What is the compound interest on a sum of ₹7500 for \(2\frac{3}{4}\) years at 8% p.a., the interest being compounded yearly? (nearest to an integer)
- ₹1248
- ₹1773
- ₹1348
- ₹1783
Answer (Detailed Solution Below)
Option 2 : ₹1773
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हिन्दी वर्णमाला सरल टेस्ट
10 Questions 10 Marks 10 Mins
Given:
Principal = Rs. 7500
Time = 2.75years = 2years 9months
Rate% = 8%
Concept:
In the case of compound interest, the rate of interest increases successively
{a + a + (a × a)/100}
Calculation:
Effective rate of interest in 2 years = {8 + 8 + (8 × 8)/100} = 16.64%
According to question:
Rate of interest in 12 months = 8%
⇒ 1month = 8/12
⇒ 9months = (8/12) × 9 = 6%
So, the effective rate of interest in 2 years 9months =
⇒ {16.64 + 6 + (16.64× 6)/100}
⇒ 23.63%
Compound interest on Rs.7500 after 2years 9moths =
⇒ 23.63% of 7500
⇒ Rs.1772.25
∴ the required compound interest = Rs.1772.25 ≈ Rs.1773
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Let's discuss the concepts related to Interest and Compound Interest. Explore more from Quantitative Aptitude here. Learn now!
At what rate \[\% \] will a sum of Rs. 7500 amount to Rs. 8427 in 2 years, compounded annually?
Answer
Verified
Hint:
Here, we need to find the rate of compound interest. We will use the formula for amount when a principal is compounded for a period of time. Then, we will simplify the equation to find the required rate of interest.
Formula used: The amount \[A\] of an investment after \[t\] years is given by $A = P{\left( {1 + \dfrac{r}{n}} \right)^{nt}}$, where $P$ is the amount invested, $n$ is the number of compounding periods in a year and $r$ is the interest rate compounded
annually.
Complete step by step solution:
The amount \[A\] of an investment after \[t\] years is given by $A = P{\left( {1 + \dfrac{r}{n}} \right)^{nt}}$, where $P$ is the amount invested, $n$ is the number of compounding periods in a year and $r$ is the interest rate compounded annually.
We will use this formula to form an equation and solve it to find the rate.
Since the sum is compounded annually, the number of compounding periods in a year is 1.
Substituting \[n = 1\],
\[t = 2\], \[P = {\text{Rs}}{\text{. }}7500\] and \[A = {\text{Rs}}{\text{. }}8427\] in the formula, we get
$ \Rightarrow 8427 = 7500{\left( {1 + \dfrac{{r\% }}{1}} \right)^{1 \times 2}}$
We have formed a linear equation in one variable in terms of \[r\]. We will solve this equation to find the value of \[r\], and hence, the rate of interest compounded annually.
Simplifying the expression, we get
$ \Rightarrow 8427 = 7500{\left( {1 + \dfrac{r}{{100}}} \right)^2}$
Dividing both
sides of the equation by 7500, we get
$ \Rightarrow \dfrac{{8427}}{{7500}} = \dfrac{{7500{{\left( {1 + \dfrac{r}{{100}}} \right)}^2}}}{{7500}} \\
\Rightarrow 1.1236 = {\left( {1 + \dfrac{r}{{100}}} \right)^2} \\
\text{Taking the square root of both sides of the equation, we get}
\Rightarrow \sqrt {1.1236} = \sqrt {{{\left( {1 + \dfrac{r}{{100}}} \right)}^2}} \\
\Rightarrow 1.06 = 1 + \dfrac{r}{{100}} \\
\text{Subtracting 1
from both sides of the equation, we get}
\Rightarrow 1.06 - 1 = 1 + \dfrac{r}{{100}} - 1 \\
\Rightarrow 0.06 = \dfrac{r}{{100}} \\ $
Multiplying both sides of the equation by 100, we get
\[ \Rightarrow 0.06 \times 100 = \dfrac{r}{{100}} \times 100\]
Therefore, we get the rate as
$ \Rightarrow r = 6\% $
Thus, we get that the sum of Rs. 7500 amounts to Rs. 8427 in 2 years, compounded annually at the rate $6\% $ per annum.
Note:
We
have formed a linear equation in one variable in terms of \[r\] in the solution using the formula $A = P{\left( {1 + \dfrac{r}{n}} \right)^{nt}}$. A linear equation in one variable is an equation that can be written in the form \[ax + b = 0\], where \[a\] is not equal to 0, and \[a\] and \[b\] are real numbers. For example, \[x - 100 = 0\] and \[100P - 566 = 0\] are linear equations in one variable \[x\] and \[P\] respectively. A linear equation in one variable has only one solution.