Proportional Reasoning 2

CHAPTER 3: PROPORTIONAL REASONING–2

3.1 Proportionality — A Quick Recap

Understanding Proportional Relationships

**Proportional relationship** occurs when two or more related quantities change by the same factor. When quantities are connected in this way, we can represent them using **ratio notation**.

**Key Definition:** A ratio is a comparison of two quantities. When we say the ratio of quantity A to quantity B is written as A : B, we mean that for every unit of A, there are that many units of B.

**Real-life Example — Idli Batter:**

In South Indian cooking, idli batter is made by mixing rice and urad dal. One traditional proportion is: 2 cups of rice mixed with 1 cup of urad dal. This is written as 2 : 1.

When Viswanath made idlis using 6 cups of rice with 3 cups of urad dal (ratio 6 : 3), and Puneet made idlis using 4 cups of rice with 2 cups of urad dal (ratio 4 : 2), both would taste the same if cooked identically because:

Viswanath's ratio 6 : 3 simplifies to 2 : 1

Puneet's ratio 4 : 2 also simplifies to 2 : 1

These ratios are **proportional** (equivalent).

Testing if Two Ratios are Proportional

**Method 1: Cross-Multiplication Method**

Two ratios a : b and c : d are proportional if and only if:

**a × d = b × c**

**Example:** Are 6 : 3 and 4 : 2 proportional?

Cross multiply: 6 × 2 = 12 and 3 × 4 = 12

Since 12 = 12, the ratios are proportional ✓

**Method 2: Fraction Equivalence Method**

Two ratios a : b and c : d are proportional if:

**a/b = c/d** (when written as fractions, they are equal)

**Example:** Are 6 : 3 and 4 : 2 proportional?

6/3 = 2 and 4/2 = 2

Since 2 = 2, the ratios are proportional ✓

Important Notes

When ratios are proportional, we use the notation a : b :: c : d (read as "a is to b as c is to d")

The order of terms in a ratio matters; 2 : 1 is different from 1 : 2

A ratio remains unchanged if both terms are multiplied or divided by the same non-zero number

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3.2 Ratios in Maps

Understanding Representative Fraction (RF)

**Representative Fraction (RF)** is a special type of ratio that shows the relationship between a distance measured on a map and the corresponding actual geographical distance on the ground.

**Key Formula:**

**RF = Map Distance : Actual Ground Distance**

The RF is usually written in the form **1 : n**, where n is a large number.

Interpreting Map Scales

**Example:** If a map has RF 1 : 60,00,000, this means:

1 cm on the map represents 60,00,000 cm on the actual ground

Convert to km: 60,00,000 cm ÷ 100,000 = 60 km

So 1 cm on the map = 60 km in real life.

Calculating Actual Distances from Maps

**Step-by-step Process:**

1. Use a ruler to measure the distance between two locations on the map (in cm)

2. Note the RF given on the map

3. Multiply the map distance by the actual distance represented by 1 cm

**Example Problem:**

If the distance between Bengaluru and Chennai on a map is 4 cm, and the map's RF is 1 : 60,00,000, what is the actual geographical distance?

**Solution:**

Map distance = 4 cm

From RF 1 : 60,00,000, we know 1 cm = 60 km

Actual distance = 4 × 60 = 240 km

Important Properties of Map Scales

Different maps with different scales (different RF values) will give approximately the same geographical distance when properly measured and calculated

A larger scale (like 1 : 5,00,000) shows more detail but covers a smaller area than a smaller scale (like 1 : 50,00,000)

Maps are always marked as "not to scale" when hand-drawn, meaning measurements might not be perfectly accurate

RF is always located in the lower right corner of a map for easy reference

Practical Application in Classroom

When creating sketch maps (like a classroom floor plan):

Choose an appropriate scale, such as 1 : 50 (1 cm on the plan = 50 cm actual)

Measure actual distances carefully

Scale down all measurements by dividing by 50

Mark locations of furniture, doors, windows, and other objects using symbols

This demonstrates how maps preserve proportional relationships while reducing size

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3.3 Ratios with More than 2 Terms

Understanding Multi-term Ratios

A **ratio with multiple terms** compares more than two quantities where each quantity changes by the same factor to maintain the proportional relationship.

