A 10​-foot piece of string is cut into two pieces so that the longer piece is 1 foot longer than twice the shorter piece. If the shorter piece is x feet​ long, find the lengths of both pieces.

Answer 1

Here’s what we know:

Short string = x

Long string = 1 foot longer than twice the short piece.

Short string plus long string = 10 feet

How can we mathematically state the length of the long piece?

“1 foot longer than twice the short piece”?

1 + 2x

Now, how can we mathematically state:

“Short plus long piece = 10”?

x + (1 + 2x) = 10

Solve for x:

3x + 1 = 10

3x = 9

x = 3

So the short board is 3 feet and the long board is 7 feet.

Answer 2

call the pieces X and Y, X being the shortest:

From the text:

X + Y = 10 (equation 1)

Y = 2X + 1 (equation 2)

Substitute #2 for Y in #1:

X + (2X + 1) = 10

3X = 9

X = 3

from the 1st equation, Y = 10 - 3 = 7

CHECK: using the 2nd equation: 7 = 2(3) + 1 ....OK!!

Answer 3

Translate the words into math:

1. 10 foot piece of string
2. X is the short piece
3. twice the short piece (2X) plus one foot (+1) is the long piece = (2X+1)

which in mathspeak is:

X + 2X+1 = 10

Now solve:

X + 2X+1 = 10

X + 2X = 10 - 1

3X = 9

X=3 and 2X + 1 = 6 +1 = 7

So, the short piece is 3 feet long and the long piece is 7 feet long.