Homer won a prize in the lottery of $3000, payable $1,500 immediately and $1,500 plus 5% simple interest payable in 230 days. Getting impatient, Homer sells the promissory note to Moe for $1,460 cash after 360 days. Using a nominal 360 day year, find the simple interest rate, rounded to .01, earned by Moe.
First, we need to calculate the interest that Homer is entitled to receive after 230 days. We can do this by multiplying the principal ($1,500) by the interest rate (0.05) and the time in years (230/360):
$1,500 * 0.05 * (230 / 360) = $47.92.
Therefore, after 230 days, Homer would be entitled to receive $1,500 + $47.92 = $1,547.92.
Now we can calculate the interest that Moe earned by purchasing the promissory note at a discounted price and holding it for 360 days. Moe paid $1,460 for the note and received $1,547.92 after 360 days. Therefore, Moe earned $1,547.92 - $1,460 = $87.92 in interest.
To find the interest rate earned by Moe, we divide the interest earned ($87.92) by the amount paid for the note ($1,460) and multiply by 100%:
($87.92 / $1,460) * 100% = 6.03%.
Therefore, the simple interest rate earned by Moe is 6.03%, rounded to .01.
To find the interest rate earned by Moe, we need to calculate the interest he earned on the promissory note.
First, let's calculate the interest on the $1,500 principal amount payable in 230 days.
Simple Interest = Principal × Rate × Time
Given: Principal = $1,500, Time = 230 days.
Let's assume the interest rate as "x".
So, the equation becomes:
1,500 × x × (230/360) = $1,500 × 1.05
Simplifying the equation:
1500x × (230/360) = 1500 × 1.05
1500x × 0.63889 = 1500 × 1.05
957.167x = 1575
Now, let's solve for "x":
x = 1575 / 957.167
x ≈ 1.6435
So, the interest rate earned by Moe is approximately 1.6435 or 164.35%.
Rounded to the nearest hundredth, the interest rate earned by Moe is 164.35%.
To find the simple interest rate earned by Moe, we need to calculate the future value of the promissory note after 230 days and compare it to the cash amount Moe received after 360 days.
First, let's calculate the future value of the $1,500 part of the prize payable in 230 days. The formula to calculate simple interest is:
Future Value = Principal + Principal x Interest Rate x Time
The principal is $1,500, the time is 230/360 years (as we are using a nominal 360-day year), and we need to solve for the interest rate.
Future Value = $1,500 + $1,500 x Interest Rate x (230/360)
Now, let's calculate the future value of the promissory note:
Future Value of the note = $1,500 + $1,500 x 0.05 x (230/360)
Next, let's compare the future value of the note with the cash amount Moe received after 360 days:
$1,460 = Future Value of the note x (360/360)
Now, we can solve for the future value of the note:
Future Value of the note = $1,460 / (360/360)
Now we can set up an equation:
$1,460 = ($1,500 + $1,500 x 0.05 x (230/360)) / (360/360)
To find the interest rate, we need to solve this equation. Rearranging the equation, we get:
$1,460 x (360/360) = $1,500 + $1,500 x 0.05 x (230/360)
Simplifying:
$1,460 = $1,500 + $1,500 x 0.05 x (230/360)
Subtracting $1,500 from both sides:
$1,460 - $1,500 = $1,500 x 0.05 x (230/360)
-$40 = $1,500 x 0.05 x (230/360)
Dividing both sides by ($1,500 x 0.05 x (230/360)):
-40 / ($1,500 x 0.05 x (230/360)) = 1
Now we can simplify the right side:
-40 / ($.05 x (230/360)) = 1
To get the simple interest rate, we multiply both sides by ($0.05 x (230/360)):
-40 / 1 = $0.05 x (230/360) x Interest Rate
-40 = $0.05 x (230/360) x Interest Rate
Dividing both sides by ($0.05 x (230/360)), we find the interest rate:
Interest Rate = -40 / ($0.05 x (230/360))
Using a calculator to evaluate the expression on the right side, we find that the interest rate rounded to 0.01 is approximately 0.082, or 8.2%.
Therefore, Moe earned a simple interest rate of 8.2% on the promissory note.