**Real-life Example — Spice Mix Powder:**

Viswanath makes spice powder by mixing:

8 spoons of coriander seeds

4 red chillies

2 spoons of toor dal

1 spoon of fenugreek seeds

The ratio is written as **8 : 4 : 2 : 1** (four-term ratio)

Each quantity in this ratio holds a fixed proportion to the others. If we want to make a different amount while maintaining the same taste, all quantities must be scaled by the same factor.

Scaling Multi-term Ratios

**Principle:** If we need to use different quantities of ingredients but want the same proportions, we multiply (or divide) ALL terms by the same factor.

**Example with Spice Mix:**

Puneet has only 2 red chillies but wants the same-tasting spice powder as Viswanath.

Original ratio: 8 : 4 : 2 : 1

Since Puneet has 2 chillies instead of 4, he has **half** the amount Viswanath used. Therefore, all ingredients must be reduced to half:

Coriander seeds: 8 ÷ 2 = 4 spoons

Red chillies: 4 ÷ 2 = 2 chillies

Toor dal: 2 ÷ 2 = 1 spoon

Fenugreek seeds: 1 ÷ 2 = 0.5 spoon

Puneet's ratio: **4 : 2 : 1 : 0.5**

Proportionality of Multi-term Ratios

When two multi-term ratios are proportional, we write:

**a : b : c : d :: p : q : r : s**

This means:

**a/p = b/q = c/r = d/s** (all fractions are equal)

This is the **fundamental property of proportional multi-term ratios**.

Worked Example 1: Purple Paint Mixture

**Problem:** To make a special shade of purple, paint must be mixed in the ratio Red : Blue : White :: 2 : 3 : 5. If Yasmin has 10 litres of white paint, how many litres of red and blue paint should she add?

**Solution:**

Step 1: Identify what each number in the ratio represents.

Red paint corresponds to 2 parts

Blue paint corresponds to 3 parts

White paint corresponds to 5 parts

Step 2: Find the value of 1 part.

If 5 parts = 10 litres

Then 1 part = 10 ÷ 5 = 2 litres

Step 3: Calculate red and blue paint needed.

Red paint = 2 parts = 2 × 2 = 4 litres

Blue paint = 3 parts = 3 × 2 = 6 litres

Step 4: Verify the ratio.

Red : Blue : White = 4 : 6 : 10 = 2 : 3 : 5 ✓

**Answer:** Yasmin needs 4 litres of red paint and 6 litres of blue paint.

**Total volume of purple paint = 4 + 6 + 10 = 20 litres**

Worked Example 2: Cement Concrete Mixture

**Problem:** Cement concrete for strong structures (pillars, beams, roofs) is mixed in the ratio 1 : 1.5 : 3 (cement : sand : gravel). If we have 3 bags of cement, how many bags of concrete mixture can we make?

**Solution:**

The ratio is: Cement : Sand : Gravel :: 1 : 1.5 : 3

Since we have 3 bags of cement and the ratio shows 1 bag of cement, we must multiply all terms by 3.

Cement needed = 1 × 3 = 3 bags

Sand needed = 1.5 × 3 = 4.5 bags

Gravel needed = 3 × 3 = 9 bags

Total concrete mixture = 3 + 4.5 + 9 = **16.5 bags**

Key Properties of Multi-term Ratios

1. **All terms must change by the same factor** — if you multiply one term by n, you multiply all terms by n

2. **The ratio can contain decimal or fractional values** — not limited to whole numbers

3. **Reducing to simplest form** — divide all terms by their HCF (Highest Common Factor)

4. **The ratio represents relative proportions, not absolute quantities** — 2 : 3 : 5 is the same as 4 : 6 : 10

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3.4 Dividing a Whole in a Given Ratio

The Principle of Division

When we need to divide a total quantity into parts according to a given ratio, we use this procedure:

**For dividing quantity x in the ratio a : b : c : ..., the parts are:**

**Part A = x × a/(a + b + c + ...)**

**Part B = x × b/(a + b + c + ...)**

**Part C = x × c/(a + b + c + ...)**

The denominator (a + b + c + ...) is the **sum of all ratio terms**.

