Question
london invested $3400 in an account paying an interest rate of 6 7/8% compounded quarterly victoria invested $3400 in an account paying an interest rate of 6 3/8% compounded monthly after 5 years how much more money would london have in her account than victoria
Answers
GPT-4o mini
To find out how much more money London would have than Victoria after 5 years, we'll first calculate the future value of each account using the compound interest formula:
\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]
where:
- \(A\) is the amount of money accumulated after n years, including interest.
- \(P\) is the principal amount (the initial investment).
- \(r\) is the annual interest rate (decimal).
- \(n\) is the number of times that interest is compounded per year.
- \(t\) is the number of years the money is invested or borrowed for.
### For London:
- Principal, \(P = 3400\)
- Interest rate, \(r = 6 \frac{7}{8}\% = 6.875\% = 0.06875\)
- Compounded quarterly, so \(n = 4\)
- Time, \(t = 5\)
Plugging in the values:
\[
A = 3400 \left(1 + \frac{0.06875}{4}\right)^{4 \times 5}
\]
Calculating:
\[
A = 3400 \left(1 + \frac{0.06875}{4}\right)^{20}
\]
\[
= 3400 \left(1 + 0.0171875\right)^{20}
\]
\[
= 3400 \left(1.0171875\right)^{20}
\]
Now calculating \(1.0171875^{20}\):
\[
A \approx 3400 \times 1.485947 = 5051.22
\]
### For Victoria:
- Principal, \(P = 3400\)
- Interest rate, \(r = 6 \frac{3}{8}\% = 6.375\% = 0.06375\)
- Compounded monthly, so \(n = 12\)
- Time, \(t = 5\)
Plugging in the values:
\[
A = 3400 \left(1 + \frac{0.06375}{12}\right)^{12 \times 5}
\]
Calculating:
\[
A = 3400 \left(1 + \frac{0.06375}{12}\right)^{60}
\]
\[
= 3400 \left(1 + 0.0053125\right)^{60}
\]
\[
= 3400 \left(1.0053125\right)^{60}
\]
Now calculating \(1.0053125^{60}\):
\[
A \approx 3400 \times 1.348248 = 4583.84
\]
### Difference in Amounts:
Now we find out how much more London has than Victoria:
\[
5051.22 - 4583.84 = 467.38
\]
Thus, after 5 years, London would have approximately **$467.38** more than Victoria.
\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]
where:
- \(A\) is the amount of money accumulated after n years, including interest.
- \(P\) is the principal amount (the initial investment).
- \(r\) is the annual interest rate (decimal).
- \(n\) is the number of times that interest is compounded per year.
- \(t\) is the number of years the money is invested or borrowed for.
### For London:
- Principal, \(P = 3400\)
- Interest rate, \(r = 6 \frac{7}{8}\% = 6.875\% = 0.06875\)
- Compounded quarterly, so \(n = 4\)
- Time, \(t = 5\)
Plugging in the values:
\[
A = 3400 \left(1 + \frac{0.06875}{4}\right)^{4 \times 5}
\]
Calculating:
\[
A = 3400 \left(1 + \frac{0.06875}{4}\right)^{20}
\]
\[
= 3400 \left(1 + 0.0171875\right)^{20}
\]
\[
= 3400 \left(1.0171875\right)^{20}
\]
Now calculating \(1.0171875^{20}\):
\[
A \approx 3400 \times 1.485947 = 5051.22
\]
### For Victoria:
- Principal, \(P = 3400\)
- Interest rate, \(r = 6 \frac{3}{8}\% = 6.375\% = 0.06375\)
- Compounded monthly, so \(n = 12\)
- Time, \(t = 5\)
Plugging in the values:
\[
A = 3400 \left(1 + \frac{0.06375}{12}\right)^{12 \times 5}
\]
Calculating:
\[
A = 3400 \left(1 + \frac{0.06375}{12}\right)^{60}
\]
\[
= 3400 \left(1 + 0.0053125\right)^{60}
\]
\[
= 3400 \left(1.0053125\right)^{60}
\]
Now calculating \(1.0053125^{60}\):
\[
A \approx 3400 \times 1.348248 = 4583.84
\]
### Difference in Amounts:
Now we find out how much more London has than Victoria:
\[
5051.22 - 4583.84 = 467.38
\]
Thus, after 5 years, London would have approximately **$467.38** more than Victoria.