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Proportional Reasoning 2

NCERT Class 8 · Mathematics Based on NCERT Class 8 Mathematics textbook · Free CBSE study kit

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Chapter Notes

Overview
This chapter builds on earlier ideas of proportionality to explore ratios with more than two terms, their use in maps (Representative Fraction), dividing a whole in a given ratio, constructing pie charts, and understanding inverse proportions. The one big idea a student must walk away with is that proportional relationships can be direct (both quantities change by the same factor) or inverse (one quantity increases while the other decreases by the same factor), and these principles apply to real-life situations like mixing ingredients, reading maps, and interpreting data visually.
Key Concepts (10)
Proportional relationship
When two or more related quantities change by the same factor, the relationship is called a proportional relationship.
e.g. If idli batter uses 2 cups of rice for every 1 cup of urad dal, then using 6 cups of rice and 3 cups of urad dal maintains the same proportion (2:1).
Cross-multiplication method
To verify if two ratios a:b and c:d are proportional, check if a × d = b × c.
e.g. For ratios 6:3 and 4:2, 6 × 2 = 12 and 3 × 4 = 12, so they are proportional.
Representative Fraction (RF)
An expression that shows the ratio between a distance on the map and the corresponding actual distance on the ground.
e.g. A map with RF 1:60,00,000 means 1 cm on the map equals 60,00,000 cm (60 km) on the ground.
Ratio with more than 2 terms
A ratio that compares three or more quantities, written as a:b:c:d..., indicating that for every a units of the first, there are b units of the second, c units of the third, and so on.
e.g. Spice mix ratio 8:4:2:1 means 8 spoons coriander, 4 red chillies, 2 spoons toor dal, 1 spoon fenugreek.
Dividing a whole in a given ratio
To divide a quantity x in the ratio a:b:c..., the parts are x × a/(a+b+c+...), x × b/(a+b+c+...), and so on.
e.g. To divide 110 units of concrete in ratio 1:1.5:3, we get 20 units cement, 30 units sand, 60 units gravel.
Pie chart
A circular chart divided into slices that show different proportions of a whole, where the angle of each slice is proportional to the quantity it represents.
e.g. Grade distribution pie chart: 12 students got A, 10 got B, etc., with angles 108°, 90°, 72°, 54°, 36°.
Inverse proportion
Two quantities x and y vary in inverse proportion if there exists a relation of the type xy = k, where k is a constant. When one quantity changes by a factor n, the other changes by the inverse 1/n.
e.g. Speed and time for a fixed distance: if speed doubles, time halves (e.g., 30 km/h takes 3 hours, 60 km/h takes 1.5 hours).
Direct proportion
Two quantities are directly proportional when they both change by the same factor, and their quotient remains constant (x/y = k).
e.g. If 5 workers move 4500 bricks, then 20 workers move 18000 bricks (since 4500:18000 :: 5:20).
Simplest form of a ratio
A ratio is in its simplest form when all terms are divided by their HCF.
e.g. Ratio 12:10:8:6:4 simplifies to 6:5:4:3:2 (HCF = 2).
Constant of proportionality (inverse)
The constant k in xy = k for inverse proportions, representing the fixed product of the two quantities.
e.g. For speed and time, distance = 90 km is constant: 5 × 18 = 15 × 6 = 30 × 3 = 60 × 1.5 = 90.
Formulas & Equations (8)
a × d = b × c
Reads as: a times d equals b times c
Variables: a, b, c, d are terms of two ratios a:b and c:d
Use when: To check if two ratios are proportional (cross-multiplication)
⚠ Students often multiply a × b and c × d instead of a × d and b × c.
a/c = b/d
Reads as: a divided by c equals b divided by d
Variables: a, b, c, d are terms of two ratios a:b and c:d
Use when: To express proportionality of two ratios
⚠ Students invert the fractions incorrectly, e.g., writing a/b = c/d instead of a/c = b/d.
x × a/(a+b+c+...), x × b/(a+b+c+...), x × c/(a+b+c+...), ...
Reads as: x multiplied by a divided by the sum of all terms, etc.
Variables: x = total quantity; a, b, c, ... = terms of the ratio
Use when: To divide a quantity x into parts in a given ratio a:b:c...
⚠ Students forget to add all terms of the ratio before dividing, or they multiply x by the ratio terms without dividing by the sum.
Angle = (term / sum of all terms) × 360°
Reads as: Angle equals the term divided by the sum of all terms, multiplied by 360 degrees
Variables: term = value for a slice; sum of all terms = total of all values; 360° = total angle in a circle
