robin invests $50 on the first of every month into a superannuation account.
His employer, Batman, pays 9% of Robin's monthly income of $2500 at the
end of the month into the same account. Interest is paid at the rate of 6% p.a.
compounded monthly.
(a) Calculate the total amount in the account after 45 years.
(b) How much more should Robin invest each month if he wishes to have a
superannuation fund value of $1 000 000 after 45 years?
Your School Subject is Math, not gscc, whatever that is.
To calculate the total amount in Robin's superannuation account after 45 years, we need to consider the monthly contributions he makes, the employer contributions, and the compound interest earned.
(a) Calculating the total amount in the account after 45 years:
1. Calculate the total number of months Robin will be contributing: 45 years * 12 months = 540 months.
2. Calculate the monthly contribution by Robin: $50/month.
3. Calculate the total contribution made by Robin: $50/month * 540 months = $27,000.
4. Calculate the employer contribution:
- Employer contribution percentage: 9%
- Monthly income of Robin: $2500
- Calculate the monthly employer contribution: 9% * $2500 = $225/month.
- Calculate the total employer contribution: $225/month * 540 months = $121,500.
Now let's calculate the total amount in the account including compound interest:
5. Calculate the annual interest rate: 6% p.a.
6. Convert the annual interest rate to a monthly interest rate: 6% / 12 = 0.5% per month.
7. Calculate the total interest rate for 45 years: 540 months * 0.5% = 270%.
8. Calculate the total amount in the account including compound interest:
- Total contributions made by Robin and his employer: $27,000 + $121,500 = $148,500.
- Calculate the compound interest using the formula: A = P * (1 + r/n)^(nt),
where A is the final amount, P is the principal (total contributions), r is the interest rate, n is the number of compounding periods per year, and t is the total number of years.
- In this case, P = $148,500, r = 0.5% per month, n = 12 (since interest is compounded monthly), and t = 45 years.
- Plugging in the values and calculating, A = $148,500 * (1 + 0.005)^ (12 * 45) = $700,989.44 (rounded to the nearest cent).
Therefore, the total amount in Robin's superannuation account after 45 years would be approximately $700,989.44.
(b) To calculate how much more Robin should invest each month to have a superannuation fund value of $1,000,000 after 45 years, we can use a similar approach.
1. Subtract the total contributions made by Robin and his employer from the desired superannuation fund value: $1,000,000 - $148,500 = $851,500.
2. Set up the equation: $851,500 = X * (1 + 0.005)^(12 * 45), where X is the additional amount Robin needs to invest each month.
3. Solve the equation for X.
- Divide both sides by [(1 + 0.005)^(12 * 45)].
- X = $851,500 / [(1 + 0.005)^(12 * 45)].
- Calculate X using this formula, which gives X = $851,500 / 2169.857154.
- X ≈ $392.37 (rounded to the nearest cent).
Therefore, Robin would need to invest approximately $392.37 more each month to reach a superannuation fund value of $1,000,000 after 45 years.