CLASS 8 · GANITA PRAKASH PART II · CHAPTER 3

Proportional Reasoning-2

Use ratios to mix and divide quantities, build pie charts, and solve direct and inverse proportion problems.

39 worked answersAll 4 exercise sets

What you will learn

Scale ratios with two or more terms.
Divide a whole into proportional parts and construct pie-chart angles.
Recognise direct and inverse proportion and solve practical problems.

Read the NCERT questions ↗← All chapters

39 worked answers · short steps first, explanations when you need them

3.1–3.4 Ratios and sharing in a given ratio

Textbook pages 56–60

Question idli-mixtures

Viswanath mixes rice and urad dal in the ratio 6:3. Puneet uses 4:2. Would the mixtures have the same proportion?

Steps only

Compare 6/3 and 4/2.

Both simplify to 2.

Answer

Yes. Both mixtures use 2 parts rice for every 1 part dal.

Question map-scale-conversion

A map scale is 1:6,000,000. What actual distance does 1 cm on the map represent?

Steps only

1 cm on map represents 6,000,000 cm on the ground.

100,000 cm=1 km; 6,000,000÷100,000=60.

Answer

60 km. This is straight-line geographical distance, not road distance.

Question map-city-distance

Using the map printed in the book, estimate the straight-line distances Bengaluru–Chennai and Mangaluru–Chennai at scale 1:6,000,000.

Steps only

Measure each line on the printed map with a ruler.

For each measured centimetre, multiply by 60 km.

Answer

Multiply each measured map length by 60 km per centimetre. The book labels this map ‘not to scale’, so ruler-based answers are only rough estimates; an accurate distance needs a true-scale map.

Question map-classroom-activity

Make a classroom sketch at scale 1:50.

Steps only

At 1:50, 1 cm on the page represents 50 cm in the room.

Measure each room length and divide by 50 to find the drawing length in centimetres.

Mark the desk, board, fans and lights at measured positions.

Answer

This is a construction activity: for example, a 6 m wall is 600÷50=12 cm on the drawing. Your layout depends on your classroom measurements.

Question spice-ratio-scale

The spice mix uses coriander:red chilli:toor dal:methi = 8:4:2:1. If only 2 chillies are used, how much of each ingredient is needed?

Steps only

The chilli amount changes from 4 to 2, a scale factor of 1/2.

Multiply every ratio term by 1/2.

Answer

4 spoons coriander, 2 chillies, 1 spoon toor dal and 1/2 spoon methi; the ratio is 4:2:1:0.5.

Question purple-paint

Red:blue:white paint is 2:3:5. If there are 10 L of white paint, how much red and blue paint is needed, and what is the total?

Steps only

5 parts correspond to 10 L, so 1 part=2 L.

Red=2×2=4 L; blue=3×2=6 L.

Total=4+6+10.

Answer

4 L red, 6 L blue; 20 L total.

Question concrete-mixture

Cement:sand:gravel=1:1.5:3. How many bags of mixture can be made from 3 bags of cement?

Steps only

Scale factor=3÷1=3.

Sand=1.5×3=4.5 bags; gravel=3×3=9 bags.

Total=3+4.5+9.

Answer

16.5 bags in total: 3 cement, 4.5 sand and 9 gravel.

Question divide-110-concrete

Divide 110 units of concrete among cement, sand and gravel in the ratio 1:1.5:3.

Steps only

Total parts=1+1.5+3=5.5.

110÷5.5=20 units per part.

Amounts: 20, 30 and 60 units.

Answer

Cement 20 units, sand 30 units, gravel 60 units.

Question purple-paint-50ml

Make 50 mL of paint using red:blue:white=2:3:5.

Steps only

There are 10 parts altogether.

One part=50÷10=5 mL.

Answer

Red 10 mL, blue 15 mL and white 25 mL.

Question triangle-angle-ratio

A triangle’s angles are in the ratio 1:3:5. Find the three angles.

Steps only

Triangle angles total 180°.

There are 1+3+5=9 parts, so each part is 180÷9=20°.

Answer

20°, 60° and 100°.

Question cricket-session

A 150-minute practice is split warm-up/cool-down:batting:bowling:fielding=3:4:3:5. How many minutes for each?

Steps only

Total parts=3+4+3+5=15.

One part=150÷15=10 minutes.

Answer

30 minutes warm-up/cool-down, 40 batting, 30 bowling and 50 fielding.

Question library-books

Odia:Hindi:English books are in the ratio 3:2:1. There are 288 Odia books. Find the other two counts.