Use when: To calculate the central angle for each slice of a pie chart
⚠ Students use the raw number instead of the fraction of the total, or forget to multiply by 360°.
xy = k
Reads as: x times y equals k
Variables: x and y are two inversely proportional quantities; k is a constant
Use when: To represent inverse proportion between two quantities
⚠ Students treat inverse proportion as direct proportion and set x1/y1 = x2/y2 instead of x1y1 = x2y2.
x1y1 = x2y2
Reads as: x1 times y1 equals x2 times y2
Variables: x1, y1 are one pair of inversely proportional values; x2, y2 are another pair
Use when: To find an unknown value in an inverse proportion problem
⚠ Students set up the equation as x1/x2 = y1/y2 (direct proportion) instead of x1y1 = x2y2.
x1/x2 = y2/y1
Reads as: x1 divided by x2 equals y2 divided by y1
Variables: x1, x2 are values of one quantity; y1, y2 are corresponding values of the other quantity in inverse proportion
Use when: To check or solve inverse proportion problems
⚠ Students write x1/x2 = y1/y2, which is the condition for direct proportion, not inverse.
d = bc/a (from a:b :: c:d)
Reads as: d equals b times c divided by a
Variables: a, b, c, d are terms of two proportional ratios a:b and c:d
Use when: To find the fourth term in a direct proportion (rule of three)
⚠ Students misplace terms and write d = ab/c or d = ac/b.
How It Works
Checking if two ratios are proportional using cross-multiplication
  1. Write the two ratios in the form a:b and c:d.
  2. Multiply the first term of the first ratio (a) by the second term of the second ratio (d).
  3. Multiply the second term of the first ratio (b) by the first term of the second ratio (c).
  4. If the two products are equal, the ratios are proportional.
💡 Always multiply a × d and b × c — not a × c and b × d.
Using a map's Representative Fraction (RF) to find actual distance
  1. Measure the distance between two points on the map using a ruler (in cm).
  2. Note the RF given on the map (e.g., 1:60,00,000).
  3. Multiply the map distance by the second number of the RF to get the actual distance in cm.
  4. Convert the actual distance from cm to km (divide by 1,00,000).
💡 The RF gives geographical distance, not road distance. Also, 1 km = 1,00,000 cm.
Dividing a whole in a given ratio (multiple terms)
  1. Add all the terms of the ratio to find the total number of parts.
  2. Divide the total quantity by the sum of the terms to find the value of one part.
  3. Multiply each term of the ratio by the value of one part to get each share.
  4. Alternatively, use the formula: share = (total quantity × term) / sum of terms.
💡 Always add all terms of the ratio — do not miss any term. For ratios with decimals (e.g., 1:1.5:3), treat them as they are.
Constructing a pie chart
  1. List all categories and their corresponding values.
  2. Find the total of all values.
  3. Calculate the central angle for each category using: angle = (value / total) × 360°.
  4. Draw a circle and mark a radius.
  5. Using a protractor, measure and draw each angle successively from the previous radius.
  6. Label and colour each slice.
💡 The sum of all angles must be 360°. Always start from the same radius and measure angles in the same direction (usually anti-clockwise).
Solving inverse proportion problems
  1. Identify the two quantities that are inversely proportional (e.g., speed and time, number of workers and days).
  2. Write the known pair of values (x1, y1) and the unknown pair (x2, y2).
  3. Set up the equation: x1 × y1 = x2 × y2.
  4. Solve for the unknown value.
💡 In inverse proportion, as one quantity increases, the other decreases. Do not set up a direct proportion equation.
Solving work-time problems (combined work)
  1. Consider the total work as 1 unit.
  2. Find the work done by each person in 1 hour (e.g., if Ram takes 1 hour, he does 1 unit/hour; if Shyam takes 1.5 hours, he does 1/1.5 = 2/3 unit/hour).
  3. Add the work done per hour by all persons to get the combined work per hour.
  4. Use direct proportion: if combined work per hour is W, then time to complete 1 unit = 1/W hours.
💡 Work done per hour is the reciprocal of time taken. Do not add the times directly — add the rates of work.
Exam Traps — Don't Make These Mistakes
⚠ Confusing cross-multiplication order
✗ For ratios 6:3 and 4:2, they write 6 × 4 = 24 and 3 × 2 = 6, and say not proportional.
→ The correct cross-multiplication is a × d and b × c, i.e., 6 × 2 and 3 × 4, not a × c and b × d.
✓ Check: 6 × 2 = 12 and 3 × 4 = 12, so the ratios are proportional.
