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Chapter 3: Problem 26

Suppose a savings account pays \(5 \%\) interest per year, compounded four times per year. If the savings account starts with \(\$ 600\), how many years would it take for the savings account to exceed \(\$ 1400 ?\)

Short Answer

Expert verified
It will take approximately 12 years for the savings account to exceed $1400 with a 5% annual interest rate compounded quarterly.

Step by step solution

01

Identify the known values

For this problem, the following values are known: - The initial amount (P) is $600. - The annual interest rate (r) is \(5 \% = 0.05. - The number of times interest is compounded per year (n) is 4 (quarterly compounding). - The final amount to exceed is $1400 (A).
02

Rewrite the compound interest formula to solve for time 't'

We want to find 't' such that A exceeds $1400. We will rewrite the equation to isolate 't'. \[\frac{A}{P} = (1 + \frac{r}{n})^{(nt)}\] Taking the natural logarithm of both sides, we get: \[ln(\frac{A}{P}) = ln((1 + \frac{r}{n})^{(nt)})\] \[ln(\frac{A}{P}) = nt \, ln(1 + \frac{r}{n})\] Finally, solving for 't': \[t = \frac{ln(\frac{A}{P})}{n \, ln(1 + \frac{r}{n})}\]
03

Plug in the known values and solve for t

Substitute the known values (P, r, n, A) into the formula: \[t = \frac{ln(\frac{1400}{600})}{4 \, ln(1 + \frac{0.05}{4})}\] Now, compute the value of 't': \[t \approx 11.2\]
04

Round up the result

Since we want to know how many years it would take for the savings account to exceed $1400, we should round up the result to the next whole number, as we can't have a fraction of a year in this case: \[t = \lceil 11.2 \rceil = 12\] So, it will take approximately 12 years for the savings account to exceed $1400.

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