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CBSE · Class 8 · Mathematics Part 2

Part 2, Chapter 3: Proportional Reasoning-2

5 concepts · 40 practice questions with answers · Free to read

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Proportion, unit rate and cross-multiplication

From daily life

Two cooks make the same sambar. One uses 6 spoons of masala for 4 litres, the other 9 spoons for 6 litres. Do they taste the same? Both use 1.5 spoons per litre, so yes.

Simple definition

Two ratios are in proportion when they are equal, a : b = c : d. Cross-multiplication checks it: a × d = b × c. The unit rate (the value for one unit) lets us scale up or down.

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Proportion, unit rate and cross-multiplication

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In the figure: Change one quantity and watch the other change by the same factor.

Rule 1: Check a proportion — Multiply across and compare the two products.

  • Easy: 3 : 5 = 12 : 20 is true: 3 × 20 = 60 = 5 × 12.
  • Medium: 6 : 9 = 10 : 16 is false: 6 × 16 = 96 but 9 × 10 = 90.
  • Exam-type: Is the table (2, 7), (4, 14), (6, 21) proportional? y ÷ x = 3.5 every time. Yes.

Rule 2: Find the missing term — Multiply the two terms that are diagonal, divide by the third.

  • Easy: 4 : 7 = 20 : ? gives 7 × 20 ÷ 4 = 35.
  • Medium: 9 : 2 = 45 : ? gives 2 × 45 ÷ 9 = 10.
  • Exam-type: If 5 : 8 = ? : 72, the missing term is 5 × 72 ÷ 8 = 45.

Rule 3: Unit rate method — First find the value of ONE, then multiply.

  • Easy: 12 notebooks cost ₹84: one costs ₹7; 5 cost ₹35.
  • Medium: Bus: 50 km each hour, so 7 hours is 350 km.
  • Exam-type: 5 kg of rice cost ₹275. Cost of 8 kg = 275 ÷ 5 × 8 = ₹440.
Remember

In a proportional relationship y ÷ x is always the same number, called the constant of proportionality or unit rate.

Common mistake

Adding the same number to both terms (3 : 5 and 5 : 7) and calling it proportional. Proportion needs multiplying by the same factor.

Shortcut

Cross, multiply, compare. Memory clue: "X marks the spot - cross products match".

Spot the difference

PointProportionalNot proportional
y ÷ xsame every timechanges
Example(2, 6), (3, 9)(2, 7), (3, 10)
Cross productsequaldifferent

Practice questions

Easy
  1. 1Is 6 : 9 = 10 : 15 true?
  2. 2Find ? in 3 : 5 = 12 : ?.
  3. 312 pens cost ₹84. Find the cost of one pen.
Medium
  1. 48 pens cost ₹120. Find the cost of 15 pens.
  2. 5Find ? in 9 : 2 = 45 : ?.
  3. 6A bus covers 50 km in 1 hour. How far in 7 hours?
Challenging
  1. 7Is the table (2, 7), (4, 15), (6, 21) proportional?
  2. 85 kg rice cost ₹275. Find the cost of 8 kg.
Show answers (try them yourself first)
  1. 1Yes (6 × 15 = 90 = 9 × 10)
  2. 220
  3. 3₹7
  4. 4₹225
  5. 510
  6. 6350 km
  7. 7No (15 ÷ 4 is not 3.5)
  8. 8₹440

Ratios in maps and scale

From daily life

On a road map 2 cm show a distance of 1 km on the ground. The map is a small copy of the real land, so every distance is shrunk by the same factor.

Simple definition

A scale 1 : n means 1 unit on the map stands for n of the same unit on the ground. Change both to the same unit first (1 km = 1,00,000 cm). A model scale works the same way.

2 cm1 : 500001 km100000 cm

In the figure: 2 cm on the map × 50000 = 100000 cm = 1 km.

Rule 1: Map to real — Multiply by the scale number, then change cm to km.

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Ratios in maps and scale

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In the figure: Change the map length and the scale to see the real distance.

