ANSWERs for figure it out Class 8 Mathematics – NCERT (Ganita Prakash) Part 2 Chapter 3: PROPORTIONAL REASONING–2

 ANSWERs for figure it out Class 8 Mathematics – NCERT (Ganita Prakash) Part 2  

Chapter 3: PROPORTIONAL REASONING–2 

Example 1: 

To make a special shade of purple, paint must be mixed in the ratio, Red : Blue : White :: 2 : 3 : 5. If Yasmin has 10 litres of white paint, how many litres of red and blue paint should she add to get the same shade of purple? 

In the ratio 2 : 3 : 5, the white paint corresponds to 5 parts. If 5 parts is 10 litres, 1 part is 10 ÷ 5 = 2 litres. Red = 2 parts = 2 × 2 = 4 litres. 

Blue = 3 parts = 3 × 2 = 6 litres. 

So, the purple paint will have 4 litres of red, 6 litres of blue, and 10 litres of white paint. 

What is the total volume of this purple paint?

 The total volume of purple paint is 4 + 6 + 10 = 20 litres.

Example 2: 

Cement concrete is a mixture of cement, sand, and gravel, and is widely used in construction. The ratio of the components in the mixture varies depending on how strong the structure needs to be. For structures that need greater strength like pillars, beams, and roofs, the ratio is 1 : 1.5 : 3, and the construction is also reinforced with steel rods. Using this ratio, if we have 3 bags of cement, how many bags of concrete mixture can we make? 

The concrete mixture is in the ratio Bags of cement : bags of sand : bags of gravel :: 1 : 1.5 : 3. 

If we have 3 bags of cement, we have to multiply the other terms by 3. 

So, the ratio is cement : sand : gravel :: 3 : 4.5 : 9. 

In total, we have 3 + 4.5 + 9 = 16.5 bags of concrete

Example 3: For some construction, 110 units of concrete are needed. How many units of cement, sand, and gravel are needed if these are to be mixed in the ratio 1 : 1.5 : 3? 

For 1 unit of cement, 

1+1.5 + 3 =5.5

110 ÷ 5.5 = 20

20. 1 × 20 = 20 units of cement, 

1.5 × 20 = 30 units of sand, 

and 3 × 20 = 60 units of gravel.

Example 4: You get a particular shade of purple paint by mixing red, blue, and white paint in the ratio 2 : 3 : 5. If you need 50 ml of purple paint, how many ml of red, blue, and white paint will you mix together? 

Example 5: Construct a triangle with angles in the ratio 1 : 3 : 5.

 The sum of the angles in a triangle is 180°. 

So the angles are 

Figure it Out Page number 60

1. 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?

Given: Ratio of time = 3 : 4 : 3 : 5, Total time = 150 minutes

Step 1: Add ratio terms 3 + 4 + 3 + 5 = 15

Step 2: Find value of 1 part 150 ÷ 15 = 10 minutes

Step 3: Calculate time for each activity

• Warm-up/Cool-down = 3 × 10 = 30 minutes

• Batting = 4 × 10 = 40 minutes

• Bowling = 3 × 10 = 30 minutes

• Fielding = 5 × 10 = 50 minutes

2. 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?

Given: Odiya : Hindi : English = 3 : 2 : 1, Odiya books = 288

Step 1: Value of 1 part 288 ÷ 3 = 96

Step 2: Find other quantities

• Hindi books = 2 × 96 = 192

• English books = 1 × 96 = 96

3. 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?

Given: Coin ratio = 4 : 3 : 2 : 1, Total coins = 100

Step 1: Sum of ratio = 10

Step 2: Value of 1 part 100 ÷ 10 = 10

Step 3: Number and value of coins

• ₹10 coins = 40 → ₹400

• ₹5 coins = 30 → ₹150

• ₹2 coins = 20 → ₹40

• ₹1 coins = 10 → ₹10

Total money = ₹600

 4. 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? 

Triangle with sides 3 : 4 : 5

Answer: Yes, such a triangle can be constructed.

Explanation: 3 + 4 > 5, 4 + 5 > 3, 3 + 5 > 4

All triangle inequalities are satisfied.

Congruency: All such triangles are not congruent, but similar, because actual side lengths may differ.

 5. Can you construct a triangle with sidelengths in the ratio 1 : 3 : 5? Why or why not?

Triangle with sides 1 : 3 : 5

Answer: Cannot be constructed.

Reason: 1 + 3 = 4 < 5

This violates the triangle inequality rule.

Figure it Out Page number 62

1. 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. 

