Question

A 45-foot wire is cut into three pieces. The second piece is 3 times as long as the first piece. The third piece is 12 feet longer than the first piece.

Which equation could be used to find the length of the first piece? Select all that apply.

Responses

p+3p+p+12=45
p plus 3 p plus p plus 12 is equal to 45

5p+12=45
5 p plus 12 is equal to 45

p+3+p+p+12=45
p plus 3 plus p plus p plus 12 is equal to 45

3p+15=45

Answer 1

To solve this problem, let's first define the length of the first piece as \( p \).

According to the information given:
- The second piece is three times as long as the first piece, which can be represented as \( 3p \).
- The third piece is 12 feet longer than the first piece, which can be represented as \( p + 12 \).

Now, we can set up an equation based on the total length of the wire, which is 45 feet:

\[
p + 3p + (p + 12) = 45
\]

Simplifying this equation:
\[
p + 3p + p + 12 = 45
\]
\[
5p + 12 = 45
\]

Now we can see which equations from the choices provided are valid:

1. **\( p + 3p + p + 12 = 45 \)**
- This simplifies to \( 5p + 12 = 45 \) (valid).

2. **\( 5p + 12 = 45 \)**
- This directly represents the simplified form of the first equation (valid).

3. **\( p + 3 + p + p + 12 = 45 \)**
- This does not represent the correct relationship as it incorrectly adds a constant 3 instead of 3 times \( p \) (not valid).

4. **\( 3p + 15 = 45 \)**
- This does not represent the correct calculations shown above, as it does not correctly account for the pieces (not valid).

### Valid Equations
Based on the analysis:
- \( p + 3p + p + 12 = 45 \)
- \( 5p + 12 = 45 \)

Thus, the valid equations are:
- \( p + 3p + p + 12 = 45 \)
- \( 5p + 12 = 45 \)