Compound Growth and Decay Revision Notes for Edexcel GCSE Maths

Compound Growth and Decay

What is compound growth and decay?

Compound growth and decay are mathematical concepts that describe how quantities change over time. Compound growth shows how something increases over time, such as money growing in a savings account through interest. Compound decay shows the opposite - how something decreases over time, like a car losing its value as it ages.

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The key difference from simple growth is that in compound situations, the change is calculated on the new amount each time period, not just the original amount.

The essential formula

The most important thing to understand about this topic is mastering the compound formula. This single formula can solve virtually any compound growth or decay problem you'll encounter.

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The Compound Formula

$N = N\_0\left(1 + \frac{r}{100}\right)^n$

Where each part represents:

N = the final amount after the time period
N₀ = the initial starting amount
r = the percentage change per time period
n = the number of time periods

Understanding this formula thoroughly will make compound problems much more manageable.

Working with percentage increases and decreases

One aspect that often confuses students is how to handle the percentage part of the formula. The key is converting percentages into decimal multipliers:

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Converting Percentages to Multipliers

For increases: Add the percentage to 100, then divide by 100

A 5% increase becomes 1.05
A 26% increase becomes 1.26

For decreases: Subtract the percentage from 100, then divide by 100

A 5% decrease becomes 0.95
A 26% decrease becomes 0.74

This conversion is crucial because it allows us to use the same formula for both growth and decay situations.

Compound interest example

Let's look at a typical compound interest problem to see how the formula works in practice.

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Worked Example: Compound Interest Calculation

When someone invests £1000 in a savings account that pays 8% compound interest per year, we can calculate how much they'll have after 6 years.

Step 1: Identify the components

Initial amount ( $N\_0$ ) = £1000
Percentage increase (r) = 8% per year
Time period (n) = 6 years

Step 2: Apply the formula $\text{Amount} = 1000(1.08)^6$

Step 3: Calculate $\text{Amount} = 1000 \times 1.586874 = £1586.87$

This shows how the money grows because each year, the interest is calculated on the new total (including previous interest), not just the original £1000.

Depreciation example

Depreciation problems work similarly but involve decreasing values. Here's how to handle them:

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Worked Example: Car Depreciation

Susan's car cost £6500 and depreciates by 9% each year. To find its value after 3 years:

Step 1: Identify the components

Initial value ( $N\_0$ ) = £6500
Percentage decrease (r) = 9% per year
Time period (n) = 3 years

Step 2: Apply the formula (note the decrease factor) $\text{Value} = 6500(0.91)^3$

Step 3: Calculate $\text{Value} = 6500 \times 0.753571 = £4898.21$

Notice how we use 0.91 instead of 1.09 because the car is losing value, not gaining it.

Biological applications

Compound growth isn't limited to financial situations. In biology, populations often grow exponentially.

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Worked Example: Bacteria Growth

If bacteria start with 500 cells and increase by 15% each day, we can create a formula relating the cell count (c) to the number of days (d).

Formula: $c = 500 \times (1.15)^d$

This shows how the same mathematical principles apply across different contexts, whether dealing with money, populations, or other quantities that change over time.

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Research has found that many biological processes follow exponential patterns, making this formula incredibly useful in science and medicine.

Key problem-solving steps

When approaching any compound growth or decay problem, follow these systematic steps:

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Problem-Solving Strategy:

Identify the components: What's the initial amount? What's the percentage change? Over how many time periods?
Determine if it's growth or decay: This affects whether you add or subtract the percentage from 1.
Apply the formula: Substitute your values into $N = N\_0\left(1 + \frac{r}{100}\right)^n$
Calculate carefully: Use a calculator for the exponential part, and round appropriately for the context.

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Key Points to Remember:

The compound formula $N = N\_0\left(1 + \frac{r}{100}\right)^n$ is your most powerful tool for these problems
For increases, add the percentage to 1; for decreases, subtract from 1
The same formula works for money, populations, depreciation, and many other real-world situations
Practice converting percentages to decimal multipliers - this is where many mistakes happen
Always check if your answer makes sense in the context of the problem

Compound Growth and Decay (Edexcel GCSE Maths): Revision Notes