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A loan of $10,000 has an annual interest rate of 8%, compounded quarte

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Bunuel
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A loan of $10,000 has an annual interest rate of 8%, compounded quarte [#permalink] A loan of $10,000 has an annual interest rate of 8%, compounded quarte   09 Mar 2022, 05:45
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A
B
C
D
E

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45% (medium)

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72% (02:20) correct 28% (01:57) wrong based on 186 sessions
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A loan of $10,000 has an annual interest rate of 8%, compounded quarterly. What will be the total due (principal + interest) if no payment is made for 12 months?

(A) $800
(B) $824
(C) $10,800
(D) $10,824
(E) $11,600
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D
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lucasd14
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Re: A loan of $10,000 has an annual interest rate of 8%, compounded quarte [#permalink] A loan of $10,000 has an annual interest rate of 8%, compounded quarte   12 Apr 2022, 16:28
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Correct procedure would be to calculate the Compounded Interest formula [P*(1+x/n)^n]. But we can save some time estimating and then comparing with the answer choices.

Let's consider 4 times the simple interest of 8%/4 (four quarters annually) * 10,000. That would be 200 * 4 = 800. Now we know that still is missing the compounded (the interest of the accumulative interest), so it must be something a bit greater than 800. Only possible answer, D.
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Re: A loan of $10,000 has an annual interest rate of 8%, compounded quarte [#permalink] A loan of $10,000 has an annual interest rate of 8%, compounded quarte   03 Apr 2022, 22:58
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is there any easier way to calculate \(1.02^4\)
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Re: A loan of $10,000 has an annual interest rate of 8%, compounded quarte [#permalink] A loan of $10,000 has an annual interest rate of 8%, compounded quarte   14 Apr 2022, 04:48
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Amount = \(10000(1 + \frac{8}{400})^4\)
= \(\frac{(10000*51*51*51*51)}{(50*50*50*50)}\)
= \(\frac{(51*51*51*51)}{(5*5*5*5)}\)
\(\frac{51}{5}\) = \(10.2\)
So, \((51*51*51*51)/(5*5*5*5)\) = \((10.2)^4 = (10.2)^2*(10.2)^2\)

\(102*102 = 102*100 + 102*2 = 10200 + 204 = 10404\)
So, \(10.2*10.2 = 104.04\)

Now, \(104*104 = 104*100 + 104*4 = 10400 + 416 = 10816\)
So, \(104.04*104.04\) is slightly greater than \(10816\)

Hence, the answer is D, \($10,824\).
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A loan of $10,000 has an annual interest rate of 8%, compounded quarte [#permalink] A loan of $10,000 has an annual interest rate of 8%, compounded quarte   27 Sep 2024, 02:57
rsrighosh
is there any easier way to calculate \(1.02^4\)
Using binomial expansion
1.02^4 = (1+ 2/100)^4
=4c0 * 1^4* (2/100)^0
+4c1 * 1^3* (2/100)^1
+4c2 * 1^2* (2/100)^2
+4c3 * 1^1* (2/100)^3
+4c3 * 1^0* (2/100)^4

last two terms can be ignored as high powers of small numbers. So negligible.
rest will give 1+.08+.0024=1.0824

We could also approximate to 1+nx=1.08. Answer will be slightly higher than this.
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A loan of $10,000 has an annual interest rate of 8%, compounded quarte [#permalink] A loan of $10,000 has an annual interest rate of 8%, compounded quarte   28 Sep 2024, 08:01
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Right off the bat you can reject A and B because the principal is not included. Next, you can reject 10,800 (C) as well because that'll be due if there is no compounding. The next 2 choices are spread apart, first quarter's interest is 200, next quarter's interest is 204, next quarter's interest is close to 208. It is clear that the due amount cannot exceed 11000 which leaves us only with Option D.
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A loan of $10,000 has an annual interest rate of 8%, compounded quarte [#permalink]
28 Sep 2024, 08:01
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