Proportional Reasoning 1 — Class 8 Mathematics NCERT Solutions (Free)
Free step-by-step NCERT solutions for Class 8 Mathematics chapter "Proportional Reasoning 1" — 8 important questions with detailed answers for CBSE board exam preparation.
TL;DR: Free step-by-step NCERT solutions for Class 8 Mathematics chapter "Proportional Reasoning 1" — 8 important questions with detailed answers for CBSE bo…
Written & reviewed by the Syllab.in Academic Team (CBSE/NCERT subject experts) · Updated
NCERT Solutions for Class 8 Mathematics — Proportional Reasoning 1. Step-by-step answers to all 8 textbook questions from this chapter, written for CBSE board preparation and free to use.
Proportional Reasoning 1 — All 8 Questions Solved
Q1. If 5 pens cost Rs. 40, how much will 12 pens cost?
This is a direct proportion problem.
Step 1: Identify the relationship
Number of pens and cost are directly proportional.
When one increases, the other increases proportionally.
Step 2: Find the cost of 1 pen
If 5 pens cost Rs. 40
Cost of 1 pen = 40/5 = Rs. 8
Step 3: Find the cost of 12 pens
Cost of 12 pens = 12 × 8 = Rs. 96
Step 4: Alternative method - using proportion
5 pens : Rs. 40 = 12 pens : x
5/40 = 12/x
5x = 40 × 12
5x = 480
x = 480/5
x = 96
Answer: 12 pens will cost Rs. 96
Q2. A car travels 240 km in 4 hours. At the same speed, how long will it take to travel 600 km?
This is a direct proportion problem (distance and time).
Step 1: Find the speed of the car
Speed = Distance/Time = 240/4 = 60 km/h
Step 2: Find the time to travel 600 km
Time = Distance/Speed = 600/60 = 10 hours
Step 3: Alternative method - using proportion
Distance and Time are in direct proportion at constant speed.
240 : 4 = 600 : t
240/4 = 600/t
60 = 600/t
60t = 600
t = 600/60
t = 10 hours
Step 4: Verification
60 km/h × 10 hours = 600 km ✓
Answer: It will take 10 hours to travel 600 km
Q3. If 6 workers can build a wall in 8 days, how long will it take 12 workers to build the same wall?
This is an inverse proportion problem.
Step 1: Identify the relationship
Number of workers and time taken are inversely proportional.
When the number of workers increases, time decreases.
Step 2: Find the total amount of work (in worker-days)
Work = Number of workers × Number of days
Work = 6 × 8 = 48 worker-days
Step 3: Find the time for 12 workers
Number of days = Work/Number of workers
Number of days = 48/12 = 4 days
Step 4: Alternative method - using inverse proportion
6 workers : 8 days = 12 workers : x days
For inverse proportion: 6 × 8 = 12 × x
48 = 12x
x = 48/12
x = 4 days
Step 5: Verification
6 workers × 8 days = 12 workers × 4 days
48 worker-days = 48 worker-days ✓
Answer: 12 workers will take 4 days to build the same wall
Q4. Write the ratio of 500 g to 2 kg in simplest form.
We need to express the ratio in simplest form with the same units.
Step 1: Convert to the same unit
500 g and 2 kg
2 kg = 2 × 1000 g = 2000 g
Step 2: Write the ratio
500 g : 2000 g or 500 : 2000
Step 3: Simplify by finding the GCD
GCD(500, 2000):
500 = 2² × 5³
2000 = 2⁴ × 5³ × 1
GCD = 2² × 5³ = 4 × 125 = 500
Step 4: Divide both terms by GCD
500 : 2000 = (500÷500) : (2000÷500) = 1 : 4
Step 5: Verification
1 : 4 means the ratio of 500 g to 2000 g
500/2000 = 1/4 ✓
Answer: The ratio is 1 : 4
Q5. A map has a scale of 1 : 100,000. If two cities are 5 cm apart on the map, what is the actual distance between them?
We use the map scale to find the actual distance.
