Question

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**