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Sagot :
Let's break down the solution step by step:
### Part (a)
Formula for finding the yearly compound interest:
The formula to calculate the compound interest when it is compounded annually is:
[tex]\[ CI = P \left[\left(1+\frac{R}{100}\right)^T - 1\right] \][/tex]
where [tex]\(P\)[/tex] is the principal amount, [tex]\(R\)[/tex] is the rate of interest per annum, and [tex]\(T\)[/tex] is the time period in years.
Thus, for this problem:
[tex]\[ CI = 50000 \left[\left(1+\frac{10}{100}\right)^2 - 1\right] \][/tex]
### Part (b)
How much profit did Ramesh get in this transaction?
1. Calculate the Simple Interest (SI):
[tex]\[ SI = \frac{P \times R \times T}{100} \][/tex]
Given:
- [tex]\(P = 50000\)[/tex]
- [tex]\(R = 10\%\)[/tex]
- [tex]\(T = 2\)[/tex] years
[tex]\[ SI = \frac{50000 \times 10 \times 2}{100} = 10000 \][/tex]
2. Calculate the Compound Interest (CI) annually:
[tex]\[ CI = 50000 \left[\left(1+\frac{10}{100}\right)^2 - 1\right] \][/tex]
Simplify the inner term first:
[tex]\[ \left(1+\frac{10}{100}\right) = 1.1 \][/tex]
[tex]\[ \left(1.1\right)^2 = 1.21 \][/tex]
[tex]\[ CI = 50000 \left[1.21 - 1\right] = 50000 \times 0.21 = 10500 \][/tex]
3. Calculate the profit:
[tex]\[ \text{Profit} = CI - SI \][/tex]
[tex]\[ \text{Profit} = 10500 - 10000 = 500 \][/tex]
Thus, the profit Ramesh gets in this transaction is Rs. 500.
### Part (c)
How much more profit would Ramesh get if it is invested at semi-annual compound interest?
1. Calculate the Compound Interest (CI) semi-annually:
When interest is compounded semi-annually, the rate and time adjust as follows:
- The rate per period becomes [tex]\( R/2 \)[/tex] (since it is twice a year), so [tex]\( \frac{10}{2} = 5\% \)[/tex]
- The number of periods doubles, so [tex]\( 2 \times T = 4 \)[/tex] periods for 2 years.
The formula for compound interest in this case is:
[tex]\[ CI_{\text{semi-annual}} = P \left[\left(1 + \frac{R/2}{100}\right)^{2T} - 1\right] \][/tex]
Plugging in the values:
[tex]\[ CI_{\text{semi-annual}} = 50000 \left[\left(1 + \frac{5}{100}\right)^4 - 1\right] \][/tex]
Simplify the inner term first:
[tex]\[ \left(1+\frac{5}{100}\right) = 1.05 \][/tex]
[tex]\[ \left(1.05\right)^4 \approx 1.21550625 \][/tex]
[tex]\[ CI_{\text{semi-annual}} = 50000 \left[1.21550625 - 1\right] = 50000 \times 0.21550625 = 10775.3125\][/tex]
2. Calculate the extra profit from semi-annual compounding:
[tex]\[ \text{Extra Profit} = CI_{\text{semi-annual}} - CI \][/tex]
[tex]\[ \text{Extra Profit} = 10775.3125 - 10500 = 275.3125 \][/tex]
Thus, the additional profit Ramesh would get if invested at semi-annual compound interest is approximately Rs. 275.
### Part (a)
Formula for finding the yearly compound interest:
The formula to calculate the compound interest when it is compounded annually is:
[tex]\[ CI = P \left[\left(1+\frac{R}{100}\right)^T - 1\right] \][/tex]
where [tex]\(P\)[/tex] is the principal amount, [tex]\(R\)[/tex] is the rate of interest per annum, and [tex]\(T\)[/tex] is the time period in years.
Thus, for this problem:
[tex]\[ CI = 50000 \left[\left(1+\frac{10}{100}\right)^2 - 1\right] \][/tex]
### Part (b)
How much profit did Ramesh get in this transaction?
1. Calculate the Simple Interest (SI):
[tex]\[ SI = \frac{P \times R \times T}{100} \][/tex]
Given:
- [tex]\(P = 50000\)[/tex]
- [tex]\(R = 10\%\)[/tex]
- [tex]\(T = 2\)[/tex] years
[tex]\[ SI = \frac{50000 \times 10 \times 2}{100} = 10000 \][/tex]
2. Calculate the Compound Interest (CI) annually:
[tex]\[ CI = 50000 \left[\left(1+\frac{10}{100}\right)^2 - 1\right] \][/tex]
Simplify the inner term first:
[tex]\[ \left(1+\frac{10}{100}\right) = 1.1 \][/tex]
[tex]\[ \left(1.1\right)^2 = 1.21 \][/tex]
[tex]\[ CI = 50000 \left[1.21 - 1\right] = 50000 \times 0.21 = 10500 \][/tex]
3. Calculate the profit:
[tex]\[ \text{Profit} = CI - SI \][/tex]
[tex]\[ \text{Profit} = 10500 - 10000 = 500 \][/tex]
Thus, the profit Ramesh gets in this transaction is Rs. 500.
### Part (c)
How much more profit would Ramesh get if it is invested at semi-annual compound interest?
1. Calculate the Compound Interest (CI) semi-annually:
When interest is compounded semi-annually, the rate and time adjust as follows:
- The rate per period becomes [tex]\( R/2 \)[/tex] (since it is twice a year), so [tex]\( \frac{10}{2} = 5\% \)[/tex]
- The number of periods doubles, so [tex]\( 2 \times T = 4 \)[/tex] periods for 2 years.
The formula for compound interest in this case is:
[tex]\[ CI_{\text{semi-annual}} = P \left[\left(1 + \frac{R/2}{100}\right)^{2T} - 1\right] \][/tex]
Plugging in the values:
[tex]\[ CI_{\text{semi-annual}} = 50000 \left[\left(1 + \frac{5}{100}\right)^4 - 1\right] \][/tex]
Simplify the inner term first:
[tex]\[ \left(1+\frac{5}{100}\right) = 1.05 \][/tex]
[tex]\[ \left(1.05\right)^4 \approx 1.21550625 \][/tex]
[tex]\[ CI_{\text{semi-annual}} = 50000 \left[1.21550625 - 1\right] = 50000 \times 0.21550625 = 10775.3125\][/tex]
2. Calculate the extra profit from semi-annual compounding:
[tex]\[ \text{Extra Profit} = CI_{\text{semi-annual}} - CI \][/tex]
[tex]\[ \text{Extra Profit} = 10775.3125 - 10500 = 275.3125 \][/tex]
Thus, the additional profit Ramesh would get if invested at semi-annual compound interest is approximately Rs. 275.
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