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Sagot :
Let's solve the given problem step-by-step.
First, we need to understand the given data:
- Principal (P) = Rs. 60,00,000
- Rate of interest (R) = 4% per annum
- Time (T) = 1 year
### Part (i): Compounded Half Yearly
When interest is compounded half yearly, the rate of interest is effectively divided by 2, because there are two compounding periods in one year.
1. Calculate the half yearly rate of interest:
[tex]\[ \text{Half yearly rate} = \frac{4\%}{2} = 2\% \][/tex]
2. Number of compounding periods in one year:
[tex]\[ n = 2 \][/tex]
3. Apply the compound interest formula:
[tex]\[ A_\text{half yearly} = P \left(1 + \frac{R/n}{100}\right)^{n \cdot T} \][/tex]
[tex]\[ A_\text{half yearly} = 60,00,000 \left(1 + \frac{2}{100}\right)^{2} \][/tex]
[tex]\[ A_\text{half yearly} = 60,00,000 \left(1 + 0.02\right)^{2} \][/tex]
[tex]\[ A_\text{half yearly} = 60,00,000 \left(1.02\right)^{2} \][/tex]
[tex]\[ A_\text{half yearly} = 60,00,000 \times 1.0404 \][/tex]
[tex]\[ A_\text{half yearly} = 62,42,400 \][/tex]
4. Calculate the compound interest (CI) for half yearly compounding:
[tex]\[ \text{CI}_\text{half yearly} = A_\text{half yearly} - P \][/tex]
[tex]\[ \text{CI}_\text{half yearly} = 62,42,400 - 60,00,000 \][/tex]
[tex]\[ \text{CI}_\text{half yearly} = 2,42,400 \][/tex]
### Part (ii): Compounded Yearly
When interest is compounded yearly, the interest is compounded only once a year.
1. The yearly rate of interest remains 4%:
[tex]\[ \text{Yearly rate} = 4\% \][/tex]
2. Number of compounding periods in one year:
[tex]\[ n = 1 \][/tex]
3. Apply the compound interest formula:
[tex]\[ A_\text{yearly} = P \left(1 + \frac{R/n}{100}\right)^{n \cdot T} \][/tex]
[tex]\[ A_\text{yearly} = 60,00,000 \left(1 + \frac{4}{100}\right)^{1} \][/tex]
[tex]\[ A_\text{yearly} = 60,00,000 \left(1 + 0.04\right) \][/tex]
[tex]\[ A_\text{yearly} = 60,00,000 \times 1.04 \][/tex]
[tex]\[ A_\text{yearly} = 62,40,000 \][/tex]
4. Calculate the compound interest (CI) for yearly compounding:
[tex]\[ \text{CI}_\text{yearly} = A_\text{yearly} - P \][/tex]
[tex]\[ \text{CI}_\text{yearly} = 62,40,000 - 60,00,000 \][/tex]
[tex]\[ \text{CI}_\text{yearly} = 2,40,000 \][/tex]
### Part (iii): Comparison of Interests
Now, we compare the compound interest obtained from (i) and (ii):
1. The difference in interest:
[tex]\[ \text{Difference} = \text{CI}_\text{half yearly} - \text{CI}_\text{yearly} \][/tex]
[tex]\[ \text{Difference} = 2,42,400 - 2,40,000 \][/tex]
[tex]\[ \text{Difference} = 2,400 \][/tex]
2. The percentage by which the interest compounded half yearly is more than the interest compounded yearly:
[tex]\[ \text{Percentage increase} = \left(\frac{\text{Difference}}{\text{CI}_\text{yearly}}\right) \times 100 \][/tex]
[tex]\[ \text{Percentage increase} = \left(\frac{2,400}{2,40,000}\right) \times 100 \][/tex]
[tex]\[ \text{Percentage increase} = 1\% \][/tex]
### Summary of Results
(i) The compound interest if the interest is compounded half yearly is Rs. 2,42,400.
(ii) The compound interest if the interest is compounded yearly is Rs. 2,40,000.
(iii) The interest compounded half yearly is more than the interest compounded yearly by Rs. 2,400, which is a 1% increase.
