Chapter 3 exercise answers: Proportional Reasoning-2

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1. The ratio 3 : 4 : 3 : 5 has 3 + 4 + 3 + 5 = 15 equal parts in all.
2. The whole session is 150 minutes, so 1 part = 150 ÷ 15 = 10 minutes.
3. Warm-up/cool-down = 3 parts = 3 × 10 = 30 minutes.
4. Batting = 4 parts = 4 × 10 = 40 minutes.
5. Bowling = 3 parts = 3 × 10 = 30 minutes.
6. Fielding = 5 parts = 5 × 10 = 50 minutes.
7. Check: 30 + 40 + 30 + 50 = 150 minutes, which matches the session length.

AnswerWarm-up/cool-down 30 min, batting 40 min, bowling 30 min, fielding 50 min.

Watch this explained “The sum of the terms counts parts”, 0:34 into Dividing a quantity in a given ratio · हिंदी में देखें

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1. 288 Odiya books is only ONE share of the ratio, not the whole library — the ratio's 3 stands for Odiya.
2. 1 share = 288 ÷ 3 = 96 books.
3. Hindi is 2 shares = 2 × 96 = 192 books.
4. English is 1 share = 1 × 96 = 96 books.

Answer192 Hindi books and 96 English books.

Watch this explained “A part, not the whole”, 4:56 into Dividing a quantity in a given ratio · हिंदी में देखें

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1. The ratio 4 : 3 : 2 : 1 has 4 + 3 + 2 + 1 = 10 parts, and there are 100 coins in all, so 1 part = 100 ÷ 10 = 10 coins.
2. ₹10 coins = 4 parts = 40 coins; ₹5 coins = 3 parts = 30 coins; ₹2 coins = 2 parts = 20 coins; ₹1 coins = 1 part = 10 coins.
3. Value: ₹10 × 40 = ₹400; ₹5 × 30 = ₹150; ₹2 × 20 = ₹40; ₹1 × 10 = ₹10.
4. Total value = 400 + 150 + 40 + 10 = ₹600.

Answer₹600 in all.

Watch this explained “Counts and values”, 8:27 into Dividing a quantity in a given ratio · हिंदी में देखें

Open NCERT p. 60One way to think about it

1. A ratio 3 : 4 : 5 only says how the sides compare. It does not say how long they are, so we may choose the size. Take 1 part = 1 cm: the sides are 3 cm, 4 cm and 5 cm. (Any other size, such as 1 part = 2 cm, would also do.)
2. Construction: draw a line segment AB = 5 cm. With centre A and radius 3 cm draw an arc. With centre B and radius 4 cm draw another arc cutting the first arc at C. Join AC and BC. Triangle ABC has sides 3 cm, 4 cm and 5 cm.
3. Now take 1 part = 2 cm instead: the sides are 6 cm, 8 cm and 10 cm. Since 6 : 8 : 10 = 3 : 4 : 5, this triangle also has its sides in the ratio 3 : 4 : 5.
4. Both triangles have the same shape, but every side of the second is twice as long, so they are not the same size.
5. Congruent triangles must be exactly the same size as well as the same shape. So two triangles with sides in the ratio 3 : 4 : 5 are congruent only if their actual side lengths are also equal.

In shortNo. All triangles with sides in the ratio 3 : 4 : 5 have the same shape, but the ratio does not fix their size — for example 3 cm, 4 cm, 5 cm and 6 cm, 8 cm, 10 cm — so they need not be congruent.

Watch this explained “The same ratio as sides”, 7:29 into Dividing a quantity in a given ratio · हिंदी में देखें

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1. Take any scale factor x, so the three sides would be x, 3x and 5x.
2. In any triangle, the sum of any two sides must be greater than the third side.
3. Here the two smaller sides add up to x + 3x = 4x, which is LESS than the third side 5x.
4. Since the two smaller sides can never reach the longest side, no triangle with this ratio of sides can exist, for any size.

