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What amount, in dollars, invested for one year at an interest rate of
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27 Nov 2023, 09:51
27 Nov 2023, 09:51
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What amount, in dollars, invested for one year at an interest rate of 2 percent compounded semiannually would produce the same final balance at the end of the year as $10,000 invested for one year at an interest rate of 4 percent compounded quarterly?
A. $10,000
B. $10,201
C. $10,500
D. $10,801
E. $20,000
2023-11-27_20-50-18.png [ 47.56 KiB | Viewed 4456 times ]
A. $10,000
B. $10,201
C. $10,500
D. $10,801
E. $20,000
Attachment:
2023-11-27_20-50-18.png [ 47.56 KiB | Viewed 4456 times ]
Re: What amount, in dollars, invested for one year at an interest rate of
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27 Nov 2023, 11:57
27 Nov 2023, 11:57
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Expert Reply
guddo wrote:
What amount, in dollars, invested for one year at an interest rate of 2 percent compounded semiannually would produce the same final balance at the end of the year as $10,000 invested for one year at an interest rate of 4 percent compounded quarterly?
A. $10,000
B. $10,201
C. $10,500
D. $10,801
E. $20,000
A. $10,000
B. $10,201
C. $10,500
D. $10,801
E. $20,000
First, we'll calculate the final balance of $10,000 invested at a 4 percent interest rate compounded quarterly. An annual rate of 4 percent compounded quarterly equates to 1 percent per quarter, applied 4 times a year (at the end of the 3rd, 6th, 9th, and 12th months). Therefore, the final balance after one year will be:
- \($10,000*(1+\frac{0.04}{4})^{1*4}=$10,000*(1.01)^{4}\)
For 2 percent compounded semiannually, which means 1 percent twice a year, the final balance at the year's end is:
- \($x*1.01^2\)
These two final balances need to be equal:
- \($x*1.01^2 = $10,000*1.01^{4}\);
\($x = $10,000*1.01^2\);
\($x = $10,000*\frac{101^2}{100^2}\);
\($x = 101^2\);
\($x = (100+1)^2\);
\($x = 100^2+2*100+1\);
\($x= 10,201\).
Answer: B.
General Discussion
Re: What amount, in dollars, invested for one year at an interest rate of
[#permalink]
28 Nov 2023, 00:36
28 Nov 2023, 00:36
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Asked: What amount, in dollars, invested for one year at an interest rate of 2 percent compounded semiannually would produce the same final balance at the end of the year as $10,000 invested for one year at an interest rate of 4 percent compounded quarterly?
Let the amount be x dollars
x (1 + 2%/2)ˆ2 = 10000 (1 + 4%/4)ˆ4
x (1.01)ˆ2 = 10000 (1.01)ˆ4
x = 10000(1.01)ˆ2 = $10,201
IMO B
Let the amount be x dollars
x (1 + 2%/2)ˆ2 = 10000 (1 + 4%/4)ˆ4
x (1.01)ˆ2 = 10000 (1.01)ˆ4
x = 10000(1.01)ˆ2 = $10,201
IMO B
