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.
| | Card A | Card B |
| :------ | :---------- | :---------- |
| Amount | $1,463.82 | $1,157.98 |
| APR | 13% | 17% |
| Monthly Payment | $24.60 | $22.14 |
After 8 years, how much will Brandon have saved in interest by consolidating the two balances?
a. $581.76
b. $194.40
c. $256.32
d. $325.44
Please select the best answer from the choices provided.
Answer (1)
Calculate the total interest paid on Card A individually: $898.27. - Calculate the total interest paid on Card B individually: $967.90.
Calculate the total interest paid without consolidation: $1866.17. - Calculate the total interest paid with consolidation: 1608.92 , an d t h e in t eres t s a v e d : \boxed{{$257.25}}$.
Explanation
Calculate interest paid on each card separately First, we need to determine the total interest paid on each card individually. We are given the balances, APRs, and monthly payments for Card A and Card B. We will calculate the number of months required to pay off each card and then determine the total interest paid.
Card A Calculations For Card A: Balance: $B_A = 1463.82 A PR : APR_A = 13% = 0.13 M o n t h l y p a y m e n t : M_A = 24.60 M o n t h l y in t eres t r a t e : r_A = \frac{0.13}{12} = 0.0108333 N u mb ero f m o n t h s t o p a yo ff C a r d A : n_A = -\frac{\log(1 - \frac{B_A _A}{M_A})}{\log(1 + r_A)} = -\frac{\log(1 - \frac{1463.82 0.0108333}{24.60})}{\log(1 + 0.0108333)} \approx 96.02 T o t a lp a y m e n t f or C a r d A : Total_A = n_A M_A = 96.02 24.60 = 2362.09 I n t eres tp ai df or C a r d A : Interest_A = Total_A - B_A = 2362.09 - 1463.82 = $898.27
Card B Calculations For Card B: Balance: $B_B = 1157.98 A PR : APR_B = 17% = 0.17 M o n t h l y p a y m e n t : M_B = 22.14 M o n t h l y in t eres t r a t e : r_B = \frac{0.17}{12} = 0.0141667 N u mb ero f m o n t h s t o p a yo ff C a r d B : n_B = -\frac{\log(1 - \frac{B_B _B}{M_B})}{\log(1 + r_B)} = -\frac{\log(1 - \frac{1157.98 0.0141667}{22.14})}{\log(1 + 0.0141667)} \approx 96.02 T o t a lp a y m e n t f or C a r d B : Total_B = n_B M_B = 96.02 22.14 = 2125.88 I n t eres tp ai df or C a r d B : Interest_B = Total_B - B_B = 2125.88 - 1157.98 = $967.90
Total Interest Without Consolidation Total interest paid without consolidation: $Interest_{noConsolidation} = Interest_A + Interest_B = 898.27 + 967.90 = $1866.17
Consolidated Balance Calculations Next, we calculate the interest paid with consolidation on Card A's APR. Consolidated balance: $B = B_A + B_B = 1463.82 + 1157.98 = 2621.80 A PR : APR_A = 13% = 0.13 N u mb ero f m o n t h s : n = 8 12 = 96 M o n t h l y in t eres t r a t e : r = \frac{0.13}{12} = 0.0108333 M o n t h l y p a y m e n t f or t h eco n so l i d a t e d ba l an ce : $M = \frac{B (r (1 + r)^n)}{((1 + r)^n - 1)} = \frac{2621.80 (0.0108333 (1 + 0.0108333)^{96})}{((1 + 0.0108333)^{96} - 1)} \approx 44.07 T o t a lp a y m e n t f or t h eco n so l i d a t e d ba l an ce : Total = M n = 44.07 96 = 4230.72 I n t eres tp ai d w i t h co n so l i d a t i o n : Interest_{Consolidation} = Total - B = 4230.72 - 2621.80 = $1608.92
Calculate Interest Saved Finally, we calculate the interest saved by consolidating the balances: $Savings = Interest_{noConsolidation} - Interest_{Consolidation} = 1866.17 - 1608.92 = $257.25
Final Answer The interest saved by consolidating the two balances is approximately $257.25. The closest answer from the choices provided is $256.32.
Examples
Consolidating debt is a common financial strategy. For example, imagine you have multiple loans with varying interest rates. By consolidating these loans into a single loan with a lower interest rate, you can reduce the total amount of interest you pay over time. This strategy is particularly useful for managing credit card debt, student loans, or personal loans, making it easier to budget and save money in the long run. Understanding the math behind debt consolidation helps you make informed decisions about your finances and optimize your savings.
