Proportional Reasoning 1

CHAPTER 7: PROPORTIONAL REASONING-1

7.1 Observing Similarity in Change

Understanding What Makes Images Similar

When we look at digital images of different sizes, we notice that some appear similar even though they are larger or smaller, while others look distorted.

**Key Observation:** Images look similar when their dimensions change by the **same factor** (through multiplication), not the same amount (through subtraction).

The Difference Between Proportional Change and Additive Change

**Proportional Change (Multiplicative):**

Both dimensions are multiplied by the same number

Images remain similar in appearance

Example: Image A has width 60 mm and height 40 mm. Image C has width 30 mm and height 40 mm.

Width changed by factor: 60 × (1/2) = 30

Height changed by factor: 40 × (1/2) = 20

Since both changed by the same factor (1/2), images A and C look similar

**Additive Change (Subtraction):**

Both dimensions are reduced by the same amount

Images do NOT remain similar

Example: Image A has width 60 mm and height 40 mm. Image B has width 40 mm and height 20 mm.

Width reduced by: 60 - 20 = 40 mm

Height reduced by: 40 - 20 = 20 mm

Both reduced by 20 mm, but the ratios are different, so image B looks distorted

**Why This Matters:**

The tiger in image B appears elongated because the width did not reduce by the same factor as the height. This breaks the proportional relationship between dimensions.

Why Shape Matters in Similarity

Image E is a square (60 × 60 mm) while A, C, D are rectangles. This fundamental difference in shape means they cannot be similar regardless of other measurements.

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7.2 Ratios

What is a Ratio?

A **ratio** is a way to compare two quantities by showing their relative sizes. It is expressed as a : b, where:

**a** is the first term

**b** is the second term

We read this as "a to b"

**Definition:** In a ratio a : b, we can say that for every **a** units of the first quantity, there are **b** units of the second quantity.

Writing and Interpreting Ratios

**Example 1 (Image Dimensions):**

Image A has width 60 mm and height 40 mm

Ratio of width to height = 60 : 40

This means: For every 60 mm of width, there are 40 mm of height

In a simplified view: For every 3 units of width, there are 2 units of height

**Example 2 (Real-Life Context - School):**

If a school has 5 teachers and 170 students

Ratio of teachers to students = 5 : 170

This means: For every 5 teachers, there are 170 students

**Example 3 (Cooking - Filter Coffee):**

Manjunath mixes 15 mL of coffee decoction with 35 mL of milk

Ratio of coffee decoction to milk = 15 : 35

This means: For every 15 mL of coffee, there are 35 mL of milk

Understanding Proportional Ratios

When the terms of two ratios change by the **same factor**, we say the ratios are **proportional**.

**Important:** The factor must be the same for both terms.

Method 1: Finding the Factor of Change

To check if two ratios are proportional:

1. Find the factor by which the first term changed

2. Check if the second term changed by the same factor

3. If yes, the ratios are proportional

**Step-by-Step:**

Image A ratio: 60 : 40

Image C ratio: 30 : 20

First term changed from 60 to 30: Factor = 30/60 = 1/2

Second term changed from 40 to 20: Factor = 20/40 = 1/2

Both changed by factor 1/2, so ratios are proportional ✓

Image A ratio: 60 : 40

Image D ratio: 90 : 60

First term changed from 60 to 90: Factor = 90/60 = 3/2

Second term changed from 40 to 60: Factor = 60/40 = 3/2

Both changed by factor 3/2, so ratios are proportional ✓

**Common Error:** Don't check if the difference is the same. Always check the multiplicative factor.

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7.3 Ratios in Their Simplest Form

What is Simplest Form?

A ratio is in its **simplest form** when both terms have no common factor other than 1 (when HCF = 1).