Why This Works

The denominator represents the total "parts" into which we're dividing. If we divide the total by this sum, we get the value of one part. Multiplying each ratio term by this value gives us the actual amount.

Worked Example 1: Concrete Mixture Division

**Problem:** For construction, 110 units of concrete are needed. How many units of cement, sand, and gravel are needed if they're mixed in the ratio 1 : 1.5 : 3?

**Solution:**

Step 1: Add all ratio terms.

Sum = 1 + 1.5 + 3 = 5.5 parts

Step 2: Find how many times we need to make this ratio to get 110 units.

Multiplier = 110 ÷ 5.5 = 20

Step 3: Multiply each ratio term by 20.

Cement = 1 × 20 = 20 units

Sand = 1.5 × 20 = 30 units

Gravel = 3 × 20 = 60 units

Step 4: Verify.

20 + 30 + 60 = 110 units ✓

Ratio check: 20 : 30 : 60 = 1 : 1.5 : 3 ✓

**Answer:** 20 units of cement, 30 units of sand, and 60 units of gravel.

Worked Example 2: Purple Paint Division

**Problem:** To make a shade of purple, mix red, blue, and white paint in the ratio 2 : 3 : 5. If you need 50 ml of purple paint, how many ml of each color will you use?

**Solution:**

Step 1: Add ratio terms.

Sum = 2 + 3 + 5 = 10

Step 2: Apply the division formula.

Red paint = 50 × 2/10 = 50 × 0.2 = 10 ml

Blue paint = 50 × 3/10 = 50 × 0.3 = 15 ml

White paint = 50 × 5/10 = 50 × 0.5 = 25 ml

Step 3: Verify.

10 + 15 + 25 = 50 ml ✓

**Answer:** 10 ml red, 15 ml blue, and 25 ml white paint.

Worked Example 3: Triangle with Given Angle Ratio

**Problem:** Construct a triangle with angles in the ratio 1 : 3 : 5.

**Solution:**

Step 1: Recall that the sum of angles in a triangle is 180°.

Step 2: Add ratio terms.

Sum = 1 + 3 + 5 = 9

Step 3: Apply the division formula to 180°.

Angle A = 180° × 1/9 = 20°

Angle B = 180° × 3/9 = 60°

Angle C = 180° × 5/9 = 100°

Step 4: Verify.

20° + 60° + 100° = 180° ✓

Ratio check: 20 : 60 : 100 = 1 : 3 : 5 ✓

**Answer:** The triangle has angles of 20°, 60°, and 100°.

Worked Example 4: Cricket Practice Session

**Problem:** A cricket coach schedules practice in the ratio — warm-up/cool-down : batting : bowling : fielding :: 3 : 4 : 3 : 5. If each session is 150 minutes long, how much time is spent on each activity?

**Solution:**

Step 1: Add all ratio terms.

Sum = 3 + 4 + 3 + 5 = 15 parts

Step 2: Find the value of 1 part.

1 part = 150 ÷ 15 = 10 minutes

Step 3: Calculate time for each activity.

Warm-up/cool-down = 3 × 10 = 30 minutes

Batting = 4 × 10 = 40 minutes

Bowling = 3 × 10 = 30 minutes

Fielding = 5 × 10 = 50 minutes

Step 4: Verify.

30 + 40 + 30 + 50 = 150 minutes ✓

**Answer:** Warm-up 30 min, batting 40 min, bowling 30 min, fielding 50 min.

Common Errors to Avoid

1. **Forgetting to add all ratio terms** — Always sum them first before dividing

2. **Not multiplying all parts by the same factor** — Each part gets the same multiplier

3. **Incorrect arithmetic** — Verify your answer by adding all parts; they should equal the total

4. **Misunderstanding the ratio** — The ratio 3 : 4 : 3 : 5 means the parts are in these proportions, not that these are the actual values

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3.5 A Slice of the Pie

Understanding Pie Charts

A **pie chart** is a circular graph divided into slices (sectors) where each slice represents a proportion of the whole. The entire circle represents 100% of the data or a total of 360°.

