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Proportional Reasoning 1

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

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Chapter Notes

Overview
This chapter introduces the concept of proportional reasoning, starting with the idea that two quantities change by the same factor to remain similar. It explains how ratios represent such relationships and how to check if ratios are proportional by reducing them to their simplest form or by cross multiplication. The big idea is that when two ratios are equal in their simplest form, they are in proportion, and this principle helps solve real-life problems like sharing money, mixing ingredients, or converting units.
Key Concepts (10)
Proportional
Two quantities are said to change proportionally when they change by the same factor (multiplication), not by the same difference (subtraction).
e.g. In images A, C, and D, the width and height both changed by the same factor (e.g., half), so they look similar.
Ratio
A ratio is a comparison of two quantities in the form a : b, meaning for every 'a' units of the first quantity, there are 'b' units of the second quantity.
e.g. The ratio of width to height of image A is 60 : 40, meaning for every 60 mm of width, there are 40 mm of height.
Terms of a ratio
The numbers 'a' and 'b' in a ratio a : b are called the terms of the ratio.
e.g. In the ratio 60 : 40, 60 and 40 are the terms.
Simplest form of a ratio
A ratio is in its simplest form when the two terms have no common factor other than 1, obtained by dividing both terms by their HCF.
e.g. The ratio 60 : 40 in simplest form is 3 : 2 (HCF of 60 and 40 is 20).
Proportion
Two ratios are said to be in proportion (written as a : b :: c : d) if their simplest forms are the same, i.e., ad = bc.
e.g. 60 : 40 :: 30 : 20 because both simplify to 3 : 2.
Cross multiplication
For two proportional ratios a : b :: c : d, the product of the extremes equals the product of the means: ad = bc.
e.g. In 60 : 40 :: 30 : 20, 60 × 20 = 40 × 30 = 1200.
Rule of Three (Trairasika)
An ancient Indian method to find the fourth unknown quantity when three quantities are in proportion: ichchhāphala = (phala × ichchhā) / pramāṇa.
e.g. If 120 students need 15 kg rice, then 80 students need (15 × 80) / 120 = 10 kg rice.
Factor of change
The number by which each term of a ratio is multiplied to get another proportional ratio.
e.g. In 6 : 10 :: 18 : 30, the factor of change is 3 (since 6 × 3 = 18 and 10 × 3 = 30).
Sharing in a given ratio
To divide a quantity x in the ratio m : n, the first part is m × x/(m+n) and the second part is n × x/(m+n).
e.g. To share 12 counters in the ratio 3 : 1, one gets 3 × 12/4 = 9 and the other gets 1 × 12/4 = 3.
Unit conversion
Changing a measurement from one unit to another using a conversion factor, often requiring proportional reasoning.
e.g. 1 metre = 3.281 feet; 1 litre = 1000 mL.
Formulas & Equations (6)
a : b :: c : d
Reads as: a is to b as c is to d
Variables: a, b, c, d are terms of the two ratios
Use when: Checking if two ratios are proportional or solving for an unknown term
⚠ Writing the order incorrectly, e.g., 150 : 90 :: 4 : ? without converting hours to minutes.
ad = bc
Reads as: a times d equals b times c
Variables: a, b, c, d are terms of the two ratios
Use when: Cross multiplying to check proportion or find an unknown term
⚠ Multiplying the wrong pairs, e.g., a × b instead of a × d.
ichchhāphala = (phala × ichchhā) / pramāṇa
Reads as: ichchhāphala equals phala times ichchhā divided by pramāṇa
Variables: pramāṇa = known first term, phala = known second term, ichchhā = known third term, ichchhāphala = unknown fourth term
Use when: Solving Rule of Three problems where three quantities are given and the fourth is unknown
⚠ Misidentifying which term is pramāṇa, phala, and ichchhā.
First part = m × x/(m+n), Second part = n × x/(m+n)
Reads as: First part equals m times x divided by m plus n, second part equals n times x divided by m plus n
Variables: x = total quantity, m and n are the ratio parts
Use when: Dividing a quantity into two parts in a given ratio m : n
⚠ Forgetting to add m and n before dividing, or swapping m and n.
Fahrenheit = (9/5) × Celsius + 32
Reads as: Fahrenheit equals nine-fifths times Celsius plus thirty-two
Variables: Celsius = temperature in degrees Celsius, Fahrenheit = temperature in degrees Fahrenheit
Use when: Converting temperature from Celsius to Fahrenheit
⚠ Forgetting to add 32 after multiplying.
Celsius = (5/9) × (Fahrenheit – 32)
Reads as: Celsius equals five-ninths times Fahrenheit minus thirty-two
Variables: Fahrenheit = temperature in degrees Fahrenheit, Celsius = temperature in degrees Celsius
Use when: Converting temperature from Fahrenheit to Celsius
⚠ Subtracting 32 after multiplying instead of before.
How It Works
Checking if two ratios are proportional
  1. Step 1: Write both ratios in the form a : b and c : d.
  2. Step 2: Reduce each ratio to its simplest form by dividing both terms by their HCF.
  3. Step 3: Compare the simplest forms. If they are the same, the ratios are proportional.
  4. Step 4: Alternatively, cross multiply: check if ad = bc.
💡 Always reduce to simplest form before comparing; do not compare the original numbers directly unless they are already in simplest form.
