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 $8535 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 Standard Annuities: Micky wants to save $450,000 ove

Micky wants to save $450,000 over the next 10 years - HSC - SSCE Mathematics Standard - Question 25 - 2023 - Paper 1

Question 25

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... show full transcript

Worked Solution & Example Answer:Micky wants to save $450,000 over the next 10 years - HSC - SSCE Mathematics Standard - Question 25 - 2023 - Paper 1

Step 1

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.

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Answer

To determine how much Micky needs to contribute each year to save $450,000 in 10 years with an interest rate of 6%, we will use the future value annuity formula:

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

Where:

$FV$ is the future value ($450,000)
$P$ is the annual payment (which we want to find)
$r$ is the interest rate (0.06)
$n$ is the number of years (10)

Rearranging for $P$ gives: $P = \frac{FV \cdot r}{(1 + r)^n - 1}$

Substituting the values: $P = \frac{450000 \cdot 0.06}{(1 + 0.06)^{10} - 1}$

Calculating: $(1 + 0.06)^{10} = 1.790847$

So, $P = \frac{450000 \cdot 0.06}{1.790847 - 1}$ $P = \frac{27000}{0.790847} \approx 34140$

Thus, Micky should contribute approximately $34,140 each year (to the nearest dollar).

Step 2

Instead, Micky decides to contribute $8535 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?

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Answer

In this scenario, Micky is contributing $8535 every three months, so we'll first determine the effective quarterly interest rate and the total number of contributions:

Interest Rate per Period:
- The annual interest rate is 6%, so the quarterly rate is: $r = \frac{6\%}{4} = 1.5\% = 0.015$
Number of Periods:
- Micky contributes quarterly for 10 years: $n = 10 \times 4 = 40$
Future Value Calculation:
- The future value formula for an annuity is: $FV = P \times \frac{(1 + r)^n - 1}{r}$
- Plugging in the values: $FV = 8535 \times \frac{(1 + 0.015)^{40} - 1}{0.015}$
Calculation:
- First, calculate $(1 + 0.015)^{40}$ : $(1.015)^{40} \approx 1.806111$ Then, $FV = 8535 \times \frac{1.806111 - 1}{0.015} \approx 8535 \times 53.7411 \approx 459,423$

Thus, Micky will have approximately $459,423 at the end of 10 years.

Micky wants to save $450,000 over the next 10 years - HSC - SSCE Mathematics Standard - Question 25 - 2023 - Paper 1

Worked Solution & Example Answer:Micky wants to save $450,000 over the next 10 years - HSC - SSCE Mathematics Standard - Question 25 - 2023 - Paper 1

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 $8535 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?

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