Step 1: Understand the scale
Scale 1 : 100,000 means
1 cm on the map represents 100,000 cm in reality
Step 2: Find the actual distance
Distance on map : Actual distance = 1 : 100,000
5 cm : Actual distance = 1 : 100,000
Step 3: Set up the proportion
5/Actual distance = 1/100,000
Actual distance = 5 × 100,000 = 500,000 cm
Step 4: Convert to a more meaningful unit (km)
500,000 cm = 500,000/100,000 m = 5 m
or
500,000 cm = 500,000/(100 × 1000) km = 5 km
Step 5: Verification
If actual distance is 5 km = 500,000 cm
On the map: 500,000/100,000 = 5 cm ✓
Answer: The actual distance between the cities is 5 km
Q6. The cost of 8 books is Rs. 320. Find the cost of 15 books at the same rate.
This is a direct proportion problem.
Step 1: Find the cost of 1 book
Cost of 8 books = Rs. 320
Cost of 1 book = 320/8 = Rs. 40
Step 2: Find the cost of 15 books
Cost of 15 books = 15 × 40 = Rs. 600
Step 3: Alternative method - using direct proportion
Number of books and cost are directly proportional.
8 books : Rs. 320 = 15 books : x
8/320 = 15/x
8x = 320 × 15
8x = 4800
x = 4800/8
x = 600
Step 4: Verification
Cost per book: 320/8 = 40, and 600/15 = 40 ✓
Answer: The cost of 15 books is Rs. 600
Q7. If y is inversely proportional to x, and y = 12 when x = 5, find y when x = 8.
We solve an inverse proportion problem using the constant of proportionality.
Step 1: Write the relationship for inverse proportion
If y is inversely proportional to x:
y = k/x, where k is the constant of proportionality
Step 2: Find k using the given values
Given: y = 12 when x = 5
12 = k/5
k = 12 × 5 = 60
Step 3: Find y when x = 8
y = k/x = 60/8
y = 7.5
Step 4: Simplify the fraction
60/8 = 15/2 = 7.5
Step 5: Verification
When x = 5: y = 60/5 = 12 ✓
When x = 8: y = 60/8 = 7.5 ✓
Answer: When x = 8, y = 7.5 (or 15/2)
Q8. In a recipe, the ratio of flour to sugar is 3 : 2. If 150 g of flour is used, how much sugar is needed?
We use the given ratio to find the required amount of sugar.
Step 1: Write the ratio
Flour : Sugar = 3 : 2
Step 2: Set up the proportion
Flour used : Sugar needed = 3 : 2
150 : x = 3 : 2
Step 3: Solve using cross multiplication
150/x = 3/2
150 × 2 = 3 × x
300 = 3x
x = 300/3
x = 100 g
Step 4: Verification
Flour : Sugar = 150 : 100
Simplify: 150 : 100 = (150÷50) : (100÷50) = 3 : 2 ✓
Answer: 100 g of sugar is needed
Proportional Reasoning 1 — Practice MCQs with Answers
Attempt these 20 multiple-choice questions after working through the solutions above. Each carries the correct option and the reasoning behind it, so a wrong answer tells you which idea to revisit.
MCQ 1. If 5 pens cost ₹50, what is the cost of 12 pens?
- A. ₹100
- B. ₹120
- C. ₹150
- D. ₹110
Answer: B. ₹120 — The cost of pens is directly proportional to the number of pens. First, find the cost of one pen (₹50 / 5 = ₹10), then multiply by the desired number (₹10 * 12 = ₹120).
MCQ 2. Which of the following situations represents an inverse proportion?
- A. The number of articles purchased and their total cost.
- B. The speed of a vehicle and the distance covered in a fixed time.
- C. The number of workers to complete a job and the time taken.
- D. The amount of petrol filled in a car and the distance it travels.
Answer: C. The number of workers to complete a job and the time taken. — In inverse proportion, as one quantity increases, the other decreases proportionally. If more workers are employed, less time will be taken to complete the same job, representing an inverse relationship.
