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Question 1
A student takes out a loan of $\$12{,}000$ to purchase a motorcycle. The loan has a nominal annual interest rate of $9%$ , compounded monthly, and is to be repaid over 3 years with equal monthly payments. What is the monthly payment amount, correct to the nearest cent?
No clue? Show me the answer
Correct answerCorrect!Incorrect
A $$381.60$
Step-by-step walkthrough
Choose a solution method
Method #1Approach 1
Step 1: Identify the Variables
Principal: $P = 12, 000$ . Annual interest rate: $9%$ , so monthly rate: $r = \frac{0.09}{12} = 0.0075$ . Number of payments: $n = 3 \times 12 = 36$ .
Step 2: Apply the Loan Payment Formula
$Payment = P \times \frac{r ( 1 + r ) ^{n}}{( 1 + r ) ^{n} - 1} = 12, 000 \times \frac{0.0075 \times ( 1.0075 ) ^{36}}{( 1.0075 ) ^{36} - 1}$
Step 3: Calculate the Compounding Factor
$(1.0075)^{36} \approx 1.30865$ . This is the key exponential term accounting for compound interest over 36 months.
Step 4: Complete the Calculation
$Payment = 12, 000 \times \frac{0.0075 \times 1.30865}{1.30865 - 1} = 12, 000 \times \frac{0.009815}{0.30865} \approx 12, 000 \times 0.031800 \approx 381.60$
Step 5: State the Answer
The monthly payment is $\$ 381.60$. Using a GDC TVM Solver with N = 36, I% = 9, PV = 12000, FV = 0, P/Y = 12, C/Y = 12 and solving for PMT confirms this result.
Method #2Approach 2
Step 1: Identify What Is Being Asked
We need the equal monthly payment for a $\$ 12{,}000 $l o ana t$ 9%$ annual interest compounded monthly over 3 years. The correct approach uses the compound interest loan payment formula, not simple interest.
Step 2: Eliminate $\$333.33$
** $\$ 333.33 $* * co m es f r o m s im pl y d i v i d in g$ $12{,}000$ by 36 payments. This ignores interest entirely and is incorrect for any interest-bearing loan.
Step 3: Eliminate $\$360.00$
** $\$ 360.00 $* * mi g h t co m e f r o ma dd in g$ 9% $o f$ $12{,}000 $($ $1{,}080$) and dividing by 36, which uses simple interest logic. Real loans use compound interest, so this underestimates the true payment.
Step 4: Eliminate $\$392.44$
** $\$ 392.44 $* * r es u l t s f r o min cor r ec tl y u s in g t h e f u l l ann u a l r a t eo f$ 0.09 $(in s t e a d o f t h e m o n t h l y r a t e$ 0.0075$) in the formula, which significantly overstates the periodic rate and inflates the payment.
Step 5: Select the Correct Answer
** $\$ 381.60 $* * i s t h er es u l t o f cor r ec tl y a ppl y in g t h e l o an p a y m e n t f or m u l a w i t h$ r = 0.0075 $an d$ n = 36$, giving the only accurate monthly payment.
Question 2
The table below shows the first two rows of an amortization schedule for a loan repaid monthly. Based on this information, what is the remaining balance after payment 2?

Payment # Payment Interest Paid Principal Paid Remaining Balance
1 $\$ 310.00$ $\$ 100.00$ $\$ 210.00$ $\$ 19{,}790.00$
2 $\$ 310.00$ ? ? ?
No clue? Show me the answer
Correct answerCorrect!Incorrect
B $$19, 578.95$
Step-by-step walkthrough
Choose a solution method
Method #1Approach 1
Step 1: Identify the Monthly Interest Rate
From payment 1, the original balance must have been $\$ 20{,}000 $(s in ce p r in c i p a l p ai d$ = $210.00 $an d r e mainin g ba l an ce$ = $19{,}790.00 $, sos t a r t in g ba l an ce$ = $20{,}000 $) . T h e in t er es t r a t e p er p er i o d i s$ r = \dfrac{100}{20{,}000} = 0.005$ per month.
Step 2: Calculate Interest for Payment 2
Interest in payment 2 $= 19{,}790.00 \times 0.005 = \$ 98.95$.
Step 3: Calculate Principal Paid in Payment 2
Principal paid in payment 2 $= 310.00 - 98.95 = \$ 211.05$.
Step 4: Calculate Remaining Balance After Payment 2
New balance $= 19{,}790.00 - 211.05 = \$ 19{,}578.95$.
Step 5: State the Answer
The remaining balance after payment 2 is $\$ 19{,}578.95$. Notice the balance decreases slightly more in period 2 than period 1 because less interest was charged.
Method #2Approach 2
Step 1: Identify the Key Calculation Required
We must find the balance after payment 2 by computing: Interest $=$ balance $\times$ rate, then Principal $=$ payment $-$ interest, then New balance $=$ old balance $-$ principal paid. The monthly rate is $r = 100/20, 000 = 0.005$ .
Step 2: Eliminate $\$19{,}579.05$
** $\$ 19{,}579.05 $* * r es u l t s f r o ma s ma l l a r i t hm e t i cr o u n d in g er r or in co m p u t in g$ 19{,}790 \times 0.005 = 98.95 $a s$ 98.90 $. T h ecor r ec t in t er es t i s$ $98.95 $, mak in g t hi s an s w er o f f b y$ $0.10$.
Step 3: Eliminate $\$19{,}580.00$
** $\$ 19{,}580.00 $* * w o u l d r es u l t f r o m s u b t r a c t in g$ $210.00$ (the period-1 principal) again in period 2, ignoring that the interest changes each period. The principal paid increases slightly each period.
Step 4: Eliminate $\$19{,}481.00$
** $\$ 19{,}481.00 $* * r es u l t s f r o min cor r ec tl y s u b t r a c t in g t h e f u l l p a y m e n t o f$ $310.00$ from the balance instead of only the principal portion, which confuses the total payment with the principal reduction.
Step 5: Select the Correct Answer
** $\$ 19{,}578.95 $* * i scor r ec t : in t er es t$ = $98.95 $, p r in c i p a l$ = $211.05 $, n e w ba l an ce$ = 19{,}790.00 - 211.05 = $19{,}578.95$.

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13 more questions in this topic

Payment #

Payment

Interest Paid

Principal Paid

Remaining Balance

\

310.00$

\

100.00$

\

210.00$

\

19{,}790.00$

\

310.00$

Payment #	Payment	Interest Paid	Principal Paid	Remaining Balance
1	$\$ 310.00$	$\$ 100.00$	$\$ 210.00$	$\$ 19{,}790.00$
2	$\$ 310.00$	?	?	?

Step-by-step walkthrough

Step 1: Identify the Variables

Step 2: Apply the Loan Payment Formula

Step 3: Calculate the Compounding Factor

Step 4: Complete the Calculation

Step 5: State the Answer

Step 1: Identify What Is Being Asked

Step 2: Eliminate $\$333.33$

Step 3: Eliminate $\$360.00$

Step 4: Eliminate $\$392.44$

Step 5: Select the Correct Answer

Step-by-step walkthrough

Step 1: Identify the Monthly Interest Rate

Step 2: Calculate Interest for Payment 2

Step 3: Calculate Principal Paid in Payment 2

Step 4: Calculate Remaining Balance After Payment 2

Step 5: State the Answer

Step 1: Identify the Key Calculation Required

Step 2: Eliminate $\$19{,}579.05$

Step 3: Eliminate $\$19{,}580.00$

Step 4: Eliminate $\$19{,}481.00$

Step 5: Select the Correct Answer

13 more questions in this topic

Step-by-step walkthrough

Step 1: Identify the Variables

Step 2: Apply the Loan Payment Formula

Step 3: Calculate the Compounding Factor

Step 4: Complete the Calculation

Step 5: State the Answer

Step 1: Identify What Is Being Asked

Step 2: Eliminate $\$333.33$

Step 3: Eliminate $\$360.00$

Step 4: Eliminate $\$392.44$

Step 5: Select the Correct Answer

Step-by-step walkthrough

Step 1: Identify the Monthly Interest Rate

Step 2: Calculate Interest for Payment 2

Step 3: Calculate Principal Paid in Payment 2

Step 4: Calculate Remaining Balance After Payment 2

Step 5: State the Answer

Step 1: Identify the Key Calculation Required

Step 2: Eliminate $\$19{,}579.05$

Step 3: Eliminate $\$19{,}580.00$

Step 4: Eliminate $\$19{,}481.00$

Step 5: Select the Correct Answer

13 more questions in this topic