⚠ Using direct proportion for inverse proportion problems
✗ For '20 workers take 4 days, how many days for 10 workers?' they write 20/10 = 4/x, giving x = 2 days.
→ Workers and days are inversely proportional — fewer workers take more days, not fewer. The correct equation is 20 × 4 = 10 × x.
✓ Since workers and days are inversely proportional, 20 × 4 = 10 × x, so x = 8 days.
⚠ Forgetting to convert units in map problems
✗ If map distance is 5 cm and RF is 1:60,00,000, they write actual distance = 5 × 60,00,000 = 3,00,00,000 cm and leave it as cm.
→ The question usually asks for distance in km. Students forget to convert cm to km (divide by 1,00,000).
✓ Actual distance = 5 × 60,00,000 = 3,00,00,000 cm = 300 km (since 1 km = 1,00,000 cm).
⚠ Adding ratio terms incorrectly when dividing a whole
✗ For dividing 110 in ratio 1:1.5:3, they add 1 + 1.5 + 3 = 5.5, then write 110/5.5 = 20, then say cement = 1 × 20 = 20, sand = 1.5 × 20 = 30, gravel = 3 × 20 = 60. They forget to check that 20+30+60 = 110.
→ The calculation is actually correct here, but the common mistake is to forget to add all terms, e.g., adding only 1+3=4 and ignoring 1.5, leading to wrong shares.
✓ Always add all terms of the ratio: 1 + 1.5 + 3 = 5.5. Then proceed.
⚠ Mistaking the constant in inverse proportion
✗ For speed and time, they write 30/60 = 3/x, giving x = 6 hours.
→ Speed and time are inversely proportional, so the product is constant, not the quotient. The correct equation is 30 × 3 = 60 × x.
✓ Since speed and time are inversely proportional, 30 × 3 = 60 × x, so x = 1.5 hours.
⚠ Incorrectly calculating pie chart angles
✗ For grade A with 12 out of 40 students, they write angle = 12/40 = 0.3° or 12 × 360 = 4320°.
→ The angle must be (value/total) × 360°, not just the fraction or the value multiplied by 360 without dividing by total.
✓ Angle for grade A = (12/40) × 360° = 108°.
Compare & Contrast
Relationship between two quantities
Direct proportion
Inverse proportion
In direct proportion, both quantities increase or decrease together by the same factor (x/y = k). In inverse proportion, one quantity increases while the other decreases by the same factor (xy = k).
Number of quantities compared
Ratio with two terms (a:b)
Ratio with more than two terms (a:b:c:d)
A two-term ratio compares only two quantities (e.g., 2:1 for rice and urad dal). A multi-term ratio compares three or more quantities (e.g., 8:4:2:1 for spice mix ingredients) and is used when multiple ingredients must be in the same proportion.
Exam Focus
Marks pattern: 2 MCQs (1 mark each) on identifying direct/inverse proportion or cross-multiplication, 1 SA (3 marks) on dividing a whole in a ratio or pie chart angle calculation, 1 LA (5 marks) on a combined inverse proportion problem or a multi-step problem involving map scale or work-time.
Most asked: Identifying and solving inverse proportion problems (e.g., workers and days, speed and time) · Dividing a whole in a given ratio (including multi-term ratios) · Constructing and interpreting pie charts · Using cross-multiplication to check proportionality · Map scale (RF) problems
Don't skip: Inverse proportion word problems (e.g., number of workers and days, speed and time); Dividing a quantity in a given ratio (especially with three or more terms); Pie chart construction and interpretation (calculating angles and drawing)
One-Line Revision (read aloud before the exam)
  1. Two ratios a:b and c:d are proportional if a × d = b × c.
  2. A Representative Fraction (RF) on a map shows the ratio of map distance to actual ground distance.
  3. To convert cm to km, divide by 1,00,000.
  4. Ratios can have more than two terms, e.g., 8:4:2:1 for a spice mix.
  5. To divide a quantity x in the ratio a:b:c, use x × a/(a+b+c), x × b/(a+b+c), x × c/(a+b+c).
  6. A pie chart's central angle for a category is (value/total) × 360°.
  7. In inverse proportion, the product of the two quantities is constant: xy = k.
  8. If one quantity doubles in inverse proportion, the other halves.
  9. In direct proportion, the quotient of the two quantities is constant: x/y = k.
  10. Work done per hour is the reciprocal of the time taken to complete the work.
  11. For combined work, add the rates of work (work per hour) of all workers.
  12. Always check if a problem involves direct or inverse proportion before setting up the equation.
  13. The sum of all angles in a pie chart is always 360°.
  14. To reduce a ratio to its simplest form, divide all terms by their HCF.
  15. Map distance × RF denominator gives actual distance in the same unit.

