Respuesta :
Answer: C. June 15th
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Explanation:
Laura starts off with a balance of $606.40
This balance is in effect for x days, where x is some positive whole number between 1 and 30.
For the remaining 30-x days, her balance is 606.40-55.25 = 551.15 dollars after making the payment of $55.25
Here's a table to keep track of everything so far
[tex]\begin{array}{|c|c|c|} \cline{1-3}& A & B\\ \cline{1-3}\text{Interval} & \text{Num Days} & \text{Balance}\\ \cline{1-3}\text{June 1st to June x} & x & 606.40\\ \cline{1-3}\text{June x+1 to June 30th} & 30-x & 551.15\\ \cline{1-3}\end{array}[/tex]
where "Num Days" is shorthand for "number of days".
What we do is multiply the A and B column to form column C like this
[tex]\begin{array}{|c|c|c|c|} \cline{1-4}& A & B & C\\ \cline{1-4}\text{Interval} & \text{Num Days} & \text{Balance} & \text{A*B}\\ \cline{1-4}\text{June 1st to June x} & x & 606.40 & 606.40x\\ \cline{1-4}\text{June x+1 to June 30th} & 30-x & 551.15 & 551.15(30-x)\\ \cline{1-4}\end{array}[/tex]
Add up the stuff in column C
606.40x+551.15(30-x)
Then divide by 30 to compute the average daily balance
[tex]\frac{606.40x+551.15(30-x)}{30}[/tex]
That average daily balance is plugged into this formula
[tex]\text{F} = \frac{\text{ADB}*\text{APR}*\text{n}}{365}[/tex]
where
- F = finance charge
- ADB = average daily balance
- APR = annual percentage rate
- n = number of days in the billing cycle
We are given the following
- F = 5.71
- APR = 0.1204
- n = 30
Plug in [tex]\text{ADB} = \frac{606.40x+515.15(30-x)}{30}[/tex] and solve for x
So,
[tex]\text{F} = \frac{\text{ADB}*\text{APR}*\text{n}}{365}\\\\5.71 = \frac{\frac{606.40x+551.15(30-x)}{30}*0.1204*30}{365}\\\\5.71 = \frac{(606.40x+551.15(30-x))*0.1204}{365}\\\\5.71*365 = (606.40x+551.15(30-x))*0.1204\\\\2,084.15 = 606.40*0.1204x+551.15*0.1204(30-x)\\\\2,084.15 = 73.01056x+66.35846(30-x)\\\\[/tex]
[tex]2,084.15 = 73.01056x+1,990.7538-66.35846x\\\\2,084.15 = 6.6521x+1,990.7538\\\\2,084.15-1,990.7538 = 6.6521x\\\\93.3962 = 6.6521x\\\\6.6521x = 93.3962\\\\x = 93.3962/6.6521\\\\x = 14.0401076351829\\\\[/tex]
That value is approximate.
When rounding to the nearest whole number, we get x = 14.
Therefore, from June 1st to June 14th, Laura has a balance of $606.40
From June 15th to June 30th, she has a balance of $551.15
The payment was made on June 15th
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Checking the answer:
Here's the updated table with x replaced with 14. So x+1 = 14+1 = 15
[tex]\begin{array}{|c|c|c|c|} \cline{1-4}& \text{A} & \text{B} & \text{C}\\ \cline{1-4}\text{Interval} & \text{Num Days} & \text{Balance} & \text{A*B}\\ \cline{1-4}\text{June 1st to June 14th} & 14 & 606.40 & 8489.60\\ \cline{1-4}\text{June 15th to June 30th} & 16 & 551.15 & 8818.40\\ \cline{1-4}\end{array}[/tex]
Adding everything in column C gets us
8489.60+8818.40 = 17,308
Divide that over 30 days
(17,308)/30 = 576.93
Her average daily balance for the month of June is $576.93
Plug that into the formula mentioned to get the finance charge.
[tex]\text{F} = \frac{\text{ADB}*\text{APR}*\text{n}}{365}\\\\\text{F} = \frac{576.93*0.1204*30}{365}\\\\\text{F} = \frac{2,083.87116}{365}\\\\\text{F} = 5.7092\\\\\text{F} = 5.71[/tex]
We get the correct finance charge of $5.71, so the answer has been confirmed.
