Proportion is used to show how quantities and amounts are related to each other. The amount that quantities change in relation to each other is governed by proportion rules.

Money invested in a bank can generate two different types of interest. Compound interest occurs when interest is added to the balance at the end of a time period and interest is then calculated on the total of these at the end of the next time period.

If £10,000 is invested into an account offering 2% compound interest each year then the balance grows as shown in the table.

Year | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|---|

Balance | 1,000.00 | 1,020.00 | 1,040.40 | 1,061.21 | 1,082.43 | 1,104.08 | 1,126.16 | 1,148.69 | 1,171.66 | 1,195.09 | 1,218.99 |

This is an example of an exponential graph in the form \(y = k^x\).

Graphs in the form of \(y = k^x\) increase in value.

The graph shows how bacteria \((y)\) grows over time \((t)\). The equation of the graph is:

\[y = 20g^t\]

From the graph, when \(t = 2\) \(y = 2,000\).

Substituting these values into \(y = 20g^t\):

\[2,000 = 20g^2\]

Dividing by 20 gives:

\[100 = g^2\]

Square rooting gives:

\[10 = g\]

So \(y = 20 \times 10^t\).