Given

Total number of people = 360
Summer = 90
Rainy = 120
Winter = 360 − (90 + 120) = 150
Total angle of a circle = 360°
$$\text{Angle} = \frac{\text{Number of people}}{\text{Total people}} \times 360^\circ$$
Summer
$$\frac{90}{360} \times 360^\circ = 90^\circ$$
Rainy
$$\frac{120}{360} \times 360^\circ = 120^\circ$$
Winter
$$\frac{150}{360} \times 360^\circ = 150^\circ$$

• Draw a circle with centre O.
• Draw a radius OA.
• Using a protractor:
• Measure 90° from OA and draw radius OB → Summer
• From OB, measure 120° and draw radius OC → Rainy
• The remaining sector (150°) represents Winter
• Summer → 90 people → 90°
• Rainy → 120 people → 120°
• Winter → 150 people → 150°
• Label each sector clearly.

• Shade or colour each sector differently.

Season	Number of People
Summer	90
Rainy	120
Winter	150
Total	360

2. Draw a pie chart based on the following information about viewers᾿ favourite type of TV channel: Entertainment — 50%, Sports — 25%, News — 15%, Information — 10%.  

3. Prepare a pie chart that shows the favourite subjects of the students in your class. You can collect the data of the number of students for Proportional Reasoning–2 each subject shown in the table (each student should choose only one subject). Then write these numbers in the table and construct a pie chart:

Subject	Number of Students
Mathematics	10
Science	8
Social Science	6
Language                     Arts                                3	3
Physical Education	5
Vocational Education	5
Total	40

$$\text{Angle} = \frac{\text{Number of students}}{40} \times 360^\circ$$

• Mathematics

$$\frac{10}{40} \times 360^\circ = 90^\circ$$

• Science

$$\frac{8}{40} \times 360^\circ = 72^\circ$$

• Social Science

$$\frac{6}{40} \times 360^\circ = 54^\circ$$

• Language Arts 

340×360∘=27∘\frac{3}{40} \times 360^\circ = 27^\circ

• Arts Education   

340×360∘=27∘\frac{3}{40} \times 360^\circ = 27^\circ

• Physical Education

$$\frac{5}{40} \times 360^\circ = 45^\circ$$

• Vocational Education : $$\frac{5}{40} \times 360^\circ = 45^\circ$$ $$90^\circ + 72^\circ + 54^\circ + 54^\circ + 45^\circ + 45^\circ = 360^\circ$$

Pie Chart Construction

1. Draw a circle with centre O.

2. Draw a radius OA.

3. Using a protractor, draw sectors with angles:

   • Mathematics – 90°

   • Science – 72°

   • Social Science – 54°

   • Language   27°

   • Arts – 27°

   • Physical Education – 45°

   • Vocational Education – 45°

4. Label each sector and colour them neatly.

Figure it Out page number 64

1. Which of these are in inverse proportion?

x	40	80	25	16
y	20	10	32	50

• $40\times 20=800$

• $80\times 10=800$

• $25\times 32=800$

• $16\times 50=800$

 All products are equal.

(i) is in inverse proportion

x	40	80	25	16
y	20	10	12.5	8

• $40\times 20=800$

• $80\times 10=800$

• $25\times 12.5=312.5$

• $16\times 8=128$

 Products are not equal.

(ii) is NOT in inverse proportion

x	30	90	150	10
y	15	5	3	45

• $30\times 15=450$

• $90\times 5=450$

• $150\times 3=450$

• $10\times 45=450$

 All products are equal.

(iii) is in inverse proportion

(i) and (iii) are in inverse proportion.

2. Fill in the empty cells if x and y are in inverse proportion

$$x\times y=16\times 9=144$$

• When $x = 12$:

  $$y = \frac{144}{12} = 12$$

• When $y = 48$:

  $$x = \frac{144}{48} = 3$$

• When $x = 36$:

  $$y = \frac{144}{36} = 4$$

Figure it Out Page number 67-68

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 bought and the price paid at a fixed rate per metre. 

(vi) The number of pages in a book and the time required to read it at a fixed reading speed.

(i) Number of taps & time to fill a tank

✔ More taps → less time
Inverse proportion

(ii) Number of painters & days to paint a wall

✔ More painters → fewer days
Inverse proportion

(iii) Distance travelled & amount of petrol

 More petrol → more distance (both increase)

Not inverse (this is direct proportion)

(iv) Speed of cyclist & time for fixed distance

✔ More speed → less time
Inverse proportion

(v) Length of cloth & price (fixed rate)

 More cloth → more price
Not inverse (direct proportion)

(vi) Number of pages & time to read

 More pages → more time
Not inverse

Answer (Q1):(i), (ii), and (iv)

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

Cost ∝ number of pencils (direct proportion)

Cost of 1 pencil =

$$120÷24=5$$

Cost of 20 pencils =

$$20\times 5=\text{₹}100$$

Answer: ₹100

3. 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? 