MCQs — 10 Questions with Answers

Q1. Two ratios 3:5 and 6:10 are proportional. Which cross-multiplication confirms this?

  • A. 3 × 10 = 5 × 6 = 30 ✓
  • B. 3 × 5 = 6 × 10
  • C. 3 + 10 = 5 + 6
  • D. 3 ÷ 5 = 6 ÷ 10

Answer: A — Cross-multiplication for proportional ratios a:b and c:d is a × d = b × c; here 3 × 10 = 5 × 6 = 30.

Q2. On a map with scale 1:50,00,000, a distance of 2 cm represents how many km on the ground?

  • A. 100 km ✓
  • B. 50 km
  • C. 25 km
  • D. 200 km

Answer: A — 2 cm on map = 2 × 50,00,000 = 1,00,00,000 cm = 100 km on ground.

Q3. A spice mix uses coriander, chillies, and dal in ratio 8:4:2. What is this ratio in simplest form?

  • A. 2:1:0.5
  • B. 4:2:1 ✓
  • C. 8:4:2
  • D. 16:8:4

Answer: B — HCF of 8, 4, and 2 is 2; dividing by 2 gives 4:2:1.

Q4. To divide 60 in the ratio 2:3:5, what is the first part?

  • A. 12 ✓
  • B. 18
  • C. 30
  • D. 20

Answer: A — First part = 60 × (2 ÷ (2+3+5)) = 60 × (2 ÷ 10) = 12.

Q5. In a class of 90 students, boys and girls are in ratio 2:1. How many boys are there?

  • A. 30
  • B. 45
  • C. 60 ✓
  • D. 75

Answer: C — Boys = 90 × (2 ÷ (2+1)) = 90 × (2 ÷ 3) = 60.

Q6. If a paint mixture requires red, blue, and white in ratio 2:3:5, and you need 50 ml total, how much red paint is needed?

  • A. 5 ml
  • B. 10 ml ✓
  • C. 15 ml
  • D. 25 ml

Answer: B — Red = 50 × (2 ÷ (2+3+5)) = 50 × (2 ÷ 10) = 10 ml.

Q7. A cricket coach allocates warm-up : batting : bowling : fielding time in ratio 3:4:3:5 for a 150-minute session. How much time is spent on batting?

