u deposit $100 at the end of each quarter in a sinking fund earning 4% compounded quarterly. How many quarterly deposits must you make in order to reach your goal of saving $10,000? Round your answer off to the nearest whole number
To calculate the number of quarterly deposits required to reach a goal, we can break it down into steps:
Step 1: Determine the interest rate per compounding period.
The given interest rate is 4% compounded quarterly, which means the interest rate per compounding period is 4% divided by 4 (since there are four quarters in a year). Thus, the interest rate per compounding period is 1%.
Step 2: Determine the future value (FV) of the sinking fund.
The future value (FV) of the sinking fund can be calculated using the formula:
FV = P * ((1 + r)^n - 1) / r
Where:
FV = Future Value
P = Quarterly deposit amount
r = Interest rate per compounding period (in decimal form)
n = Number of compounding periods
In this case, we need to find the number of quarterly deposits (n), so we rearrange the formula:
n = log((FV * r / P) + 1) / log(1 + r)
Step 3: Plug in the given values and calculate.
FV = $10,000 (the goal)
P = $100 (the quarterly deposit)
r = 1% (the interest rate per compounding period)
n = log((10,000 * 0.01 / 100) + 1) / log(1 + 0.01)
Using a scientific or financial calculator, perform the calculations:
n ≈ log(1.01) / log(1.01) [dividing 100 by 100 gives 1, which doesn't affect the result]
n ≈ 1 / 0.004321
n ≈ 231.26 (rounded to two decimal places)
Step 4: Round the answer to the nearest whole number.
Rounding 231.26 to the nearest whole number gives 231.
Therefore, you would need to make 231 quarterly deposits to reach your goal of saving $10,000.