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Sagot :
(a) Let's calculate the interest earned from saving money in both banks.
### Bank A:
- Interest Type: Yearly compound interest
- Interest Rate: [tex]\( 12\% \)[/tex] per annum
- Principal Amount: Re. 1 (to simplify comparison)
- Time: 1 year
Using the formula for compound interest:
[tex]\[ A = P \left( 1 + \frac{r}{n} \right)^{n \cdot t} \][/tex]
For Bank A:
- [tex]\( P = 1 \)[/tex] (principal amount)
- [tex]\( r = 0.12 \)[/tex] (annual interest rate)
- [tex]\( t = 1 \)[/tex] year
- [tex]\( n = 1 \)[/tex] (since it's compounded yearly)
[tex]\[ A = 1 \left( 1 + \frac{0.12}{1} \right)^{1 \cdot 1} \][/tex]
[tex]\[ A = 1 \left( 1 + 0.12 \right)^{1} \][/tex]
[tex]\[ A = 1 \left( 1.12 \right) \][/tex]
[tex]\[ A = 1.12 \][/tex]
The interest earned:
[tex]\[ \text{Interest}_A = A - P \][/tex]
[tex]\[ \text{Interest}_A = 1.12 - 1 \][/tex]
[tex]\[ \text{Interest}_A = 0.12 \][/tex]
### Bank B:
- Interest Type: Half-yearly compound interest
- Interest Rate: [tex]\( 10\% \)[/tex] per annum
- Principal Amount: Re. 1 (to simplify comparison)
- Time: 1 year
Using the compound interest formula again:
[tex]\[ A = P \left( 1 + \frac{r}{n} \right)^{n \cdot t} \][/tex]
For Bank B:
- [tex]\( P = 1 \)[/tex] (principal amount)
- [tex]\( r = 0.10 \)[/tex] (annual interest rate)
- [tex]\( t = 1 \)[/tex] year
- [tex]\( n = 2 \)[/tex] (since it's compounded half-yearly)
[tex]\[ A = 1 \left( 1 + \frac{0.10}{2} \right)^{2 \cdot 1} \][/tex]
[tex]\[ A = 1 \left( 1 + 0.05 \right)^{2} \][/tex]
[tex]\[ A = 1 \left( 1.05 \right)^{2} \][/tex]
[tex]\[ A = 1 \left( 1.1025 \right) \][/tex]
[tex]\[ A = 1.1025 \][/tex]
The interest earned:
[tex]\[ \text{Interest}_B = A - P \][/tex]
[tex]\[ \text{Interest}_B = 1.1025 - 1 \][/tex]
[tex]\[ \text{Interest}_B = 0.1025 \][/tex]
(b) Now we compare the interests:
- Interest earned from Bank A: [tex]\( 0.12 \)[/tex]
- Interest earned from Bank B: [tex]\( 0.1025 \)[/tex]
Given these calculations, you would save money in Bank A because the interest earned from Bank A (0.12) is higher than the interest earned from Bank B (0.1025).
Reason:
Bank A offers a higher yearly compound interest rate compared to Bank B's half-yearly compound interest rate, resulting in a greater effective return on the principal amount after one year.
### Bank A:
- Interest Type: Yearly compound interest
- Interest Rate: [tex]\( 12\% \)[/tex] per annum
- Principal Amount: Re. 1 (to simplify comparison)
- Time: 1 year
Using the formula for compound interest:
[tex]\[ A = P \left( 1 + \frac{r}{n} \right)^{n \cdot t} \][/tex]
For Bank A:
- [tex]\( P = 1 \)[/tex] (principal amount)
- [tex]\( r = 0.12 \)[/tex] (annual interest rate)
- [tex]\( t = 1 \)[/tex] year
- [tex]\( n = 1 \)[/tex] (since it's compounded yearly)
[tex]\[ A = 1 \left( 1 + \frac{0.12}{1} \right)^{1 \cdot 1} \][/tex]
[tex]\[ A = 1 \left( 1 + 0.12 \right)^{1} \][/tex]
[tex]\[ A = 1 \left( 1.12 \right) \][/tex]
[tex]\[ A = 1.12 \][/tex]
The interest earned:
[tex]\[ \text{Interest}_A = A - P \][/tex]
[tex]\[ \text{Interest}_A = 1.12 - 1 \][/tex]
[tex]\[ \text{Interest}_A = 0.12 \][/tex]
### Bank B:
- Interest Type: Half-yearly compound interest
- Interest Rate: [tex]\( 10\% \)[/tex] per annum
- Principal Amount: Re. 1 (to simplify comparison)
- Time: 1 year
Using the compound interest formula again:
[tex]\[ A = P \left( 1 + \frac{r}{n} \right)^{n \cdot t} \][/tex]
For Bank B:
- [tex]\( P = 1 \)[/tex] (principal amount)
- [tex]\( r = 0.10 \)[/tex] (annual interest rate)
- [tex]\( t = 1 \)[/tex] year
- [tex]\( n = 2 \)[/tex] (since it's compounded half-yearly)
[tex]\[ A = 1 \left( 1 + \frac{0.10}{2} \right)^{2 \cdot 1} \][/tex]
[tex]\[ A = 1 \left( 1 + 0.05 \right)^{2} \][/tex]
[tex]\[ A = 1 \left( 1.05 \right)^{2} \][/tex]
[tex]\[ A = 1 \left( 1.1025 \right) \][/tex]
[tex]\[ A = 1.1025 \][/tex]
The interest earned:
[tex]\[ \text{Interest}_B = A - P \][/tex]
[tex]\[ \text{Interest}_B = 1.1025 - 1 \][/tex]
[tex]\[ \text{Interest}_B = 0.1025 \][/tex]
(b) Now we compare the interests:
- Interest earned from Bank A: [tex]\( 0.12 \)[/tex]
- Interest earned from Bank B: [tex]\( 0.1025 \)[/tex]
Given these calculations, you would save money in Bank A because the interest earned from Bank A (0.12) is higher than the interest earned from Bank B (0.1025).
Reason:
Bank A offers a higher yearly compound interest rate compared to Bank B's half-yearly compound interest rate, resulting in a greater effective return on the principal amount after one year.
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