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The table shows the percentage changes in profit for a shop from 2019 to 2021:

- 2019: +12%
- 2020: -5%
- 2021: -8%

### Part (a): Find the percentage change in profit of the shop from 2018 to 2021.

The overall percentage change over multiple years is not simply the sum of the percentage changes for each year. We need to apply the percentage change year by year using the formula for compounded growth.

Let’s assume the profit in 2018 is $ P_{2018} $.

The profit in each subsequent year can be calculated using the following steps:

1. **For 2019**:  
   Profit increases by 12%:  
   \[
   P_{2019} = P_{2018} \times (1 + 12\%) = P_{2018} \times 1.12
   \]

2. **For 2020**:  
   Profit decreases by 5%:  
   \[
   P_{2020} = P_{2019} \times (1 - 5\%) = P_{2019} \times 0.95 = P_{2018} \times 1.12 \times 0.95
   \]

3. **For 2021**:  
   Profit decreases by 8%:  
   \[
   P_{2021} = P_{2020} \times (1 - 8\%) = P_{2020} \times 0.92 = P_{2018} \times 1.12 \times 0.95 \times 0.92
   \]

Now, let’s calculate the total percentage change over these years. The overall multiplier is:

\[
\text{Multiplier} = 1.12 \times 0.95 \times 0.92
\]

After finding this multiplier, the total percentage change can be determined by:

\[
\text{Total percentage change} = (\text{Multiplier} - 1) \times 100\%
\]

### Part (b): Find the profit in 2021, given that the profit was $24.2 million in 2018.

We will use the profit in 2018, $ P_{2018} = 24.2 $ million, and apply the same calculations we did for Part (a) to determine the profit in 2021:

\[
P_{2021} = 24.2 \times 1.12 \times 0.95 \times 0.92
\]

Let me now compute both parts.### Part (a):  
The total percentage change in the profit of the shop from 2018 to 2021 is approximately **-2.11%**. This means the overall profit decreased by about 2.11% over the three-year period.

### Part (b):  
Given that the profit in 2018 was $24.2 million, the profit in 2021 would be approximately **$23.69 million**.

Do you need further explanation on any part of the solution?

Here are 5 related questions that might help deepen your understanding:
1. How would the overall percentage change be different if all the yearly changes were positive?
2. What would be the profit in 2021 if there was no change in profit for 2020 or 2021?
3. How does compounding affect percentage changes over multiple years?
4. How would you calculate the profit for 2022, assuming a further 10% increase?
5. What if the profit in 2018 had been $30 million—how would the 2021 profit change?

**Tip**: When dealing with percentage increases or decreases over multiple years, always apply the changes sequentially instead of adding or subtracting percentages directly.

The table below shows the percentage changes in the profit of a shop, as compared with the previous years, from 2019 to 2021. Find the percentage change in the profit of the shop from 2018 to 2021. If the profit of the shop was $24.2 million in 2018, find the profit in 2021.

Learn how to calculate the compound percentage changes in profit for a business over multiple years. Using initial data from 2018 and annual percentage changes, find out how the profit evolved by 2021, starting with an initial profit of $24.2 million.

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