Find the compound interest on Rs 12000 for one and half years at 8% per annum compounded half yearly

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

Verified

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.

Solution : Principle`=` Rs .`12000`<br>Time`(n) = 2`year<br>Rate`= 20%`<br>Now , `A= P(1+R/100)^n`<br>`A= 12000 ( 1+20/100)^2`<br>`A= 12000 (1+1/5)^2`<br>`A= 12000 (6/5)^2`<br>`A= 12000xx 36/25`<br>`A=17280`

The compound interest (in Rs.) on a sum of Rs. 12,000 at 10% per annum for 1.5 years, interest compounded half-yearly, is:

1. Rs. 1,750
2. Rs. 1,900
3. Rs. 1,821.50
4. Rs. 1,891.50

Answer (Detailed Solution Below)

Option 4 : Rs. 1,891.50

Free

10 Questions 10 Marks 7 Mins

Given:

A sum = Rs. 12,000

Rate = 10% per annum

And time = 1.5 years.

Formula used:

Amount = Principal{1 + (R/100)}n

Where n = time in years and R = rate percentage/annum

Compund interest = Amount - Principal

If time = n years, rate = R% and interest compounded half-yearly

Then time = 2n  and rate = (R/2)%

Calculation:

Interest compounded half-yearly:

Then time = 2 × 1.5

⇒ 3 years

And the rate = 10/2

⇒ 5% 

Amount = 12000 × {1 + (5/100)}3

⇒ 12000 × {1 + (1/20)}3

⇒ 12000 × {(20 + 1)/20}3

⇒ 12000 × (21/20)3

⇒ 12000 × (9261/8000)

⇒ 12 × 9261/8

⇒ 1,11,132/8

⇒ Rs. 13891.5

Compound interest = amount - principal

⇒ 13891.5 - 12000

⇒ Rs. 1891.5

∴ The compound interest is Rs. 1891.5

Last updated on Sep 21, 2022

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