Direct and Inverse Proportion: GCSE Mathematics Guide

Introduction

Direct and inverse proportions are fundamental concepts in GCSE Mathematics. They describe how two variables change in relation to each other, and they have numerous real-world applications.

Key Concepts and Definitions

Direct Proportion:
Two variables are directly proportional if they both increase or decrease together at the same rate.
Formula: y = kx
y is the dependent variable that changes in proportion to x
x is the independent variable
k is the constant of proportionality
Inverse Proportion:
Two variables are inversely proportional if one decreases as the other increases at the same rate.
Formula: y = k/x
y is the dependent variable that changes inversely to x
x is the independent variable
k is the constant of proportionality

Common Mistakes to Avoid

Using the wrong formula (direct vs. inverse)
Assuming all proportional relationships are linear
Not using consistent units in the equation
Forgetting to find the constant of proportionality

Step-by-Step Explanations

Direct Proportion:

1. Find the constant of proportionality (k) by dividing the change in y by the change in x.

2. Use the formula y = kx to find the value of y for any given value of x.

Inverse Proportion:

1. Find the constant of proportionality (k) by multiplying the values of x and y that are inversely proportional.

2. Use the formula y = k/x to find the value of y for any given value of x.

Practice Problems with Solutions

Problem 1 (Direct Proportion):

The speed of a car is directly proportional to its engine power. If a car with a 120bhp engine can reach a speed of 140mph, what speed can a car with a 180bhp engine reach?

Solution: k = 140/120 = 1.17
y = kx = 1.17 x 180 = 210.6mph
Problem 2 (Inverse Proportion):

The volume of a gas is inversely proportional to its pressure. If a gas occupies 10 liters at a pressure of 2 atmospheres, what volume will the same gas occupy at a pressure of 5 atmospheres?

Solution: k = 10 x 2 = 20
y = k/x = 20/5 = 4 liters

Conclusion

Understanding direct and inverse proportion will significantly enhance your performance in GCSE Mathematics. Remember the following tips for exam success: