Question

brandon has two credit cards and would like to consolidate the two balances into one balance on the card with the lower interest rate. the table below shows the information about the two credit cards brandon currently uses.

	amount	apr	monthly payment
card b	$1,476.12	15%	$31.21

after 6 years, approximately how much will brandon have saved in interest by consolidating the two balances?
a. $224.64
b. $935.28
c. $901.20
d. $145.44
please select the best answer from the choices provided.

Explanation:

Step1: Calculate total interest for Card A

First, find the total amount paid for Card A over 6 years (72 months). The monthly payment is $18.22. So total payment for Card A is \( 18.22\times72 \).
\( 18.22\times72 = 1311.84 \)
The principal amount for Card A is $957.00. So interest for Card A is total payment minus principal: \( 1311.84 - 957.00 = 354.84 \)

Step2: Calculate total interest for Card B

The monthly payment for Card B is $31.21. Total payment over 72 months is \( 31.21\times72 \).
\( 31.21\times72 = 2247.12 \)
Principal for Card B is $1476.12. Interest for Card B is \( 2247.12 - 1476.12 = 771.00 \)

Step3: Total interest before consolidation

Sum the interests of Card A and Card B: \( 354.84 + 771.00 = 1125.84 \)

Step4: Calculate total amount after consolidation

First, find the consolidated principal: \( 957.00 + 1476.12 = 2433.12 \)
The monthly payment on the new card (assuming it's Card A's rate? Wait, no, the problem says "the card with the lower interest rate", which is Card A with 11% APR. Wait, no, wait: the new card is the one with lower interest rate, which is Card A's rate? Wait, no, the monthly payment for the consolidated balance: wait, maybe I misread. Wait, the problem is to consolidate into one balance on the card with lower interest rate (Card A, 11% APR). Wait, no, the consolidated balance's monthly payment: wait, no, the total amount to pay on the new card (Card A's terms? Wait, no, the consolidated principal is \( 957 + 1476.12 = 2433.12 \). The monthly payment on the new card (with lower interest rate, which is Card A's 11% APR? Wait, no, the monthly payment for the consolidated balance: wait, maybe the new monthly payment is calculated? Wait, no, the problem is to find the savings, so we need to find the total interest paid if he keeps both cards, and the total interest paid if he consolidates (so one payment, with the lower rate, i.e., Card A's 11% APR? Wait, no, the lower interest rate is Card A's 11% vs Card B's 15%, so the consolidated balance is on Card A, so the monthly payment for the consolidated principal of $2433.12 at 11% APR. Wait, but the monthly payment formula for a loan: \( M = P\frac{r(1 + r)^n}{(1 + r)^n - 1} \), where \( r = \frac{0.11}{12} \), \( n = 72 \), \( P = 2433.12 \)

First, calculate \( r = \frac{0.11}{12} \approx 0.0091667 \)
\( (1 + r)^n = (1 + 0.0091667)^{72} \approx e^{72\times0.0091667} \approx e^{0.66} \approx 1.9348 \) (or calculate directly: \( 1.0091667^{72} \approx 1.9348 \))
Then \( M = 2433.12\times\frac{0.0091667\times1.9348}{1.9348 - 1} \)
Calculate numerator: \( 0.0091667\times1.9348 \approx 0.01773 \)
Denominator: \( 0.9348 \)
So \( M \approx 2433.12\times\frac{0.01773}{0.9348} \approx 2433.12\times0.01897 \approx 46.16 \)? Wait, no, that can't be. Wait, no, I think I made a mistake. Wait, the original monthly payments are $18.22 (Card A) and $31.21 (Card B), total monthly payment $49.43. But if he consolidates, the monthly payment would be based on the consolidated principal at the lower rate (11% APR). Wait, no, maybe the problem is simpler: the total amount paid over 6 years (72 months) for both cards is \( (18.22 + 31.21)\times72 = 49.43\times72 = 3558.96 \). The total principal is \( 957 + 1476.12 = 2433.12 \). So total interest before consolidation is \( 3558.96 - 2433.12 = 1125.84 \), which matches steps 1-3.

Now, for the consolidated balance: the new principal is 2433.12, and the monthly payment is calculated at 11% APR. Wait, but the problem says "the card with the lower interest rate" is Card A, so the monthly payment on the consolidated balance would be using Card A's APR (11%) and the total amount. Wait, but maybe the new monthly payment is not given, but wait, the answer choices are about savings, so maybe the consolidated total payment is (monthly[LLM SSE On Failure]

Answer:

Step1: Calculate total interest for Card A

First, find the total amount paid for Card A over 6 years (72 months). The monthly payment is $18.22. So total payment for Card A is \( 18.22\times72 \).
\( 18.22\times72 = 1311.84 \)
The principal amount for Card A is $957.00. So interest for Card A is total payment minus principal: \( 1311.84 - 957.00 = 354.84 \)

Step2: Calculate total interest for Card B

The monthly payment for Card B is $31.21. Total payment over 72 months is \( 31.21\times72 \).
\( 31.21\times72 = 2247.12 \)
Principal for Card B is $1476.12. Interest for Card B is \( 2247.12 - 1476.12 = 771.00 \)

Step3: Total interest before consolidation

Sum the interests of Card A and Card B: \( 354.84 + 771.00 = 1125.84 \)

Step4: Calculate total amount after consolidation

First, find the consolidated principal: \( 957.00 + 1476.12 = 2433.12 \)
The monthly payment on the new card (assuming it's Card A's rate? Wait, no, the problem says "the card with the lower interest rate", which is Card A with 11% APR. Wait, no, wait: the new card is the one with lower interest rate, which is Card A's rate? Wait, no, the monthly payment for the consolidated balance: wait, maybe I misread. Wait, the problem is to consolidate into one balance on the card with lower interest rate (Card A, 11% APR). Wait, no, the consolidated balance's monthly payment: wait, no, the total amount to pay on the new card (Card A's terms? Wait, no, the consolidated principal is \( 957 + 1476.12 = 2433.12 \). The monthly payment on the new card (with lower interest rate, which is Card A's 11% APR? Wait, no, the monthly payment for the consolidated balance: wait, maybe the new monthly payment is calculated? Wait, no, the problem is to find the savings, so we need to find the total interest paid if he keeps both cards, and the total interest paid if he consolidates (so one payment, with the lower rate, i.e., Card A's 11% APR? Wait, no, the lower interest rate is Card A's 11% vs Card B's 15%, so the consolidated balance is on Card A, so the monthly payment for the consolidated principal of $2433.12 at 11% APR. Wait, but the monthly payment formula for a loan: \( M = P\frac{r(1 + r)^n}{(1 + r)^n - 1} \), where \( r = \frac{0.11}{12} \), \( n = 72 \), \( P = 2433.12 \)

First, calculate \( r = \frac{0.11}{12} \approx 0.0091667 \)
\( (1 + r)^n = (1 + 0.0091667)^{72} \approx e^{72\times0.0091667} \approx e^{0.66} \approx 1.9348 \) (or calculate directly: \( 1.0091667^{72} \approx 1.9348 \))
Then \( M = 2433.12\times\frac{0.0091667\times1.9348}{1.9348 - 1} \)
Calculate numerator: \( 0.0091667\times1.9348 \approx 0.01773 \)
Denominator: \( 0.9348 \)
So \( M \approx 2433.12\times\frac{0.01773}{0.9348} \approx 2433.12\times0.01897 \approx 46.16 \)? Wait, no, that can't be. Wait, no, I think I made a mistake. Wait, the original monthly payments are $18.22 (Card A) and $31.21 (Card B), total monthly payment $49.43. But if he consolidates, the monthly payment would be based on the consolidated principal at the lower rate (11% APR). Wait, no, maybe the problem is simpler: the total amount paid over 6 years (72 months) for both cards is \( (18.22 + 31.21)\times72 = 49.43\times72 = 3558.96 \). The total principal is \( 957 + 1476.12 = 2433.12 \). So total interest before consolidation is \( 3558.96 - 2433.12 = 1125.84 \), which matches steps 1-3.

Now, for the consolidated balance: the new principal is 2433.12, and the monthly payment is calculated at 11% APR. Wait, but the problem says "the card with the lower interest rate" is Card A, so the monthly payment on the consolidated balance would be using Card A's APR (11%) and the total amount. Wait, but maybe the new monthly payment is not given, but wait, the answer choices are about savings, so maybe the consolidated total payment is (monthly[LLM SSE On Failure]