At what rate of percent compound interest a sum of money becomes 4 times itself in 2 years?

• Aptitude
• Simple and compound interest

Correct Answer:

Description for Correct answer:
\( \Large Principal\ \ Amount \)

\( \Large 1 \rightarrow\ \ 4 \)

\( \Large 4=1 \left(1+\frac{r}{100}\right)^{2}\)

\( \Large 4= \left(1+\frac{r}{100}\right)^{2}\)

r = 100 %

Part of solved Simple and compound interest questions and answers : >> Aptitude >> Simple and compound interest

At what rate percent compound interest, does a sum of money become 1.44 times of itself in 2 years?

Answer

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Hint: First, we will let the principal sum of money as ‘P’ and the rate of interest as ‘R’. we will use the conditions given in the question and formula of compound interest to form a different equation. And by solving those equations we will find the rate of interest.

Complete step-by-step solution:
Let the principal sum of money be ‘P’.
Let the rate of interest compounded annually be ‘R’.
Given: the amount becomes 1.44 times the principal amount in the span of 2 years.
So, \[A = 1.44 \times P\]----- (1)
By using compound interest formulas. We get,
$A = P \times {\left( {1 + \dfrac{R}{{100}}} \right)^2}$----- (2)
From equation 1 and 2. We get,
$1.44 \times P = P \times {\left( {1 + \dfrac{R}{{100}}} \right)^2}$
$\Rightarrow 1.44 = {\left( {1 + \dfrac{R}{{100}}} \right)^2}$
Squaring on both sides.
$\sqrt {1.44} = \left( {1 + \dfrac{R}{{100}}} \right)$
Value of square root 1.44 is 1.2.
$1.2 = \left( {1 + \dfrac{R}{{100}}} \right)$
We can also write 1.2 as 1 + 0.2.
$1 + 0.2 = \left( {1 + \dfrac{R}{{100}}} \right)$
We can write 0.2 as 2/10.
$1 + \dfrac{2}{{10}} = \left( {1 + \dfrac{R}{{100}}} \right)$
$\Rightarrow \dfrac{2}{{10}} = \dfrac{R}{{100}}$
$\Rightarrow 2 = \dfrac{R}{{10}}$
$\Rightarrow R = 2 \times 10$
$\Rightarrow R = 20\% $
So, the rate percent compound interest is $20\%.$

Note: Compound interest (or compounding interest) is the interest on a loan or deposit calculated based on both the initial principal and the accumulated interest from previous periods. The rate at which compound interest accrues depends on the frequency of compounding, such that the higher the number of compounding periods, the greater the compound interest. Compound interest is calculated by multiplying the initial principal amount by one plus the annual interest rate raised to the number of compound periods minus one. The total initial amount of the loan is then subtracted from the resulting value.

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Find the rate at which a sum of money will become four times the original amount in 2 years, if the interest is compounded half yearly.

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Solution

Let Principal (P) = Rs. 100
then Amount (A) = Rs. 400
Period (n) = 2 years or 4 half years.
Let R be the rate % half-yearly, then
AP=(1+R100)n⇒400100=(1+R100)4
⇒(1+R100)4=41
⇒[(1+R100)2]2=(2)2
⇒(1+R100)2=2⇒(1+R100)=√2
⇒1+R100=1.4142
⇒R100=1.4142−1.0000
⇒R100=0.4142⇒R=0.4142×100
⇒R=41.42
∴ Rate %=41.42% half yearly and 82.84% p.a.

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