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Paying Off a Credit Card with Payments that Form an Annuity

After spending too much money on credit card interest last year, Usha applied for a new credit card that had a lower interest rate of $16\mathrm{.}9\%$ compounded daily. The terms of her new credit card are similar to her old one:

For any payment she makes within a month from the date of purchase, she will be charged no interest.

However, if she does not pay off the full balance, interest is charged daily on the remaining balance effective from the date of purchase.

$=$ Every month she must make a minimum payment.

On November $30,$ Usha completed all of her Christmas shopping using her new credit card. Also, she did not make any other purchases in the following months.

a$.$ On December $31,$ Usha made a payment of $$100,$ and on the last days of January to May, respectively, she made the minimum payment of $$20$ that was charged on her credit card. After the May $31$ payment, the balance on her card was $$610\mathrm{.}06.$ What was the value of all the purchases she made on November $30?($Hint: Remember, interest is not charged on any payments she made within the first month.$)$

b$.$ How much longer would it take Usha to pay off her debt if she continued to only make the minimum payment of $$20$ at the end of each month? Round your answer up to the next month.

c$.$ How much quicker could she pay off her debt by making payments of $$40$ at the end of each month instead? Round your answer up to the next month.

d$.$ Usha was shocked by how long it would take for her to pay off her debt if she continued to only make the minimum payment of $$20,$ and so she made a plan to be debt$-$free in six months. What equal payments would she have to make at the end of each month $($starting on June $30)$ in order to achieve this goal?

e$.$ If she followed her payment plan laid out in $($d$),$ what would the balance be on Usha's credit card after her third payment $($halfway through her repayment plan$)?$

f$.$ If on November $30,$ after making her final payment, her credit card balance is $$0,$ how much interest did she pay towards her credit card since November $30$ the previous year?