  • Easy: 3 cm at 1 : 100000 is 3 km.
  • Medium: 8 cm at 1 : 50000 = 4,00,000 cm = 4 km.
  • Exam-type: 6 cm at 1 : 250000 = 15,00,000 cm = 15 km.

Rule 2: Real to map — Change km to cm, then divide by the scale number.

  • Easy: 12 km at 1 : 200000: 12,00,000 ÷ 200000 = 6 cm.
  • Medium: 2.5 km at 1 : 50000 = 5 cm.
  • Exam-type: 30 km at 1 : 1000000 = 3 cm.

Rule 3: Find the scale or a model — Scale = map length : real length in the same unit, then make the first term 1.

  • Easy: 5 cm = 25 km: 5 : 25,00,000 = 1 : 500000.
  • Medium: Model car 1 : 25, real length 450 cm: model is 18 cm.
  • Exam-type: Plan 1 : 1000, plot 3 cm × 2 cm is 30 m × 20 m on the ground.
Remember

A smaller second number (1 : 10000) means a bigger drawing with more detail than 1 : 100000.

Common mistake

Forgetting to change units: 3 cm at 1 : 100000 is 3,00,000 cm, which is 3 km, not 3,00,000 km.

Shortcut

Map x scale = real; real ÷ scale = map. Memory clue: "Map to Multiply, Real to Reduce" (divide).

Spot the difference

PointMap to realReal to map
Operation× scale number÷ scale number
Example3 cm → 3 km12 km → 6 cm
Result isbiggersmaller

Practice questions

Easy
  1. 1At 1 : 100000 a road is 3 cm long. Find its real length.
  2. 2At 1 : 200000 how long is 12 km on the map?
  3. 3At 1 : 100000, 1 cm stands for how many km?
Medium
  1. 4At 1 : 250000 find the real length of 6 cm.
  2. 55 cm on a map is 25 km. Find the scale.
  3. 6A model car has scale 1 : 25. The real car is 450 cm long. Find the model length.
Challenging
  1. 7Which is more detailed, 1 : 10000 or 1 : 100000?
  2. 8At 1 : 1000 a plot is drawn 3 cm by 2 cm. Find its real size.
Show answers (try them yourself first)
  1. 13 km
  2. 26 cm
  3. 31 km
  4. 415 km
  5. 51 : 500000
  6. 618 cm
  7. 71 : 10000
  8. 830 m × 20 m

Ratios with more than two terms

From daily life

A shop mixes dry fruits: 2 parts cashew, 3 parts almond and 5 parts raisin. One ratio 2 : 3 : 5 describes all three at once.

Simple definition

A ratio a : b : c compares three quantities. Divide every term by a common factor to simplify; multiply every term by the same number to get an equivalent ratio. Total parts = a + b + c.

2352 : 3 : 5 (10 parts)₹120 → 24 + 36 + 60

In the figure: Ten equal parts: 120 ÷ 10 = 12 per part, so 24, 36 and 60.

Rule 1: Simplify and share — Simplify, add the terms, find one part, multiply.

  • Easy: 12 : 20 : 28 simplifies to 3 : 5 : 7 (divide by 4).
  • Medium: ₹180 in 1 : 2 : 3: parts 6, one part ₹30, shares 30, 60, 90.
  • Exam-type: ₹200 in 2 : 5 : 3: parts 10, one part ₹20, shares 40, 100, 60.

Rule 2: Find the total from one part — Find the value of one part, then multiply by the total number of parts.

  • Easy: 3 : 4 : 5 with the second part 36: one part 9, total 9 × 12 = 108.
  • Medium: 2 : 3 : 7 with the first part 8: one part 4, total 48.
  • Exam-type: 4 : 5 : 6 with the last part 30: one part 5, total 75.

Rule 3: Join two ratios — Make the common quantity equal in both ratios, then write a : b : c.

