🔢 Class 8 Mathematics — Direct and Inverse Proportions
Interactive Visual Revision Notes
Aligned with the Class 8 Mathematics syllabus. Independently created study material. Not affiliated with or endorsed by NCERT.
🌎 Real-World Connection: A bus uses 4 litres of diesel to travel 40 km. To travel 100 km it needs 10 litres — the distance and diesel are in direct proportion. A factory has enough food for 50 workers for 30 days. If 10 more workers join, the food will last fewer days — that is inverse proportion. Understanding these two types of proportion helps you plan journeys, manage resources, and solve everyday problems efficiently.
🔢 Topic 1: What Is Direct Proportion?
Two things are linked together.
When one goes up, the other goes up.
When one goes down, the other goes down.
They change in the same direction.
Example: more chocolates = more money.
Example: more hours = more distance.
🔢 Topic 2: Direct Proportion Formula
We divide y by x every time.
If y ÷ x is always the same, it is direct.
We call this same number the constant.
We write it as k.
So the rule is: y ÷ x = k.
Or we can write: y = k × x.
y ÷ x = k (constant)
🔢 Topic 3: Testing for Direct Proportion
Make a table of x and y values.
Divide each y by its x.
If all answers are the same, it is direct.
If answers are different, it is NOT direct.
Example: 2,4,6 and 10,20,30.
10÷2=5, 20÷4=5, 30÷6=5. Same! Direct!
🔢 Topic 4: Solving Direct Proportion Problems
Use the rule: y₁/x₁ = y₂/x₂.
Put the known values in the formula.
One value will be unknown.
Cross multiply to find it.
x₁ × y₂ = x₂ × y₁
Then divide to get the answer.
🔢 Topic 5: What Is Inverse Proportion?
Two things are linked again.
But now one goes up, other goes down.
One goes down, other goes up.
They change in opposite directions.
Example: more workers = less time.
Example: faster speed = less time.
🔢 Topic 6: Inverse Proportion Formula
We multiply x and y every time.
If x × y is always the same, it is inverse.
We call this the constant too.
We write: x × y = k.
Or we can write: y = k ÷ x.
The product stays fixed.
x × y = k (constant)
🔢 Topic 7: Testing for Inverse Proportion
Make a table of x and y values.
Multiply each x with its y.
If all products are the same, it is inverse.
If products are different, it is NOT inverse.
Example: 2,4,8 and 12,6,3.
2×12=24, 4×6=24, 8×3=24. Same! Inverse!
🔢 Topic 8: Solving Inverse Proportion Problems
Use the rule: x₁ × y₁ = x₂ × y₂.
Put the known values in the formula.
One value will be unknown.
Multiply the two known numbers.
Then divide by the third known number.
That gives you the answer.
🔢 Topic 9: Real Life — Direct Proportion
More items you buy = more you pay.
More petrol = more distance you travel.
More hours you work = more money you earn.
More cloth you buy = more the cost.
More weight on spring = more it stretches.
Both go up together. That is the key!
🔢 Topic 10: Real Life — Inverse Proportion
More workers = less days to finish work.
Faster speed = less time for a trip.
More pipes filling a tank = less time.
Wider road = less traffic jam time.
More people share food = less for each.
One goes up, other goes down. That is the key!
🎨 Section 2 — Easy Illustrations
Picture 1: Direct Proportion Graph
Direct proportion always gives a straight line that passes through (0, 0).
Picture 2: Inverse Proportion Graph
Inverse proportion gives a curve. As x goes up, y comes down.
Picture 3: Direct vs Inverse — Comparison
Direct = same direction. Inverse = opposite direction. Remember this!
Picture 4: Real Life Direct — Shopping
More items you buy, more you pay. Cost per item stays the same.
Picture 5: Real Life Inverse — Workers and Days
More workers join, the work finishes faster. Product stays fixed.
✏️ Section 3 — Worked Examples
Example 1: Direct Proportion — Cost of Cloth
Question: If 6 metres of cloth costs ₹420, find the cost of 14 metres.
1 This is direct proportion. More cloth = more cost.
2 We know: 6 metres = ₹420.
3 We need: 14 metres = ₹?
4 Use: cost₁/metres₁ = cost₂/metres₂
5 Put values: 420/6 = ?/14
6 Cross multiply: 420 × 14 = 6 × ?
7 420 × 14 = 5880
8 5880 ÷ 6 = 980
Cost of 14 metres = ₹ 980
Check: 420÷6 = 70 per metre. 70×14 = 980 ✓
Example 2: Inverse Proportion — Speed and Time
Question: A train at 60 km/h takes 5 hours. How long at 75 km/h?
1 This is inverse proportion. Faster speed = less time.
2 We know: speed = 60, time = 5 hours.
3 We need: speed = 75, time = ?
4 Use: speed₁ × time₁ = speed₂ × time₂
5 Put values: 60 × 5 = 75 × ?
6 60 × 5 = 300
7 300 ÷ 75 = 4
Time at 75 km/h = 4 hours
Check: 60×5 = 300 and 75×4 = 300. Same! ✓
Example 3: Inverse Proportion — Workers and Food
Question: A factory has food for 150 workers for 28 days. How many days for 200 workers?
