Merchant's rule

[mmaulick]

I am having trouble understanding how to find the amount needed to settle a loan after 4 years using the merchant's rule. I have the answer ($900), but I don't quite know how to get there. I have tried several ways, but they don't compute. Here is the specific problem:

Draw an appropriate time diagram. An amount of $6000 was borrowed at 6%. Payments of $6,000 each were made after two years and after three years. Find the amount needed to settle the loan after for years using the merchant's rule.

Here is what I've tried:
$6000----------------------------------> $6000*1.06
l-------l-----------l--------------l
X
l____________________>$3000*1.045
l___________>$3000*1.03

X=$6000(1+.06)-$3000(1+.06*3/4)-$3000(1+.06*1/2)

I am really struggling trying to figure out what the time would be and whether I am using the appropriate formula, which I used S=P(1+rt). Additionally, I am struggling with the concept of the focal date. Please help!!!

 

[Ishuda]

I am having trouble understanding how to find the amount needed to settle a loan after 4 years using the merchant's rule. I have the answer ($900), but I don't quite know how to get there. I have tried several ways, but they don't compute. Here is the specific problem:

Draw an appropriate time diagram. An amount of $6000 was borrowed at 6%. Payments of $6,000 each were made after two years and after three years. Find the amount needed to settle the loan after for years using the merchant's rule.

Here is what I've tried:
$6000----------------------------------> $6000*1.06
l-------l-----------l--------------l
X
l____________________>$3000*1.045
l___________>$3000*1.03

X=$6000(1+.06)-$3000(1+.06*3/4)-$3000(1+.06*1/2)

I am really struggling trying to figure out what the time would be and whether I am using the appropriate formula, which I used S=P(1+rt). Additionally, I am struggling with the concept of the focal date. Please help!!!

I assume the $6000 (see the red above) is meant to be $3000.

Given the answer, this is a simple interest problem
Borrow 6000 at 6% simple interest for 4 years: 6000 * (1+4*0.06)=7440
Pay 3000 in two years (so 2 years remaining): -3000 * (1+2*.06)=-3360
Pay 3000 in three years (so 1 year remaining): -3000 * (1+1*.06)=-3180

Add to get the balance owed.

 

[Ishuda]

6000 * .06 * 2 = 720 (1st 2 years)

3000 * .06 * 1 = 180 (3rd year)

720 + 180 = 900

That's the only way you'll get your expected 900 solution.

Your problem is very badly worded.

Denis,
That's equivalent to the Merchant Rule but that rule states things a little differently. The Merchant Rule says something to the effect of
All payments made before the settlement date shall be treated as though they had earned interest at the prevailing rate of the note from the date of payment to the settlement date.
Thus the way I gave my solution. The problem the OP ran into is they did not take the proper length of time into account. Their solution properly should have been
X=$6000(1+.06*4)-$3000(1+.06*2)-$3000(1+.06*1)

EDIT: To try to explain the equivalence we have
Simple interest first two years on 6K = 720 leaving 3K owing
Simple interest year 3 on 3K = 180 leaving 0 owing
Simple interest remaining years to settlement date on 0K = 0
Total interest owing on settlement date $900

 

[jonah2.0]

Beer soaked ramblings follow.

Denis,
That's equivalent to the Merchant Rule but that rule states things a little differently. The Merchant Rule says something to the effect of
All payments made before the settlement date shall be treated as though they had earned interest at the prevailing rate of the note from the date of payment to the settlement date.
Thus the way I gave my solution. The problem the OP ran into is they did not take the proper length of time into account. Their solution properly should have been
X=$6000(1+.06*4)-$3000(1+.06*2)-$3000(1+.06*1)

EDIT: To try to explain the equivalence we have
Simple interest first two years on 6K = 720 leaving 3K owing
Simple interest year 3 on 3K = 180 leaving 0 owing
Simple interest remaining years to settlement date on 0K = 0
Total interest owing on settlement date $900

Alternatively, we could also say that the so called merchant rule is basically the simple interest version of the retrospective method for determining the outstanding balance of a loan in compound interest problems as applied by Sir Denis in other posts.

 

[mmaulick]

6000 * .06 * 2 = 720 (1st 2 years)

3000 * .06 * 1 = 180 (3rd year)

720 + 180 = 900

That's the only way you'll get your expected 900 solution.

Your problem is very badly worded.

So then, you would get $999.79 with the U.S. Rule for this problem if you recalculate the amounts after each payment?