently drawn radius, NOT from the original starting line each time. Otherwise all your slices will overlap!
📺
Reading a Pie Chart
If a pie chart shows TV channel preferences — Entertainment 50%, Sports 25%, News 15%, Info 10% — and the circle is 360°, then Entertainment slice = 50% of 360° = 180°. The biggest slice is always the most popular category.
If a pie chart shows TV channel preferences — Entertainment 50%, Sports 25%, News 15%, Info 10% — and the circle is 360°, then Entertainment slice = 50% of 360° = 180°. The biggest slice is always the most popular category.
3.6 Inverse Proportions
🔵 Direct ProportionWhen one quantity increases, the other also increases by the same factor. Their ratio stays constant.
🔴 Inverse ProportionWhen one quantity increases, the other decreases by the same factor. Their product stays constant.
📌 Key Definition
x and y are in inverse proportion if: x × y = k (constant)
Equivalently: x₁/x₂ = y₂/y₁ and x₁y₁ = x₂y₂
Speed & Time — Classic Inverse Proportion
Lucknow to Kanpur journey: the faster you go, the less time it takes.
Speed × Time = Distance = constant (90 km).
Walk: 5 × 18 = 90. Bicycle: 15 × 6 = 90. Car: 60 × 1.5 = 90. ✓
Lucknow to Kanpur journey: the faster you go, the less time it takes.
Speed × Time = Distance = constant (90 km).
Walk: 5 × 18 = 90. Bicycle: 15 × 6 = 90. Car: 60 × 1.5 = 90. ✓
🧱 Example 3 — Workers & Days
20 workers complete a road in 4 days. How many days for 10 workers?
Workers × Days = constant (work done)
20 × 4 = 10 × y₂
y₂ = (20 × 4) ÷ 10 = 8 days
Workers halved (20→10), so days doubled (4→8) ✓
20 × 4 = 10 × y₂
y₂ = (20 × 4) ÷ 10 = 8 days
Workers halved (20→10), so days doubled (4→8) ✓
🚰 Example 4 — Pumps & Time
2 pumps fill a tank in 18 hours. Adding 2 more (total 4 pumps) — how long?
2 × 18 = 4 × x
x = 36 ÷ 4 = 9 hours
x = 36 ÷ 4 = 9 hours
🍱 Example 5 — Food Provisions
Food for 80 students lasts 15 days. If 20 more students join (total 100):
80 × 15 = 100 × x
x = 1200 ÷ 100 = 12 days
(More students → fewer days → inverse proportion)
x = 1200 ÷ 100 = 12 days
(More students → fewer days → inverse proportion)
🥕 Example 6 — Work Together (Combined Work)
Ram cuts vegetables in 1 hour; Shyam takes 1.5 hours. Time if working together?
Ram’s rate = 1 unit/hour
Shyam’s rate = 1/1.5 = 2/3 unit/hour
Combined rate = 1 + 2/3 = 5/3 units/hourTime to finish 1 unit of work together:
(5/3) × x = 1 × 1 → x = 3/5
Together they finish in 3/5 hours = 36 minutes
Shyam’s rate = 1/1.5 = 2/3 unit/hour
Combined rate = 1 + 2/3 = 5/3 units/hourTime to finish 1 unit of work together:
(5/3) × x = 1 × 1 → x = 3/5
Together they finish in 3/5 hours = 36 minutes
How to Identify Inverse Proportion
Ask yourself: “If I increase one quantity, does the other decrease?” If YES, and by the same factor — it’s inverse proportion!
Examples: More workers → fewer days. Higher speed → less time. More taps → less time to fill.
Ask yourself: “If I increase one quantity, does the other decrease?” If YES, and by the same factor — it’s inverse proportion!
Examples: More workers → fewer days. Higher speed → less time. More taps → less time to fill.
Common Mistake — Don’t Mix Up!
Distance vs. Petrol: more petrol → more distance (DIRECT).
Speed vs. Time: more speed → less time (INVERSE).
Always think about the relationship before applying the formula!
Distance vs. Petrol: more petrol → more distance (DIRECT).
Speed vs. Time: more speed → less time (INVERSE).
Always think about the relationship before applying the formula!
