February 2014
Problem of the Month

Contracting Costs
by Sam Lloyd

A contractor planning the construction of a house found that he would have to pay:

$1,100 to the Paper Hanger and the Painter,

$1,700 to the Plumber and the Painter,

$1,100 to the Plumber and the Electrician,

$3,300 to the Electrician and the Carpenter,

$5,300 to the Carpenter and the Mason,

$3,200 to the Mason and the Painter.

What does each man charge for his services?

Solution to the Problem:

The answer is:
$900 to the Painter,
$200 to the Paper Hanger,
$800 to the Plumber,
$300 to the Electrician,
$3,000 to the Carpenter,
$2,300 to the Mason.

Let PH = amount charged by the Paper Hanger
Let PA = amount charged by the Painter
Let PL = amount charged by the Plumber
Let C = amount charged by the Carpenter
Let E = amount charged by the Electrician
Let M = amount charged by the Mason

Then we can set up the following six equations with 6 unknowns:
1) PH + PA = 1100
2) PL + PA = 1700
3) PL + E = 1100
4) E + C = 3300
5) C + M = 5300
6) M + PA = 3200

Now, solve these six equations (the numbers of the equations which are added or subtracted are in parentheses).
M - E = 2000 (5 - 4)
M + PA + PL + E = 4300 (3 + 6)
M + E = 2600 (3 + 6 - 2)
2M = 2600 (adding the results of 5 - 4 and 3 + 6 - 2)
So, M = 2300.
Now, substitute back to get the other five values.

Correctly solved by:

1. James Alarie	Flint, Michigan
2. Shaylynn Martin	Mountain View High School, Mountain View, Wyoming

Send any comments or questions to: David Pleacher