20 families → 6 days
Total water = $20 × 6 = 120$ family-days

New families = $20 + 10 = 30$

Days water will last =

$$120 ÷ 30 = 4 \text{ days}$$

Assumptions:

• Each family uses the same amount of water per day

• No water is wasted or added

Answer: 4 days

4. Fill in the average number of hours each living being sleeps in a day by looking at the charts. Select the appropriate hours from this list : 15, 2.5, 20, 8, 3.5, 13, 10.5, 18

Using the given list: 15, 2.5, 20, 8, 3.5, 13, 10.5, 18

Typical matches:

Living Being (Image)	Average Sleeping Hours (per day)	Reason / Explanation
Human	8 hours	Normal adult sleep
Dog	10.5 hours	Dogs sleep long hours
Cat	13 hours	Cats sleep most of the day
Bat	20 hours	Nocturnal animal
Elephant	3.5 hours	Sleeps very little
Giraffe	2.5 hours	Light sleeper, short naps
Snake	18 hours	Cold-blooded, inactive most of the time
Squirrel	15 hours	Moderate sleeper

5. The pie chart on the right shows the result of a survey carried out to find the modes of transport used by children to go to school. Study the pie chart and answer the following questions

(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 which two modes of transport are equal numbers of children travelling? 

Given angles in the pie chart

• Bus = 120°
• Walk = 90°
• Cycle = 60°
• Two-wheeler = 60°
• Car = 30°
• (Total = 360°)

(i) What is the most common mode of transport?

The most common mode is the one with the largest angle.
Largest angle = 120° (Bus)
Answer: Bus

(ii) What fraction of children travel by car?

Fraction of children travelling by car

$$= \frac{\text{Angle for car}}{\text{Total angle}} = \frac{30^\circ}{360^\circ} = \frac{1}{12}$$

Answer: $\boxed{\tfrac{1}{12}}$

(iii) If 18 children travel by car, how many children took part in the survey?How many children use taxis (two-wheelers)?

Since car = $\frac{1}{12}$ of total children:

Total children

$$= 18 \times 12 = 216$$

Now, two-wheeler angle = 60°

Fraction for two-wheelers:

$$\frac{60}{360} = \frac{1}{6}$$

Children using two-wheelers:

$$\frac{1}{6} \times 216 = 36$$

Answers:

• Total children in survey = 216

• Children using two-wheelers = 36

(iv) By which two modes of transport are equal numbers of children travelling?

Equal numbers correspond to equal angles.

Cycle = 60°

Two-wheeler = 60°

Answer: Cycle and Two-wheeler

Question	Answer
(i)	Bus
(ii)	$\frac{1}{12}$
(iii)	Total = 216, Two-wheelers = 36
(iv)	Cycle and Two-wheeler

6. 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?

3 workers → 4 days

Total work = $3\times 4=12$ worker-days

New workers = 4

Days needed = $$12÷4=3\text{ days}$$

Assumptions:

• All workers work at the same speed

• Same working hours each day

Answer: 3 days

7. 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? 

2 tanks → 6 hours

1 tank → 3 hours

5 tanks → $$5\times 3=15\text{ hours}$$

 Answer: 15 hours

8. 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? 

Initial chair = 25×12=300

New rows = $$300÷20=15$$

Answer: 15 rows

9. 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? 

Original time = $$8\times 45=360\text{ minutes}$$

New period duration = $$360÷9=40\text{ minutes}$$

Answer: 40 minutes

10. 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? 

Small pump: 3 hrs → rate = $\frac{1}{3}$ tank/hr

Large pump: 2 hrs → rate = $\frac{1}{2}$ tank/hr

Combined rate = $$\frac{1}{3} + \frac{1}{2} = \frac{5}{6}$$

Time =$$1 ÷ \frac{5}{6} = \frac{6}{5} = 1.2 \text{ hours}$$

= 1 hour 12 minutes

Answer: 1 hour 12 minutes

11. 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? 

42 machines → 63 days

Work = $42\times 63$

Required machines for 54 days:

$$\frac{42 × 63}{54} = 49$$

Answer: 49 machines

12. 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?

Distance = $$60\times 2=120\text{ km}$$

New time = $$120÷80=1.5\text{ hours =}$$ 1 hour 30 minutes

Answer: 1.5 hours