Class 8 Maths Direct and Inverse Proportions Important Questions with Solutions

Direct and Inverse Proportions Class 8 Questions are available in this Maths article. Direct and Inverse Proportions Class 8 Questions are very useful to solve problems easily. This article helps the students to know the key questions and answers about Direct and Inverse Proportions. Direct proportion shows how two quantities increase or decrease together, while inverse proportion shows how one quantity increases as the other decreases.These problems have been solved in detail by our subject experts as per the CBSE syllabus and NCERT textbook. This material helps students to revise the chapter with ease and do well in the final examination.

Table of Contents

• Direct and Inverse Proportion
• Exercise 13.1: Understanding Proportions - Revision and Introduction
• Exercise 13.2: Direct Proportion - Detailed Problems
• Exercise 13.3: Inverse Proportion - Detailed Problems
• Exercise 13.4: Applications of Direct and Inverse Proportions
• Tips for Understanding Direct and Inverse Proportions
• Most Repeated Board Questions on Direct and Inverse Proportions
• Direct and Inverse Proportion importance for Class 8 exams
• Most Common Examination Questions (Board Exams)

Direct and Inverse Proportion

Direct proportion is when two things go up or down together in the same ratio. For example, if 2 pens cost ₹30, then 4 pens cost ₹60 the cost goes up as the number of pens goes up, and the ratio stays the same.

Inverse proportion is when one thing goes up while the other goes down, so their product stays constant. For example, if 4 workers finish a job in 6 days, then 6 workers will finish it in 4 days more workers mean fewer days, but workers × days stays the same (4×6 = 6×4 = 24).

Exercise 13.1: Understanding Proportions - Revision and Introduction

Exercise 13.2: Direct Proportion - Detailed Problems

Question 9: If 8 Workers Can Build a Wall in 12 Days, How Many Workers Are Needed to Build the Same Wall in 6 Days?

Solution: This is actually INVERSE PROPORTION. More workers means less time needed.

Let's verify: 8 workers × 12 days = 96 worker-days (This represents the total amount of work to be done)

For 6 days: Number of workers × 6 days = 96 worker-days Number of workers = 96 ÷ 6 = 16 workers

Answer: 16 workers are needed to build the wall in 6 days

8 workers × 12 days = 96 worker-days 16 workers × 6 days = 96 worker-days

Question 10: The Cost of 5 kg of Rice is Rs. 250. Find the Cost of 12 kg of Rice

Solution: This is direct proportion. More rice means more cost.

Cost per kg = Rs. 250 ÷ 5 = Rs. 50 per kg

Cost of 12 kg = Rs. 50 × 12 = Rs. 600

Or using proportion: 5 kg costs Rs. 250 12 kg costs x Rs.

5/250 = 12/x 5x = 250 × 12 5x = 3000 x = 600

Answer: The cost of 12 kg of rice is Rs. 600

Question 11: If 20 Workers Take 15 Days to Paint a Building, How Many Days Will 25 Workers Take?

Solution: This is INVERSE PROPORTION. More workers take less time.

Total work = 20 workers × 15 days = 300 worker-days

For 25 workers: 25 × d = 300 d = 300 ÷ 25 d = 12 days

Answer: 25 workers will take 12 days to paint the building

20 workers × 15 days = 300 worker-days 25 workers × 12 days = 300 worker-days

Question 12: The Speed of a Car and Time Taken to Cover 240 km Are Related. If the Car Travels at 80 km/h, It Takes 3 Hours. How Long Will It Take at 60 km/h?

Solution: This is INVERSE PROPORTION. Higher speed takes less time for the same distance.

Distance = Speed × Time (constant distance = 240 km)

Verification with given data: 80 km/h × 3 hours = 240 km

For 60 km/h: 60 × t = 240 t = 240 ÷ 60 t = 4 hours

Answer: At 60 km/h, it will take 4 hours to cover 240 km

Exercise 13.3: Inverse Proportion - Detailed Problems

Question 13: If 4 Taps Can Fill a Tank in 6 Hours, How Long Will 6 Taps Take to Fill the Same Tank?

Solution: This is INVERSE PROPORTION. More taps fill the tank faster (less time).

Number of taps × Time = Constant (total filling capacity)

With 4 taps: 4 × 6 = 24 (total work units)

With 6 taps: 6 × t = 24 t = 24 ÷ 6 t = 4 hours

Answer: 6 taps will fill the tank in 4 hours

4 taps × 6 hours = 24 units of work 6 taps × 4 hours = 24 units of work

Question 14: A Person Walks at 5 km/h and Takes 2 Hours to Reach Destination. How Long Will It Take If He Walks at 4 km/h?