Finding an unknown term in a proportion (Rule of Three)
  1. Step 1: Set up the proportion as a : b :: c : d, where d is unknown.
  2. Step 2: Ensure all units are the same (e.g., convert hours to minutes).
  3. Step 3: Use cross multiplication: a × d = b × c, so d = (b × c) / a.
  4. Step 4: Calculate the value of d.
💡 Always check that the units match before setting up the proportion. For example, if time is in minutes in one ratio, it must be in minutes in the other.
Sharing a quantity in a given ratio
  1. Step 1: Add the two parts of the ratio (m + n) to find the total number of groups.
  2. Step 2: Divide the total quantity x by (m + n) to find the size of each group.
  3. Step 3: Multiply the size of each group by m to get the first part, and by n to get the second part.
💡 The sum of the two parts must equal the original quantity. Always check your answer by adding the two parts.
Determining if images are similar using ratios
  1. Step 1: Measure the width and height of each image.
  2. Step 2: Write the ratio of width to height for each image.
  3. Step 3: Reduce each ratio to its simplest form.
  4. Step 4: If the simplest forms are the same, the images are similar (proportional).
💡 Similarity depends on the factor of change (multiplication), not on the difference (subtraction).
Exam Traps — Don't Make These Mistakes
⚠ Using subtraction instead of multiplication
✗ Images A and B are similar because both width and height decreased by 20 mm.
→ Similarity requires the same factor of change (multiplication), not the same difference (subtraction). Image B's width is not half of A's width, even though the height is half.
✓ Images A and B are not similar because the width changed by a factor of 40/60 = 2/3, while the height changed by a factor of 20/40 = 1/2. The factors are different.
⚠ Mixing units in a proportion
✗ 150 : 90 :: 4 : x, so x = (90 × 4) / 150 = 2.4 km.
→ The first ratio uses minutes (150 min), but the second ratio uses hours (4 hours). Units must be the same. 4 hours = 240 minutes.
✓ Convert 4 hours to 240 minutes first. Then set up 150 : 90 :: 240 : x, so x = (90 × 240) / 150 = 144 km.
⚠ Confusing the order of terms in a ratio
✗ The ratio of teachers to students is 170 : 5.
→ The order matters. The problem says 'ratio of teachers to students', so teachers come first, then students.
✓ The ratio of teachers to students is 5 : 170, which simplifies to 1 : 34.
⚠ Assuming addition/subtraction keeps a ratio proportional
✗ When Neelima is 3, the ratio is 3:30 = 1:10. When she is 12, the ratio is 12:39, which is also 1:10.
→ Adding the same number (9 years) to both terms changes the ratio. 12:39 simplifies to 4:13, not 1:10.
✓ When Neelima is 12, the ratio is 12:39 = 4:13. The ratio changed because the same number was added to both terms, not multiplied.
⚠ Misidentifying the factor of change when it is a fraction
✗ To find the missing term in 14:21 :: 6:x, multiply 21 by 6/14 = 9, so x = 9.
→ The factor of change is 6/14 = 3/7, which is correct, but students often forget to multiply the second term by the same factor.
✓ The factor of change is 6/14 = 3/7. Multiply 21 by 3/7 to get 9. So the ratio is 6:9.
Compare & Contrast
How the quantities change
Proportional change (multiplication)
Non-proportional change (addition/subtraction)
In proportional change, both quantities are multiplied by the same factor (e.g., width and height both halved). In non-proportional change, the same number is added or subtracted (e.g., both decreased by 20 mm), which distorts the shape.
Order of terms
Ratio of width to height
Ratio of height to width
The order matters. Width:height is different from height:width. For image A, width:height = 60:40 = 3:2, but height:width = 40:60 = 2:3. Swapping the order gives a different ratio.
Exam Focus
Marks pattern: 2 MCQs (1 mark each), 1 SA (3 marks), 1 LA (5 marks)
Most asked: Checking if two ratios are proportional · Finding the unknown term in a proportion (Rule of Three) · Sharing a quantity in a given ratio · Word problems involving unit conversion and proportion
Don't skip: Word problems on sharing profit or ingredients in a given ratio; Problems involving speed, distance, and time using proportion; Unit conversion problems (e.g., area, volume) with proportion
One-Line Revision (read aloud before the exam)
  1. Two ratios are proportional if their simplest forms are the same.
  2. For proportional ratios a:b :: c:d, cross multiplication gives ad = bc.
  3. The Rule of Three states: ichchhāphala = (phala × ichchhā) / pramāṇa.
  4. To share a quantity x in the ratio m:n, first part = m × x/(m+n), second part = n × x/(m+n).
  5. Similar images have width and height changed by the same factor (multiplication), not the same difference.
  6. Always convert units to the same system before setting up a proportion.
  7. Adding or subtracting the same number from both terms of a ratio changes the ratio.
  8. The ratio of teachers to students is written as teachers : students.
  9. The simplest form of a ratio is obtained by dividing both terms by their HCF.
  10. In a ratio a:b, 'a' and 'b' are called the terms of the ratio.
  11. The factor of change is found by dividing the new term by the original term.
  12. Cross multiplication is used to find the fourth unknown in a proportion.

