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On Alice’s 11th birthday she started to receive an annual allowance - Edexcel - A-Level Maths Pure - Question 9 - 2006 - Paper 1
Question 9
On Alice’s 11th birthday she started to receive an annual allowance. The first annual allowance was £500 and on each following birthday the allowance was increased b... show full transcript
Worked Solution & Example Answer:On Alice’s 11th birthday she started to receive an annual allowance - Edexcel - A-Level Maths Pure - Question 9 - 2006 - Paper 1
Step 1
Show that, immediately after her 12th birthday, the total of the allowances that Alice had received was £1200.
Answer
To find the total allowances received immediately after her 12th birthday, we first calculate the allowances for 11th and 12th birthdays:
- On her 11th birthday, Alice received £500.
- On her 12th birthday, she received £200 more than the previous year, which totals £500 + £200 = £700.
Thus, the total she received after her 12th birthday is:
[ 500 + 700 = 1200 ]
This confirms that the total allowance received immediately after her 12th birthday is indeed £1200.
Step 2
Find the amount of Alice’s annual allowance on her 18th birthday.
Answer
The allowance increases by £200 each year after her 11th birthday. Therefore, by her 18th birthday (which is 7 years later), we can calculate:
- Initial amount = £500.
- Number of increases = 7 (from 12th to 18th birthday).
- Increase amount per year = £200.
Calculating the total allowance on her 18th birthday:
[ 500 + 7 \times 200 = 500 + 1400 = 1900 ]
Thus, Alice’s annual allowance on her 18th birthday is £1900.
Step 3
Find the total of the allowances that Alice had received up to and including her 18th birthday.
Answer
To find the total amount Alice received by her 18th birthday, we use the formula for the sum of an arithmetic series:
[ S_n = \frac{n}{2} \left(2a + (n-1)d\right) ]
Where:
- n = total number of terms (8 years from 11th to 18th birthday)
- a = first term (£500)
- d = common difference (£200)
Plugging in the values:
[ S_8 = \frac{8}{2} \left(2 \times 500 + (8-1) \times 200\right) ]
Calculating gives:
[ S_8 = 4 \left(1000 + 1400\right) = 4 \times 2400 = 9600 ]
Thus, the total amount received by Alice up to and including her 18th birthday is £9600.
Step 4
When the total of the allowances that Alice had received reached £32,000 the allowance stopped.
Answer
To determine when the allowances stopped, we set the sum equal to £32,000:
[ S_n = \frac{n}{2} \left(2 \times 500 + (n-1) \times 200\right) = 32000 ]
This simplifies to:
[ n(500 + 100(n-1)) = 32000 ]
Solving the quadratic gives:
[ (n(200n - 200 + 500 ) = 32000) ]
This results in:
[ n^2 + 20n - 320 = 0 ]
Using the quadratic formula:
[ n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ]
Substituting:
- a = 1, b = 20, c = -320
Calculating gives us:
- n = 16 (the positive root)
Alice received allowances for 16 years. Thus, after her 11th birthday, she was receiving her last allowance at her 27th birthday.