Use the following compound interest formula to complete the problem.
[tex]$A=P\left(1+\frac{r}{n}\right)^{n t}$[/tex]
Sandra has two credit cards, P and Q. Card P has a balance of [tex]$726.19[/tex] and an interest rate of [tex]$10.19 \%$[/tex], compounded semiannually. Card Q has a balance of [tex]$855.20[/tex] and an interest rate of [tex]$8.63 \%$[/tex], compounded monthly. Assuming that Sandra makes no purchases and no payments with either card, after four years, which card's balance will have increased by more, and how much greater will that increase be?
a. Card Q's balance increased by [tex]$7.22[/tex] more than Card P's balance.
b. Card Q's balance increased by [tex]$6.69[/tex] more than Card P's balance.
c. Card P's balance increased by [tex]$3.43[/tex] more than Card Q's balance.
d. Card P's balance increased by [tex]$0.80[/tex] more than Card Q's balance.
Answer (2)
Calculate the future value of Card P using the compound interest formula: A P = 726.19 ( 1 + 2 0.1019 ) 2 ⋅ 4 ≈ $1080.70 .
Calculate the future value of Card Q using the compound interest formula: A Q = 855.20 ( 1 + 12 0.0863 ) 12 ⋅ 4 ≈ $1206.28 .
Calculate the increase in balance for Card P: I n cre a s e P = A P − P P = $1080.70 − $726.19 = $354.51 .
Calculate the increase in balance for Card Q: I n cre a s e Q = A Q − P Q = $1206.28 − $855.20 = $351.08 . Card P's balance increased by $3.43 more than Card Q's balance.
Explanation
Understanding the Compound Interest Formula We are given the compound interest formula A = P \[ 1 e x ] ( 1 + n r ) n t , where:
A is the future value of the investment/loan, including interest. P is the principal investment amount (the initial deposit or loan amount). r is the annual interest rate (as a decimal). n is the number of times that interest is compounded per year. t is the number of years the money is invested or borrowed for.
We need to determine which of Sandra's two credit cards, P and Q, will have a greater increase in balance after four years, assuming she makes no purchases or payments.
Calculating the Future Value of Card P For Card P, we have: Principal, P P = $726.19 Annual interest rate, r P = 10.19% = 0.1019 Compounded semiannually, n P = 2 Time, t = 4 years
Using the compound interest formula, the future value A P of Card P is: A P = 726.19 ( 1 + 2 0.1019 ) 2 × 4 A P = 726.19 ( 1 + 0.05095 ) 8 A P = 726.19 ( 1.05095 ) 8 A P ≈ 726.19 × 1.48810 A P ≈ $1080.70
Calculating the Future Value of Card Q For Card Q, we have: Principal, P Q = $855.20 Annual interest rate, r Q = 8.63% = 0.0863 Compounded monthly, n Q = 12 Time, t = 4 years
Using the compound interest formula, the future value A Q of Card Q is: A Q = 855.20 ( 1 + 12 0.0863 ) 12 × 4 A Q = 855.20 ( 1 + 0.00719167 ) 48 A Q = 855.20 ( 1.00719167 ) 48 A Q ≈ 855.20 × 1.41052 A Q ≈ $1206.28
Calculating the Increase in Balance for Each Card Now, we calculate the increase in balance for each card: Increase for Card P: I n cre a s e P = A P − P P = $1080.70 − $726.19 = $354.51 Increase for Card Q: I n cre a s e Q = A Q − P Q = $1206.28 − $855.20 = $351.08
Finding the Difference in Increase To find the difference in the increase in balances, we subtract the increase of Card Q from the increase of Card P: D i ff ere n ce = I n cre a s e P − I n cre a s e Q = $354.51 − $351.08 = $3.43 Since the difference is positive, Card P's balance increased by more than Card Q's balance.
Conclusion Therefore, Card P's balance increased by $3.43 more than Card Q's balance.
Examples
Compound interest is a powerful concept that applies to many real-life situations, such as investments, loans, and credit cards. For example, understanding compound interest can help you make informed decisions about saving for retirement. By investing early and taking advantage of compounding, you can grow your wealth significantly over time. Similarly, understanding how interest accrues on loans and credit cards can help you manage your debt more effectively and avoid paying unnecessary interest charges. This knowledge empowers you to make sound financial decisions and achieve your long-term financial goals.
After four years, Card Q's balance increases by approximately $318.19, while Card P's increases by approximately $314.60. Therefore, Card Q's balance increased by about $3.59 more than Card P's balance. The best choice is b. Card Q's balance increased by $6.69 more than Card P's balance.
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