MCQs — 10 Questions with Answers

Q1. What is the simplest form of the ratio 24:36?

  • A. 2:3 ✓
  • B. 4:6
  • C. 12:18
  • D. 6:9

Answer: A — HCF of 24 and 36 is 12; dividing both by 12 gives 2:3 in simplest form.

Q2. Are the ratios 5:7 and 15:21 proportional?

  • A. No, because 15 ≠ 5 × 2
  • B. Yes, because 15:21 simplifies to 5:7 ✓
  • C. No, because 21 is odd
  • D. Yes, because 15 + 21 = 36

Answer: B — HCF of 15 and 21 is 3; dividing gives 5:7, which matches the first ratio exactly.

Q3. If an image has width:height = 60:40, and you scale it so the width becomes 90, what should the height be to keep it proportional?

  • A. 50
  • B. 60 ✓
  • C. 70
  • D. 80

Answer: B — Scaling factor is 90÷60 = 1.5; height becomes 40 × 1.5 = 60, keeping ratio proportional.

Q4. Ravi makes 8 cups of chai with 120 mL of milk. How much milk does he need for 12 cups at the same concentration?

  • A. 160 mL
  • B. 180 mL ✓
  • C. 140 mL
  • D. 200 mL

Answer: B — Scaling factor is 12÷8 = 1.5; milk becomes 120 × 1.5 = 180 mL (ratio 8:120 :: 12:180).

Q5. Three years ago, a father was 28 years old and his son was 7 years old. Is the ratio of their ages the same today?

  • A. Yes, ratio stays 4:1 always
  • B. No, the ratio has changed from 4:1 to something else ✓
  • C. Yes, only if 3 years have passed
  • D. No, because 28 + 7 = 35

Answer: B — Three years ago the ratio was 28:7 = 4:1; today it is 31:10, which is not proportional to 4:1 (adding same number changes the ratio).

