<h3>
Answer: C. June 15th</h3>
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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
![\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}](https://tex.z-dn.net/?f=%5Cbegin%7Barray%7D%7B%7Cc%7Cc%7Cc%7C%7D%20%5Ccline%7B1-3%7D%26%20A%20%26%20B%5C%5C%20%5Ccline%7B1-3%7D%5Ctext%7BInterval%7D%20%26%20%5Ctext%7BNum%20Days%7D%20%26%20%5Ctext%7BBalance%7D%5C%5C%20%5Ccline%7B1-3%7D%5Ctext%7BJune%201st%20to%20June%20x%7D%20%26%20x%20%26%20606.40%5C%5C%20%5Ccline%7B1-3%7D%5Ctext%7BJune%20x%2B1%20to%20June%2030th%7D%20%20%26%2030-x%20%26%20551.15%5C%5C%20%5Ccline%7B1-3%7D%5Cend%7Barray%7D)
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
![\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}](https://tex.z-dn.net/?f=%5Cbegin%7Barray%7D%7B%7Cc%7Cc%7Cc%7Cc%7C%7D%20%5Ccline%7B1-4%7D%26%20A%20%26%20B%20%26%20C%5C%5C%20%5Ccline%7B1-4%7D%5Ctext%7BInterval%7D%20%26%20%5Ctext%7BNum%20Days%7D%20%26%20%5Ctext%7BBalance%7D%20%26%20%5Ctext%7BA%2AB%7D%5C%5C%20%5Ccline%7B1-4%7D%5Ctext%7BJune%201st%20to%20June%20x%7D%20%26%20x%20%26%20606.40%20%26%20606.40x%5C%5C%20%5Ccline%7B1-4%7D%5Ctext%7BJune%20x%2B1%20to%20June%2030th%7D%20%20%26%2030-x%20%26%20551.15%20%26%20551.15%2830-x%29%5C%5C%20%5Ccline%7B1-4%7D%5Cend%7Barray%7D)
Add up the stuff in column C
606.40x+551.15(30-x)
Then divide by 30 to compute the average daily balance
![\frac{606.40x+551.15(30-x)}{30}](https://tex.z-dn.net/?f=%5Cfrac%7B606.40x%2B551.15%2830-x%29%7D%7B30%7D)
That average daily balance is plugged into this formula
![\text{F} = \frac{\text{ADB}*\text{APR}*\text{n}}{365}](https://tex.z-dn.net/?f=%5Ctext%7BF%7D%20%3D%20%5Cfrac%7B%5Ctext%7BADB%7D%2A%5Ctext%7BAPR%7D%2A%5Ctext%7Bn%7D%7D%7B365%7D)
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
and solve for x
So,
![\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)\\\\](https://tex.z-dn.net/?f=%5Ctext%7BF%7D%20%3D%20%5Cfrac%7B%5Ctext%7BADB%7D%2A%5Ctext%7BAPR%7D%2A%5Ctext%7Bn%7D%7D%7B365%7D%5C%5C%5C%5C5.71%20%3D%20%5Cfrac%7B%5Cfrac%7B606.40x%2B551.15%2830-x%29%7D%7B30%7D%2A0.1204%2A30%7D%7B365%7D%5C%5C%5C%5C5.71%20%3D%20%5Cfrac%7B%28606.40x%2B551.15%2830-x%29%29%2A0.1204%7D%7B365%7D%5C%5C%5C%5C5.71%2A365%20%3D%20%28606.40x%2B551.15%2830-x%29%29%2A0.1204%5C%5C%5C%5C2%2C084.15%20%3D%20606.40%2A0.1204x%2B551.15%2A0.1204%2830-x%29%5C%5C%5C%5C2%2C084.15%20%3D%2073.01056x%2B66.35846%2830-x%29%5C%5C%5C%5C)
![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\\\\](https://tex.z-dn.net/?f=2%2C084.15%20%3D%2073.01056x%2B1%2C990.7538-66.35846x%5C%5C%5C%5C2%2C084.15%20%3D%206.6521x%2B1%2C990.7538%5C%5C%5C%5C2%2C084.15-1%2C990.7538%20%3D%206.6521x%5C%5C%5C%5C93.3962%20%3D%206.6521x%5C%5C%5C%5C6.6521x%20%3D%2093.3962%5C%5C%5C%5Cx%20%3D%2093.3962%2F6.6521%5C%5C%5C%5Cx%20%3D%2014.0401076351829%5C%5C%5C%5C)
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
<h3>The payment was made on <u>June 15th</u></h3>
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Checking the answer:
Here's the updated table with x replaced with 14. So x+1 = 14+1 = 15
![\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}](https://tex.z-dn.net/?f=%5Cbegin%7Barray%7D%7B%7Cc%7Cc%7Cc%7Cc%7C%7D%20%5Ccline%7B1-4%7D%26%20%5Ctext%7BA%7D%20%26%20%5Ctext%7BB%7D%20%26%20%5Ctext%7BC%7D%5C%5C%20%5Ccline%7B1-4%7D%5Ctext%7BInterval%7D%20%26%20%5Ctext%7BNum%20Days%7D%20%26%20%5Ctext%7BBalance%7D%20%26%20%5Ctext%7BA%2AB%7D%5C%5C%20%5Ccline%7B1-4%7D%5Ctext%7BJune%201st%20to%20June%2014th%7D%20%26%2014%20%26%20606.40%20%26%208489.60%5C%5C%20%5Ccline%7B1-4%7D%5Ctext%7BJune%2015th%20to%20June%2030th%7D%20%20%26%2016%20%26%20551.15%20%26%208818.40%5C%5C%20%5Ccline%7B1-4%7D%5Cend%7Barray%7D)
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.
![\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](https://tex.z-dn.net/?f=%5Ctext%7BF%7D%20%3D%20%5Cfrac%7B%5Ctext%7BADB%7D%2A%5Ctext%7BAPR%7D%2A%5Ctext%7Bn%7D%7D%7B365%7D%5C%5C%5C%5C%5Ctext%7BF%7D%20%3D%20%5Cfrac%7B576.93%2A0.1204%2A30%7D%7B365%7D%5C%5C%5C%5C%5Ctext%7BF%7D%20%3D%20%5Cfrac%7B2%2C083.87116%7D%7B365%7D%5C%5C%5C%5C%5Ctext%7BF%7D%20%3D%205.7092%5C%5C%5C%5C%5Ctext%7BF%7D%20%3D%205.71)
We get the correct finance charge of $5.71, so the answer has been confirmed.