How to Reduce a Ratio to Simplest Form

**Method:**

1. Find the HCF (Highest Common Factor) of both terms

2. Divide both terms by the HCF

3. Write the result as a simplified ratio

Worked Example: Simplifying Ratios of Images

**Image A: 60 : 40**

Factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60

Factors of 40: 1, 2, 4, 5, 8, 10, 20, 40

HCF of 60 and 40 = 20

Simplified ratio = 60÷20 : 40÷20 = **3 : 2**

**Image D: 90 : 60**

Factors of 90: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90

Factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60

HCF of 90 and 60 = 30

Simplified ratio = 90÷30 : 60÷30 = **3 : 2**

**Image B: 40 : 20**

HCF of 40 and 20 = 20

Simplified ratio = 40÷20 : 20÷20 = **2 : 1**

**Image E: 60 : 60**

HCF of 60 and 60 = 60

Simplified ratio = 60÷60 : 60÷60 = **1 : 1**

Using Simplest Form to Check Proportionality

**Key Rule:** Two ratios are proportional if and only if their simplest forms are identical.

**Examples:**

Are 3 : 4 and 72 : 96 proportional?

Ratio 1: 3 : 4 (already in simplest form)

Ratio 2: 72 : 96

HCF of 72 and 96 = 24

Simplified = 72÷24 : 96÷24 = 3 : 4

Since both simplify to 3 : 4, they ARE proportional ✓

Are 4 : 7 and 12 : 21 proportional?

Ratio 1: 4 : 7 (already in simplest form, HCF = 1)

Ratio 2: 12 : 21

HCF of 12 and 21 = 3

Simplified = 12÷3 : 21÷3 = 4 : 7

Since both simplify to 4 : 7, they ARE proportional ✓

Are 8 : 3 and 24 : 6 proportional?

Ratio 1: 8 : 3 (already in simplest form)

Ratio 2: 24 : 6

HCF of 24 and 6 = 6

Simplified = 24÷6 : 6÷6 = 4 : 1

Since 8 : 3 ≠ 4 : 1, they are NOT proportional ✗

The Proportion Symbol

When two ratios are proportional, we use the symbol **::** to denote this relationship.

**Notation:** a : b :: c : d means the ratio a : b is proportional to the ratio c : d

**Examples:**

60 : 40 :: 30 : 20 (both simplify to 3 : 2)

60 : 40 :: 90 : 60 (both simplify to 3 : 2)

3 : 4 :: 72 : 96 (both simplify to 3 : 4)

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7.4 Problem Solving with Proportional Reasoning

Strategy 1: Using Factor of Change

**Problem:** Kesang made 6 glasses of lemonade with 10 spoons of sugar. Now she needs to make 18 glasses with the same sweetness. How much sugar?

**Solution:**

Original ratio: 6 : 10 (glasses : sugar)

New ratio: 18 : ? (glasses : sugar)

We need: 6 : 10 :: 18 : ?

**Finding the Factor:**

First term changed from 6 to 18

Factor = 18 ÷ 6 = 3

Second term must also multiply by 3

New second term = 10 × 3 = 30

**Answer:** 6 : 10 :: 18 : 30, so she needs **30 spoons of sugar**

Strategy 2: Using Simplest Form

**Problem:** Nitin built a 60 ft wall with 3 bags of cement. Hari built a 40 ft wall with 2 bags. Is Hari's wall weaker?

**Solution:**

Nitin's ratio: 60 : 3

Simplified (HCF = 3): 60÷3 : 3÷3 = 20 : 1

This means 20 ft per 1 bag of cement

Hari's ratio: 40 : 2

Simplified (HCF = 2): 40÷2 : 2÷2 = 20 : 1

This means 20 ft per 1 bag of cement

**Conclusion:** Both walls use cement in the same proportion (20 : 1), so **both walls are equally strong**. Nitin's worry is unfounded.

