Ten foot board cut into three pieces -- need to find the three lengths

Question

A board ten feet long is cut into three pieces. One piece is one foot longer than the shortest piece and two feet shorter than the longest piece. How long is each piece?

Answer 1

Pick variables to represent the unknowns:

L = length of longest piece

m = length of middle piece

s = length of shortest piece

 

L + m + s = 10   {equation 1}

 

The middle length is 2 ft shorter than the longest length so

m = L - 2

L = m + 2

 

The middle length is one foot longer than the shortest piece so:

m = s + 1

s = m - 1

 

In equation 1 replace L with m+2 and replace s with m-1

 

(m+2) + m + (m-1) = 10

3m + 1 = 10

3m = 9

m = 3

 

The middle piece is 3 ft long

The longest piece is m+2 = 5 ft long

the shortest piece is m-1 = 2 ft long

 

 

Answer 2

If x is the length of the shortest piece, then the middle piece is x+1 long and the longest is (x+1)+2 long. Since all 3 were cut from a 10' board then x+(x+1)+((x+1)+2) = 10 -> 3x+4=10 -> 3x=6 -> x=2. So the 3 boards are 2,3 and 5 feet long.