AnswerNo — such a triangle cannot be constructed, because 1 + 3 = 4 is less than 5, so the triangle inequality fails.

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Open NCERT p. 62One way to think about it

1. Winter voters = total − summer − rainy = 360 − 90 − 120 = 150.
2. A full circle is 360°, and here the total number of people is also 360, so each person is worth exactly 360° ÷ 360 = 1°.
3. Angle for summer = 90 × 1° = 90°.
4. Angle for rainy = 120 × 1° = 120°.
5. Angle for winter = 150 × 1° = 150°.
6. Check: 90° + 120° + 150° = 360°, a full circle.

7. Season	People	Angle
   Summer	90	90°
   Rainy	120	120°
   Winter	150	150°

8. Draw a circle, mark the centre, draw one radius. Using a protractor at the centre, mark off 90° for summer, then 120° for rainy, then the remaining 150° for winter, and label each sector.

In shortSummer 90°, Rainy 120°, Winter 150° — three sectors that close up to a full circle.

Watch this explained “Nine degrees a student”, 1:20 into Building a pie chart: turning counts into angles · हिंदी में देखें

Open NCERT p. 62One way to think about it

1. The percentages already add to 50 + 25 + 15 + 10 = 100%, so they cover the whole circle.
2. Each 1% of viewers is worth 360° ÷ 100 = 3.6° of the circle.
3. Entertainment = 50 × 3.6° = 180°.
4. Sports = 25 × 3.6° = 90°.
5. News = 15 × 3.6° = 54°.
6. Information = 10 × 3.6° = 36°.

7. Channel type	Percentage	Angle
   Entertainment	50%	180°
   Sports	25%	90°
   News	15%	54°
   Information	10%	36°

8. Draw a circle with one radius, then mark off 180°, 90°, 54° and 36° in turn with a protractor, labelling each sector as you go.

In shortEntertainment 180°, Sports 90°, News 54°, Information 36°.

Watch this explained “Whatever the whole is”, 8:40 into Building a pie chart: turning counts into angles · हिंदी में देखें

Open NCERT p. 62One way to think about it

1. Ask every student in your class to choose exactly ONE favourite subject from the seven in the table, and write the counts in the table.
2. Add all seven counts. The total should be the number of students in your class.
3. For each subject, angle = (number of students who chose it ÷ total number of students) × 360°. If an angle does not come out a whole number, round it to the nearest degree.
4. Your class's numbers will be different, so there is no single answer. Here is one example, for a class of 40 students (360° ÷ 40 = 9° for each student):

5. Subject	Number of students	Angle
   Language	6	54°
   Arts Education	4	36°
   Vocational Education	3	27°
   Social Science	5	45°
   Physical Education	7	63°
   Maths	9	81°
   Science	6	54°
   Total	40	360°

6. Draw a circle, mark its centre and draw one radius. With the protractor at the centre, measure the first angle from this radius and draw the next radius. Measure each following angle from the radius you have just drawn.
7. The last radius should land back on the first one, because the angles add up to 360°. Label each slice with its subject.

In shortThere is no single answer — it depends on your class's data. Find each angle as (count ÷ total) × 360°; for example, in a class of 40 with 9 students choosing Maths, the Maths slice is 81°.

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1. x and y are in inverse proportion only if the product x × y is the SAME for every pair in the table.

2. Table	Products x × y
   (i)	40×20=800, 80×10=800, 25×32=800, 16×50=800
   (ii)	40×20=800, 80×10=800, 25×12.5=312.5, 16×8=128
   (iii)	30×15=450, 90×5=450, 150×3=450, 10×45=450

3. (i) All four products equal 800, so x and y ARE in inverse proportion.
4. (ii) The first two products are 800, but the third and fourth are 312.5 and 128 — not all equal, so x and y are NOT in inverse proportion, even though the first two columns look the same as in (i).
5. (iii) All four products equal 450, so x and y ARE in inverse proportion.

Answer(i) Yes. (ii) No. (iii) Yes.