  • Easy: a : b = 2 : 3 and b : c = 3 : 4 gives 2 : 3 : 4 (b already same).
  • Medium: a : b = 2 : 3 and b : c = 4 : 5: make b = 12: 8 : 12 : 15.
  • Exam-type: a : b = 5 : 2 and b : c = 3 : 4: make b = 6: 15 : 6 : 8.
Remember

The angles of a triangle in the ratio 2 : 3 : 4: parts 9, one part 20°, angles 40°, 60°, 80°.

Common mistake

Using the ratio terms as the actual amounts, or using only two of the three terms when adding the parts.

Shortcut

Total parts first, one part next. Memory clue: "Add, Divide, Multiply" - ADM.

Spot the difference

PointTwo-term ratioThree-term ratio
Example3 : 52 : 3 : 5
Total parts810
Shares of 12045, 7524, 36, 60

Practice questions

Easy
  1. 1Simplify 9 : 36 : 45.
  2. 2Share ₹120 in 2 : 3 : 5. Find the largest share.
  3. 3Find an equivalent ratio of 2 : 3 : 5 with first term 4.
Medium
  1. 4In 3 : 4 : 5 the middle part is 36. Find the total.
  2. 5a : b = 2 : 3 and b : c = 4 : 5. Find a : b : c.
  3. 6A drink has liquids in 3 : 2 : 1. How much of the second liquid is in 24 litres?
Challenging
  1. 7The angles of a triangle are in the ratio 2 : 3 : 4. Find the largest angle.
  2. 8a : b = 3 : 5 and b : c = 10 : 7. Find a : b : c.
Show answers (try them yourself first)
  1. 11 : 4 : 5
  2. 2₹60
  3. 34 : 6 : 10
  4. 4108
  5. 58 : 12 : 15
  6. 68 litres
  7. 780°
  8. 86 : 10 : 7

Dividing a whole in a given ratio and pie charts

From daily life

Two friends put money into a shop in the ratio 4 : 5 and must share ₹450 profit in the same ratio. A pie chart shows such shares as slices of a circle.

Simple definition

To divide a whole in a ratio: share = whole ÷ total parts × the part. In a pie chart the whole circle is 360°, so a slice has angle = (fraction or percentage) × 360°, and percentage = angle ÷ 360 × 100.

72°108°180°

In the figure: Shares 2 : 3 : 5 become 20%, 30% and 50%, that is 72°, 108° and 180°.

Rule 1: Divide in a ratio — Add the parts, divide the whole by the total, multiply.

  • Easy: ₹450 in 4 : 5: ₹50 per part; shares ₹200 and ₹250.
  • Medium: ₹840 in 3 : 4: ₹120 per part; shares ₹360 and ₹480.
  • Exam-type: 600 m² in 7 : 3: 60 per part; areas 420 m² and 180 m².

Rule 2: Fraction or percentage to angle — Multiply the fraction (or percentage) by 360°.

Live figure

Dividing a whole in a given ratio and pie charts

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In the figure: See how a percentage of the whole is shaded.

  • Easy: 1/4 of 360° = 90°; 1/6 of 360° = 60°.
  • Medium: 40% of 360° = 144°.
  • Exam-type: 30 of 120 students like cricket: 30/120 × 360° = 90°.

Rule 3: Angle to percentage — Divide the angle by 360 and multiply by 100. All angles add up to 360°.

  • Easy: 72° is 72/360 × 100 = 20%.
  • Medium: Two slices 100° and 120°: the third is 360° − 220° = 140°.
  • Exam-type: Slices 20%, 35%, 25% and the rest: rest 20% = 72°.
Remember

Check a pie chart: the percentages add to 100% and the angles add to 360°.

Common mistake

Using 100 instead of 360 for the angle of a slice. 25% of the circle is 90°, not 25°.

Shortcut

Whole circle = 360°. Memory clue: "360 for a full circle; 1% = 3.6°".

