Compound Interest Growth Calculation
The problem asks for the time it takes for an investment to grow to 9 times its initial value, given that it triples in 3 years under compound interest.
Understanding Compound Interest Growth
Let the principal amount be \(P\). The formula for the amount \(A\) after \(t\) years with compound interest is \(A = P(1+r)^t\), where \(r\) is the annual interest rate.
We are given that the sum triples itself in 3 years. This means:
\( 3P = P(1+r)^3 \)Dividing both sides by \(P\), we get:
\( 3 = (1+r)^3 \)This equation tells us the growth factor over a 3-year period is 3.
Calculating Time for 9 Times the Amount
We need to find the time \(T\) when the amount becomes 9 times the principal, i.e., \(A = 9P\).
\( 9P = P(1+r)^T \)Dividing by \(P\) gives:
\( 9 = (1+r)^T \)We know that \(9\) can be expressed as \(3^2\). So, the equation becomes:
\( 3^2 = (1+r)^T \)Now, substitute the value of \(3\) from the first condition, which is \(3 = (1+r)^3\):
\( ((1+r)^3)^2 = (1+r)^T \)Using the power rule \((a^m)^n = a^{mn}\), we simplify the left side:
\( (1+r)^{3 \times 2} = (1+r)^T \) \( (1+r)^6 = (1+r)^T \)By comparing the exponents, since the base \((1+r)\) is the same and greater than 1 (as it represents growth), the time periods must be equal:
\( T = 6 \)Therefore, the sum will amount to 9 times itself in 6 years.
Similar Questions
If a sum amounts to \(2 \frac{1}{4}\) times the sum after 2 years on a certain rate of compound interest, compounded annually, then the rate of interest per annum is:
A sum of money, on compound interest, amounts to ₹5,290 in 2 years and to ₹6,083.50 in 3 years. If interest is compounded annually, then the rate of interest per annum is:
If a sum of money doubles itself in 10 years on compound interest, then in how many years will it become 16 times itself at the same rate of interest?
A sum of money, on compound interest, amounts to ₹578.40 in 2 years and to ₹614.55 in 3 years. The rate of interest per annum, in case of annual compounding, is:
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- A certain sum amounts to ₹1,323 in 2 years and to ₹1389.15 in 3 years at r% p.a., when the interest is compounded yearly. What is the value of r?
A sum of money doubles itself at a certain rate of compound interest in 12 years when the interest is compounded annually. In how many years will it become eight times itself?
Jaspreet deposited a sum of ₹12,800 at 5% rate of interest per annum, compounded annually. The total amount (in ₹) received by Jaspreet after 2 years will be:
Important Questions from Compound Interest
A person borrowed Rs. 10000 on compound interest at the rate of 40 percent per annum. If the interest is compounded half yearly, then what will be the amount to be paid after 1.5 years?
The difference between the compound interest (compounding annually) and the simple interest on a sum of money at the rate of 40 per cent per annum for 2 years is Rs. 2400. What is the amount?
In how many years will a sum of Rs.1875 amount to Rs.2187 at 8 percent p.a. compound interest?
A sum of money has increased by 45% in 9 years at simple interest. What will be the compound interest of Rs. 12,000 after 3 years at the same rate?
At a certain rate of compound interest a certain sum amounts to Rs. 64800 in 4 years and Rs. 93312 in 6 years. What is the compound interest earned in fifth year?
