Question

A savings account earns 4% annual interest compounded quarterly. How much interest would $500 earn if it was invested for one year?

Answer 1

Amount obtained in Compound interest is given by :

$\bigstar\;\;\boxed{\mathsf{Amount = Principal\bigg(1 + \dfrac{Rate\;of \;interest}{100}\bigg)^{Conversion\;periods}}}$

Note : Conversion period is the time from one interest period to the next interest period. If the interest is compounded annually then there is one conversion period in an year. If the interest is compounded semi-annually then there are two conversion periods in an year. if the interest is compounded quarterly then there are four conversion periods in an year.

Problem :

Given : $500 is invested for one year at 4% annual interest

$\implies\boxed{\begin{minipage}{4 cm}\bigstar\;\;\textsf{Principal = 500}\\\\\bigstar\;\;\textsf{Time period = 1 year}\\\\\bigstar\;\;\textsf{Rate of interest = 4\%}\end{minipage}}$

As the question mentions the term ''compounded quarterly'', there are 4 conversion periods in a year.

If the interest is compounded quarterly, then the rate of interest per conversion period (quarter) will be :

$\implies \mathsf{\left(\dfrac{1}{4} \times 4\%\right) = 1\%}$

Substituting all the values in the Amount formula of C.I, We get :

$\mathsf{\implies Amount = 500\bigg(1 + \dfrac{1}{100}\bigg)^4}$

$\mathsf{\implies Amount = 500\left(1 + 0.01\right)^4}$

$\mathsf{\implies Amount = 500\left(1.01\right)^4}$

$\mathsf{\implies Amount = 520.30}$

We know that : Interest = Amount - Principal

$:\implies$  Interest = 520.30 - 500

$:\implies$  Interest = $20.30