Strategy 3: Finding Missing Terms in Proportional Ratios

**Problem:** Fill in missing numbers for ratios proportional to 14 : 21

**Finding the Pattern:**

Simplest form of 14 : 21

HCF of 14 and 21 = 7

Simplified = 14÷7 : 21÷7 = **2 : 3**

Any proportional ratio must also simplify to 2 : 3

**Case 1: __ : 42**

Second term is 42

Factor from 21 to 42 = 42 ÷ 21 = 2

First term = 14 × 2 = **28**

Answer: 28 : 42

**Case 2: 6 : __**

First term is 6

To find factor: We need to find what multiplies 14 to give 6

14 × (?) = 6

Factor = 6/14 = 3/7

Second term = 21 × (3/7) = 9

Answer: 6 : 9

**Case 3: 2 : __**

First term is 2

Notice that 2 is 14 divided by 7

If we divide 14 by 7 to get 2, divide 21 by 7 to get 3

Answer: 2 : 3

**Quick Verification:**

28 : 42 = (28÷14) : (42÷21) = 2 : 3 ✓

6 : 9 = (6÷3) : (9÷3) = 2 : 3 ✓

2 : 3 is already in form 2 : 3 ✓

Real-Life Application: Filter Coffee Example

**Scenario:** Manjunath prepares filter coffee in different strengths by varying the ratio of coffee decoction to milk.

**Regular Coffee:** 15 mL decoction : 35 mL milk

Simplest form: HCF = 5, so 3 : 7

This is the standard ratio

**Stronger Coffee:** 20 mL decoction : 30 mL milk

Simplest form: HCF = 10, so 2 : 3

More decoction relative to milk (2 : 3 has higher coffee proportion than 3 : 7)

To see why: 2/3 ≈ 0.667 and 3/7 ≈ 0.429

Stronger coffee has higher decoction-to-milk ratio

**Lighter Coffee:** 10 mL decoction : 40 mL milk

Simplest form: HCF = 10, so 1 : 4

Less decoction relative to milk

1/4 = 0.25, which is less than 3/7

Lighter coffee has lower decoction-to-milk ratio

**General Rule:** When comparing ratios, convert to decimal form or check which has more of the first quantity relative to the second.

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7.5 Understanding When Ratios Don't Change (Important!)

What Happens When We Add the Same Number

**Problem:** When Neelima was 3 years old, her mother was 30 (10 times her age). When Neelima is 12, what is the ratio of their ages?

**Solution:**

At age 3: Ratio = 3 : 30, simplified to **1 : 10**

At age 12: Neelima is 12, mother is 39 (9 years later), Ratio = 12 : 39, simplified to **4 : 13**

These ratios are NOT the same!

**Why?** When we **add or subtract the same number** from both terms of a ratio, the ratio changes and does NOT remain proportional to the original.

**Mathematical Explanation:**

Original ratio: a : b

After adding n to both: (a+n) : (b+n)

These are NOT proportional because the factor of change is different for each term

For a : b to equal (a+n) : (b+n), we would need: a/(a+n) = b/(b+n)

This is only true in special cases, not generally

**Practical Example:**

3 : 30 means for every 3 of Neelima's age, mother is 30

12 : 39 means for every 12 of Neelima's age, mother is 39

The relationship changed!

**Important Rule:** Only multiplication (or division) by the same factor preserves proportionality. Addition and subtraction break it.

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7.6 Trairasika (The Rule of Three)

What is the Rule of Three?

The **Rule of Three** is an ancient method for solving proportion problems where three quantities are known and we need to find the fourth unknown quantity.

**Historical Context:**

Named in ancient Indian mathematics (199 CE)

Āryabhaṭa and other Indian mathematicians formulated this rule

Sanskrit terminology:

**Pramāṇa** = measure (first given quantity) = a

**Phala** = fruit/result (second quantity) = b

**Ichchhā** = requisition/requirement (third quantity) = c

**Ichchhāphala** = desired result (unknown fourth quantity) = d

The Cross Multiplication Principle

**Mathematical Foundation:**

When a : b :: c : d, the ratios are proportional.

This means:

c = f × a (where f is the factor of change)

d = f × b (same factor applied to b)

Therefore:

f = c/a

f = d/b

So: c/a = d/b

Cross-multiplying both sides:

c/a = d/b

c × b = d × a

**ad = bc**

This is the **Cross Multiplication Rule**.

To find the unknown d:

**d = bc/a**

Āryabhaṭa's Formulation

Āryabhaṭa stated:

"Multiply the phala by the ichchhā and divide the resulting product by the pramāṇa."

In algebraic form:

**ichchhāphala = (phala × ichchhā) / pramāṇa**

Or in our notation:

**d = (b × c) / a**

This is equivalent to the cross multiplication formula.