First, we need to understand the given data:
- Principal (P) = Rs. 60,00,000
- Rate of interest (R) = 4% per annum
- Time (T) = 1 year
### Part (i): Compounded Half Yearly
When interest is compounded half yearly, the rate of interest is effectively divided by 2, because there are two compounding periods in one year.
1. Calculate the half yearly rate of interest:
[tex]\[ \text{Half yearly rate} = \frac{4\%}{2} = 2\% \][/tex]
2. Number of compounding periods in one year:
[tex]\[ n = 2 \][/tex]
3. Apply the compound interest formula:
[tex]\[ A_\text{half yearly} = P \left(1 + \frac{R/n}{100}\right)^{n \cdot T} \][/tex]
[tex]\[ A_\text{half yearly} = 60,00,000 \left(1 + \frac{2}{100}\right)^{2} \][/tex]
[tex]\[ A_\text{half yearly} = 60,00,000 \left(1 + 0.02\right)^{2} \][/tex]
[tex]\[ A_\text{half yearly} = 60,00,000 \left(1.02\right)^{2} \][/tex]
[tex]\[ A_\text{half yearly} = 60,00,000 \times 1.0404 \][/tex]
[tex]\[ A_\text{half yearly} = 62,42,400 \][/tex]
4. Calculate the compound interest (CI) for half yearly compounding:
[tex]\[ \text{CI}_\text{half yearly} = A_\text{half yearly} - P \][/tex]
[tex]\[ \text{CI}_\text{half yearly} = 62,42,400 - 60,00,000 \][/tex]
[tex]\[ \text{CI}_\text{half yearly} = 2,42,400 \][/tex]
### Part (ii): Compounded Yearly
When interest is compounded yearly, the interest is compounded only once a year.
1. The yearly rate of interest remains 4%:
[tex]\[ \text{Yearly rate} = 4\% \][/tex]
2. Number of compounding periods in one year:
[tex]\[ n = 1 \][/tex]
3. Apply the compound interest formula:
[tex]\[ A_\text{yearly} = P \left(1 + \frac{R/n}{100}\right)^{n \cdot T} \][/tex]
[tex]\[ A_\text{yearly} = 60,00,000 \left(1 + \frac{4}{100}\right)^{1} \][/tex]
[tex]\[ A_\text{yearly} = 60,00,000 \left(1 + 0.04\right) \][/tex]
[tex]\[ A_\text{yearly} = 60,00,000 \times 1.04 \][/tex]
[tex]\[ A_\text{yearly} = 62,40,000 \][/tex]
4. Calculate the compound interest (CI) for yearly compounding:
[tex]\[ \text{CI}_\text{yearly} = A_\text{yearly} - P \][/tex]
[tex]\[ \text{CI}_\text{yearly} = 62,40,000 - 60,00,000 \][/tex]
[tex]\[ \text{CI}_\text{yearly} = 2,40,000 \][/tex]
### Part (iii): Comparison of Interests
Now, we compare the compound interest obtained from (i) and (ii):
1. The difference in interest:
[tex]\[ \text{Difference} = \text{CI}_\text{half yearly} - \text{CI}_\text{yearly} \][/tex]
[tex]\[ \text{Difference} = 2,42,400 - 2,40,000 \][/tex]
[tex]\[ \text{Difference} = 2,400 \][/tex]
2. The percentage by which the interest compounded half yearly is more than the interest compounded yearly:
[tex]\[ \text{Percentage increase} = \left(\frac{\text{Difference}}{\text{CI}_\text{yearly}}\right) \times 100 \][/tex]
[tex]\[ \text{Percentage increase} = \left(\frac{2,400}{2,40,000}\right) \times 100 \][/tex]
[tex]\[ \text{Percentage increase} = 1\% \][/tex]
### Summary of Results
(i) The compound interest if the interest is compounded half yearly is Rs. 2,42,400.
(ii) The compound interest if the interest is compounded yearly is Rs. 2,40,000.
(iii) The interest compounded half yearly is more than the interest compounded yearly by Rs. 2,400, which is a 1% increase.
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