Official NCERT 2026-27 Solutions
NCERT Solutions for Class 8 Maths Chapter 10: Proportional Reasoning-2 (Ganita Prakash Part II)
Authentic handwritten solutions in Topper Notebook style. Features complete inverse proportion formulations, combined rates of work, school transport pie chart angles, water provision tanks, and speed-time relations from Pages 66 to 69 of Ganita Prakash Part II.
Quick Chapter Navigation & Blueprint
- Part A: Inverse vs. Direct Proportion Classifications (Question 1) — Rigorous mathematical criteria distinguishing constant-product (xy = k) from constant-ratio (y/x = k) quantities.
- Part B: Resource Depletion & Manpower Scheduling (Questions 2 to 4) — Pencil cost unit rates, residential building water tank depletion (20 vs 30 families), and biological sleep cycles.
- Part C: Circular Data & Work Efficiency (Questions 5 to 7) — Transport survey central angle fractions (Walk, Car, Cycle, Bus), workforce expansion fence painting (3 → 4 workers), and multi-tank pump filling.
- Part D: Factory Production & Kinematics (Questions 8 to 12) — Chair grid rearranging, school bell duration scaling (8 → 9 periods), dual pump rate addition (72 mins), factory machinery (42 → 49 machines), and speed-time inverse scaling (60 → 80 km/h).
Official Textbook PDF Reference: Download NCERT Class 8 Maths Chapter 10 (hegp203.pdf)
Part A: Proportionality Classifications (Pages 66–67)
Question 1 (Page 67):
Which of the following pairs of quantities are in inverse proportion?
(i) Number of taps filling a water tank and time taken to fill it.
(ii) Number of painters hired and days needed to paint a wall of fixed size.
(iii) Distance a car can travel and amount of petrol in the tank.
(iv) Speed of a cyclist and time taken to cover a fixed route.
(v) Length of cloth bought and price paid at a fixed rate per metre.
(vi) Number of pages in a book and time required to read it at fixed reading speed.
Which of the following pairs of quantities are in inverse proportion?
(i) Number of taps filling a water tank and time taken to fill it.
(ii) Number of painters hired and days needed to paint a wall of fixed size.
(iii) Distance a car can travel and amount of petrol in the tank.
(iv) Speed of a cyclist and time taken to cover a fixed route.
(v) Length of cloth bought and price paid at a fixed rate per metre.
(vi) Number of pages in a book and time required to read it at fixed reading speed.
Topper’s Criteria: Constant Product vs. Constant Ratio:
- (i) Inverse Proportion: More taps running simultaneously deliver water faster, so the time taken decreases proportionally (Product = Taps × Time = Constant Tank Volume).
- (ii) Inverse Proportion: More workers working at the same pace complete a fixed task in fewer days (Product = Painters × Days = Constant Total Work).
- (iii) Direct Proportion: More petrol in the tank allows the car to travel a greater distance (Ratio = Distance / Petrol = Fuel Mileage).
- (iv) Inverse Proportion: Cycling faster reduces the travel time over a fixed distance (Product = Speed × Time = Fixed Distance).
- (v) Direct Proportion: Buying more metres of cloth costs more money (Ratio = Cost / Length = Fixed Rate per Metre).
- (vi) Direct Proportion: A thicker book with more pages takes proportionally more time to read (Ratio = Pages / Time = Reading Speed).
Pairs in Inverse Proportion: (i), (ii), and (iv).
Question 2 (Page 67):
If 24 pencils cost &inr;120, how much will 20 such pencils cost?
If 24 pencils cost &inr;120, how much will 20 such pencils cost?
Direct Unitary Calculation:
- Quantity and price are in direct proportion.
- Cost of 1 pencil = &inr;120 / 24 = &inr;5.
- Cost of 20 pencils = 20 × &inr;5 = &inr;100.
Final Answer: 20 pencils will cost &inr;100.
Question 3 (Page 67 – Building Water Supply):
A water tank on a building has enough water to supply 20 families for 6 days. If 10 more families move in, how long will the water last? State assumptions.
A water tank on a building has enough water to supply 20 families for 6 days. If 10 more families move in, how long will the water last? State assumptions.
Inverse Proportionality Formulation:
- Initial families f1 = 20, initial days d1 = 6.
- Total water capacity = 20 × 6 = 120 family-days of water.
- New number of families = 20 + 10 = 30 families.
- Since number of families and days are inversely proportional:
f1 × d1 = f2 × d2
20 × 6 = 30 × d2
120 = 30 × d2 → d2 = 4 days. - Assumptions Made:
• All families consume water at the exact same uniform average daily rate.
• No water leaks, evaporates, or is replenished during the 4-day period.
Final Answer: The water will last for 4 days.
Part B: Pie Charts & Combined Work (Pages 67–68)
Question 5 (Page 68 – Transport Pie Chart):
A school survey pie chart shows: Walk = 120°, Cycle = 90°, Bus = 60°, Two-wheeler = 60°, Car = 30°.
(i) What is the most common mode of transport?
(ii) What fraction of children travel by car?
(iii) If 18 children travel by car, how many took part in the survey? How many use a taxi/bus?
(iv) Which two modes of transport have equal numbers of children?
A school survey pie chart shows: Walk = 120°, Cycle = 90°, Bus = 60°, Two-wheeler = 60°, Car = 30°.
(i) What is the most common mode of transport?
(ii) What fraction of children travel by car?
(iii) If 18 children travel by car, how many took part in the survey? How many use a taxi/bus?
(iv) Which two modes of transport have equal numbers of children?