Worked Examples Using the Rule of Three

#### Example 1: Cooking Proportion

**Problem:** For the mid-day meal in a school with 120 students, the cook makes 15 kg of rice. On a rainy day, only 80 students came. How much rice should be made?

**Setting up the Proportion:**

The ratio of students to rice should remain proportional

120 : 15 :: 80 : x

Here: a = 120, b = 15, c = 80, d = x

**Using Cross Multiplication:**

ad = bc

120 × x = 15 × 80

120x = 1200

x = 1200/120 = 10

**Answer:** The cook should make **10 kg of rice**

**Verification:**

Original: 120 students : 15 kg rice

Simplified: 8 students : 1 kg rice

New: 80 students : 10 kg rice

Simplified: 8 students : 1 kg rice ✓

#### Example 2: Distance and Time

**Problem:** A car travels 90 km in 150 minutes. At the same speed, what distance will it cover in 4 hours?

**Important Note:** Convert units to be consistent!

4 hours = 4 × 60 = 240 minutes

**Setting up the Proportion:**

Time (in minutes) : Distance (in km) should be proportional

150 : 90 :: 240 : x

Here: a = 150, b = 90, c = 240, d = x

**Using Cross Multiplication:**

ad = bc

150 × x = 90 × 240

150x = 21,600

x = 21,600 / 150

x = 144

**Answer:** The car will cover **144 km** in 4 hours

**Alternative Method (Unitary Method):**

Distance in 1 minute = 90/150 = 0.6 km

Distance in 240 minutes = 0.6 × 240 = 144 km

#### Example 3: Price Comparison

**Problem:** A farmer in Himachal Pradesh sells 200 g of tea for ₹200. An estate in Meghalaya sells 1 kg for ₹800. Which is more expensive?

**Step 1: Convert to Same Units**

Himachal: 200 g : ₹200

Meghalaya: 1 kg = 1000 g : ₹800

**Step 2: Check if Proportional**

Himachal simplified: 200 : 200 = 1 : 1 (meaning ₹1 per gram)

Meghalaya simplified: 1000 : 800 = 5 : 4 (meaning ₹4 per 5 grams, or ₹0.8 per gram)

These are NOT proportional (1 : 1 ≠ 5 : 4)

**Step 3: Find Price of 1 kg in Both Places**

For Himachal:

200 g costs ₹200

1 kg = 1000 g = 5 × 200 g

Price of 1 kg = 5 × ₹200 = ₹1000

For Meghalaya:

1 kg costs ₹800

**Answer:** **Himachal tea is more expensive** (₹1000 per kg vs ₹800 per kg)

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7.7 Important Rules and Theorems

Rule 1: Proportional Change is Multiplicative

**Statement:** Two quantities change proportionally if they are both multiplied (or divided) by the same factor.

**Mathematical Form:**

If a : b :: c : d

Then c = f × a and d = f × b (for the same factor f)

Rule 2: Cross Multiplication Test

**Statement:** Two ratios a : b and c : d are proportional if and only if ad = bc.

**Proof Outline:**

If a : b :: c : d, then a/b = c/d

Cross-multiplying: a × d = b × c

Therefore: ad = bc (and vice versa)

Rule 3: Simplest Form Check

**Statement:** Two ratios are proportional if their simplest forms are identical.

**Process:**

1. Find HCF of both terms in first ratio, divide both by HCF

2. Find HCF of both terms in second ratio, divide both by HCF

3. If results are equal, ratios are proportional

Rule 4: Addition/Subtraction Doesn't Preserve Proportionality

**Statement:** Adding or subtracting the same number from both terms of a ratio changes the ratio and breaks proportionality with the original.

**Example:**

3 : 30 (simplifies to 1 : 10)

Add 9 to both: 12 : 39 (simplifies to 4 : 13)

1 : 10 ≠ 4 : 13, so NOT proportional

Rule 5: Finding Missing Term in Proportion

**Statement:** To find a missing term d in a : b :: c : d, use:

**d = (b × c) / a**

This comes directly from cross multiplication: ad = bc, so d = bc/a

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7.8 Common Mistakes to Avoid