Pie charts help us quickly visualize and compare proportions of different categories.

**Common uses:**

Grade distribution in a class

Preference surveys

Budget allocation

Market share distribution

Converting Data to Angles for Pie Charts

**Key Formula:**

**Angle for a category = (Frequency of category / Total frequency) × 360°**

Alternatively:

**Angle = (Ratio term / Sum of all ratio terms) × 360°**

Worked Example: Grade Distribution

**Problem:** A class of 40 students received grades as follows: A (12), B (10), C (8), D (6), E (4). Create a pie chart.

**Solution:**

Step 1: Create a ratio from the frequencies.

Ratio = 12 : 10 : 8 : 6 : 4

Step 2: Simplify the ratio (optional, makes calculations easier).

Find HCF of all terms = 2

Simplified ratio = 6 : 5 : 4 : 3 : 2

Step 3: Add all ratio terms.

Sum = 6 + 5 + 4 + 3 + 2 = 20

Step 4: Calculate angles for each grade.

Grade A: (6/20) × 360° = 6 × 18° = 108°

Grade B: (5/20) × 360° = 5 × 18° = 90°

Grade C: (4/20) × 360° = 4 × 18° = 72°

Grade D: (3/20) × 360° = 3 × 18° = 54°

Grade E: (2/20) × 360° = 2 × 18° = 36°

Step 5: Verify.

108° + 90° + 72° + 54° + 36° = 360° ✓

Step-by-Step Construction of a Pie Chart

**Step 1:** Draw a circle using a compass.

**Step 2:** Draw a radius (line from center to edge) and label it as the starting point.

**Step 3:** Using a protractor, measure the first angle (108°) from the starting radius, going counter-clockwise. Draw a new radius.

**Step 4:** From this new radius, measure the second angle (90°) counter-clockwise and draw another radius.

**Step 5:** Continue this process for all remaining angles, ensuring each new angle is measured from the previous radius.

**Step 6:** After all angles are marked, you'll have created all the slices.

**Step 7:** Label each slice with its category name and either the percentage or number of items it represents. Color each slice differently for clarity.

Important Notes on Pie Chart Construction

Always use a **protractor** to measure angles accurately

Measure angles **counter-clockwise** (or consistently in one direction)

All angles should sum to exactly 360°

Each slice should be labeled clearly with category name and value

If using percentages: (Category value / Total) × 100% = Percentage

Converting Percentages to Angles

When given percentages instead of frequencies:

**Angle = Percentage × 360° / 100 = Percentage × 3.6°**

**Example:** If Entertainment is 50% of TV channel preferences:

Angle = 50 × 3.6° = 180° (which makes sense — half the circle)

Worked Example 2: TV Channel Preferences

**Problem:** Create a pie chart for TV channel preferences: Entertainment 50%, Sports 25%, News 15%, Information 10%.

**Solution:**

Calculate angles:

Entertainment: 50 × 3.6° = 180°

Sports: 25 × 3.6° = 90°

News: 15 × 3.6° = 54°

Information: 10 × 3.6° = 36°

Verify: 180° + 90° + 54° + 36° = 360° ✓

Construct the pie chart using these angles with a protractor.

Key Properties of Pie Charts

1. **Proportional representation** — The size of each slice is directly proportional to its value

2. **Total is always 360°** — All angles must sum to exactly 360°

3. **Best for comparing parts to a whole** — Shows how each category contributes to the total

4. **Not ideal for many categories** — More than 6-7 categories make the chart cluttered

5. **Always label clearly** — Include category names and either values or percentages

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3.6 Inverse Proportions

Understanding Direct vs. Inverse Proportions

**Direct Proportion:** When one quantity increases, the other increases by the same factor. When one decreases, the other decreases by the same factor.

**Inverse Proportion:** When one quantity increases, the other decreases by the same factor, and vice versa. They change in opposite directions.

The Concept of Inverse Proportionality

**Real-life Example — Travel Speed and Time:**

Puneeth's father travels from Lucknow to Kanpur (a fixed distance) using different modes:

| Mode of Transport | Speed (km/h) | Time (hours) |

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

| Walking | 5 | 18 |

| Bicycle | 15 | 6 |

| Motorcycle | 30 | 3 |

| Car | 60 | 1.5 |

**Key Observation:**

When speed is multiplied by 3 (from 5 to 15), time is divided by 3 (from 18 to 6)

When speed is multiplied by 2 (from 15 to 30), time is divided by 2 (from 6 to 3)

When speed is multiplied by 4 (from 15 to 60), time is divided by 4 (from 6 to 1.5)

**The Pattern:** Speed and time change by inverse (reciprocal) factors.

The Mathematical Definition

Two quantities **x and y are in inverse proportion** if there exists a constant **k** such that:

**x × y = k** (or equivalently, **y = k/x**)

Where k is the constant of proportionality.

In the travel example, k represents the fixed distance between the cities. No matter which mode of transport is used, distance = speed × time always equals this constant.

Distance = 5 × 18 = 90 km

Distance = 15 × 6 = 90 km

Distance = 30 × 3 = 90 km

Distance = 60 × 1.5 = 90 km

Property of Inverse Proportions

If quantities x and y are inversely proportional with constant k, and we have two pairs of values (x₁, y₁) and (x₂, y₂), then:

**x₁ × y₁ = x₂ × y₂ = k**

This can be rearranged to:

**x₁/x₂ = y₂/y₁**

**Verification with Travel Example:**

Using walking (x₁ = 5, y₁ = 18) and car (x₂ = 60, y₂ = 1.5):

x₁/x₂ = 5/60 = 1/12

y₂/y₁ = 1.5/18 = 1/12 ✓

Identifying Inverse Proportions

To check if two quantities are inversely proportional:

1. Calculate x × y for each pair of values

2. If all products equal the same constant, then x and y are inversely proportional

3. Alternatively, check if x₁/x₂ = y₂/y₁

**Example:** Are these values inversely proportional?

| x | 40 | 80 | 25 | 16 |

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

| y | 20 | 10 | 12.5 | 8 |

Check products:

40 × 20 = 800

80 × 10 = 800

25 × 12.5 = 312.5

16 × 8 = 128

Products are NOT all equal, so these are **not inversely proportional**.

Worked Example 1: Workers and Days

**Problem:** 20 workers can lay a road in 4 days. How many days will 10 workers take to lay the same road?

**Solution:**

Step 1: Identify the relationship.

More workers → fewer days needed (inverse proportion)

Step 2: Set up the inverse proportion equation.

Let the time for 10 workers be y days.

20 × 4 = 10 × y

Step 3: Solve for y.

80 = 10y

y = 8 days

Step 4: Verify the pattern.

When workers decreased from 20 to 10 (decreased by factor of 2), days increased from 4 to 8 (increased by factor of 2). ✓

**Answer:** 10 workers will take 8 days to lay the road.

Worked Example 2: Pumps Filling a Tank

**Problem:** 2 pumps can fill a tank in 18 hours. How long will it take 4 pumps of the same kind to fill the tank?

**Solution:**

Step 1: Recognize the inverse relationship.

More pumps → less time needed (inverse proportion)

Step 2: Apply inverse proportion formula.

2 × 18 = 4 × x (where x is time for 4 pumps)

Step 3: Solve.

36 = 4x

x = 9 hours

Step 4: Verify.

When pumps doubled (2 to 4), time was halved (18 to 9). ✓

**Answer:** 4 pumps will fill the tank in 9 hours.

Worked Example 3: School Provisions

**Problem:** A school has food provisions to feed 80 students for 15 days. If 20 more students join, for how many days will the provisions last?

**Solution:**

Step 1: Identify what changed.

Original: 80 students, 15 days

New: 80 + 20 = 100 students, x days

Step 2: Recognize the inverse relationship.

More students → provisions last fewer days (inverse proportion)

Step 3: Set up equation.

80 × 15 = 100 × x

Step 4: Solve.

1200 = 100x

x = 12 days

Step 5: Verify.