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1. Since x and y are in inverse proportion, x × y is the same constant k for every column.
2. The first column gives k: 16 × 9 = 144.
3. When x = 12: y = 144 ÷ 12 = 12.
4. When y = 48: x = 144 ÷ 48 = 3.
5. When x = 36: y = 144 ÷ 36 = 4.

6. x	16	12	3	36
   y	9	12	48	4

Answery = 12 when x = 12; x = 3 when y = 48; y = 4 when x = 36.

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Open NCERT p. 67Checked by computer

1. A pair is in inverse proportion when one quantity going up makes the other come down by the same factor, i.e. their PRODUCT stays fixed.
2. A pair is in direct proportion instead when their QUOTIENT (rate) stays fixed — then it is NOT inverse.
3. (i) More taps fill the tank in less time (each tap has a fixed rate) — product (taps × time) is fixed. Inverse: yes.
4. (ii) More painters finish the fixed wall in fewer days — product (painters × days) is fixed. Inverse: yes.
5. (iii) More petrol lets the car travel further at the same mileage — it is the QUOTIENT (distance ÷ petrol) that stays fixed. Inverse: no (direct).
6. (iv) A faster cyclist covers the same fixed route in less time — product (speed × time) is fixed. Inverse: yes.
7. (v) More cloth costs proportionately more at the same rate per metre — the QUOTIENT (price ÷ length) stays fixed. Inverse: no (direct).
8. (vi) More pages take proportionately more time at the same reading speed — the QUOTIENT (time ÷ pages) stays fixed. Inverse: no (direct).

Answer(i) Yes. (ii) Yes. (iii) No. (iv) Yes. (v) No. (vi) No.

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1. This is direct proportion: the cost per pencil stays fixed, so fewer pencils cost proportionately less.
2. Cost of 1 pencil = ₹120 ÷ 24 = ₹5.
3. Cost of 20 pencils = 20 × ₹5 = ₹100.

Answer₹100.

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Open NCERT p. 67One way to think about it

1. More families sharing the same tank means the water runs out sooner, so families and days are in inverse proportion: (families) × (days) stays constant.
2. 20 families × 6 days = 120 family-days of water in the tank.
3. With 10 more families, there are 20 + 10 = 30 families.
4. New number of days = 120 ÷ 30 = 4 days.
5. Assumption: every family uses the same, constant amount of water each day, and the tank is fully topped up with no extra refills or leaks.

In shortThe water will last 4 days, assuming every family uses the same fixed amount of water per day.

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1. Each ring is a pie chart of one whole day. A full turn, 360°, stands for 24 hours, so 360° ÷ 24 = 15° stands for 1 hour.
2. The book does not say which colour is sleep. Look at the dark blue part of each ring: it is smallest for the giraffe and largest for the bat, and read as hours it comes close to the numbers in the list. The orange parts would give about 21 hours for the giraffe, which is not in the list. So the dark blue part shows sleep.
3. Put the rings in order of their blue part, smallest first: giraffe, elephant, child, dog, cat, squirrel, snake, bat. Put the list in order: 2.5, 3.5, 8, 10.5, 13, 15, 18, 20. Match them in that order.
4. Check with angles (hours × 15°): the child's blue part is 8 × 15° = 120°, one third of the ring; the snake's is 18 × 15° = 270°, three quarters of the ring; the cat's 13 hours is 195°, just over half.

5. Ring	Hours of sleep	Blue angle for those hours (hours × 15°)
   Top row, 1st (giraffe)	2.5	37.5°
   Top row, 2nd (elephant)	3.5	52.5°
   Top row, 3rd (child)	8	120°
   Top row, 4th (dog)	10.5	157.5°
   Bottom row, 1st (cat)	13	195°
   Bottom row, 2nd (squirrel)	15	225°
   Bottom row, 3rd (snake)	18	270°
   Bottom row, 4th (bat)	20	300°

AnswerGiraffe 2.5 hours, elephant 3.5, child 8, dog 10.5; cat 13, squirrel 15, snake 18, bat 20.