Mistake 1: Confusing Additive vs. Multiplicative Change

**Wrong:** Image width decreased by 20 mm and height decreased by 20 mm, so the image should look the same.

**Correct:** Changes should be compared by division (multiplicative factor), not subtraction.

**Example:**

If width changes from 60 to 40 (decrease of 20), factor = 40/60 = 2/3

If height changes from 40 to 20 (decrease of 20), factor = 20/40 = 1/2

Factors are different (2/3 ≠ 1/2), so image looks distorted

Mistake 2: Not Converting Units in Proportions

**Wrong:** Comparing 150 minutes with 4 hours directly without converting to same units.

**Correct:** Convert all quantities to the same unit before setting up proportions.

150 minutes remains 150 minutes

4 hours = 240 minutes

Now compare: 150 : 90 :: 240 : x

Mistake 3: Not Reducing Ratios When Checking Proportionality

**Wrong:** Claiming 4 : 7 is proportional to 12 : 24 without simplifying.

**Correct:**

4 : 7 (already simplified)

12 : 24 simplifies to 1 : 2

Since 4 : 7 ≠ 1 : 2, they are NOT proportional

Mistake 4: Incorrectly Finding the Factor of Change

**Wrong:** Using subtraction instead of division.

18 - 6 = 12, so factor is 12? ✗

**Correct:** Use division.

18 ÷ 6 = 3, so factor is 3 ✓

Mistake 5: Applying the Same Factor to Different Ratios Without Care

**Wrong:** If first term of a ratio changes by factor 2, assuming second term should also change by 2.

**Correct:** Multiply the second term by the same factor as the first term.

6 : 10, if first term becomes 18 (multiplied by 3), second term becomes 10 × 3 = 30

New ratio: 18 : 30

Mistake 6: Forgetting That Ratio Order Matters

**Wrong:** Assuming 3 : 4 is the same as 4 : 3

**Correct:** Ratio order indicates which quantity is first and which is second

3 : 4 means first quantity is 3, second is 4

4 : 3 means first quantity is 4, second is 3

These are different ratios (reciprocals of each other)

**Example:**

Image width to height: 60 : 40

Image height to width: 40 : 60

These are different!

Mistake 7: Misunderstanding Inverse Proportionality

**Wrong:** Using the Rule of Three when quantities are inversely related.

**Example Problem:** Puneeth's father rides at 50 km/h and takes 2 hours. At 75 km/h, how long will it take?

**Incorrect:** 50 : 2 :: 75 : __ (this gives 3 hours, which is wrong!)

**Why?:** Speed and time are **inversely proportional** (as speed increases, time decreases). This cannot be solved using the Rule of Three in this form.

**Correct Approach:**

Distance = Speed × Time = 50 × 2 = 100 km

At 75 km/h: Time = Distance ÷ Speed = 100 ÷ 75 = 4/3 hours = 80 minutes

Time actually **decreases** when speed increases

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7.9 Real-World Applications

Application 1: Cooking and Recipe Scaling

**Concept:** Scaling recipes up or down while maintaining taste requires proportional reasoning.

**Example Activity:**

Choose your favorite dish. List all ingredients for your family. If you need to cook for 15 extra guests, scale all quantities proportionally.

**Process:**

1. Find the ratio of people: New people / Original people

2. Multiply each ingredient amount by this ratio

3. Keep all flavors balanced by maintaining proportional relationships

**Real Example:**

Original recipe for 4 people: 2 cups rice, 3 cups water, 1 tablespoon salt

For 10 people: ratio = 10/4 = 2.5

Rice: 2 × 2.5 = 5 cups

Water: 3 × 2.5 = 7.5 cups

Salt: 1 × 2.5 = 2.5 tablespoons

Application 2: Market Price Comparisons

**Concept:** Comparing prices of different product sizes requires proportional reasoning.

**Real Activity:** Go to market and collect prices of different shampoo bottle sizes.

| Container | Volume (mL) | Price (₹) |

|-----------|------------|----------|

| Sachet | 6 | 2 |

| Small Bottle | 180 | 154 |

| Medium Bottle | 340 | 276 |

| Large Bottle | 1000 | 540 |

**Analysis:**

To find which is cheapest, find price per mL (simplest form):

Sachet: 2 : 6 = 1 : 3 (₹1 per 3 mL = ₹0.33 per mL)

