Answer:
To solve this problem, we need to find the principal sum (the initial amount lent out). We'll use the formula for compound interest to calculate the difference between interest compounded annually and interest compounded semi-annually.
Let's denote:
- \( P \) as the principal sum (the amount lent out)
- \( r \) as the annual interest rate (20% or 0.20)
- \( n_1 \) as the number of years when interest is compounded annually (1.5 years)
- \( n_2 \) as the number of years when interest is compounded semi-annually (1.5 years, which is equivalent to 3 half-years)
- \( A_1 \) as the amount with interest compounded annually
- \( A_2 \) as the amount with interest compounded semi-annually
The formula for compound interest is:
\[ A = P \times \left(1 + \frac{r}{n}\right)^{nt} \]
Where:
- \( A \) is the amount (including interest)
- \( P \) is the principal sum
- \( r \) is the annual interest rate (in decimal)
- \( n \) is the number of times interest is compounded per year
- \( t \) is the time the money is invested for (in years)
First, let's calculate the amount with interest compounded annually:
\[ A_1 = P \times \left(1 + \frac{0.20}{1}\right)^{1.5} \]
Next, let's calculate the amount with interest compounded semi-annually:
\[ A_2 = P \times \left(1 + \frac{0.20}{2}\right)^{3 \times 1.5} \]
The difference between the two amounts is given as $178.75:
\[ A_1 - A_2 = 178.75 \]
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Step-by-step explanation:
Answer:
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Step-by-step explanation:
To find the difference in compound interest between yearly and half-yearly compounding, we can use the formula for compound interest:
A = P \left(1 + \frac{r}{n}\right)^{nt}
Where:
- \( A \) is the amount of money accumulated after \( n \) years, including interest.
- \( P \) is the principal amount (the initial amount of money).
- \( r \) is the annual interest rate (in decimal).
- \( n \) is the number of times that interest is compounded per year.
- \( t \) is the time the money is invested for, in years.
Given:
- Annual interest rate (\( r \)) = 20% per annum = 0.20
- Time (\( t \)) = 1.5 years
Let's denote:
- \( P_y \) as the amount with yearly compounding
- \( P_h \) as the amount with half-yearly compounding
For yearly compounding, \( n = 1 \), and for half-yearly compounding, \( n = 2 \).
So, we have:
\[ P_y = P \left(1 + \frac{0.20}{1}\right)^{1 \times 1.5} \]
\[ P_h = P \left(1 + \frac{0.20}{2}\right)^{2 \times 1.5} \]
The difference between these two amounts is given as $178.75:
\[ P_y - P_h = 178.75 \]
Now, we can solve for \( P \):
\[ P \left(1 + 0.20\right)^{1.5} - P \left(1 + \frac{0.20}{2}\right)^{3} = 178.75 \]
\[ P \left(1.20^{1.5} - \left(1 + \frac{0.10}{1}\right)^{3}\right) = 178.75 \]
\[ P \left(1.20^{1.5} - 1.10^3\right) = 178.75 \]
Now, we can solve for \( P \):
\[ P \left(1.20^{1.5} - 1.10^3\right) = 178.75 \]
\[ P = \frac{178.75}{1.20^{1.5} - 1.10^3} \]
\[ P \approx \frac{178.75}{1.728 - 1.331} \]
\[ P \approx \frac{178.75}{0.397} \]
\[ P \approx 450 \]
So, the principal sum is $450.