  • A. 30 minutes
  • B. 40 minutes ✓
  • C. 45 minutes
  • D. 60 minutes

Answer: B — Batting time = 150 × (4 ÷ (3+4+3+5)) = 150 × (4 ÷ 15) = 40 minutes.

Q8. In a pie chart showing grade distribution (A:B:C = 12:10:8), what angle represents grade A?

  • A. 108°
  • B. 120°
  • C. 144° ✓
  • D. 90°

Answer: C — Angle for A = (12 ÷ (12+10+8)) × 360° = (12 ÷ 30) × 360° = 144°.

Q9. A library has books in ratio Odiya:Hindi:English = 3:2:1. If there are 288 Odiya books, how many English books are there?

  • A. 48
  • B. 96 ✓
  • C. 144
  • D. 192

Answer: B — If Odiya (ratio 3) = 288, then 1 ratio unit = 288 ÷ 3 = 96; English (ratio 1) = 96 × 1 = 96. Wait, that's 96, not 48. Let me recheck: English should be 96. But answer C is 96. Rechecking: 3 parts = 288, so 1 part = 96. English = 1 part = 96. Answer should be B (96). Let me revise: if 3:2:1 and Odiya is 288, then English is 288/3 = 96. But the question asks for English, and 3:2:1 means if Odiya is 3 units and English is 1 unit, then English = 288 × (1/3) = 96. Hmm, that's still 96. Let me trust the working: answer is B.

Q10. To construct a triangle with side lengths in ratio 3:4:5, which property must be checked?

  • A. Sum of any two sides > third side (triangle inequality) ✓
  • B. Sum of all three sides = 360°
  • C. All sides are equal
  • D. The ratio simplifies to 1:2:3

Answer: A — Triangle inequality states sum of any two sides must be greater than the third; for 3:4:5, this holds (3+4=7>5), so triangle can be formed.

Flashcards

If two ratios a:b and c:d are proportional, what multiplication rule must hold?

a × d = b × c (cross-multiplication test for proportional ratios).

What does RF 1:60,00,000 on a map mean?

One cm on the map represents 60,00,000 cm (or 60 km) on actual ground.

How do you verify that ratios 6:3 and 4:2 are proportional?

Cross-multiply: 6 × 2 = 12 and 3 × 4 = 12; products are equal, so ratios are proportional.

In a ratio a:b:c, what is the total number of parts?

The total number of parts is a + b + c (sum of all terms).

When dividing 100 in the ratio 2:3:5, what is the value of the first part?

100 × (2 ÷ 10) = 20 (multiply whole by ratio term divided by sum of all terms).

What is the total angle in a pie chart?

360° (full circle angle).

If 12 Odiya books represent a ratio of 3, how many books represent a ratio of 1?

4 books (divide 12 by 3 to find the value of one ratio unit).

What does it mean to reduce a ratio to simplest form?

Divide all terms of the ratio by their HCF (Highest Common Factor).

In a concrete mixture with ratio 1:1.5:3, if cement is 20 units, how much is sand?

Sand is 30 units (multiply all ratio terms by the same factor: 20 ÷ 1 = 20, so 1.5 × 20 = 30).

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

No, because the sum of any two sides (1 + 3 = 4) is not greater than the third side (5), violating the triangle inequality rule.

Important Board Questions

Define a Representative Fraction (RF) on a map. [1 mark]

State that RF is a ratio showing the relationship between distance on map and actual geographical distance (e.g., 1:60,00,000 means 1 cm map = 60 km ground).

Two batches of idli batter are made: Batch A uses rice:urad dal = 6:3 and Batch B uses 4:2. Are these proportional? Verify using cross-multiplication and state whether both batches will taste the same. [2 marks]

Cross-multiply: 6 × 2 = 12 and 3 × 4 = 12. Since products are equal, ratios are proportional. Both batches have the same proportions, so they will taste the same (if other ingredients are also proportional).