📊 Comparison Table
| Situation | Type | Why? |
|---|---|---|
| More taps → less time to fill tank | Inverse | Product constant |
| More painters → fewer days to paint | Inverse | Product constant |
| More petrol → more distance | Direct | Ratio constant |
| Higher speed → less time for fixed route | Inverse | Product = distance |
| More cloth → higher price | Direct | Ratio constant |
| More pages → more reading time | Direct | Ratio constant |
Chapter Summary — All Key Points
Proportional Ratios
a:b :: c:d when a×d = b×c. Fractions a/c = b/d.
a:b :: c:d when a×d = b×c. Fractions a/c = b/d.
Map Scale (RF)
1 cm on map = RF × 1 cm on ground. Convert to km by dividing by 1,00,000.
1 cm on map = RF × 1 cm on ground. Convert to km by dividing by 1,00,000.
Multi-Term Ratios
a:b:c:d :: p:q:r:s → a/p = b/q = c/r = d/s
a:b:c:d :: p:q:r:s → a/p = b/q = c/r = d/s
Dividing a Whole
Part = Total × (term) ÷ (sum of all terms)
Part = Total × (term) ÷ (sum of all terms)
Pie Charts
Angle = (value ÷ total) × 360°. All angles must sum to 360°.
Angle = (value ÷ total) × 360°. All angles must sum to 360°.
Inverse Proportion
x₁y₁ = x₂y₂ = k. One increases, other decreases by same factor.
x₁y₁ = x₂y₂ = k. One increases, other decreases by same factor.
Master Formula Sheet
• Cross-multiply check: a×d = b×c
• Dividing x in a:b:c → parts are x·a/S, x·b/S, x·c/S where S = a+b+c
• Pie slice angle = (frequency ÷ total) × 360°
• Direct proportion: x/y = k (constant ratio)
• Inverse proportion: x×y = k (constant product)
• For inverse: x₁/x₂ = y₂/y₁
• Cross-multiply check: a×d = b×c
• Dividing x in a:b:c → parts are x·a/S, x·b/S, x·c/S where S = a+b+c
• Pie slice angle = (frequency ÷ total) × 360°
• Direct proportion: x/y = k (constant ratio)
• Inverse proportion: x×y = k (constant product)
• For inverse: x₁/x₂ = y₂/y₁
Exam Practice Questions
Q1. A cricket practice session of 150 minutes is divided in the ratio Warm-up : Batting : Bowling : Fielding = 3 : 4 : 3 : 5. Find time for each activity.
Sum = 3+4+3+5 = 15. Each part = 150÷15 = 10 min.
Warm-up = 30 min, Batting = 40 min, Bowling = 30 min, Fielding = 50 min.
Warm-up = 30 min, Batting = 40 min, Bowling = 30 min, Fielding = 50 min.
Q2. A library has Odiya : Hindi : English books in ratio 3:2:1. If there are 288 Odiya books, find Hindi and English books.
1 part = 288÷3 = 96. Hindi = 2×96 = 192 books. English = 1×96 = 96 books.
Q3. If RF on a map is 1 : 50,00,000 and the map distance between two cities is 3 cm, find the actual geographical distance in km.
Actual = 3 × 50,00,000 cm = 1,50,00,000 cm = 150 km.
Q4. 20 workers take 4 days to lay a road. How many days will 16 workers take?
Inverse proportion: 20 × 4 = 16 × x → x = 80÷16 = 5 days.
Q5. Draw a pie chart for seasons: Summer = 90, Rainy = 120, Winter = 150 (total 360 people).
Summer angle = (90/360)×360° = 90°. Rainy = (120/360)×360° = 120°. Winter = (150/360)×360° = 150°. Total = 360° ✓
Q6. A school has enough food for 80 students for 15 days. If 20 more students join, how many days will the food last?
Inverse proportion: 80×15 = 100×x → x = 1200÷100 = 12 days.
Q7. A small pump fills a tank in 3 hours, large pump fills in 2 hours. How long if both are used together?
Small pump rate = 1/3, Large pump rate = 1/2. Combined = 1/3+1/2 = 5/6 per hour. Time = 6/5 hours = 1 hour 12 minutes.
Q8. 100 coins are in ratio ₹10:₹5:₹2:₹1 = 4:3:2:1. Find the total money.
Sum = 10 parts. Each part = 10 coins. ₹10×40=₹400, ₹5×30=₹150, ₹2×20=₹40, ₹1×10=₹10. Total = ₹600.
Exam Strategy
Always write the type of proportion (direct/inverse) before solving. Show all steps clearly. Verify your answer by checking that it satisfies the original condition.
Always write the type of proportion (direct/inverse) before solving. Show all steps clearly. Verify your answer by checking that it satisfies the original condition.
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