You are considering purchasing some new clothes today. You know that you

will receive a $$500$ gift from a relative in exactly one year. You also have

access to a new credit card that charges an introductory rate of $10\%$ APR for

the first $6$ months, after which it charges an APR of $20\%.$ Both APR$$s are

quoted with daily compounding $($based on a $30$ day month, $360$ day year

compounding convention$).$

a$.$ What is the monthly periodic compound rate of interest $($the effective

monthly rate$)$ charged by the card for the first $6$ months $($expressed as

a percentage with three decimal places$)?$

b$.$ What is the monthly periodic compound rate of interest $($the effective

monthly rate$)$ charged by the card after the first $6$ months $($expressed

as a percentage with three decimal places$)?$

c$.$ What is the present value as of $6$ months from now of $$500$ to be

delivered $1$ year from now, discounted at the interest rate charged by

the credit card after the first $6$ months $($expressed as a dollar amount

with two decimal places$)?$

d$.$ What is the most you could spend on clothes using the credit card

today if you finance the entire payment with the card, make no

payments until you receive the $$500$ gift, then pay off the entire credit

card loan with the $$500$ gift $1$ year from now $($expressed as a dollar

amount with two decimal places$)?$