The profit made by a shop increases each year. The profit made by the shop in year n is $E_n$. Given that the profit made by the shop in the next year is $E_{n+1}$, then $E_{n+1} = aE_n + 800$ where a is a constant. The table shows the profit made by the shop in 2018 and 2019. Year | 2018 | 2019 -------|---------|--------- Profit | £24000 | £29600 Work out the profit predicted to be made by the shop in 2021.

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The profit made by a shop increases each year - Edexcel - GCSE Maths - Question 22 - 2022 - Paper 3

Question 22

The profit made by a shop increases each year. The profit made by the shop in year n is $E_n$. Given that the profit made by the shop in the next year is $E_{n+1}$, ... show full transcript

Worked Solution & Example Answer:The profit made by a shop increases each year - Edexcel - GCSE Maths - Question 22 - 2022 - Paper 3

Step 1

Find the constant a using the profit in 2018 and 2019

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Answer

From the table, we know:

For 2018: $E_{2018} = 24000$
For 2019: $E_{2019} = 29600$

Using the formula for profit:
$E_{2019} = aE_{2018} + 800$
Substituting the values:
$29600 = a(24000) + 800$
Subtracting 800 from both sides:
$28800 = 24000a$
Now solving for a:
$a = \frac{28800}{24000} = 1.2$ .

Step 2

Calculate the profit for 2020 using the found value of a

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Answer

Now we can find the profit for 2020 ( $E_{2020}$ ):
Using the formula:
$E_{2020} = aE_{2019} + 800$
Substituting the known values:
$E_{2020} = 1.2(29600) + 800$
Calculating:
$E_{2020} = 35520 + 800 = 36320$ .

Step 3

Predict the profit for 2021

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Answer

Finally, we can calculate the profit for 2021 ( $E_{2021}$ ):
Using the formula:
$E_{2021} = aE_{2020} + 800$
Substituting the known values:
$E_{2021} = 1.2(36320) + 800$
Calculating:
$E_{2021} = 43584 + 800 = 44384$ .

Thus, the predicted profit for 2021 is £44384.

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