A boy purchased​ (bought) a​ party-length sandwich 50 in. long. He wants to cut it into three pieces so that the middle piece is 6 in. longer than the shortest piece and the shortest piece is 8 in. shorter than the longest piece. How long should the three pieces​ be?

Answer 1

Let's denote the lengths of the three pieces of the sandwich as follows:

• Let \( x \) be the length of the shortest piece.
• The middle piece is 6 inches longer than the shortest piece, so it can be represented as \( x + 6 \).
• The longest piece is 8 inches longer than the shortest piece, so it can be represented as \( x + 8 \).

Now we can set up an equation based on the total length of the sandwich, which is 50 inches:

\[ x + (x + 6) + (x + 8) = 50 \]

Now, simplify the left side of the equation:

\[ x + x + 6 + x + 8 = 50 \] \[ 3x + 14 = 50 \]

Next, subtract 14 from both sides of the equation:

\[ 3x = 50 - 14 \] \[ 3x = 36 \]

Now, divide both sides by 3 to find \( x \):

\[ x = 12 \]

Now that we have the length of the shortest piece, we can find the lengths of the other two pieces:

• The shortest piece: \( x = 12 \) in.
• The middle piece: \( x + 6 = 12 + 6 = 18 \) in.
• The longest piece: \( x + 8 = 12 + 8 = 20 \) in.

Thus, the lengths of the three pieces are:

• Shortest piece: 12 inches
• Middle piece: 18 inches
• Longest piece: 20 inches

To verify, we can check that the total length equals 50 inches and that the conditions are satisfied:

• \( 12 + 18 + 20 = 50 \)
• The middle piece (18) is 6 inches longer than the shortest piece (12).
• The longest piece (20) is 8 inches longer than the shortest piece (12).

All conditions are met. Therefore, the final lengths of the three pieces are:

• Shortest piece: 12 inches
• Middle piece: 18 inches
• Longest piece: 20 inches