1 This is inverse proportion. More workers = less days.
2 We know: 150 workers, 28 days.
3 We need: 200 workers, ? days.
4 Use: workers₁ × days₁ = workers₂ × days₂
5 Put values: 150 × 28 = 200 × ?
6 150 × 28 = 4200
7 4200 ÷ 200 = 21
Food will last 21 days for 200 workers
Check: 150×28 = 4200 and 200×21 = 4200. Same! ✓
📊 Section 4 — Cheat Sheet
Term
What It Means
Key Fact
Proportion
Two things linked together
They change together
Direct Proportion
Both go same way
y ÷ x = constant
Inverse Proportion
They go opposite way
x × y = constant
Constant (k)
The number that stays same
Does not change
Direct Formula
y₁/x₁ = y₂/x₂
Cross multiply
Inverse Formula
x₁×y₁ = x₂×y₂
Product is same
Direct Graph
Straight line
Through (0, 0)
Inverse Graph
Curved line
Never touches axis
Direct Example
More items, more cost
Same direction
Inverse Example
More workers, less time
Opposite direction
Test Direct
Divide y by x
Same answer each time
Test Inverse
Multiply x and y
Same answer each time
📄 Section 5 — Take-Home Handout
📝 A. Fill in the Blanks
1. In direct proportion, y ÷ x = _________.
2. In inverse proportion, x × y = _________.
3. More speed means _________ time for the trip.
4. More items you buy means _________ total cost.
5. A direct proportion graph is a _________ line.
✓ B. True or False
1. More workers means more days to finish. _______
2. In direct proportion, y/x is always the same. _______
3. The graph of inverse proportion is a straight line. _______
4. More petrol means you can travel more distance. _______
5. In inverse proportion, the product x × y changes. _______
✏️ C. Solve These
1. If 8 notebooks cost ₹240, how much do 15 notebooks cost?
Show your work here
2. 12 workers can build a wall in 10 days. How many days for 15 workers?
Show your work here
🔗 D. Match the Columns
Column A
Column B
1. More cloth, more cost
a. Inverse proportion
2. More speed, less time
b. y ÷ x = constant
3. Direct proportion test
c. Direct proportion
4. x × y = constant
d. Inverse proportion test
🎨 E. Make Your Own Example
Write one example of direct proportion and one example of inverse proportion from your daily life.
Write your examples here
✅ Detailed Solutions — Section 5 Handout
A. Fill in the Blanks — Answers
1. In direct proportion, y ÷ x = constant (k).
2. In inverse proportion, x × y = constant (k).
3. More speed means less time for the trip. (Inverse proportion.)
4. More items you buy means more total cost. (Direct proportion.)
5. A direct proportion graph is a straight line. (Through the origin.)
B. True or False — Answers
1. More workers means more days to finish. — False. More workers = fewer days (inverse proportion).
2. In direct proportion, y/x is always the same. — True. That is the definition. ✓
3. The graph of inverse proportion is a straight line. — False. It is a curved line (hyperbola).
4. More petrol means you can travel more distance. — True. Direct proportion. ✓
5. In inverse proportion, the product x × y changes. — False. The product x × y = constant (that is the rule).
C1. If 8 notebooks cost ₹240, how much do 15 notebooks cost?
1Direct proportion: more notebooks = more cost.
2 Cost of 1 notebook = 240 ÷ 8 = ₹30.
3 Cost of 15 notebooks = 15 × 30 = ₹450.
Cost of 15 notebooks = ₹450
C2. 12 workers build a wall in 10 days. How many days for 15 workers?
1Inverse proportion: more workers = fewer days.
2 Total work = 12 × 10 = 120 worker-days.
3 Days for 15 workers = 120 ÷ 15 = 8 days.
4 Check: 15 × 8 = 120 worker-days. ✓
15 workers will complete the wall in 8 days.
D. Match the Columns — Answers
1. More cloth, more cost → c. Direct proportion
2. More speed, less time → a. Inverse proportion
3. Direct proportion test → b. y ÷ x = constant
4. x × y = constant → d. Inverse proportion test
🃏 Section 6 — Revision Flashcards
👉
🔢
What is direct proportion?
When one thing goes up, the other also goes up. They change in the same direction.
🔎
What is inverse proportion?
When one thing goes up, the other goes down. They change in opposite directions.
📈
What is the direct proportion formula?
y ÷ x = k (constant)
Also written as y₁/x₁ = y₂/x₂
🔗
What is the inverse proportion formula?
x × y = k (constant)
Also written as x₁×y₁ = x₂×y₂
❓
How do you test for direct proportion?
Divide y by x for each pair. If the answer is always the same, it is direct proportion.
💡
How do you test for inverse proportion?
Multiply x and y for each pair. If the answer is always the same, it is inverse proportion.
📊
What does a direct proportion graph look like?
A straight line that passes through the point (0, 0) which is the origin.
🎯
What does an inverse proportion graph look like?
A curved line (hyperbola) that never touches either axis. It goes down as you move right.
🧠 Section 7 — Mind Map
Hover on any branch to focus it. All key ideas connected together.
Class 8 Mathematics — Direct and Inverse Proportions
Original Visual Revision Notes — Different Learner Friendly
Aligned with the Class 8 Mathematics syllabus. Independently created study material. Not affiliated with or endorsed by NCERT.
💜 Created with love by a father, to make learning easier for his daughter 💜