Question

You wish to travel overseas for an extended holiday in four years' time and you estimate you will require $35,000 to pay for the trip. If your bank offers to pay you 5% per annum on a savings account, with interest credited quarterly, how much must you deposit each quarter over the next four years to reach this target?

a) $2187.50
b) $7653.40
c) $1987.64
d) $1989.64

Answer 1

To reach a target of $35,000 in four years with 5% annual interest compounded quarterly, the approximate quarterly deposit is $2018.33, which is not an option given in the multiple-choice answers. the closest provided answer to the correct amount is $1987.64.

Step-by-step explanation:

When preparing for an extended holiday, understanding compound interest is essential in planning your savings. in this scenario, we wish to accumulate $35,000 in four years with an interest of 5% per annum, compounded quarterly, to pay for the trip. We'll calculate the necessary quarterly deposit using the formula for the future value of an ordinary annuity:

Given:

• Future Value (FV) = $35,000
• Annual interest rate (r) = 5% or 0.05
• Number of times the interest is compounded yearly (n) = 4
• Number of years (t) = 4

FV = P × {[(1 + r/n)^(nt) - 1] / (r/n)}

We need to rearrange the formula to solve for the periodic deposit (P):

P = FV / {[(1 + r/n)^(nt) - 1] / (r/n)}

So,

P = $35,000 / {[(1 + 0.05/4)^(4*4) - 1] / (0.05/4)}

By calculating the value inside the brackets first:

(1 + 0.05/4)^(4*4) - 1 = (1 + 0.0125)^16 - 1 ≈ 0.219395

Now plug that back into the equation for P:

P = $35,000 / (0.219395 / 0.0125) ≈ $35,000 / 17.552 ≈ $1993.86

By adding an extra deposit at the beginning of each quarter:

P = $1993.86 + ($1993.86 * 0.0125) ≈ $2018.33

So, the correct quarterly deposit to be made is roughly $2018.33; therefore, none of the options a) $2187.50, b) $7653.40, c) $1987.64, d) $1989.64 are correct. however, if the answer had to be selected from the given options, the closest amount to the correct calculation would be c) $1987.64.