Steps only

3 parts=288, so 1 part=96.

Hindi=2×96; English=1×96.

Answer

192 Hindi books and 96 English books.

Question coin-value

There are 100 coins. ₹10:₹5:₹2:₹1 coins are in the ratio 4:3:2:1. Find their total value.

Steps only

Total ratio parts=10, so 100÷10=10 coins per part.

Counts are 40, 30, 20 and 10.

Value=40×10+30×5+20×2+10×1.

Answer

₹600.

Question side-ratio-triangles

If triangles have side lengths in the ratio 3:4:5, must they all be congruent?

Steps only

The side lengths can be 3,4,5 or 6,8,10 or 9,12,15.

These have the same ratio but different sizes.

Answer

No. They are similar in shape, but they need not be the same size, so they are not necessarily congruent.

Question impossible-triangle-ratio

Can a triangle have side lengths in the ratio 1:3:5?

Steps only

For a triangle, the sum of the two shorter sides must be greater than the longest side.

Here 1+3=4, which is less than 5.

Answer

No. The triangle inequality fails.

3.5 Pie charts

Textbook pages 60–62

Question grade-pie-angles

Forty students get grades A–E in counts 12, 10, 8, 6 and 4. Find each sector angle.

Steps only

A full circle is 360°. The total count is 40, so each student represents 360÷40=9°.

Multiply each grade count by 9°.

Answer

A:108°, B:90°, C:72°, D:54°, E:36°. The angles add to 360°.

Question season-pie

Of 360 people, 90 choose summer, 120 rainy and the rest winter. Find each sector angle.

Steps only

Winter count=360−90−120=150.

Since 360 people represent 360°, one person represents 1°.

Answer

Summer 90°, rainy 120°, winter 150°.

Question channel-pie

TV channel choices are entertainment 50%, sports 25%, news 15%, information 10%. Find sector angles.

Steps only

Multiply each percentage by 360°.

50%: 180°; 25%: 90°; 15%: 54°; 10%: 36°.

Answer

Entertainment 180°, sports 90°, news 54°, information 36°.

Question class-subject-pie

Collect each classmate’s favourite subject and draw a pie chart.

Steps only

Count each subject and find the total responses.

For each subject, calculate count÷total×360°.

Use a protractor to draw the sectors and label them.

Answer

Answers depend on your class survey. The sector angles should total 360°.

3.6 Direct and inverse proportion

Textbook pages 63–68

Question bricks-workers

5 workers move 4,500 bricks in one day. How many workers are needed to move 18,000 bricks in a day at the same rate?

Steps only

18,000÷4,500=4 times as many bricks.

Workers must also be multiplied by 4: 5×4.

Answer

20 workers.

Question travel-time-speed

A trip takes 3 hours at 30 km/h. How long at 60 km/h for the same route?

Steps only

Distance=30×3=90 km.

Time=90÷60.

Answer

1.5 hours.

Question inverse-tables-classify

For each table, test whether every x×y product is the same. (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.

Steps only

(i) products: 800,800,800,800.

(ii) products: 800,800,312.5,128.

(iii) products: 450 in every pair.

Answer

Tables (i) and (iii) show inverse proportion; table (ii) does not.

Question complete-inverse-table

x and y are inversely proportional. The table gives x=16, 12, 36 and y=9, blank, 48, blank. Fill the missing values.

Steps only

Constant product=16×9=144.

For x=12, y=144÷12=12. For y=48, x=144÷48=3.

Answer

The missing values are y=12 and x=3.

Question road-workers-days

20 workers lay a road in 4 days. How long will 10 workers take at the same rate?

Steps only

Total work=20×4=80 worker-days.

80÷10=8 days.

Answer

8 days.

Question tank-pumps

2 pumps fill a tank in 18 hours. How long do 4 identical pumps take?

Steps only

Pump-hours=2×18=36.

36÷4=9 hours.

Answer

9 hours.

Question food-students

Food feeds 80 students for 15 days. How many days will it feed 100 students?

Steps only

Food amount=80×15=1,200 student-days.

1,200÷100=12 days.

Answer

12 days.

Question working-together-cutting

Ram alone cuts a set of vegetables in 1 hour; Shyam alone takes 1.5 hours. How long together?

Steps only

Ram rate=1 job/hour.

Shyam rate=1÷1.5=2/3 job/hour.

Together they do 1+2/3=5/3 jobs per hour.

Time for one job=1÷(5/3)=3/5 hour.

Answer

3/5 hour, or 36 minutes.

