A sum of money doubles itself in 5 years in how many years it will become 4 times

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Hint: To solve the given question, we will make use of the formula for calculating the compound interest which is given below.
\[A=P{{\left( 1+\dfrac{r}{n} \right)}^{nt}}\]
where A is the final amount, P is the initial principal balance, r is the interest rate, n is the number of times interest applied and t is the total time elapsed. We will use this formula to find the relation between r and n when the interest is calculated after four years. Similarly, we will find the relation of r and n after t years when the amount has become 8 times the principal. After this, we will find the value of t with these two relations.Complete step-by-step answer:
We are given the question that a sum of money doubles itself in 4 years at compound interest. Let us assume that the principal is P, then the amount will be 2P. We know that according to the formula of compound interest, we have,
\[A=P{{\left( 1+\dfrac{r}{n} \right)}^{nt}}\]
Here, A is the amount, P is the initial principal, r is the rate, n is the number of times the interest is applied and t is the total time elapsed. In our case, A = 2P, P = P, r = R, t = 4 and we assume the value of n as 1, i.e. it is compounded annually. Thus, we will get,
\[2P=P{{\left( 1+\dfrac{R}{1} \right)}^{1\times 4}}\]
\[\Rightarrow 2P=P{{\left( 1+R \right)}^{4}}\]
\[\Rightarrow 2={{\left( 1+R \right)}^{4}}........\left( i \right)\]
Now, we have to find the total time after which the amount will become 8 times the principal. Thus, we have,
\[8P=P{{\left( 1+R \right)}^{1\times T}}\]
\[\Rightarrow 8={{\left( 1+R \right)}^{T}}......\left( ii \right)\]
Here, T is the time assumed after which it will become 8 times. Now, we will cube the equation (i) on both sides. Thus, we will get,
\[\Rightarrow {{\left( 2 \right)}^{3}}={{\left[ {{\left( 1+R \right)}^{4}} \right]}^{3}}\]
\[\Rightarrow 8={{\left( 1+R \right)}^{4\times 3}}\]
\[\Rightarrow 8={{\left( 1+R \right)}^{12}}....\left( iii \right)\]
From (ii) and (iii), we have,
\[{{\left( 1+R \right)}^{12}}={{\left( 1+R \right)}^{T}}\]
Therefore, T = 12 years.
Hence, option (a) is the right answer.

Note: We have assumed n = 1 while solving. Let us take the value of n as N and solve it. Thus, we will get,
\[2P=P{{\left( 1+\dfrac{R}{N} \right)}^{4N}}\]
\[\Rightarrow 2={{\left( 1+\dfrac{R}{N} \right)}^{4N}}\]
\[\Rightarrow {{2}^{\dfrac{1}{4N}}}=1+\dfrac{R}{N}\]
\[\Rightarrow R=\left( {{2}^{\dfrac{1}{4N}}}-1 \right)N....\left( i \right)\]
Now, another information given is that the amount has become 8 times the principal after T years. Thus,
\[8P=P{{\left( 1+\dfrac{R}{N} \right)}^{NT}}\]
\[\Rightarrow 8={{\left( 1+\dfrac{R}{N} \right)}^{NT}}\]
Now, we will put the value of R from (i) to the above equation.
\[\Rightarrow 8={{\left( 1+\dfrac{\left( {{2}^{\dfrac{1}{4N}}}-1 \right)}{N} \right)}^{NT}}\]
\[\Rightarrow 8={{\left( 1+{{2}^{\dfrac{1}{4N}}}-1 \right)}^{NT}}\]
\[\Rightarrow 8={{\left( {{2}^{\dfrac{1}{4N}}} \right)}^{NT}}\]
\[\Rightarrow 8={{2}^{\dfrac{T}{4}}}\]
\[\Rightarrow {{2}^{3}}={{2}^{\dfrac{T}{4}}}\]
\[\Rightarrow \dfrac{T}{4}=3\]
\[\Rightarrow T=12\text{ years}\]

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A sum of money doubles itself in 5 years. In how many years will it become fourfold (if interest is compounded)?a)15b)10c)20d)12Correct answer is option 'B'. Can you explain this answer? for JEE 2022 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about A sum of money doubles itself in 5 years. In how many years will it become fourfold (if interest is compounded)?a)15b)10c)20d)12Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for JEE 2022 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A sum of money doubles itself in 5 years. In how many years will it become fourfold (if interest is compounded)?a)15b)10c)20d)12Correct answer is option 'B'. Can you explain this answer?.

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⇒ 4 = [21/5]T⇒ 22 = 2T/5⇒ T/5 = 2∴ T = 10 yrs.

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