Question

How much time will be needed for $39,000 to grow to $43,278.39 if deposited at 7% compounded quarterly?Do not round until the final answer. Then round to the nearest tenth as needed.

Answer 1

The compound interest formula is given by

`A=P(1+(r)/(n))^(nt)`

where A is the final amount including the principal P, r is the rate, n is the number or times interes and t is the time.

In our case, A= $43,278.39, P=$39,000, n=4 (for quarterly) and t is the unknown time.

We must find t in our formula. First, if we move P to the left hand side, we ger

`(A)/(P)=(1+(r)/(n))^(nt)`

If we apply logarithm in both sides, we obtain

`\log (A)/(P)=\log (1+(r)/(n))^(nt)`

which gives,

`\log (A)/(P)=nt\cdot\log (1+(r)/(n))`

then, t is equal to

`t=(\log (A)/(P))/(n\log (1+(r)/(n)))`

Therefore, by means of this formula, we can find t. If we substitute the given values into this formula ,we get

`t=(\log (43278.39)/(39000))/(4\cdot\log (1+(0.07)/(4)))`

then, we have

`t=(\log 1.1097)/(4\cdot\log 1.0175)`

which is equal to

`\begin{gathered} t=(0.0173)/(0.06939) \\ t=0.249 \end{gathered}`

that is, by rounding up, the times is 0.3 years.