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# monthly payment

Margaret obtains a personal loan of \$1,200 at 834 % interest for 1 year. Given that the bank follows the rule of 78, construct the payment schedule showing five columns: payment number, payment fraction, monthly interest portion, monthly principal portion, and monthly payment.

### 1 Answer by Expert Tutors

Felice R. | Need Help in Math and Science - Engineer willing to helpNeed Help in Math and Science - Engineer...
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Michelle

I think you have the interest rate incorrect, or Margaret is borrowing from a loan shark as well as using a loan rate that is sneaky as you pay most of your interest off at the beginning of the loan so paying off early isn't beneficial to the consumer.

If you don't terminate the loan early, simple interest loans and Rule of 78 loans will be equivalent. You will pay the same amount and get the interest rate quoted. However, if you pay off the loan early, you will end up paying more interest with a Rule of 78 loan than with the corresponding "simple interest" loan. For that reason, you should not take loans computed on the "Rule of 78".

In a loan using the rule of 78 the amount of interest paid each month is determined using a fixed linear scale, and the annualized interest rate actually changes each month.

The denominator of a Rule of 78 loan is the sum of the digits, the sum of the number of monthly payments in the loan. For a 12 month loan, the sum of numbers from 1 to 12 is 78 (1 + 2 + 3 + . . . +12 = 78).

The sum of the numbers from 1 to n is given by the equation n * (n+1) / 2.

For a 12 month loan, 12/78s of the finance charge is assessed as the first month’s portion of the finance charge, 11/78s of the finance charge is assessed as the second month’s portion of the finance charge and so on until the 12th month at which time 1/78s of the finance charge is assessed as that month’s portion of the finance charge.

I'm going to assume that your rate is 8.34% per year.

In a simple rate loan, each month you pay 1/12 of the yearly interest rate against the money you still owe.

so using a bank loan calculator

Month / Year   Payment   Principal Paid   Interest Paid   Total   Acrued Interest   Balance
Oct. 2013   \$104.57   \$96.23   \$8.34    \$8.34   \$1,103.77
Nov. 2013  \$104.57   \$96.90   \$7.67   \$16.01   \$1,006.86
Dec. 2013  \$104.57   \$97.58   \$7.00   \$23.01    \$909.28
Jan. 2014   \$104.57   \$98.26   \$6.32   \$29.33    \$811.03
Feb. 2014   \$104.57   \$98.94   \$5.64   \$34.97    \$712.09
Mar. 2014   \$104.57   \$99.63   \$4.95   \$39.91    \$612.46
April 2014   \$104.57  \$100.32  \$4.26   \$44.17    \$512.15
May 2014    \$104.57  \$101.02  \$3.56   \$47.73    \$411.13
June 2014   \$104.57  \$101.72  \$2.86   \$50.59    \$309.41
July 2014    \$104.57  \$102.42  \$2.15   \$52.74    \$206.99
Aug. 2014   \$104.57  \$103.14  \$1.44   \$54.18    \$103.85
Sept. 2014  \$104.57  \$103.85  \$0.72   \$54.90       \$0.00

They estimate that your loan payments are \$104.57 which would be a total of \$54.90 interest over the whole loan period

Now for a rule of 78 loan you would have the same amount of interest accrued over the year but you would pay more of the interest in the beginning

Month   Start Balance   Interest   Monthly Rate   Principal   End Balance   Cum. Int. Paid   Interest left to Pay
1 1200      8.45  8.4462 %   96.13   1103.87     8.45   46.45
2 1103.87  7.74  8.4165 %   96.84   1007.03   16.19   38.71
3 1007.03  7.04  8.3872 %   97.54     909.49   23.23   31.67
4 909.49    6.33  8.3580 %   98.25     811.24   29.56   25.34
5 811.24    5.63  8.3291 %   98.95     712.29   35.19   19.71
6 712.29    4.93  8.3004 %   99.65     612.64   40.12   14.78
7 612.64    4.22  8.2719 %   100.36   512.28   44.34   10.56
8 512.28    3.52  8.2437 %   101.06   411.22   47.86    7.04
9 411.22    2.82  8.2157 %   101.76   309.46   50.68    4.22
10 309.46  2.11  8.1880 %   102.47   206.99   52.79    2.11
11 206.99  1.41  8.1609 %   103.17   103.82   54.20    0.70
12 103.82  0.70  8.1354 %   103.82       0.00   54.90       0

You can see from these two tables that you are paying more interest in a rule 78 loan up until the 5th month.

I know I didn't totally answer your question but I was trying to show you how to calculate the interest and payment info. From there using your interest you can calculate the fractions.

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