Small challenge
09-20-2017, 06:43 PM (This post was last modified: 09-20-2017 07:21 PM by Pekis.)
Post: #1
 Pekis Member Posts: 115 Joined: Aug 2014
Small challenge
- "Hi, old chap, can you lend me 6000 $?" - "Of course, yes, you can pay me back in two years, with a monthly payment, without any interests but linearly decreasing" - "Thanks a lot !" he said, putting the 6000$ in his pocket

But back home, he was wondering, "How much shall I have to pay him each month ? ... Sure it won't be 6000$/ 24 = 250$ each month and it will be more than that the first month ..."

How can you help him ? Can you propose a formula for each month ?

Have fun.
09-20-2017, 07:33 PM (This post was last modified: 09-20-2017 07:56 PM by Dieter.)
Post: #2
 Dieter Senior Member Posts: 2,397 Joined: Dec 2013
RE: Small challenge
(09-20-2017 06:43 PM)Pekis Wrote:  But back home, he was wondering, "How much shall I have to pay him each month ? ... Sure it won't be 6000 / 24 = 480 each month and it will be more the first month ..."

How can you help him ? Can you propose a formula for each month ?

First of all: 6000/24 is not 480 but 250. ;-)
And now let me see if I correctly understood the idea.

Every month a certain amount is payed. This amount decreases from month do month by the same delta d.

So in the last month the payment is p.
In the month before the payment is p + d.
In the month before that the payment is p + d + d.
etc.

This can be written as follows:
p + p+d + p+2d + p+3d ... + p+23d = 6000

In other words:
24*p + d*(1+2+3+...+23) = 6000

The sum from 1 to 23 is 23*24/2 = 276.
Or, generally spoken with n months:

n*p + d*(n*(n–1))/2 = 6000

Now you can write
d = (6000 – n*p) / (n*(n–1)) * 2

Or the other way round:
p = (6000 – d*n*(n–1)/2) / n

Since the monthly payments are supposed to decrease, d must be > 0. Looking at the formula for d this means that p must be less than 6000/n = 250.

Otherwise you are free to choose what basic amount (p) you want to pay and by how much (d) the payments differ from month to month. You may start with a high payment which is substantially decreased each month, or you may start with a lower initial payment which descreass only gradually. In any case your last payment will be p and you start with p+(n–1)*d.

Example:
As shown above, p must be less than 250. Let's say p=200. This means that according to the formula above d is 4,3478, or roughly 4,35.

The payment in month i (i=1...n) is p + (n–i)*d

Here is a table (based on the exact d-value of 4,347826...)

Code:
month  initial     monthly   remaining  no.   debt        payment   debt --------------------------------------  1     6.000,00     300,00    5.700,00  2     5.700,00     295,65    5.404,35  3     5.404,35     291,30    5.113,04  4     5.113,04     286,96    4.826,09  5     4.826,09     282,61    4.543,48  6     4.543,48     278,26    4.265,22  7     4.265,22     273,91    3.991,30  8     3.991,30     269,57    3.721,74  9     3.721,74     265,22    3.456,52 10     3.456,52     260,87    3.195,65 11     3.195,65     256,52    2.939,13 12     2.939,13     252,17    2.686,96 13     2.686,96     247,83    2.439,13 14     2.439,13     243,48    2.195,65 15     2.195,65     239,13    1.956,52 16     1.956,52     234,78    1.721,74 17     1.721,74     230,43    1.491,30 18     1.491,30     226,09    1.265,22 19     1.265,22     221,74    1.043,48 20     1.043,48     217,39      826,09 21       826,09     213,04      613,04 22       613,04     208,70      404,35 23       404,35     204,35      200,00 24       200,00     200,00        0,00

So you start with a 300,- payment and end up with 200,-.

So the procedure is as follows:
Define your last payment p and calculate d from this. Here d = (6000 – n*p) / (n*(n–1)) * 2.
- or -
Define the payment delta d and calculate p from this. Here p = (6000 – d*n*(n–1)/2) / n.

Then your payment in month i is p + (n–i)*d.

Special case:
In the above formulas you may increase n by 1 and set p=0. Then the last payment is close to zero (actually the n+1st would be zero).
For the example this leads to d = 6000/(24*25/2) = 20 and the payments are 24*20=480, 23*20=460, ... , 2*20=40, 1*20=20.