Spot the difference

PointRatio sharingPie chart
Whole isthe amount360°
One partwhole ÷ total parts360° ÷ total parts
Example₹450, 4 : 52 : 3 : 5 → 72°, 108°, 180°

Practice questions

Easy
  1. 1Divide ₹450 in the ratio 4 : 5. Find the larger share.
  2. 2Find the angle for 1/4 of a pie chart.
  3. 3Find the angle of a 40% slice.
Medium
  1. 4Divide ₹840 in 3 : 4. Find the smaller share.
  2. 5A slice has angle 72°. What percentage is it?
  3. 618 of 60 students like football. Find the angle of that slice.
Challenging
  1. 7Two slices measure 100° and 120°. Find the third.
  2. 8A garden of 600 m² is divided 7 : 3. Find the smaller area.
Show answers (try them yourself first)
  1. 1₹250
  2. 290°
  3. 3144°
  4. 4₹360
  5. 520%
  6. 6108°
  7. 7140°
  8. 8180 m²

Inverse proportion

From daily life

A school hall is cleaned by 6 people in 10 days. If 12 people work, the same job takes only 5 days. More people means fewer days, but the total work, 60 person-days, stays the same.

Simple definition

Two quantities x and y are in inverse proportion when one grows as the other shrinks by the same factor, so x × y stays constant. If x becomes 2 times, y becomes half.

workersdays24 = workers × days

In the figure: Every point has workers × days = 24.

Rule 1: Workers and days — Find the total worker-days first, then divide by the new number of workers.

Live figure

Inverse proportion

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In the figure: Change one value and see how the other changes while the product stays the same.

  • Easy: 6 workers, 10 days; 4 workers need 60 ÷ 4 = 15 days.
  • Medium: 3 pipes fill a tank in 20 hours; 5 pipes: 60 ÷ 5 = 12 hours.
  • Exam-type: Food for 40 students lasts 15 days; for 50 students 600 ÷ 50 = 12 days.

Rule 2: Speed and time — For a fixed distance, speed × time is constant.

  • Easy: 60 km/h takes 5 h; 75 km/h: 300 ÷ 75 = 4 h.
  • Medium: 90 km/h takes 4 h; 60 km/h: 360 ÷ 60 = 6 h.
  • Exam-type: 30 km/h takes 8 h; 40 km/h: 240 ÷ 40 = 6 h.

Rule 3: Constant product rule — If x1 × y1 = x2 × y2, then y2 = x1 × y1 ÷ x2.

  • Easy: x = 4, y = 15; when x = 6, y = 60 ÷ 6 = 10.
  • Medium: y = 24 ÷ x: when x = 8, y = 3.
  • Exam-type: Table (2, 10), (4, 5), (5, 4), (10, 2): product always 20.
Remember

Direct proportion: y ÷ x is constant. Inverse proportion: x × y is constant. Ask: when one goes up, does the other go up (direct) or down (inverse)?

Common mistake

Treating "more workers, fewer days" like a direct proportion and multiplying instead of dividing.

Shortcut

Direct: divide to compare. Inverse: multiply to compare. Memory clue: "Inverse = Interchange" (the product stays).

Spot the difference

PointDirect proportionInverse proportion
Constanty ÷ xx × y
When x doublesy doublesy halves
Examplecost and weightspeed and time

Practice questions

Easy
  1. 16 workers finish a job in 10 days. How many days will 4 workers need?
  2. 2At 60 km/h a trip takes 5 hours. How long at 75 km/h?
  3. 3x × y = 24. Find y when x = 8.
Medium
  1. 43 pipes fill a tank in 20 hours. How long will 5 pipes take?
  2. 5Food for 40 students lasts 15 days. How long for 50 students?
  3. 6x = 4, y = 15. Find y when x = 6 (inverse proportion).
Challenging
  1. 7If one quantity becomes 3 times and the other one third, how are they related?
  2. 84 painters finish in 9 days. How long will 6 painters take?
Show answers (try them yourself first)
  1. 115 days
  2. 24 hours
  3. 33
  4. 412 hours
  5. 512 days
  6. 610
  7. 7Inverse proportion
  8. 86 days