Solution: This is INVERSE PROPORTION. Slower speed takes more time for the same distance.

Speed × Time = Distance (constant)

Distance = 5 km/h × 2 hours = 10 km

At 4 km/h: 4 × t = 10 t = 10 ÷ 4 t = 2.5 hours

Answer: At 4 km/h, it will take 2.5 hours (or 2 hours 30 minutes)

Question 15: 6 Machines Can Produce 180 Items in One Day. How Many Machines Are Needed to Produce 360 Items in One Day?

Solution: This is DIRECT PROPORTION. More machines produce more items.

Production per machine = 180 ÷ 6 = 30 items per machine per day

For 360 items: Number of machines = 360 ÷ 30 = 12 machines

Answer: 12 machines are needed to produce 360 items in one day

Question 16: If 3 Painters Can Paint a House in 8 Days, How Many Days Will 6 Painters Take?

Solution: This is INVERSE PROPORTION. More painters complete work faster.

Number of painters × Number of days = Total work (constant)

With 3 painters: 3 × 8 = 24 (painter-days)

With 6 painters: 6 × d = 24 d = 24 ÷ 6 d = 4 days

Answer: 6 painters can paint the house in 4 days

3 painters × 8 days = 24 painter-days 6 painters × 4 days = 24 painter-days

Question 17: 5 Copies of a Book Cost Rs. 175. How Much Will 8 Copies Cost?

Solution: This is DIRECT PROPORTION. More copies cost more.

Cost per copy = Rs. 175 ÷ 5 = Rs. 35 per copy

Cost of 8 copies = Rs. 35 × 8 = Rs. 280

Answer: 8 copies will cost Rs. 280

Question 18: A Truck Traveling at 60 km/h Takes 5 Hours to Complete a Journey. How Long Will It Take at 50 km/h?

Solution: This is INVERSE PROPORTION. Slower speed takes more time.

Speed × Time = Distance

Distance = 60 × 5 = 300 km

At 50 km/h: 50 × t = 300 t = 300 ÷ 50 t = 6 hours

Answer: At 50 km/h, it will take 6 hours

Exercise 13.4: Applications of Direct and Inverse Proportions

Question 19: A School Organizes a Picnic. If 200 Students Are on the Bus and They Have Enough Food for 4 Days, How Many Days Will the Food Last If 150 Students Go Instead?

Solution: This is INVERSE PROPORTION. Fewer students means food lasts longer.

Number of students × Number of days = Total food (constant)

With 200 students: 200 × 4 = 800 (student-days of food)

With 150 students: 150 × d = 800 d = 800 ÷ 150 d = 5.33 days (approximately 5 days and 8 hours)

Answer: The food will last approximately 5.33 days (or 5 days and 8 hours)

Question 20: A Recipe for 4 People Uses 2 Cups of Flour. How Much Flour Is Needed for 10 People?

Solution: This is DIRECT PROPORTION. More people need more flour.

Flour per person = 2 cups ÷ 4 people = 0.5 cups per person

For 10 people: Flour needed = 0.5 × 10 = 5 cups

Or using proportion: 4 people need 2 cups 10 people need x cups

4/2 = 10/x 4x = 2 × 10 4x = 20 x = 5 cups

Answer: 5 cups of flour are needed for 10 people

Question 21: A Factory Takes 2 Hours to Produce 300 Items with 5 Machines. How Long Will It Take with 8 Machines?

Solution: This is INVERSE PROPORTION. More machines take less time.

Number of machines × Time = Production capacity per hour (constant)

With 5 machines: 5 × 2 = 10 (machine-hours for 300 items)

With 8 machines: 8 × t = 10 t = 10 ÷ 8 t = 1.25 hours

Answer: With 8 machines, it will take 1.25 hours (or 1 hour 15 minutes)

Question 22: A Farmer Has Enough Grain to Feed 20 Cows for 30 Days. For How Many Days Can He Feed 15 Cows?

Solution: This is INVERSE PROPORTION. Fewer cows means grain lasts longer.

Number of cows × Number of days = Total grain (constant)

With 20 cows: 20 × 30 = 600 (cow-days of grain)

With 15 cows: 15 × d = 600 d = 600 ÷ 15 d = 40 days

Answer: He can feed 15 cows for 40 days

Verification: 20 cows × 30 days = 600 cow-days 15 cows × 40 days = 600 cow-days

Question 23: 12 Workers Can Complete a Project in 18 Days Working 8 Hours a Day. If They Work 9 Hours a Day, How Many Days Will It Take?