Code:
month  initial     monthly   remaining  no.   debt        payment   debt --------------------------------------  1     6.000,00    480,00    5.520,00  2     5.520,00    460,00    5.060,00  3     5.060,00    440,00    4.620,00  4     4.620,00    420,00    4.200,00  5     4.200,00    400,00    3.800,00  6     3.800,00    380,00    3.420,00  7     3.420,00    360,00    3.060,00  8     3.060,00    340,00    2.720,00  9     2.720,00    320,00    2.400,00 10     2.400,00    300,00    2.100,00 11     2.100,00    280,00    1.820,00 12     1.820,00    260,00    1.560,00 13     1.560,00    240,00    1.320,00 14     1.320,00    220,00    1.100,00 15     1.100,00    200,00      900,00 16       900,00    180,00      720,00 17       720,00    160,00      560,00 18       560,00    140,00      420,00 19       420,00    120,00      300,00 20       300,00    100,00      200,00 21       200,00     80,00      120,00 22       120,00     60,00       60,00 23        60,00     40,00       20,00 24        20,00     20,00        0,00

Is this what you were looking for?

Dieter
09-20-2017, 07:49 PM
Post: #3
 Didier Lachieze Senior Member Posts: 1,238 Joined: Dec 2013
RE: Small challenge
(09-20-2017 06:43 PM)Pekis Wrote:  - "Hi, old chap, can you lend me 6000 $?" - "Of course, yes, you can pay me back in two years, with a monthly payment, without any interests but linearly decreasing" So the total amount over 24 month is 6000 and it's decreasing by a constant amount a each month. So 6000=a*(24*(24+1)/2) or 6000=a*300 or a=20. The first month payment is$480 (20*24), decreasing by $20 every month. Code:  1 480 480 2 460 940 3 440 1380 4 420 1800 5 400 2200 6 380 2580 7 360 2940 8 340 3280 9 320 3600 10 300 3900 11 280 4180 12 260 4440 13 240 4680 14 220 4900 15 200 5100 16 180 5280 17 160 5440 18 140 5580 19 120 5700 20 100 5800 21 80 5880 22 60 5940 23 40 5980 24 20 6000 09-20-2017, 07:54 PM Post: #4  Pekis Member Posts: 115 Joined: Aug 2014 RE: Small challenge Thanks for your enlighting answer ! I had imagined only one solution: M=amount to pay=6000 n=number of payments=24 i=number of the payment: i=1..n Payement number i= 2/n*(1-i/(n+1)) * M So it began with 480 and ended up with 20 Thanks ! 09-20-2017, 08:00 PM (This post was last modified: 09-20-2017 08:17 PM by Dieter.) Post: #5  Dieter Senior Member Posts: 2,397 Joined: Dec 2013 RE: Small challenge (09-20-2017 07:54 PM)Pekis Wrote: Thanks for your enlighting answer ! I had imagined only one solution: M=amount to pay=6000 n=number of payments=24 i=number of the payment: i=1..n Payement number i= 2/n*(1-i/(n+1)) * M So it began with 480 and ended up with 20 I finally added this as a "special case". Before I read Didier's and your reply. Really. ;-) Dieter 09-20-2017, 08:05 PM Post: #6  Joe Horn Senior Member Posts: 1,687 Joined: Dec 2013 RE: Small challenge My favorite solution is: first payment is$250.12, and each month's payment after that is one penny less.

<0|ɸ|0>
-Joe-
09-20-2017, 10:51 PM (This post was last modified: 09-20-2017 10:58 PM by AlexFekken.)
Post: #7
 AlexFekken Member Posts: 151 Joined: May 2016
RE: Small challenge
No need to sum the series: because of the linearity the average payment needs to be 250 and that will be the "virtual payment" at month 12.5. So if the delta is d then the payments of month 12 and 13 should be 250 + d/2 and 250 - d/2 etc... Just need to make sure that the last payment will be positive then, i.e. 250 - 11.5 *d > 0.
09-20-2017, 10:57 PM (This post was last modified: 09-20-2017 11:00 PM by AlexFekken.)
Post: #8
 AlexFekken Member Posts: 151 Joined: May 2016
RE: Small challenge
(09-20-2017 08:05 PM)Joe Horn Wrote:  My favorite solution is: first payment is $250.12, and each month's payment after that is one penny less. You'll be paying too much then: you'd have to start with$250.115. Had to fix more or less the same mistake in my solution :-)
09-20-2017, 11:52 PM
Post: #9
 SlideRule Senior Member Posts: 1,253 Joined: Dec 2013
RE: Small challenge
(09-20-2017 08:05 PM)Joe Horn Wrote:  ... first payment is $250.12, and each month's payment after that is one penny less... OR, skip the central payment ($ 250.00) with +12 above and -12 below, as in the following table:
$250.12$250.11
$250.10$250.09
$250.08$250.07
$250.06$250.05
$250.04$250.03
$250.02$250.01...∑ (1-12)=$3,000.78$249.99
$249.98$249.97
$249.96$249.95
$249.94$249.93
$249.92$249.91
$249.90$249.89
$249.88...∑ (13-24)=$2,999.22

therefore.∑ (01-24)=\$6,000.00

I do like the approach, it has a simple elegance.

BEST!
SlideRule
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