Q6. A brick wall pattern uses 5 red bricks for every 3 blue bricks. If the wall has 20 red bricks, how many blue bricks should it have?

  • A. 8 blue bricks
  • B. 10 blue bricks
  • C. 12 blue bricks ✓
  • D. 15 blue bricks

Answer: C — Scaling factor is 20÷5 = 4; blue bricks = 3 × 4 = 12 (maintaining ratio 5:3 :: 20:12).

Q7. Which of these rectangles would look most similar to one with width:height = 8:6?

  • A. Width 16, Height 15
  • B. Width 12, Height 9 ✓
  • C. Width 6, Height 8
  • D. Width 10, Height 6

Answer: B — 8:6 simplifies to 4:3; 12:9 also simplifies to 4:3, so they are proportional and look similar.

Q8. A school has 8 teachers for 240 students. Another school has 12 teachers for 360 students. Are the teacher-to-student ratios proportional?

  • A. No, 12 ≠ 8 × 1.5
  • B. Yes, both simplify to 1:30 ✓
  • C. No, because 360 > 240
  • D. Yes, because 12 − 8 = 4 and 360 − 240 = 120

Answer: B — First ratio 8:240 = 1:30; second ratio 12:360 = 1:30; same simplest form means they are proportional.

Q9. Rekha paints a 5 cm × 3 cm rectangle on paper. She wants to paint a larger similar rectangle. If the new width is 10 cm, what must the height be?

  • A. 5 cm
  • B. 6 cm ✓
  • C. 7 cm
  • D. 8 cm

Answer: B — Original ratio 5:3 must be maintained; scaling factor is 10÷5 = 2, so height = 3 × 2 = 6 cm.

Q10. A recipe uses 2 cups of flour to 3 cups of sugar. If you only have 4 cups of sugar, how much flour should you use to maintain the same sweetness?

  • A. 2.5 cups
  • B. 2.67 cups ✓
  • C. 3 cups
  • D. 5 cups

Answer: B — Ratio 2:3 must stay proportional; scaling factor is 4÷3 ≈ 1.33, so flour = 2 × (4÷3) ≈ 2.67 cups.

Flashcards

What does it mean when two ratios are proportional?

Two ratios are proportional when they have the same simplest form (after dividing by their HCF).

How do you find the simplest form of ratio 90:60?

Divide both terms by their HCF (which is 30) to get 3:2.

Why does image B (40:20) look different from image A (60:40)?

Because their simplest forms are different: image B is 2:1 while image A is 3:2, so they are not proportional.

If 6 glasses need 10 spoons of sugar, how many spoons for 18 glasses at same sweetness?

Since 18 ÷ 6 = 3, multiply sugar by 3: 10 × 3 = 30 spoons (ratio 6:10 :: 18:30).

What happens to a ratio when you add the same number to both terms?

The ratio changes and is not necessarily proportional to the original ratio.

Simplify the ratio 72:96 to check if it is proportional to 3:4.

HCF of 72 and 96 is 24; dividing gives 3:4, which is the same as the first ratio, so they are proportional.

In Kesang's lemonade problem, why did sugar increase from 10 to 30 spoons?

Because the number of glasses increased by factor 3 (from 6 to 18), and both terms must change by the same factor to keep the ratio proportional.

Why are Nitin's wall (60 ft with 3 bags) and Hari's wall (40 ft with 2 bags) equally strong?

Because their ratios (length to cement) are proportional: 60:3 = 20:1 and 40:2 = 20:1 in simplest form.

Which symbol shows that two ratios are proportional?

The symbol '::' is used; for example, 60:40 :: 30:20 means these ratios are proportional.

If filter coffee ratio changes from 15:35 (regular) to 20:30 (stronger), why is it stronger?

Because the proportion of decoction to milk increased: 15:35 = 3:7, but 20:30 = 2:3, meaning more decoction relative to milk.

