# Growth and Decay Problems

## Growth and Decay

Exponential growth and decay problems are often associated with compound interest problems, or looking at the rate of virus’ increasing or decreasing.

They use an exponential function such as y=k^{x} or y=k^{-x} k stands for a constant.

This graph shows y=3^{x}

This graph shows y=3^{-x}

In the higher paper you may be asked to use an exponential graph to find its’ equation.

For example: This graph shows the rate at which a virus (y) increases over time (x).

The formula given is y=3g^{x}. Using the graph work out the value of g.

From the graph we can see that when y=6 x=1

We can use these values to solve the equation and work out:

y=3g^{x}

6=3g^{1}

2=g^{1}

2=g

## Compound Interest

**Simple Interest**

If I have some money in the bank, each year I may be able to earn interest on it.

For example, if I have £1000 in the bank and I earn 5% interest each year after the first year I will have:

£1000 +5% of £1000 =

£1000 + £50 = £1050

Or this can be worked out using multipliers:

£1000 x 1.05 = £1050

This is an example of simple interest. I simply add the percentage increase onto the amount.

**Compound interest**

Compound interest is when we look at how much interest will be added over a longer period of time.

For example, after one year with an interest rate of 5% I no longer have just £1000 in the bank, I have £1050, and I need to take this into account for the next year.

Year 1:

- Start: £1000
- End: £1050

Year 2:

- Start: £1050
- End: 1,102.5

The interest is compounding. Now, we can work out each yearly value, or take a short-cut and use this formula:

Eg. If I start with £1000 and get a fixed interest rate of 5% for five years I could work out the total by:

£1000 x (1+5%)^{5}

=£1000 x ( 1+0.05)^{5}

=£1,276.28

**Compound decay**

Equally, sometimes things may steadily decrease every year. In this case I would use this formula

= Original value x (1-%)

So questions may require you to rearrange the formula so be prepared!

- 1. A car decreases in value by 4% each year. It was initially worth £3000, how much is it worth after 6 years?
- Your answer should include: £2348.27 / 2348.27
- The graph y=6h^x has the point ( 2,216) what is the value of h?
- 6
- I have £400 in my bank account, I earn 3.5% interest each year, how much will I have earnt after 5 years?
- Your answer should include: £75.07 / 75.07