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

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

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Answer

To determine the annual contribution needed to save $450,000 over 10 years with a 6% interest rate compounded annually, we first need to find the annuity factor from the table for 10 years at 6%:

From the table:

Annuity Factor = 13.181

Now, we can establish the equation:

ext{Annual Contribution} = \frac{ ext{Future Value}}{ ext{Annuity Factor}} = \frac{450000}{13.181} \\ = 34,140

Thus, Micky needs to contribute $34,140 each year.

Step 2

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

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Answer

Micky contributes $835 every three months, meaning he contributes quarterly. The annual contribution will be calculated as follows:

Contribution per period = $835
Number of contributions per year = 4
Total number of years = 10
Total number of contributions = 10 * 4 = 40

Next, we use the quarterly compounding interest rate:

Annual interest rate = 6%
Quarterly interest rate, $r = \frac{6}{4} = 1.5\% = 0.015$

Referencing the factor from the table:

Annuity Factor (for 10 years, compounding quarterly) = 54.268

Now we multiply the contribution by the annuity factor to find the future value:

ext{Future Value} = ext{Contribution per period} \times ext{Annuity Factor} \\ = 835 \times 54.268 = 45,130.78

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

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

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

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