Direct & Inverse Proportion

Unit: Factorization Of Expressions

Chapter: Direct & Inverse Proportion

Reference: – What is Proportion, Direct Proportion Definition, Direct Proportion Formula (y = kx), Constant of Proportionality (k), Real-Life Examples of Direct Proportion, Graph of Direct Proportion, Inverse Proportion Definition, Inverse Proportion Formula (y = k/x), Real-Life Examples of Inverse Proportion, Graph of Inverse Proportion, Solving Proportion Problems, Solved Examples, Odd-One-Out Problems, Common Mistakes

After studying this chapter, you should be able to understand:

• What is Direct Proportion and How to Identify It
• What is Inverse Proportion and How to Identify It
• How to Find the Constant of Proportionality
• How to Solve Direct and Inverse Proportion Problems
• How to Recognize the Graphs of Direct and Inverse Proportion

Introduction to Direct and Inverse Proportion

Definition

Proportion describes the relationship between two quantities. When two quantities change in relation to each other, they are either in direct proportion or inverse proportion.

When we study proportion, we essentially ask:

"As one quantity increases, what happens to the other quantity? Does it also increase, or does it decrease?"

The answer determines whether the relationship is direct or inverse.

Importance of Proportion

• Used in everyday situations (speed, time, cost, recipes)
• Essential for scaling and resizing (maps, blueprints)
• Helps solve problems without complex equations
• Foundation for ratio and percentage concepts
• Used in science (density, pressure, gas laws)

Example – Direct Proportion: The more hours you work, the more money you earn. As one increases, the other increases.

Example – Inverse Proportion: The faster you drive, the less time a trip takes. As one increases, the other decreases.

Subtopics

1. Direct Proportion

Two quantities are in direct proportion when they increase or decrease together at the same rate. If one doubles, the other doubles. If one triples, the other triples.

Key Property: The ratio between the two quantities is always constant.

Formula: y = kx

Where y and x are the two quantities, and k is the constant of proportionality.

Constant of Proportionality (k): k = y/x (the same value for all pairs)

Examples of Direct Proportion:

• Cost of apples and number of apples (cost = price per apple × number)
• Distance travelled and time at constant speed (distance = speed × time)
• Weight and mass (weight = gravity × mass)
• Circumference and diameter of a circle (C = π × d)
• Amount of ingredients and number of servings in a recipe

How to Solve Direct Proportion Problems:

Step 1: Identify the two quantities and write the relationship y = kx

Step 2: Use the given pair of values to find k

Step 3: Use k to find the unknown value

Example 1: If 3 pencils cost $6, how much do 8 pencils cost?

Here cost ∝ number of pencils. Let c = cost, n = number. c = k × n

k = c/n = 6/3 = 2 (cost per pencil = $2)

For 8 pencils: c = 2 × 8 = $16

Answer: $16

Example 2: A car travels 120 miles in 2 hours. How far will it travel in 5 hours at the same speed?

Distance ∝ time. d = k × t

k = d/t = 120/2 = 60 miles per hour

In 5 hours: d = 60 × 5 = 300 miles

Answer: 300 miles

Graph of Direct Proportion: A straight line passing through the origin (0,0). As x increases, y increases.

2. Inverse Proportion

Two quantities are in inverse proportion when one increases and the other decreases at the same rate. If one doubles, the other halves. If one triples, the other becomes one-third.

Key Property: The product of the two quantities is always constant.

Formula: y = k/x (or xy = k)

Constant of Proportionality (k): k = x × y (the same value for all pairs)

Examples of Inverse Proportion:

• Speed and time for a fixed distance (faster speed = less time)
• Number of workers and time to complete a job (more workers = less time)
• Price per item and number purchased with a fixed budget (higher price = fewer items)
• Pressure and volume of a gas at constant temperature (Boyle's Law)

How to Solve Inverse Proportion Problems:

Step 1: Identify the two quantities and write the relationship xy = k

Step 2: Use the given pair of values to find k

Step 3: Use k to find the unknown value

Example 1: 6 workers can build a wall in 10 days. How many days will 15 workers take to build the same wall?

Workers ∝ 1/time, so workers × days = k

k = 6 × 10 = 60 (total worker-days)

For 15 workers: 15 × d = 60 → d = 60/15 = 4 days

Answer: 4 days

Example 2: A car traveling at 50 mph takes 6 hours to reach a destination. How long will it take at 75 mph?