MCQ 3. A car travels 240 km on 15 litres of petrol. How much distance can it travel on 25 litres of petrol?
- A. 300 km
- B. 400 km
- C. 450 km
- D. 500 km
Answer: B. 400 km — The distance travelled is directly proportional to the amount of petrol consumed. The car travels 240 km / 15 litres = 16 km/litre. So, for 25 litres, it can travel 16 km/litre * 25 litres = 400 km.
MCQ 4. A train covers a certain distance in 6 hours at a speed of 80 km/h. How long will it take to cover the same distance at a speed of 120 km/h?
- A. 3 hours
- B. 4 hours
- C. 5 hours
- D. 4.5 hours
Answer: B. 4 hours — Speed and time are inversely proportional when the distance is constant. The total distance is 80 km/h * 6 h = 480 km. At 120 km/h, the time taken will be 480 km / 120 km/h = 4 hours.
MCQ 5. Which of the following ratios is equivalent to 3:4?
- A. 6:9
- B. 9:12
- C. 12:15
- D. 15:18
Answer: B. 9:12 — Equivalent ratios are obtained by multiplying or dividing both parts of the ratio by the same non-zero number. Multiplying both parts of 3:4 by 3 gives 9:12.
MCQ 6. A worker is paid ₹1200 for 5 days of work. How much will he be paid for 13 days of work?
- A. ₹2880
- B. ₹3120
- C. ₹3600
- D. ₹2400
Answer: B. ₹3120 — The payment received is directly proportional to the number of days worked. The worker is paid ₹1200 / 5 days = ₹240 per day. For 13 days, he will be paid ₹240 * 13 = ₹3120.
MCQ 7. A fort has enough food for 100 soldiers for 20 days. How many days will the food last if 25 more soldiers join the fort?
- A. 12 days
- B. 16 days
- C. 18 days
- D. 20 days
Answer: B. 16 days — The number of soldiers and the number of days the food lasts are inversely proportional. The total 'soldier-days' of food is 100 soldiers * 20 days = 2000. If 25 more soldiers join, the total is 125 soldiers, so the food will last 2000 / 125 = 16 days.
MCQ 8. In a direct proportion, if one quantity increases, how does the other quantity change?
- A. It increases proportionally.
- B. It decreases proportionally.
- C. It remains constant.
- D. It decreases disproportionately.
Answer: A. It increases proportionally. — In direct proportion, two quantities increase or decrease together such that their ratio remains constant. If one quantity doubles, the other also doubles.
MCQ 9. In an inverse proportion, if one quantity increases, how does the other quantity change?
- A. It increases proportionally.
- B. It decreases proportionally.
- C. It remains constant.
- D. It increases disproportionately.
Answer: B. It decreases proportionally. — In inverse proportion, two quantities change in opposite directions. If one quantity increases, the other decreases such that their product remains constant.
MCQ 10. A tap fills a tank in 8 hours. What fraction of the tank is filled in 3 hours?
- A. 3/8
- B. 1/8
- C. 5/8
- D. 1/3
Answer: A. 3/8 — The fraction of the tank filled is directly proportional to the time duration. In 1 hour, 1/8 of the tank is filled, so in 3 hours, 3 * (1/8) = 3/8 of the tank is filled.
MCQ 11. If a car travels at a speed of 60 km/h, it takes 4 hours to reach a destination. How much time will it take if the speed is increased to 80 km/h?
- A. 2 hours
- B. 2.5 hours
- C. 3 hours
- D. 3.5 hours
Answer: C. 3 hours — For a fixed distance, speed and time are inversely proportional. The total distance is 60 km/h * 4 h = 240 km. At 80 km/h, the time taken will be 240 km / 80 km/h = 3 hours.
MCQ 12. 15 workers can build a wall in 48 hours. How many workers will be required to build the same wall in 30 hours?
