Direct and Inverse Proportion
⭐ Higher Tier Content
Proportion describes a relationship between two variables. In GCSE mathematics, the two main types are direct proportion and inverse proportion. Understanding these relationships allows equations to be constructed and used to solve problems.
Direct Proportion
Two variables are in direct proportion if, as one increases, the other increases at a constant rate, and if one decreases, the other decreases at the same rate.
If \( y \) is directly proportional to \( x \), this is written as:
$$
y \propto x
$$
This means that \( y \) is equal to a constant multiple of \( x \). Introducing a constant of proportionality \( k \) gives the equation:
$$
y = kx
$$
Constructing an Equation for Direct Proportion
To find the value of \( k \), substitute a known pair of values into the equation.
For example, suppose \( y \) is directly proportional to \( x \) and when \( x = 4 \), \( y = 20 \).
Substitute into \( y = kx \):
$$
20 = 4k
$$
$$
k = 5
$$
So the equation is:
$$
y = 5x
$$
Using an Equation for Direct Proportion
Once the equation is known, it can be used to find other values.
Using \( y = 5x \), when \( x = 7 \):
$$
y = 5 \times 7
$$
$$
y = 35
$$
Inverse Proportion
Two variables are in inverse proportion if, as one increases, the other decreases so that their product stays constant.
If \( y \) is inversely proportional to \( x \), this is written as:
$$
y \propto \frac{1}{x}
$$
Introducing a constant of proportionality \( k \) gives:
$$
y = \frac{k}{x}
$$
Constructing an Equation for Inverse Proportion
To find \( k \), substitute known values.
For example, suppose \( y \) is inversely proportional to \( x \) and when \( x = 3 \), \( y = 8 \).
Substitute into the equation:
$$
8 = \frac{k}{3}
$$
Multiply both sides by \( 3 \):
$$
k = 24
$$
So the equation is:
$$
y = \frac{24}{x}
$$
Using an Equation for Inverse Proportion
Using \( y = \frac{24}{x} \), when \( x = 6 \):
$$
y = \frac{24}{6}
$$
$$
y = 4
$$
As \( x \) increases, \( y \) decreases, showing inverse proportion.
Comparing Direct and Inverse Proportion
In direct proportion, the ratio \( \frac{y}{x} \) is constant.
In inverse proportion, the product \( xy \) is constant.
Recognising which type of proportion applies is essential before forming an equation.
Key Points to Remember
Direct proportion has the form \( y = kx \).
Inverse proportion has the form \( y = \frac{k}{x} \).
Always find the constant of proportionality using known values.
Use the equation to calculate unknown values.
Being able to construct and use equations for direct and inverse proportion is an important skill in algebra and in many real world applications.