Speed × time = k

k = 50 × 6 = 300

At 75 mph: 75 × t = 300 → t = 300/75 = 4 hours

Answer: 4 hours

Graph of Inverse Proportion: A curve (hyperbola) that never touches the axes. As x increases, y decreases.

4. Word Problems – Step by Step

Direct Proportion Word Problem Strategy:

Step 1: Identify the two variables and write the proportion statement (y ∝ x)

Step 2: Set up the equation y = kx

Step 3: Find k using the given pair

Step 4: Substitute the new value to find the unknown

Inverse Proportion Word Problem Strategy:

Step 1: Identify the two variables and write the proportion statement (y ∝ 1/x)

Step 2: Set up the equation xy = k

Step 3: Find k using the given pair

Step 4: Substitute the new value to find the unknown

5. Direct Proportion in Tables

In a table showing direct proportion, the ratio y/x is the same for all pairs.

Example – Direct Proportion Table:

Check ratios: 6/2 = 3, 12/4 = 3, 18/6 = 3, 24/8 = 3 ✓ Constant

Example – Not Direct Proportion:

Ratios: 5/2 = 2.5, 9/4 = 2.25 – not constant

6. Inverse Proportion in Tables

In a table showing inverse proportion, the product xy is the same for all pairs.

Example – Inverse Proportion Table:

Check products: 2×30=60, 4×15=60, 5×12=60, 10×6=60 ✓ Constant

Solved Examples

Example 1 – Direct Proportion: If 4 kg of rice costs $20, how much do 7 kg cost?

Solution: Cost ∝ weight → C = k × w

k = C/w = 20/4 = 5

For 7 kg: C = 5 × 7 = $35

Answer: $35

Example 2 – Inverse Proportion: 5 taps can fill a tank in 12 minutes. How long will 8 taps take?

Solution: Taps × time = k

k = 5 × 12 = 60

For 8 taps: 8 × t = 60 → t = 60/8 = 7.5 minutes

Answer: 7.5 minutes

Example 3 – Find k in Direct Proportion: y varies directly with x. When x = 3, y = 21. Find y when x = 8.

Solution: y = kx → k = y/x = 21/3 = 7

When x = 8: y = 7 × 8 = 56

Answer: y = 56

Example 4 – Find k in Inverse Proportion: y varies inversely with x. When x = 4, y = 9. Find y when x = 6.

Solution: xy = k → k = 4 × 9 = 36

When x = 6: 6 × y = 36 → y = 36/6 = 6

Answer: y = 6

Common Mistakes to Avoid

Mistake 1 – Confusing direct and inverse proportion
Thinking "more workers means more time" is wrong. More workers actually mean less time (inverse).
Correct understanding: Identify whether the quantities move together (direct) or opposite (inverse).

Mistake 2 – Using the wrong formula
Using y = kx for inverse proportion or xy = k for direct proportion leads to wrong answers.
Correct understanding: Direct → y/x = k; Inverse → xy = k.

Mistake 3 – Forgetting to find k first
Trying to solve proportion problems without finding the constant of proportionality leads to errors.
Correct understanding: Always find k using the given pair before solving for the unknown.

Mistake 4 – Not checking if the graph passes through the origin
A direct proportion graph must pass through (0,0). If it doesn't, it is not direct proportion.
Correct understanding: y = kx always gives (0,0). If there is a fixed starting value, it is not direct proportion.

Mistake 5 – Assuming all increasing relationships are direct
A relationship can be increasing but not proportional (like y = 2x + 1).
Correct understanding: Direct proportion requires y/x to be constant and the line to pass through the origin.

Mistake 6 – Dividing instead of multiplying for inverse proportion
In inverse proportion, if x doubles, y halves (divide by 2), not subtract something.
Correct understanding: Use the formula xy = k to find new values.

Quick Reference Summary

Direct Proportion: y = kx (y/x = k constant)
As x increases, y increases
Graph: straight line through origin (0,0)

Inverse Proportion: y = k/x (xy = k constant)
As x increases, y decreases
Graph: curve (hyperbola)

Constant of Proportionality (k):
Direct: k = y/x
Inverse: k = xy

To Solve Direct Proportion: Find k = y/x, then y = k × new x

To Solve Inverse Proportion: Find k = x × y, then new y = k ÷ new x

Real-Life Examples:
Direct: cost and quantity, distance and time (constant speed)
Inverse: speed and time (fixed distance), workers and time (fixed job)