Question

Find the annual interest rate. (Round your answer to the nearest whole number.)

Principal
Balance
$4000
$62,702.95
Enter a number.
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X%
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Time
20 years
Compounding
Quarterly
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Answer 1

`\text{Annual interest rate = 14 \%}`

Explanation:

The formula for computing accrued amount(principal + interest) is given by

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

where

P =principal amount
r = annual interest rate as a decimal
n = number of times compounded per year
t = number of years
A = amount accrued at the end of t years

We are given
A = 62702.95
P = 4000
n = 4 since we are compounding quarterly, i.e. 4 times a year
t = 20

Plugging in known values into the equation we get

`62702.95 = 4000\left(1 +(r)/(4)\right)^(4 \cdot 20)\\\\62702.95 = 4000\left(1 +(r)/(4)\right)^(80)\\\\`

Switch sides and divide both sides by 4000 to get

`\left(1 +(r)/(4)\right)^(80) = (62702.95)/(4000)`

`1 +(r)/(4)\right) = \left(\frac {62,702.95}{4000}\right)^{(1)/(80)}\\\\\text{Using a calculator,}\\1 +(r)/(4)\right) = 1.03499\dots \approx 1.035`

`(r)/(4) = 1.035 - 1 = 0.035\\\\r = 4 * 0.035\\\\\\r = 0.14\\\\\\`

As a percentage that would be

`0.14 * 100 = 14\%`