Question inverse-examples

Which pairs are inversely proportional: (i) taps and time to fill a tank; (ii) painters and days for a fixed wall; (iii) distance travelled and petrol used at fixed mileage; (iv) cyclist speed and time for a fixed route; (v) cloth length and price at fixed rate; (vi) pages read and time at fixed reading speed?

Steps only

For a fixed task, increasing the first quantity reduces the second: (i), (ii), (iv).

For (iii), (v), and (vi), both quantities increase together.

Answer

Inverse pairs:

(i) ,
(ii) and

(iv).

Question pencils-cost

24 pencils cost ₹120. How much do 20 pencils cost?

Steps only

Unit price=120÷24=₹5.

20×5=₹100.

Answer

₹100.

Question water-families

Water feeds 20 families for 6 days. If 10 more families join, how long will it last, assuming each family uses the same amount each day?

Steps only

Total supply=20×6=120 family-days.

There are now 30 families, so 120÷30=4 days.

Answer

4 days, assuming daily use per family stays the same and there is no other water supply.

Question sleep-chart

Use the textbook’s animal sleep chart and place the listed values 15, 2.5, 20, 8, 3.5, 13, 10.5 and 18 hours beside the matching animals.

Steps only

One full ring represents 24 hours.

Hours asleep = shaded fraction × 24.

Match each ring from the smallest shaded part to the largest.

Living being	Approximate shaded fraction	Sleep in 24 hours
Giraffe	about 1/10	2.5 h
Elephant	about 1/7	3.5 h
Child	1/3	8 h
Dog	about 7/16	10.5 h
Cat	about 13/24	13 h
Monkey	5/8	15 h
Snake	3/4	18 h
Bat	5/6	20 h

Answer

Giraffe 2.5 h; elephant 3.5 h; child 8 h; dog 10.5 h; cat 13 h; monkey 15 h; snake 18 h; bat 20 h.

Question transport-pie

Use the school-transport pie chart to find the most common mode, the fraction travelling by car, the total surveyed when 18 children travel by car, the taxi count, and the two modes with equal counts.

Steps only

Known sectors: bus 120°, walk 90°, cycle 60°, two-wheeler 60°.

Car=360°−(120°+90°+60°+60°)=30°; car fraction=30/360=1/12.

Total=18×12=216; cycle and two-wheeler are equal (60° each).

Mode shown in chart	Sector angle
Bus	120°
Walk	90°
Cycle	60°
Two-wheeler	60°
Car	30° (remaining angle)
Taxi	Not shown

Answer

Bus is most common; car=1/12; total=216 children; taxi=0 in the chart; cycle and two-wheeler are equal.

Question fence-workers

3 workers paint a fence in 4 days. How long will 4 workers take, assuming equal work rates?

Steps only

Total work=3×4=12 worker-days.

12÷4=3 days.

Answer

3 days, assuming each worker paints at the same rate and the whole fence can be divided among them.

Question tanks-same-pump

A pump fills 2 equal tanks in 6 hours. How long for 5 such tanks?

Steps only

The pump fills 2 tanks in 6 hours, or 1/3 tank per hour.

5÷(1/3)=15 hours.

Answer

15 hours, assuming the pump rate remains constant and the tanks are filled one after another or continuously at that rate.

Question chair-rows

Chairs are arranged in 25 rows of 12. How many rows if each row has 20 chairs?

Steps only

Total chairs=25×12=300.

300÷20=15.

Answer

15 rows.

Question school-period-length

A school has 8 periods of 45 minutes. How long is each period with 9 periods over the same total time?

Steps only

Total class time=8×45=360 minutes.

360÷9=40 minutes.

Answer

40 minutes per period.

Question two-pumps-rates

A small pump fills a tank in 3 hours and a large pump in 2 hours. How long together?

Steps only

Small-pump rate=1/3 tank/hour; large-pump rate=1/2 tank/hour.

Combined rate=1/3+1/2=5/6 tank/hour.

Time=1÷(5/6)=6/5 hours.

Answer

1.2 hours, or 1 hour 12 minutes.

Question factory-machines

42 machines make a fixed number of toys in 63 days. How many machines are needed in 54 days at the same rate?

Steps only

Machine-days=42×63=2,646.

2,646÷54=49.

Answer

49 machines.

Question car-speed

A car takes 2 hours at 60 km/h. How long for the same journey at 80 km/h?

Steps only

Distance=60×2=120 km.

Time=120÷80=1.5 hours.

Answer

1.5 hours.