Class 8 Mathematics

Chapter 10 Solutions — Proportional Reasoning-2

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Solutions

Overview

Step-by-step NCERT solutions for Proportional Reasoning-2 (Chapter 10, NCERT Class 8 Mathematics) — every question and answer worked out in full, not just the final result. You can also read the Proportional Reasoning-2 textbook chapter.

Solved

What these solutions cover

All 21 questions in Proportional Reasoning-2 are solved in the PDF. Here's what's inside, exercise by exercise:

Dividing a Whole in a Given Ratio

A cricket coach schedules practice sessions that include different activities in a specific ratio — time for warm-up/cool-down : time for batting : time for bowling : time for fielding :: 3 : 4 : 3 : 5. If each session is 150 minutes long, how much time is spent on each activity?
A school library has books in different languages in the following ratio — no. of Odiya books : no. of Hindi books : no. of English books :: 3 : 2 : 1. If the library has 288 Odiya books, how many Hindi and English books does it have?
I have 100 coins in the ratio — no. of ₹10 coins : no. of ₹5 coins : no. of ₹2 coins : no. of ₹1 coins :: 4 : 3 : 2 : 1. How much money do I have in coins?
Construct a triangle with sidelengths in the ratio 3 : 4 : 5. Will all the triangles drawn with this ratio of sidelengths be congruent to each other? Why or why not?
Can you construct a triangle with sidelengths in the ratio 1 : 3 : 5? Why or why not?

A Slice of the Pie

A group of 360 people were asked to vote for their favourite season from the three seasons — rainy, winter and summer. 90 liked the summer season, 120 liked the rainy season, and the rest liked the winter. Draw a pie chart to show this information.
Draw a pie chart based on the following information about viewers' favourite type of TV channel: Entertainment — 50%, Sports — 25%, News — 15%, Information — 10%.
Prepare a pie chart that shows the favourite subjects of the students in your class. Collect the data of the number of students for each subject (Language, Arts Education, Vocational Education, Social Science, Physical Education, Maths, Science), write these numbers in the table, and construct a pie chart.

Inverse Proportions

1.
Which of these are in inverse proportion?
- (i) x: 40, 80, 25, 16 | y: 20, 10, 32, 50
- (ii) x: 40, 80, 25, 16 | y: 20, 10, 12.5, 8
- (iii) x: 30, 90, 150, 10 | y: 15, 5, 3, 45
Fill in the empty cells if x and y are in inverse proportion. x: 16, 12, ___, 36 y: 9, ___, 48, ___

Inverse Proportions (Applications)

1.
Which of the following pairs of quantities are in inverse proportion?
- (i) The number of taps filling a water tank and the time taken to fill it.
- (ii) The number of painters hired and the days needed to paint a wall of fixed size.
- (iii) The distance a car can travel and the amount of petrol in the tank.
- (iv) The speed of a cyclist and the time taken to cover a fixed route.
- (v) The length of cloth…
If 24 pencils cost ₹120, how much will 20 such pencils cost?
A tank on a building has enough water to supply 20 families living there for 6 days. If 10 more families move in there, how long will the water last? What assumptions do you need to make to work out this problem?
4.
The pie chart shows the result of a survey on modes of transport used by children to go to school (Walk: 90°, Bus: 120°, Cycle: 60°, Car: 60°, Two-wheeler: 30°). Answer:
- (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 children took part in the survey? How many children use taxis to travel to school?
- (iv) By…
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 the work? What are the assumptions you need to make?
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?
A given set of chairs are arranged in 25 rows, with 12 chairs in each row. If the chairs are rearranged with 20 chairs in each row, how many rows does this new arrangement have?
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 that the number of school hours per day stays the same?
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 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 car takes 2 hours to reach a destination, travelling at a speed of 60 km/h. How long will the car take if it travels at a speed of 80 km/h?

Sample

Worked examples

3 questions from Proportional Reasoning-2, solved step by step.

Dividing a Whole in a Given Ratio

A cricket coach schedules practice sessions that include different activities in a specific ratio — time for warm-up/cool-down : time for batting : time for bowling : time for fielding :: 3 : 4 : 3 : 5. If each session is 150 minutes long, how much time is spent on each activity?

The ratio of activities is 3 : 4 : 3 : 5.

Sum of ratio terms = 3 + 4 + 3 + 5 = 15.

Time for warm-up/cool-down = 150 × (3/15) = 150 × (1/5) = 30 minutes.

Time for batting = 150 × (4/15) = 40 minutes.

Time for bowling = 150 × (3/15) = 30 minutes.

Time for fielding = 150 × (5/15) = 50 minutes.

Verification: 30 + 40 + 30 + 50 = 150 minutes. ✓

Answer: Warm-up/cool-down: 30 min, Batting: 40 min, Bowling: 30 min, Fielding: 50 min

Inverse Proportions

Which of these are in inverse proportion? (i) x: 40, 80, 25, 16 | y: 20, 10, 32, 50 (ii) x: 40, 80, 25, 16 | y: 20, 10, 12.5, 8 (iii) x: 30, 90, 150, 10 | y: 15, 5, 3, 45

Two quantities are in inverse proportion if their product x × y is constant for all pairs.

(i) Check products:

40 × 20 = 800, 80 × 10 = 800, 25 × 32 = 800, 16 × 50 = 800.

All products are equal (800). ⇒ (i) IS in inverse proportion.

(ii) Check products:

40 × 20 = 800, 80 × 10 = 800, 25 × 12.5 = 312.5, 16 × 8 = 128.

Products are NOT all equal. ⇒ (ii) is NOT in inverse proportion.

(iii) Check products:

30 × 15 = 450, 90 × 5 = 450, 150 × 3 = 450, 10 × 45 = 450.

All products are equal (450). ⇒ (iii) IS in inverse proportion.

Answer: (i) and (iii) are in inverse proportion; (ii) is not.

Inverse Proportions (Applications)

If 24 pencils cost ₹120, how much will 20 such pencils cost?

Cost is directly proportional to the number of pencils (at a fixed price per pencil).

Cost per pencil = ₹120 ÷ 24 = ₹5.

Cost of 20 pencils = 20 × ₹5 = ₹100.

Answer: ₹100

All 21 questions in Proportional Reasoning-2 are solved step by step in the Solutions PDF above.

Keep solving

Chapter 10 Solutions — Proportional Reasoning-2

Overview

What these solutions cover

Dividing a Whole in a Given Ratio

A Slice of the Pie

Inverse Proportions

Inverse Proportions (Applications)

Worked examples

A cricket coach schedules practice sessions that include different activities in a specific ratio — time for warm-up/cool-down : time for batting : time for bowling : time for fielding :: 3 : 4 : 3 : 5. If each session is 150 minutes long, how much time is spent on each activity?

Which of these are in inverse proportion? (i) x: 40, 80, 25, 16 | y: 20, 10, 32, 50 (ii) x: 40, 80, 25, 16 | y: 20, 10, 12.5, 8 (iii) x: 30, 90, 150, 10 | y: 15, 5, 3, 45

If 24 pencils cost ₹120, how much will 20 such pencils cost?

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