Question

What will be the compound interest on Rs. 5000 if it is compounded half-yearly for 1 year 6 months at 8 % per annum.

Answer 1

Hint: The amount can be calculated using the given data and the formula:
$A = P{\left( {1 + \dfrac{R}{{100}}} \right)^t}$ where,
A = Amount
P = Principal
R = Rate
T = Time
Remember to half the rate and double the time as the principal is compounded half-yearly.
Then, the compound interest can be calculated using the relationship:
Amount = Principal + Compound Interest

Complete step-by-step answer:
Given:
Principal (P) = Rs. 5000
Rate (R) = 8 %
As it is compounded half yearly, the rate will reduce to half
$R = \dfrac{8}{2}\% $
R = 4 %
Time (t) = 1 year 6 months
= $\left( {1 + \dfrac{1}{2}} \right)yrs$
 $\left(
 \because 12m = 1yr \\
 6m = \dfrac{1}{{12}} \times 6 \\
 6m = \dfrac{1}{2}yrs \\
 \right)$
= $\dfrac{3}{2}yrs$
As it is compounded half yearly, the time will be doubled:
$t = 2 \times \dfrac{3}{2}yrs$
T = 3 yrs.
Substituting these values in the formula of Amount, we get:
$A = P{\left( {1 + \dfrac{R}{{100}}} \right)^t}$
$A = 5000{\left( {1 + \dfrac{4}{{100}}} \right)^3}$
$A = 5000{\left( {\dfrac{{104}}{{100}}} \right)^3}$
$A = 5000 \times \dfrac{{104}}{{100}} \times \dfrac{{104}}{{100}} \times \dfrac{{104}}{{100}}$
A = 5624.32
The amount is equal to Rs. 5624.32
Now,
Amount = Principal + Compound Interest
Compound Interest = Amount – Principal
Substituting the values, we get:
Compound Interest = 5624.32 – 5000
Compound Interest = 624.32
Therefore, the compound interest is Rs. 624.32 on Rs. 5000 if it is compounded half yearly for 1 year 6 months at 8 % per annum.

Note: We make the respective changes when compounded half-yearly because:
Rate is halved. The rate is generally given for a year (per annum) but when we require for half-yearly, the per annum rate is also reduced to half.
Time is doubled.When we talk about half a year (6 months), it occurs twice every year (6 + 6 = 12) and hence the time is doubled.