1. Let m
= the number of men (an integer)
2. Let w
= the number of women (an integer)
3. Let c
= the number of children (an integer)
4. Number
of bushels = 3m+2w+.5c = 100
5. Number
of people = m+w+c=100
6. 0<m<100
7. 0<w<100
8. 0<c<100
If we assume that there are 10 children, then we can solve for
two equations and two unknowns:
1. 3m+2w+5
= 100
2. m+w+10=100
We can solve this by subtracting two times the second equation
from the first equation:
1. 3m+2w+5
= 100
2. 2m+2w+20=200
(this is two times the second equation shown above)
3. m
- 15 = -100 (this is the result of subtracting two times the second equation
from the first equation)
4. m =
-85
5. w=100-10-
(-85) = 175
The problem with this result is that the values for m and w are
outside the allowable range, so clearly the assumption that there are 10
children is wrong. Let’s try increasing the number of children to 80 and see
how that impacts the result from m:
1. 3m+2w+40
= 100
2. 2m+2w+160=200
3. m
-125 = -100
4. m
= 25
5. w =
100-80-25=-5
Here, the result for m is within the range, but the value for w
is outside the range. Let’s try reducing the number of children to 70:
1. 3m+2w+35
= 100
2. 2m+2w+140=200
3. m
-105 = -100
4. m
= 5
5. w =
100-70-5= 25
Now we have a solution that makes sense (m=5, w=25, c=70).
However, there may be other answers that work, so let’s try increasing the
number of students to 72 and see what happens:
1. 3m+2w+36
= 100
2. 2m+2w+142=200
3. m
-106 = -100
4. m
= 6
5. w =
100-72-6= 22
This solution also makes sense (m=6, w=22, c=72). Clearly, there
can be several correct solutions to this question.