A student borrows $800 at 5.3% annual interest compounded semiannually. How much does he owe after 4 years?

2 Answers
Feb 28, 2018

$186.19

Explanation:

For this type of question, we would use Compound Interest which is gaining interest continuously over a certain amount of years, instead of working them all out separately. The formula for Compound Interest is:

A=P(1+r// nxx100)^t

Where A is the amount of interest, P is the original amount, R is the interest rate, N is how many times interested per year and T is the time.

Plugging in values:

A=$800(1+5.3/200)^8

We change this to ^8 as semiannually means twice a year, and as this is after 4 years, we double it to get 8.

Plugging into calculator:

=$986.1923212

Rounding to 2 d.p:

$986.19

Minusing the before value from the after value:

$986.19-800=$186.19

Feb 28, 2018

Rounding to 2 decimal places gives $186.18 being owed in interest

Explanation:

color(blue)("Some thoughts")

Note that 'semiannually' means twice per year. So each year has 2 calculation cycles

You have to 'split up' the annual percentage into a proportion that reflects the calculation cycle.

So the annual interest of 5.3% becomes 5.3/2% at each calculation cycle.

The general form for this context is

color(white)("dddddddd")P(1+x/(2xx100))^(2n)

where x=5.3 and n is the count in years
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
color(blue)("Answering the question")

P(1+x/(2xx100))^(2n)color(white)("ddd")->color(white)("ddd")$800(1+5.3/(2xx100))^(2xx4)

color(white)("dddddddddddddddddd")->color(white)("ddd")$800(205.3/200)^8 ~~986.1923...

The amount owed is
$986.1923...
ul($800.0000 larr" Subtract")
$186.1923...

Rounding to 2 decimal places gives $186.18 being owed in interest