Question

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Part 1
Translate the problem to an algebraic equation and solve.
A 9797​-inch length of ribbon is to be cut into three pieces. The longest piece is to be 2828 inches longer than the shortest​ piece, and the third piece is to be half the length of the longest piece. Find the length of each piece of ribbon.

Answer 1

Let's define the lengths of the three pieces of ribbon.

1. Let \( x \) be the length of the shortest piece.
2. The longest piece is then \( x + 2828 \) inches.
3. The third piece is half the length of the longest piece, which is \( \frac{1}{2}(x + 2828) \).

According to the problem, the total length of all three pieces must equal 9797 inches:

\[
x + (x + 2828) + \frac{1}{2}(x + 2828) = 9797
\]

Now, let's simplify the left-hand side:

- Combine the terms involving \( x \):
\[
x + x + 2828 + \frac{1}{2}(x + 2828) = 2x + 2828 + \frac{1}{2}x + 1414
\]

Combine like terms:
\[
2x + \frac{1}{2}x + 2828 + 1414 = 2x + 0.5x + 4242
\]

Combine \( 2x + 0.5x \):
\[
2.5x + 4242
\]

Now, set up the equation:
\[
2.5x + 4242 = 9797
\]

Subtract 4242 from both sides:
\[
2.5x = 9797 - 4242
\]
\[
2.5x = 5555
\]

Now, divide both sides by 2.5:
\[
x = \frac{5555}{2.5}
\]
\[
x = 2222
\]

Now that we have the length of the shortest piece, we can find the lengths of the other pieces:
1. The shortest piece \( x = 2222 \) inches.
2. The longest piece \( x + 2828 = 2222 + 2828 = 5050 \) inches.
3. The third piece \( \frac{1}{2}(x + 2828) = \frac{1}{2}(5050) = 2525 \) inches.

To summarize:
- The shortest piece is **2222 inches**.
- The longest piece is **5050 inches**.
- The third piece is **2525 inches**.

Finally, let's verify the total:
\[
2222 + 5050 + 2525 = 9797 \text{ inches}
\]

The calculations are correct. The lengths of the three pieces of ribbon are:
- Shortest piece: **2222 inches**
- Longest piece: **5050 inches**
- Third piece: **2525 inches**