a loan for $400 is to be paid off in 66 months witha payment of $11.62 The borrowee pays it off in 18 months use rule 78 to find the interest saved
To find the interest saved using the Rule of 78, we first need to calculate the total interest that would have been paid if the loan was paid off in the full 66 months.
The Rule of 78 is a simplified formula used to calculate the allocation of interest charges over the life of a loan, assuming that the interest is front-loaded. It states that the interest charges for each month are based on the sum of the digits representing the remaining months of the loan.
Here's how you can find the interest using the Rule of 78:
Step 1: Calculate the sum of the digits.
The loan term is 66 months. We can calculate the sum of the digits using the formula:
Sum = (n * (n + 1)) / 2
where n is the number of digits, starting from 1.
In this case, n = 6, since we have digits from 1 to 6. Plugging in this value:
Sum = (6 * (6 + 1)) / 2
Sum = (6 * 7) / 2
Sum = 42 / 2
Sum = 21
Step 2: Calculate the proportion of the loan used by each month.
To find the proportion of the loan used by each month, divide the remaining number of months by the sum of the digits. In this case, the loan was paid off in 18 months out of 66.
Proportion = Remaining Months / Sum of Digits
Proportion = 18 / 21
Step 3: Calculate the total interest that would have been paid.
To calculate the total interest, multiply the loan amount by the proportion from Step 2.
Total Interest = Loan Amount * Proportion
Total Interest = $400 * (18 / 21)
Step 4: Calculate the interest saved.
The interest saved is the difference between the total interest paid over 66 months and the total interest paid over 18 months.
Interest Saved = Total Interest (66 months) - Total Interest (18 months)
Now, let's plug in the values and calculate the answer:
Interest Saved = ($400 * (18 / 21)) - ($400 * (66 / 21))
Interest Saved = ($400 * 0.857) - ($400 * 3.143)
Interest Saved = $342.8 - $1257.2
Interest Saved = -$914.4
The negative value indicates that the borrower did not save any interest by paying off the loan in 18 months instead of the full 66 months. In fact, the borrower ended up paying more interest than anticipated.