Some problems involve more than one rate being applied to a value. This can include percentages such as discounts, price increases, VAT, interest rates or combinations of these. The order in which the rates are applied is important.

A clear and efficient way to deal with multiple rates is to use multipliers. Each rate is converted into a multiplier, and these multipliers are applied step by step.

For a percentage increase, the multiplier is found by adding the percentage increase to 1.
For a percentage decrease, the multiplier is found by subtracting the percentage decrease from 1.

For example, a price is reduced by \( 10\% \) and then increased by \( 20\% \).

The multiplier for a \( 10\% \) reduction is:

$$
0.90
$$

The multiplier for a \( 20\% \) increase is:

$$
1.20
$$

If the original price is 80, the final price is found by applying both multipliers:

$$
80 \times 0.90 \times 1.20
$$

$$
= 86.4
$$

This shows that applying a decrease and then an increase does not return the value to its original amount.

Multiple rates are common in money problems. For example, an item may be discounted and then have VAT added. If an item costs 50, is discounted by \( 20\% \) and then has VAT added at \( 20\% \), the calculation is:

$$
50 \times 0.80 \times 1.20
$$

$$
= 48
$$

So the final price is 48.

It is important to apply each rate in the correct order and to use multipliers carefully. Combining multipliers helps reduce rounding errors and makes calculations involving multiple rates quicker and more reliable.