Class 8 Mathematics — Ratio and Proportional Reasoning

📝 Section 1 — Classroom Notes

🌎 Real-World Connection: You mix orange juice with water in the ratio 1 : 3 — one part juice to three parts water. A recipe that serves 4 people uses 2 cups of flour. To serve 10 people, how many cups do you need? That is direct proportion. A tap fills a tank in 6 hours. With two taps it fills in 3 hours — that is inverse proportion. Maps, recipes, speed, wages — ratios and proportions are in everything we measure!

⚖ Topic 1: What Is a Ratio?

A ratio compares two quantities.
We write it as a : b.
Read it as “a is to b.”
Example: 3 apples and 5 oranges.
Ratio of apples to oranges = 3 : 5.
Order matters in a ratio.
3 : 5 is NOT the same as 5 : 3.

⚖ Topic 2: Equivalent Ratios

Equivalent ratios are equal ratios.
They work like equivalent fractions.
Multiply or divide both parts equally.
Example: 2 : 3 = 4 : 6 = 6 : 9.
We multiplied both sides by 2, then 3.
To check: cross multiply.
If 2 × 6 = 3 × 4, they are equal.

⚖ Topic 3: What Is Proportion?

A proportion means two ratios are equal.
We write it as a : b = c : d.
Read it as “a is to b as c is to d.”
Example: 2 : 5 = 6 : 15.
Check: 2 × 15 = 30. And 5 × 6 = 30.
Both products are equal. So it is a proportion.
This check is called cross multiplication.

If a : b = c : d, then a × d = b × c

⚖ Topic 4: Unitary Method

The unitary method finds the value of one unit.
Step 1: Find cost of 1 item.
Step 2: Multiply for the required number.
Example: 4 pens cost ₹60.
Cost of 1 pen = 60 ÷ 4 = ₹15.
Cost of 7 pens = 15 × 7 = ₹105.
This method works in many problems.

⚖ Topic 5: Direct Proportion

Two things are in direct proportion.
When one increases, the other also increases.
When one decreases, the other also decreases.
Example: more chocolates = more money.
Buy 2 chocolates for ₹40.
Buy 4 chocolates for ₹80.
The ratio stays the same.

If x ÷ y = constant, it is direct proportion.

⚖ Topic 6: Inverse Proportion

Two things are in inverse proportion.
When one increases, the other decreases.
Example: more workers = less days.
4 workers finish a job in 12 days.
8 workers finish the same job in 6 days.
Workers doubled, days became half.
The product stays the same.

If x × y = constant, it is inverse proportion.

⚖ Topic 7: Time and Work Problems

More workers = less time to finish.
This is inverse proportion.
Use the rule: Workers × Days = constant.
Example: 6 painters paint a wall in 10 days.
Total work = 6 × 10 = 60 worker-days.
For 15 painters: 60 ÷ 15 = 4 days.
Always find total work first.

⚖ Topic 8: Speed, Distance, Time

Speed = Distance ÷ Time.
Distance = Speed × Time.
Time = Distance ÷ Speed.
If speed is constant, distance and time are in direct proportion.
More time = more distance covered.
If distance is constant, speed and time are in inverse proportion.
More speed = less time needed.

Speed = Distance ÷ Time

⚖ Topic 9: Scale Drawings and Maps

Maps use a scale to show real distances.
A scale is a ratio.
Example: Scale = 1 cm : 5 km.
1 cm on map = 5 km in real life.
3 cm on map = 3 × 5 = 15 km in real life.
Scale drawings use the same idea.
A building plan uses a scale ratio.

⚖ Topic 10: Word Problems with Ratios

Read the problem carefully.
Find what is being compared.
Write the ratio or proportion.
Decide: is it direct or inverse?
Direct: both go up or both go down.
Inverse: one goes up, other goes down.
Solve using unitary method or cross multiplication.

🎨 Section 2 — Easy Illustrations

Picture 1: Ratio Visual — Apples to Oranges

A ratio compares two groups. Here we compare apples to oranges as 3 to 5.

Picture 2: Direct Proportion Graph

In direct proportion, the graph is always a straight line passing through the origin.

Picture 3: Inverse Proportion Graph

In inverse proportion, the graph is a curve. As one value goes up, the other goes down.

Picture 4: Unitary Method Steps

The unitary method: find the value of one, then multiply for as many as you need.

Picture 5: Map Scale Visual

A map scale is a ratio. Multiply map distance by the scale to get real distance.

✏️ Section 3 — Worked Examples

Example 1: Cost of Notebooks (Unitary Method)

Question: If 5 notebooks cost ₹125, find the cost of 8 notebooks.

1 We know: 5 notebooks = ₹125.

2 Find cost of 1 notebook.

3 Cost of 1 notebook = 125 ÷ 5 = ₹25.

4 Now find cost of 8 notebooks.

5 Cost of 8 notebooks = 25 × 8 = ₹200.

Cost of 8 notebooks = ₹200

This is direct proportion: more notebooks = more cost.

Example 2: Time for a Journey (Direct Proportion)

Question: A car travels 240 km in 4 hours. How long to travel 390 km at the same speed?

1 We know: 240 km takes 4 hours.

2 Find speed: 240 ÷ 4 = 60 km per hour.

3 Time = Distance ÷ Speed.

4 Time = 390 ÷ 60 = 6.5 hours.

5 6.5 hours = 6 hours 30 minutes.

Time for 390 km = 6 hours 30 minutes

Speed is constant, so distance and time are in direct proportion.

Example 3: Workers and Days (Inverse Proportion)

Question: 12 workers finish a job in 15 days. How many days for 20 workers?

1 This is inverse proportion.

2 More workers = fewer days.

3 Total work = Workers × Days = 12 × 15 = 180.

4 For 20 workers: Days = 180 ÷ 20.

5 Days = 9 days.

20 workers will finish in 9 days

Check: 20 × 9 = 180 = 12 × 15 ✓

Term	What It Means	Key Fact
Ratio	Compares two quantities	a : b
Equivalent Ratios	Ratios that are equal	2:3 = 4:6
Proportion	Two equal ratios	a:b = c:d
Cross Multiply	Check if proportion is true	a×d = b×c
Unitary Method	Find value of one unit	Total ÷ number
Direct Proportion	Both increase together	x/y = constant
Inverse Proportion	One up, other down	x×y = constant
Speed	How fast you travel	Distance ÷ Time
Time & Work	Workers and days	Inverse proportion
Map Scale	Map distance : real distance	1 cm : 5 km

📄 Section 5 — Take-Home Handout

📝 A. Fill in the Blanks

1. A ratio compares _________ quantities.
2. In direct proportion, both values go _________.
3. In inverse proportion, one goes up and the other goes _________.
4. Speed = Distance ÷ _________.
5. If 6 pens cost ₹90, one pen costs ₹_________.

✓ B. True or False

1. 3 : 5 and 5 : 3 are the same ratio. _______
2. More workers means more days to finish a job. _______
3. 2 : 4 = 3 : 6 is a proportion. _______
4. In direct proportion, the ratio stays the same. _______
5. A map scale of 1 cm : 2 km means 3 cm = 6 km. _______

✏️ C. Solve These

1. If 8 chocolates cost ₹120, find the cost of 15 chocolates.

Show your work here

2. 10 farmers harvest a field in 6 days. How many days for 15 farmers?

Show your work here

🔗 D. Match the Columns

Column A	Column B
1. More items, more cost	a. Inverse proportion
2. More speed, less time	b. Unitary method
3. Find value of one unit	c. Scale
4. Map distance : real distance	d. Direct proportion

🎨 E. Draw and Solve

A car moves at 50 km per hour. Draw a table showing distance covered in 1, 2, 3, 4, and 5 hours. Is this direct or inverse proportion?

Draw your table and write your answer here

✅ Detailed Solutions — Section 5 Handout

A. Fill in the Blanks — Answers

1. A ratio compares two quantities. (e.g. 3 : 5 compares 3 and 5.)

2. In direct proportion, both values go up (or both go down) together.

3. In inverse proportion, one goes up and the other goes down.

4. Speed = Distance ÷ Time.

5. If 6 pens cost ₹90, one pen costs ₹15. (90 ÷ 6 = 15.)

B. True or False — Answers

1. 3 : 5 and 5 : 3 are the same ratio. — False. 3/5 ≠ 5/3. Order matters in a ratio.

2. More workers means more days to finish a job. — False. More workers means fewer days (inverse proportion).

3. 2 : 4 = 3 : 6 is a proportion. — True. 2/4 = 1/2 and 3/6 = 1/2. ✓

4. In direct proportion, the ratio stays the same. — True. y/x = constant. ✓

5. A map scale of 1 cm : 2 km means 3 cm = 6 km. — True. 3 × 2 km = 6 km. ✓

C1. If 8 chocolates cost ₹120, find the cost of 15 chocolates.

1 This is direct proportion (more chocolates = more cost).

2 Cost of 1 chocolate = 120 ÷ 8 = ₹15.

3 Cost of 15 chocolates = 15 × 15 = ₹225.

Cost of 15 chocolates = ₹225

C2. 10 farmers harvest a field in 6 days. How many days for 15 farmers?

1 This is inverse proportion (more workers = fewer days).

2 Total work = 10 farmers × 6 days = 60 farmer-days.

3 Days for 15 farmers = 60 ÷ 15 = 4 days.

4 Check: 15 × 4 = 60 farmer-days. ✓

15 farmers will finish in 4 days.

D. Match the Columns — Answers

1. More items, more cost → d. Direct proportion

2. More speed, less time → a. Inverse proportion

3. Find value of one unit → b. Unitary method

4. Map distance : real distance → c. Scale

E. Car at 50 km/h — Distance Table

Time 1 hr → Distance = 50 × 1 = 50 km.

Time 2 hr → Distance = 50 × 2 = 100 km.

Time 3 hr → Distance = 50 × 3 = 150 km.

Time 4 hr → Distance = 50 × 4 = 200 km.

Time 5 hr → Distance = 50 × 5 = 250 km.

As time increases, distance increases proportionally. Distance ÷ Time = 50 (constant).

This is direct proportion. (Distance / Time = constant = 50 km/h.)

🃏 Section 6 — Revision Flashcards

👉

⚖

What is a ratio?

A ratio compares two quantities. We write it as a : b. Order matters!

⚖

What is a proportion?

A proportion means two ratios are equal. Example: 2 : 3 = 4 : 6.

🔎

What is the unitary method?

Find the value of one unit first. Then multiply by the number you need.

📈

What is direct proportion?

When one value increases, the other also increases. The ratio stays the same.

📉

What is inverse proportion?

When one value increases, the other decreases. The product stays the same.

🚗

What is the speed formula?