The Dynamics of Growth and Decay

Growth and decay describe how things increase or decrease over time. It’s like watching a plant grow (growth) or seeing a snowball melt (decay). Let’s dive into how we can use maths to measure these changes

What Are Growth and Decay?

• Growth: When something grows, it increases over time. Imagine money in a bank account that earns interest each year—each year, the amount grows a bit more!

• Decay: When something decays, it decreases over time. Think about food going stale or a battery running out. Each day, there’s a little less until it’s gone.

The Formula for Growth and Decay

For both growth and decay, we use a similar formula:

$$New Amount = Original Amount \times ( 1 \pm \frac{Rate}{100} )^{t}$$

where:

• New Amount is the amount after growth or decay.
• Original Amount is the starting amount.
• Rate is the percentage growth or decay rate.
• $t$ is the time (usually in years).

Tip: Use $+$ for growth and $-$ for decay in the formula

 

Understanding Growth and Decay with Examples

Example 1: Growth

Question: You invest $\pounds 200$ in an account that grows by $5\%$ each year. What will it be worth after 3 years?

1. Identify the values:

   • Original Amount = $\pounds 200$
   • Rate = $5\%$
   • Time $t = 3$ 
2. Plug into the growth formula: $$New Amount  = 200 \times (1 + \frac{5}{100})^{3}$$
3. Calculate: $$= 200 \times (1.05)^3$$
4. Solve: $$= 200 \times 1.157625 = 231.53$$

So the investment is worth $\pounds 231.53$ 

 

 

Example 2: Decay

Question: A car valued at $\pounds 10,000$ loses $12\%$ of its value each year. What will it be worth after 4 years?

1. Identify the values:

   • Original Amount = $\pounds 10,000$
   • Rate = $12\%$
   • Time $t = 4$
2. Plug into the growth formula: $$New Amount  = 200 \times (1 + \frac{5}{100})^{3}$$

    

3. Calculate: $$= 200 \times (1.05)^3$$

    

4. Solve: $$= 200 \times 1.157625 = 231.53$$

So the cars value is worth $\pounds 5996.95$