A cricket practice session of 120 minutes is divided among warm-up, batting, and fielding in the ratio 2:3:5. Calculate the time allocated to each activity. [3 marks]

Sum of ratio terms = 2+3+5 = 10. Use formula: time = 120 × (ratio term ÷ 10). Warm-up = 120 × 2/10 = 24 min; Batting = 120 × 3/10 = 36 min; Fielding = 120 × 5/10 = 60 min. Verify: 24 + 36 + 60 = 120 min.

A school library has books in the ratio English:Hindi:Marathi = 5:3:2. If the total number of books is 600, find the number of books in each language. Also, draw a pie chart to represent this data, showing the angle for each language. [5 marks]

Total parts = 5+3+2 = 10. English = 600 × 5/10 = 300 books; Hindi = 600 × 3/10 = 180 books; Marathi = 600 × 2/10 = 120 books. For pie chart: English angle = (5/10) × 360° = 180°; Hindi angle = (3/10) × 360° = 108°; Marathi angle = (2/10) × 360° = 72°. Draw circle, mark radii at these angles, and label each section.

Assertion-Reason Questions for Proportional Reasoning 2

Q1. Assertion (A):

The ratio 2 : 3 is proportional to the ratio 4 : 6.

Reason (R):

When all terms of a ratio are multiplied by the same number, the resulting ratio is proportional to the original ratio.

Answer: A — Both A and R are true, and R is the correct explanation of A — Both the assertion and reason are true. The ratio 2:3 becomes 4:6 when multiplied by 2, which is the definition of proportional ratios. The reason correctly explains why the assertion is true — multiplying all terms by the same factor preserves proportionality.

Q2. Assertion (A):

On a map with RF 1 : 80,00,000, a distance of 2 cm represents 160 km of actual geographical distance.

Reason (R):

Representative Fraction shows the ratio between map distance and actual ground distance in centimetres.

Answer: A — Both A and R are true, and R is the correct explanation of A — Both assertion and reason are true. 2 cm × 80,00,000 cm = 1,60,00,000 cm = 160 km. The reason correctly explains that RF compares map distance to actual ground distance. Both parts work together perfectly.

Q3. Assertion (A):

To divide 100 units in the ratio 2 : 3 : 5, the first part is 20 units.

Reason (R):

The first part equals 100 × 2 / (2 + 3 + 5) = 100 × 2 / 10 = 20 units.

Answer: B — Both A and R are true, but R is NOT the correct explanation of A — The assertion is true (first part is indeed 20 units), and the reason correctly shows the calculation. However, the reason is not the explanation FOR why this method works — it's just a restatement of the calculation. The real reason would be about dividing in a given ratio formula. Both are true but reason doesn't explain the assertion.

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Case-Based Questions (CBQ) for Proportional Reasoning 2

Case Study:

A local bakery produces three types of traditional sweets: laddu, barfi, and jalebi. To maintain consistent quality and taste, the bakery uses ingredients in fixed ratios. For 50 kg batch of mixed sweets, the ratio of laddu : barfi : jalebi is 3 : 2 : 5. The head baker, Meera, needs to prepare sweets for a wedding where 200 kg of mixed sweets are needed. She must scale up her recipe proportionally to ensure all three types maintain their original flavour and quality characteristics.

Q1. What is the sum of the ratio terms for this sweet mixture?

  • A. 5
  • B. 8
  • C. 10 ✓
  • D. 15

Answer: C

Q2. For the 200 kg batch, how many kg of laddu should Meera make?

  • A. 30 kg
  • B. 40 kg
  • C. 60 kg ✓
  • D. 80 kg

Answer: C

Q3. How many kg of barfi is needed for 200 kg of mixed sweets?

  • A. 20 kg
  • B. 40 kg ✓
  • C. 60 kg
  • D. 80 kg

Answer: B

Q4. If Meera has only 80 kg of jalebi available, can she make the full 200 kg batch while maintaining the correct ratio?

  • A. Yes, because 80 kg is enough for the jalebi portion
  • B. No, because she would need 100 kg of jalebi (5 parts of 20) ✓
  • C. Yes, if she reduces the total batch size proportionally
  • D. No, because jalebi requires twice the amount of other ingredients

Answer: B

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