Small: 154 : 180 (divide by HCF) ≈ ₹0.86 per mL

Medium: 276 : 340 ≈ ₹0.81 per mL

Large: 540 : 1000 = 27 : 50 (₹0.54 per mL)

**Conclusion:** The sachet is cheapest per mL, but the large bottle offers better value for bulk buyers.

Application 3: Construction and Scale Drawings

**Concept:** Architectural drawings use proportional ratios to represent real buildings.

**Example Problem:** A mason needs 1450 bricks for 10 feet of wall. How many for a complete house?

Using Rule of Three:

10 : 1450 :: (total length) : (total bricks needed)

The proportional relationship ensures the walls have consistent strength.

Application 4: Measurement and Scale in Models

**Example Activity:** Measure a blackboard's width and height. Draw a proportional rectangle in your notebook.

**Process:**

1. Measure blackboard width and height

2. Find ratio: width : height

3. Scale down proportionally (divide both by same factor)

4. Draw rectangle with new dimensions

5. Your drawing should have the same proportion as the blackboard

**Why?** If width : height ratio is maintained, the drawing looks the same as the original, just smaller.

Application 5: Human Proportions in Art

**Concept:** Realistic human drawings maintain proportions between body parts.

**Example Activity:** Measure a friend's body:

Head length : Torso length

Torso length : Arm length

Torso length : Leg length

Then draw a figure with proportional measurements. A realistic figure maintains these ratios!

Application 6: Travel and Speed

**Concept:** Distance, speed, and time are related proportionally (only if speed is constant).

**Example:** If a motorcycle travels 100 km in 2 hours at constant speed:

In 3 hours: (100 : 2) :: (? : 3)

Distance = (100 × 3) / 2 = 150 km

**Important:** This works only when speed is constant. If speed changes, the relationship changes.

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7.10 Key Formulas and Their Uses

Formula 1: Cross Multiplication

**When:** To check if ratios are proportional or find missing term

**Formula:** If a : b :: c : d, then **ad = bc**

**To Find Missing d:**

**d = bc/a**

**Example:** 6 : 10 :: 18 : x

Cross multiply: 6x = 10 × 18

6x = 180

x = 30

Formula 2: Finding the Factor of Change

**When:** To find by what multiple one ratio changed to another

**Formula:** If first ratio is a : b and second is c : d, then **factor = c/a** (or d/b)

**Example:** 6 : 10 becomes 18 : ?

Factor = 18/6 = 3

New second term = 10 × 3 = 30

Formula 3: Finding HCF for Simplification

**When:** To reduce a ratio to simplest form

**Formula:** Find **HCF(a, b)**, then **simplified ratio = a/HCF : b/HCF**

**Example:** 60 : 40

HCF(60, 40) = 20

Simplified = 60/20 : 40/20 = 3 : 2

Formula 4: Āryabhaṭa's Rule of Three

**When:** To find the fourth quantity in a proportion

**Formula:** If a : b :: c : d, then **d = (b × c) / a**

**Also Known As:** Cross Multiplication Method

**Example:** 120 students : 15 kg rice :: 80 students : x kg rice

x = (15 × 80) / 120 = 1200 / 120 = 10 kg

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7.11 Summary of Key Concepts

When Ratios are Proportional

Two ratios are proportional when:

1. Their simplest forms are identical, OR

2. One ratio can be obtained from the other by multiplying both terms by the same factor, OR

3. Their cross products are equal (ad = bc)

When Ratios are NOT Proportional

Two ratios are NOT proportional when:

1. Their simplest forms are different, OR

2. The terms changed by different factors, OR

3. The same amount was added to or subtracted from both terms

Steps to Solve a Proportion Problem

1. **Identify** the quantities and their relationship

2. **Set up** the proportion: a : b :: c : d

3. **Ensure** all units are the same

4. **Apply** cross multiplication: ad = bc

5. **Solve** for the unknown

6. **Verify** by checking if the ratios are actually proportional

When NOT to Use Rule of Three

When quantities are **inversely related** (as one increases, the other decreases)

When the relationship is **additive** (change by same amount, not same factor)

When the relationship is **non-linear** (more complex mathematical relationship