Watch this explained “Whatever the whole is”, 8:40 into Building a pie chart: turning counts into angles · हिंदी में देखें

Open NCERT p. 68Checked by computerReads two ways: both answers shown

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

1. The angle for Car is not printed. The angles at the centre add up to 360°, so Car = 360° − (90° + 120° + 60° + 60°) = 360° − 330° = 30°.
2. Bus has the biggest angle, 120°.

AnswerBus

(ii) What fraction of children travel by car?

1. Fraction travelling by car = 30/360 = 1/12.

Answer1/12 of the children travel by car.

(iii) If 18 children travel by car, how many children took part

1. 1/12 of all the children = 18, so the number of children = 18 × 12 = 216.
2. Walk: 216 × 90/360 = 54, Bus: 72, Two-wheeler: 36, Cycle: 36, Car: 18. Together 54 + 72 + 36 + 36 + 18 = 216, the whole survey.
3. Reading 1 (the chart covers every child surveyed, and there is no taxi sector). We lead with this because it is what the chart literally shows; the book gives no answer key. Number using taxis = 216 − 216 = 0.
4. Reading 2 (taxi was not one of the choices in the survey; a taxi ride might even have been counted under Car): the chart cannot tell us a taxi number.

Answer216 children took part. Taxis: read as the chart covering every child, 0; read as taxi not being a choice in the survey, the number is not shown by the chart.

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

1. Two-wheeler and Cycle both have 60°, so each has 36 children.

AnswerTwo-wheeler and Cycle

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Open NCERT p. 68One way to think about it

1. More workers finish the same fence sooner, so workers and days are in inverse proportion: (workers) × (days) is fixed.
2. 3 workers × 4 days = 12 worker-days of painting needed for the fence.
3. With one more worker, there are 3 + 1 = 4 workers.
4. New number of days = 12 ÷ 4 = 3 days.
5. Assumption: every worker paints at the same steady rate, and they all work full days with nothing slowing them down by working together.

In short4 workers will finish the fence in 3 days, assuming every worker paints at the same constant rate.

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1. The same pump fills tanks at a fixed rate, so more tanks need proportionately more time — this is direct proportion.
2. Time for 1 tank = 6 hours ÷ 2 = 3 hours.
3. Time for 5 tanks = 5 × 3 = 15 hours.

Answer15 hours.

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1. The total number of chairs stays the same, so (rows) × (chairs per row) is fixed — this is inverse proportion.
2. Total chairs = 25 × 12 = 300.
3. New number of rows = 300 ÷ 20 = 15.

Answer15 rows.

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1. The total minutes of school in a day stays fixed, so (number of periods) × (minutes per period) is constant — inverse proportion.
2. Total school minutes = 8 × 45 = 360 minutes.
3. New duration of each period = 360 ÷ 9 = 40 minutes.

Answer40 minutes.

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1. Work out what fraction of the tank each pump fills in 1 hour: the small pump does 1/3 of the tank per hour, the large pump does 1/2 of the tank per hour.
2. Working together, in 1 hour they fill 1/3 + 1/2 = 2/6 + 3/6 = 5/6 of the tank.
3. If 5/6 of the tank takes 1 hour, the whole tank takes 1 ÷ (5/6) = 6/5 hours.

Answer6/5 hours (that is, 1 hour 12 minutes).

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1. The same number of toys is made either way, so (machines) × (days) is a fixed 'machine-days' amount of work — inverse proportion.
2. Machine-days needed = 42 × 63 = 2646.
3. Machines needed for 54 days = 2646 ÷ 54 = 49.

Answer49 machines.

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1. The distance to the destination stays fixed, so (speed) × (time) is constant — inverse proportion.
2. Distance = 60 km/h × 2 h = 120 km.
3. Time at 80 km/h = 120 ÷ 80 = 3/2 hours.

Answer3/2 hours (that is, 1 hour 30 minutes).

Watch this explained “One line each”, 7:34 into When one quantity rises and the other falls by the inverse factor · हिंदी में देखें