In proportion problems, two quantities are linked so that a change in one causes a predictable change in the other. The relationship can be direct or inverse.

 

In direct proportion, as one quantity increases, the other increases at the same rate (and vice versa).
This is written as:

$$
y \propto x
$$

 

This means \(y\) is directly proportional to \(x\).
We turn this into an equation by introducing a constant of proportionality \(k\):

$$
y = kx
$$

 

To solve problems:

1. Use given values to find \(k\)
2. Substitute \(k\) to find the unknown value

For example, if \(y = 20\) when \(x = 5\):

$$
20 = k \times 5
$$

$$
k = 4
$$

 

So the equation is:

$$
y = 4x
$$

 

If \(x = 9\), then:

$$
y = 4 \times 9 = 36
$$

 

In inverse proportion, as one quantity increases, the other decreases so that the product stays constant.
This is written as:

$$
y \propto \frac{1}{x}
$$

 

Which gives the equation:

$$
y = \frac{k}{x}
$$

 

or equivalently:

$$
xy = k
$$

 

For example, if \(y = 6\) when \(x = 4\):

$$
k = 6 \times 4 = 24
$$

 

So the relationship is:

$$
y = \frac{24}{x}
$$

If \(x = 8\), then:

$$
y = \frac{24}{8} = 3
$$

 

A useful check:

• In direct proportion, doubling \(x\) doubles \(y\)
• In inverse proportion, doubling \(x\) halves \(y\)

 

Always identify the type of proportion first. Using the wrong model will give a sensible-looking answer that is mathematically incorrect.