Solution: This involves inverse proportion. More hours per day means fewer days needed.

Total work = 12 workers × 18 days × 8 hours = 1728 worker-hours

For 9 hours per day: 12 workers × d days × 9 hours = 1728 worker-hours 108d = 1728 d = 1728 ÷ 108 d = 16 days

Answer: Working 9 hours a day, it will take 16 days

Question 24: The Ratio of Boys to Girls in a School is 3:2. If There Are 600 Students in Total, How Many Are Boys and How Many Are Girls?

Solution: This involves proportions but not direct/inverse proportion.

Total parts = 3 + 2 = 5

Boys = (3/5) × 600 = 360 boys Girls = (2/5) × 600 = 240 girls

Answer: There are 360 boys and 240 girls

Verification: 360 + 240 = 600

Question 25: A Car Uses 5 Liters of Petrol to Travel 60 km. How Much Petrol Will It Need to Travel 180 km?

Solution: This is DIRECT PROPORTION. More distance requires more petrol.

Petrol per km = 5 liters ÷ 60 km = 0.0833 liters per km

For 180 km: Petrol needed = 0.0833 × 180 = 15 liters

Or using proportion: 5 liters for 60 km x liters for 180 km

5/60 = x/180 60x = 5 × 180 60x = 900 x = 15 liters

Answer: The car will need 15 liters of petrol to travel 180 km

Tips for Understanding Direct and Inverse Proportions

1. Use real examples: think of speed and distance or workers and time to see how things change.

2. Remember the rule: direct means "both go same way" (if one up, other up), inverse means "one goes up, other goes down."

3. Make a small table of pairs to check: direct keeps ratios same, inverse keeps product same.

4. Draw a quick picture: direct looks like a straight line from origin, inverse looks like a falling curve.

5. Practice simple word problems, turn them into equations, and check your answer makes sense.

Most Repeated Board Questions on Direct and Inverse Proportions

Question 1: Direct Proportion Cost-Quantity

If 8 kg of rice costs Rs. 400, find the cost of 15 kg

Solution :

1. Find cost per kg: 400 ÷ 8 = 50

2. Cost of 15 kg: 50 × 15 = 750

Question 2: Inverse Proportion Workers-Time

If 6 workers complete work in 10 days, how many days for 15 workers?

Solution :

1. Total work = 6 × 10 = 60 worker-days

2. For 15 workers: 15 × d = 60, so d = 4 days

Question 3: Direct Proportion Distance-Time

A car travels 240 km in 4 hours. How far in 7 hours at same speed?

Solution :

1. Speed = 240 ÷ 4 = 60 km/h

2. Distance in 7 hours = 60 × 7 = 420 km

Question 4: Inverse Proportion Speed-Time

A truck at 60 km/h takes 5 hours. How long at 50 km/h?

Solution :

1. Distance = 60 × 5 = 300 km

2. Time at 50 km/h: 300 ÷ 50 = 6 hours

Question 5: Direct Proportion Recipe-Servings

Recipe for 6 people uses 3 cups flour. How much for 10 people?

Solution :

1. Per person: 3 ÷ 6 = 0.5 cups

2. For 10 people: 0.5 × 10 = 5 cups

Question 6: Inverse Proportion Taps-Time

3 taps fill tank in 8 hours. How long for 4 taps?

Solution :

1. Total work = 3 × 8 = 24 tap-hours

2. For 4 taps: 4 × t = 24, so t = 6 hours

Question 7: Direct Proportion Machine-Production

5 machines produce 250 items daily. How many items for 8 machines?

Solution :

1. Per machine: 250 ÷ 5 = 50 items

2. For 8 machines: 50 × 8 = 400 items

Question 8: Inverse Proportion Animal-Food Days

Food for 20 animals lasts 15 days. How long for 12 animals

Solution :

1. Total food = 20 × 15 = 300 animal-days

2. For 12 animals: 12 × d = 300, so d = 25 days

Question 9: Identify Type and Solve

Identify if direct or inverse, then solve the problem

Solution :

1. Analyze relationship between quantities

2. Identify if both increase or one increases while other decreases

3. Apply correct formula and solve

Question 10: Real-World Application Problem

A school trip has enough food for 100 students for 5 days. If only 80 students go, how long will food last?

Solution :

1. Total food = 100 × 5 = 500 student-days

2. For 80 students: 80 × d = 500, so d = 6.25 days

Direct and Inverse Proportion importance for Class 8 exams

• Frequent in exams: Expect 2-4 questions ranging from 1-mark MCQs to 4-mark word problems.

• Real-life application: Problems on work-time, speed-distance, pipes-cisterns, food provisions, and map scales commonly appear.

• Builds foundation: It’s essential for later topics in classes 9-10 (ratios, variations, algebra, and linear relationships).

• HOTS focus: Many questions are Higher Order Thinking Skill (HOTS) type, requiring setup of a/b = constant or ab = constant and careful unit handling.

Most Common Examination Questions (Board Exams)

1 - 2 Mark Questions (Very Frequently Asked):

1. If 10 pens cost Rs. 100, find cost of 25 pens Answer: Rs. 250 (direct proportion)

2. Define direct proportion Answer: Relationship where quantities increase/decrease together with constant ratio

3. If 5 workers take 12 days, how many days for 10 workers? Answer: 6 days (inverse proportion)

4. Define inverse proportion Answer: Relationship where one increases as other decreases, product is constant

5. A car travels 150 km in 3 hours. Distance in 5 hours? Answer: 250 km (direct proportion)

6. What is the constant k in direct proportion y = 3x? Answer: k = 3

7. What is the constant k in inverse proportion xy = 24? Answer: k = 24

8. If y is directly proportional to x and y = 20 when x = 5, find k Answer: k = 4

9. Identify: More workers means less time taken Answer: Inverse proportion

10. Identify: More items means more cost Answer: Direct proportion

3 - 4 Mark Questions (Frequently Asked):

1. If 6 kg sugar costs Rs. 180, find cost of 11 kg Solution: Cost/kg = 30, Cost = 30 × 11 = Rs. 330

2. 4 pumps fill tank in 6 hours. How long for 6 pumps? Solution: Work = 24 pump-hours, Time = 24/6 = 4 hours

3. y is inversely proportional to x. If y = 12 when x = 5, find y when x = 4 Solution: k = 60, y = 60/4 = 15

4. A recipe for 8 people needs 2 cups rice. How much for 12 people? Solution: Per person = 0.25 cups, For 12 = 3 cups

5. Distance-time graph at constant speed shows direct proportion. Why? Solution: d = kt, as time increases, distance increases proportionally

6. Identify and solve: 3 workers, 20 days; 5 workers, ? days Solution: Inverse, Work = 60 worker-days, Time = 12 days

7. If 8 litres petrol covers 160 km, how much for 360 km? Solution: Per km = 0.05 litres, For 360 = 18 litres

8. y is directly proportional to x. Table shows x: 2,4,6 and y: 10,20,30. Verify Solution: Ratio 10/2=5, 20/4=5, 30/6=5, constant k=5, so yes

9. Food for 50 animals lasts 20 days. For 40 animals, ? days Solution: Inverse, Total = 1000 animal-days, Time = 25 days

10. Write equation: y is directly proportional to x, and y = 30 when x = 6 Answer: y = 5x

5 - 6 Mark Questions (Less Frequent but Important):

1. Two quantities in direct proportion. When first is 8, second is 24. Find when first is 15 Solution: k = 3, when first = 15, second = 45

2. Compare two scenarios: 10 workers, 20 days vs 15 workers, ? days. Show inverse relationship Solution: 200 worker-days both; 15 workers take 13.33 days

3. A trip has food for 200 people for 6 days. If 50 less people go, how long? Solution: Inverse, Total = 1200 person-days, For 150: 8 days

4. Three quantities related: Speed, Distance, Time. Explain direct and inverse proportions Solution: d = st (direct with t), t = d/s (inverse with s)

5. Project: 12 workers, 8 hours/day, 20 days. Same project with 15 workers, 8 hours/day, ? days Solution: Work = 1920 worker-hours, Time = 16 days

6. Create real-world inverse proportion problem, solve, verify Solution: Student-created with solution and verification

7. Table of values showing direct proportion: 2-10, 4-20, 6-30. Verify and find for x=10 Solution: k = 5, for x=10, y=50

8. Speed of vehicle varies inversely with time for 300 km distance. If speed = 75 km/h takes 4 hours, verify and find time for 60 km/h Solution: Verify 75×4=300, For 60: 300/60 = 5 hours

9. Combination: 6 identical articles cost Rs. 90. With 10% discount on each, total cost for 8 articles? Solution: Cost/article = 15, with discount = 13.5, Total = 108

10. Investigation: Analyze if two quantities in table are in direct/inverse/no proportion Solution: Create table, calculate ratios/products, conclude