Central Angle Proportion Analysis:
- Total angle of a circle = 360°.
- (i) Most Common Mode: The mode with the largest sector angle is Walking (120°).
- (ii) Fraction Traveling by Car:
Fraction = 30° / 360° = 1/12. - (iii) Total Children:
If (1/12) of total children = 18:
Total Children = 18 × 12 = 216 children.
• Children traveling by Bus (60°): (60/360) × 216 = (1/6) × 216 = 36 children. - (iv) Equal Modes: Both Bus (60°) and Two-wheeler (60°) have identical angles, hence equal children (36 each).
Summary Answers:
(i) Most common: Walking (120°).
(ii) Fraction by car: 1/12.
(iii) Total children = 216 | Bus users = 36.
(iv) Equal modes: Bus and Two-wheeler (36 children each).
(i) Most common: Walking (120°).
(ii) Fraction by car: 1/12.
(iii) Total children = 216 | Bus users = 36.
(iv) Equal modes: Bus and Two-wheeler (36 children each).
Question 6 (Page 68):
Three workers can paint a fence in 4 days. If one more worker joins the team, how many days will it take them to finish? State assumptions.
Three workers can paint a fence in 4 days. If one more worker joins the team, how many days will it take them to finish? State assumptions.
Worker-Days Calculation:
- Total work required = 3 workers × 4 days = 12 worker-days.
- New team size = 3 + 1 = 4 workers.
- Days required = Total Work ÷ Team Size = 12 / 4 = 3 days.
- Assumptions: All workers work at the same uniform rate and efficiency.
Final Answer: The team will take 3 days to paint the fence.
Question 7 (Page 68):
It takes 6 hours to fill 2 tanks of the same size with a pump. How long will it take to fill 5 such tanks with the same pump?
It takes 6 hours to fill 2 tanks of the same size with a pump. How long will it take to fill 5 such tanks with the same pump?
Direct Time Scaling:
- Time to fill 1 tank = 6 hours / 2 = 3 hours.
- Time to fill 5 tanks = 5 × 3 hours = 15 hours.
Final Answer: It will take 15 hours to fill 5 tanks.
Part C: Production, Dual Rates & Travel Kinematics (Pages 68–69)
Question 8 (Page 68):
Chairs are arranged in 25 rows with 12 chairs in each row. If rearranged with 20 chairs in each row, how many rows will there be?
Chairs are arranged in 25 rows with 12 chairs in each row. If rearranged with 20 chairs in each row, how many rows will there be?
Constant Total chairs Equation:
- Total number of chairs = 25 × 12 = 300 chairs.
- New number of rows = 300 / 20 = 15 rows.
Final Answer: The new arrangement will have 15 rows.
Question 9 (Page 68):
A school has 8 periods a day, each of 45 minutes duration. How long is each period if the school has 9 periods a day, assuming the total school instructional time remains unchanged?
A school has 8 periods a day, each of 45 minutes duration. How long is each period if the school has 9 periods a day, assuming the total school instructional time remains unchanged?
Inverse Proportionality Formulation:
- Total daily school instructional time = 8 × 45 minutes = 360 minutes (6 hours).
- If divided into 9 equal periods:
Duration of 1 period = 360 / 9 = 40 minutes.
Final Answer: Each period will be 40 minutes long.
Question 10 (Page 68 – Dual Pump Rate Addition):
A small pump can fill a tank in 3 hours, while a large pump can fill the same tank in 2 hours. If both pumps are used together, how long will the tank take to fill?
A small pump can fill a tank in 3 hours, while a large pump can fill the same tank in 2 hours. If both pumps are used together, how long will the tank take to fill?
Topper’s Reciprocal Rate Method:
- Let the volume of the tank be 1 unit.
- Rate of small pump = 1/3 tank per hour.
- Rate of large pump = 1/2 tank per hour.
- Combined rate when running together:
Rate = 1/3 + 1/2 = 2/6 + 3/6 = 5/6 tank per hour. - Time required = Total Work ÷ Combined Rate = 1 ÷ (5/6) = 6/5 hours.
- Converting to hours and minutes:
6/5 hours = 1.2 hours = 1 hour + (0.2 × 60) minutes = 1 hour 12 minutes (72 minutes).
Final Answer: Both pumps together will fill the tank in 1 hour 12 minutes (or 1⅕ hours).
Question 11 (Page 68 – Factory Toy Machinery):
A factory requires 42 machines to produce a given number of toys in 63 days. How many machines are required to produce the same number of toys in 54 days?
A factory requires 42 machines to produce a given number of toys in 63 days. How many machines are required to produce the same number of toys in 54 days?
Machine-Days Constant Formulation:
- Total workload = 42 machines × 63 days = 2,646 machine-days.
- Target time = 54 days.
- Number of machines required = 2646 / 54:
2646 ÷ 54 = 49 machines.
Final Answer: Exactly 49 machines are required.
Question 12 (Page 68 – Speed vs. Time):
A car takes 2 hours to reach a destination travelling at 60 km/h. How long will the car take if it travels at 80 km/h?
A car takes 2 hours to reach a destination travelling at 60 km/h. How long will the car take if it travels at 80 km/h?
Distance Invariant Calculation:
- Distance = Speed × Time = 60 km/h × 2 h = 120 km.
- At new speed of 80 km/h:
Time = Distance / New Speed = 120 / 80 = 1.5 hours. - 1.5 hours = 1 hour 30 minutes.
Final Answer: The car will take 1.5 hours (1 hour 30 minutes).