+4 Assume that you have a balance of $2800 on your Discover credit card and that you make no more charges. Assume that Discover charges 21% APR and that each month you make only the minimum payment of 2% of the balance. Find how many months it will take to bring the remaining balance down to $2500. (Round your answer to the nearest whole number.)
= months
+2440 Solution:
\(\text {Preface:}\\ \text {Normally these payments are tabulated by using a spreadsheet; however, because there are no additional }\\ \text {charges to the card, it is easy to calculate a fixed formula by adjusting the payment percentage }\\ \text {rate of (2.0%) by the periodic interest rate.}\\ \text {The monthly interest rate is $\frac{21.0\%} {12} = 1.75\% \; (0.0175)$}\\ \text {The balance before the payment is found by multiplying the previous month's balance by (1.0175). }\\ \text {The new balance after payment is 98% of the previous balance and accrued interest. }\\ \text {Balance after payment is $(B * (1.0175 * 0.98))$ or $(B * (0.99715)$. The adjusted fixed payment } \\ \text {rate complement is (0.99715). By taking integer multiples of the adjusted fixed payment rate }\\ \text {complement and multiplying by the starting balance, the balance for any month(t) is given by:}\\ B(t) = 2800 * (0.99715^t)\\ \)
\(\text {Using this formula, solve for $\lceil {t}\rceil $ when $(2800 * (0.99715^t)) = 2500$}\\ \dfrac {2800 \cdot \:0.99715^t}{2800}=\dfrac {2500} {2800}\\ 0.99715^t=\dfrac{25}{28}\\ \log \left(0.99715^t\right)=\log \left(\dfrac{25}{28}\right)\\ t \log \left(0.9972\right)=\ log \left(\frac{25}{28}\right)\\ t=\frac {\log \tiny \left(\dfrac{25}{28}\right)}{\log \left(0.99715\right)}\\ t = 39.71 \text { and } \lceil {t}\rceil = 40\\ \text { }\\ \text {After the 40th payment, the balance will be below 2500 dollars.}\\ \)
GA