When students increased by a factor of 100/80 = 1.25, days decreased by a factor of 15/12 = 1.25. ✓

**Answer:** The provisions will last for 12 days.

Worked Example 4: Combined Work Problem

**Problem:** Ram takes 1 hour to cut vegetables and Shyam takes 1.5 hours to cut the same quantity. How long will they take working together?

**Solution:**

Step 1: Find each person's work rate.

Work to be completed = 1 unit

Ram's rate: completes 1 unit in 1 hour → rate = 1 unit/hour

Shyam's rate: completes 1 unit in 1.5 hours → rate = 1/1.5 = 2/3 unit/hour

Step 2: Find combined work rate.

Combined rate = 1 + 2/3 = 5/3 units/hour

Step 3: Find time to complete 1 unit.

Time = Work / Rate = 1 ÷ (5/3) = 1 × 3/5 = 3/5 hour

**Answer:** Working together, they will take 3/5 hour = 36 minutes.

Identifying Inverse Proportions in Real Life

**These pairs are in inverse proportion:**

Number of taps filling a tank and time to fill it

Number of workers and days to complete a fixed job

Number of painters and time to paint a fixed wall

Speed of a vehicle and time to cover a fixed distance

Number of machines and time to complete fixed production

Amount of food per person and number of people fed by fixed provisions

**These are NOT in inverse proportion:**

Distance a car travels and amount of petrol (direct proportion)

Length of cloth bought and price paid at fixed rate (direct proportion)

Number of pages in a book and time to read (direct proportion when reading speed is fixed)

Common Errors to Avoid

1. **Confusing direct and inverse proportions** — If one quantity increases and the other also increases, it's direct; if one increases while the other decreases, it's inverse

2. **Forgetting the constant of proportionality** — Always check that x × y equals the same value for all pairs

3. **Not recognizing real-life inverse relationships** — Familiarize yourself with common examples

4. **Arithmetic mistakes** — Carefully multiply and divide; verify by checking if the pattern makes sense

5. **Assuming all relationships are proportional** — Some quantities don't maintain proportional relationships at all

Special Case: Inverse of Inverse Proportions

If y is inversely proportional to x, then x is also inversely proportional to y. This is a symmetric relationship.

If x₁y₁ = x₂y₂, then the inverse (x₁/x₂ = y₂/y₁) is also true.

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Summary of Key Concepts

Proportionality Review

**Proportional Relationship:** When quantities change by the same factor

**Notation:** a : b :: c : d means a is to b as c is to d

**Test:** a × d = b × c (cross-multiplication method)

Multiple-term Ratios

Used for comparing more than two quantities

All terms must scale by the same factor

Formula for division: Part = (Total) × (Ratio term) / (Sum of all terms)

Maps and Scales

**RF** (Representative Fraction) shows map distance to actual distance ratio

Always shown as 1 : n format

Used to calculate actual geographical distances

Pie Charts

Circular graphs showing proportions of a whole

**Angle formula:** Angle = (Category value / Total) × 360°

Sum of all angles = 360°

Inverse Proportions

One quantity increases while the other decreases by the same factor

**Formula:** x × y = k (constant)

Test: Calculate x × y for all pairs; if all equal, they're inversely proportional

Property: x₁/x₂ = y₂/y₁

Common Application Patterns

Recipes and cooking mixtures (multi-term ratios)

Distance calculations from maps (RF and ratios)

Budget and resource allocation (dividing in ratios)

Survey and preference data (pie charts)

Work completion and time management (inverse proportions)

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Practice Problem Summary

**Type 1: Identifying Proportionality**

Given two ratios, determine if they're proportional using cross-multiplication

Simplify ratios to simplest form

**Type 2: Multi-term Ratio Division**

Divide a total quantity into parts according to given ratio

Application: paint mixing, angle construction, ingredient measurement

**Type 3: Map Distance Calculation**

Use RF to convert map measurements to actual distances

Set up proportions and solve

**Type 4: Pie Chart Construction**

Convert frequencies to angles using the 360° formula

Construct using protractor and verify sum equals 360°

**Type 5: Inverse Proportion Problems**

Identify inverse relationships in real-life scenarios

Apply the formula x₁y₁ = x₂y₂ to solve for unknown values

Verify answers using the ratio property