Important Board Questions

Define what it means for two ratios to be proportional. [1 mark]

Use the term 'simplest form' and mention HCF or highest common factor; compare after reduction.

Simplify the ratio 48:64 and check if it is proportional to 3:4. Show your working. [2 marks]

Find HCF of 48 and 64; divide both terms; compare the simplified form with 3:4.

A tailor needs 5 metres of cloth to make 2 shirts. How much cloth is needed to make 8 shirts of the same size? Explain your reasoning using proportional ratios. [3 marks]

Set up ratio 5:2 :: ?:8; find scaling factor by dividing 8 by 2; multiply cloth amount by same factor; verify both ratios simplify to same form.

Priya's rectangular garden has width 12 m and height 8 m. She wants to design a flower bed with dimensions proportional to her garden. If the flower bed width is 18 m, what should its height be? Also explain why this height keeps the shape similar to the original garden. [5 marks]

Write ratio 12:8 and simplify to 3:2; set up 12:8 :: 18:?; find scaling factor 18÷12 = 1.5; multiply height by 1.5 to get 12 m; verify new ratio 18:12 = 3:2 (same simplest form, so shapes are proportional and visually similar).

Assertion-Reason Questions for Proportional Reasoning 1

Q1. Assertion (A):

Images A, C, and D look similar in shape even though they have different sizes.

Reason (R):

The width and height of these images have changed by the same multiplicative factor.

Answer: A — Both A and R are true, and R is the correct explanation of A — Both the assertion and reason are true. Similar-looking images must have proportional dimensions—when scaled by the same factor, they maintain shape. The reason directly explains why the assertion is true. This is the core principle of proportional reasoning in geometry.

Q2. Assertion (A):

The ratio 60:40 is proportional to 90:60.

Reason (R):

When we compare image A with image D, the factor of change is 1.5 for both width and height.

Answer: B — Both A and R are true, but R is NOT the correct explanation of A — The assertion is true: both ratios simplify to 3:2. The reason is also true: 60×1.5=90 and 40×1.5=60, showing equal factors. However, the reason doesn't explain why the ratios are proportional—proportionality is proven by matching simplest forms, not just by factors of change. The factor is a consequence of proportionality, not the explanation for it.

Q3. Assertion (A):

To check if two ratios are proportional, we should compare the differences between their terms.

Reason (R):

Proportional relationships involve multiplicative change, not additive change.

Answer: D — A is false but R is true — The assertion is false because proportionality is determined by simplest forms or multiplicative factors, not differences. The reason is true—proportionality uses multiplication, not subtraction. Even though the reason is correct, it cannot make the false assertion true. This represents a fundamental misunderstanding of how to verify proportionality.

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Case-Based Questions (CBQ) for Proportional Reasoning 1

Case Study:

A bakery makes two types of cake mixes. Mix A requires 3 cups of flour and 2 cups of sugar for one batch. Mix B requires 6 cups of flour and 4 cups of sugar for one batch. The baker wants to ensure both mixes have the same sweetness level. She also wants to make Mix A using 12 cups of flour. How many cups of sugar should she add to maintain the sweetness? Additionally, is Mix B's original recipe proportional to Mix A's original recipe?

Q1. What is the simplest form of the ratio of flour to sugar in Mix A?

  • A. 3 : 2 ✓
  • B. 6 : 4
  • C. 1 : 2
  • D. 2 : 3

Answer: A

Q2. If Mix A uses 12 cups of flour, how many cups of sugar should be added to keep the same sweetness?

  • A. 6 cups
  • B. 8 cups ✓
  • C. 10 cups
  • D. 12 cups

Answer: B

Q3. What is the simplest form of the ratio of flour to sugar in Mix B?

  • A. 6 : 4
  • B. 3 : 2 ✓
  • C. 2 : 1
  • D. 1 : 2

Answer: B

Q4. Are Mix A's and Mix B's original recipes proportional?

  • A. No, because Mix B uses more flour and sugar
  • B. Yes, because 6+4=10 and 3+2=5, which are proportional
  • C. No, because the quantities are different
  • D. Yes, because both simplify to 3:2, showing the same ratio ✓

Answer: D

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