Micky wants to save $450,000 over the next 10 years. If the interest rate is 6% per annum compounding annually, how much should Micky contribute each year? Give your answer to the nearest dollar. Instead, Micky decides to contribute $835 every three months for 10 years to an annuity paying 6% per annum, compounding quarterly. How much will Micky have at the end of 10 years?

SSCE HSC Mathematics Advanced Absolute value functions: Micky wants to save $450,000 ove

Micky wants to save $450,000 over the next 10 years - HSC - SSCE Mathematics Advanced - Question 15 - 2023 - Paper 1

Question 15

Micky wants to save $450,000 over the next 10 years. If the interest rate is 6% per annum compounding annually, how much should Micky contribute each year? Give you... show full transcript

Worked Solution & Example Answer:Micky wants to save $450,000 over the next 10 years - HSC - SSCE Mathematics Advanced - Question 15 - 2023 - Paper 1

Step 1

If the interest rate is 6% per annum compounding annually, how much should Micky contribute each year?

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Answer

To determine Micky's annual contribution, we use the future value formula for an annuity:

$FV = C \times \frac{(1 + r)^n - 1}{r}$

Where:

$FV$ = future value ($450,000)
$C$ = annual contribution
$r$ = annual interest rate (0.06)
$n$ = number of periods (10)

Using the future value factor from the table for 10 years at 6%, we have:

$FV = C \times 13.181$

Thus,

$450,000 = C \times 13.181$

Solving for $C$ gives:

$C = \frac{450,000}{13.181} \approx 34,140$

Therefore, Micky should contribute approximately $34,140 each year.

Step 2

How much will Micky have at the end of 10 years?

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Answer

To find the future value of Micky's quarterly contributions, we need to calculate the total contributions compounded quarterly:

Given:

Contribution per period = $835
Total periods over 10 years = $10 \times 4 = 40$
Quarterly interest rate = $rac{6}{4} = 1.5\\% = 0.015$

Using the future value factor for 40 quarters at 6% annual interest:

$FV = 835 \times 54.268$

Calculating this gives:

$FV \approx 463,177.38$

Thus, Micky will have approximately $463,177.38 at the end of 10 years.

Micky wants to save $450,000 over the next 10 years - HSC - SSCE Mathematics Advanced - Question 15 - 2023 - Paper 1

Worked Solution & Example Answer:Micky wants to save $450,000 over the next 10 years - HSC - SSCE Mathematics Advanced - Question 15 - 2023 - Paper 1

If the interest rate is 6% per annum compounding annually, how much should Micky contribute each year?

How much will Micky have at the end of 10 years?

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