- A. 20 workers
- B. 24 workers
- C. 28 workers
- D. 32 workers
Answer: B. 24 workers — The number of workers and the time taken to complete a job are inversely proportional. The total 'worker-hours' needed is 15 * 48 = 720. To complete in 30 hours, 720 / 30 = 24 workers are required.
MCQ 13. Which of the following is an example of direct proportion?
- A. Number of days to complete a work and the number of workers.
- B. Speed of a vehicle and the time taken to cover a fixed distance.
- C. Amount of money spent and the number of articles purchased at a fixed price.
- D. Age of a person and their height.
Answer: C. Amount of money spent and the number of articles purchased at a fixed price. — If the price per article is fixed, as the amount of money spent increases, the number of articles purchased also increases proportionally, making it a direct proportion.
MCQ 14. If 3 notebooks cost ₹45, what is the cost of 7 such notebooks?
- A. ₹95
- B. ₹105
- C. ₹115
- D. ₹125
Answer: B. ₹105 — The cost of notebooks is directly proportional to the number of notebooks. The cost of one notebook is ₹45 / 3 = ₹15. So, 7 notebooks will cost ₹15 * 7 = ₹105.
MCQ 15. A car takes 2 hours to cover a distance at a speed of 50 km/h. If it travels at a speed of 40 km/h, how much time will it take?
- A. 2.5 hours
- B. 3 hours
- C. 3.5 hours
- D. 4 hours
Answer: A. 2.5 hours — Speed and time are inversely proportional for a fixed distance. The total distance is 50 km/h * 2 h = 100 km. At 40 km/h, the time taken will be 100 km / 40 km/h = 2.5 hours.
MCQ 16. If y is directly proportional to x, and y = 20 when x = 5, what is the value of y when x = 12?
- A. 36
- B. 40
- C. 48
- D. 52
Answer: C. 48 — In direct proportion, the ratio y/x is constant (y = kx). From the given values, 20 = k * 5, so k = 4. When x = 12, y = 4 * 12 = 48.
MCQ 17. If y is inversely proportional to x, and y = 10 when x = 4, what is the value of y when x = 8?
- A. 2.5
- B. 5
- C. 10
- D. 20
Answer: B. 5 — In inverse proportion, the product y * x is constant (xy = k). From the given values, 10 * 4 = 40, so k = 40. When x = 8, y * 8 = 40, which means y = 40 / 8 = 5.
MCQ 18. A recipe calls for 2 cups of flour to make 12 cookies. How many cups of flour are needed to make 30 cookies?
- A. 3 cups
- B. 4 cups
- C. 5 cups
- D. 6 cups
Answer: C. 5 cups — The amount of flour needed is directly proportional to the number of cookies. For 12 cookies, 2 cups are needed, which means 1 cup for 6 cookies. To make 30 cookies, 30 / 6 = 5 cups of flour are needed.
MCQ 19. On a map, 1 cm represents 20 km. If the distance between two cities on the map is 8 cm, what is the actual distance between them?
- A. 100 km
- B. 120 km
- C. 160 km
- D. 180 km
Answer: C. 160 km — The actual distance is directly proportional to the distance shown on the map. For every 1 cm on the map, the actual distance is 20 km. So, for 8 cm, the distance is 8 * 20 km = 160 km.
MCQ 20. A pipe can fill a tank in 10 hours. How much of the tank will be filled by the pipe in 4 hours and 30 minutes?
- A. 4/10
- B. 9/20
- C. 1/2
- D. 3/5
Answer: B. 9/20 — The fraction of the tank filled is directly proportional to the time it operates. 4 hours and 30 minutes is 4.5 hours. In 4.5 hours, the pipe will fill (4.5 / 10) = 45/100 = 9/20 of the tank.
How to Use These Solutions
Attempt each question yourself first and only then compare with the worked answer. Marks in the CBSE board exam are awarded for the METHOD as much as the final result, so reproduce the steps rather than memorising the last line. Where a solution states a law, a formula or a definition, learn that wording — examiners look for it.
Related practice for this chapter: chapter MCQs, previous-year questions, revision notes and sample papers.