77 cm long is cut into three pieces such that the second piece is twice as long as the first and the third is 2 cm longer than the second.

Find the length of the shorter piece.

Assign variables to each piece:

1st piece = x

2nd piece = y

3rd piece = z

We are given that the total length of these three pieces is 77 cm: x + y + z = 77

Since we are given that the 2nd piece (y) is twice as long as the first piece (x), then we arrive at the following: y = 2x

Also, we are given that the 3rd piece (z) is 2 cm longer than the second piece (y). This leads us to the following: z = y + 2 ==> z = 2x + 2

Now, we substitute what we've found y and z to equal into the original equations. That is,

x + y + z = 77

x + (2x) + (2x + 2) = 77

x + 2x + 2x + 2 = 77

After we combine the like terms on the left hand side of the equation, we get the following:

5x + 2 = 77

Subtract 2 from both sides of the equation:

5x + 2 - 2 = 77 - 2

5x = 75

Divide both sides of the equation by 5 to solve for x:

5x/5 = 75/5

x = 15

Therefore,

1st piece = x = 15 cm

2nd piece = y = 2x = 2·15 = 30 cm

3rd piece = z = 2x + 2 = 2·15 + 2 = 30 + 2 = 32 cm

Thus, the shortest piece is the 1st piece with a length of 15 cm.