Math  /  Algebra

QuestionA 56 -inch board is to be cut into three pieces so that the second piece is 3 times as long as the first piece and the third piece is 4 times as long as the first piece. If xx represents the length of the first piece, find the lengths of all three pieces.
What is the length of the first piece? \square \square in2i n^{2}. in.

Studdy Solution

STEP 1

1. The total length of the board is 56 inches.
2. The length of the first piece is represented by x x inches.
3. The second piece is 3 times the length of the first piece.
4. The third piece is 4 times the length of the first piece.
5. The sum of the lengths of the three pieces equals the total length of the board.

STEP 2

1. Set up the equation representing the total length of the board in terms of x x .
2. Solve the equation for x x .
3. Determine the lengths of the second and third pieces using the value of x x .
4. Verify the solution by ensuring the sum of the three pieces equals the total length of the board.

STEP 3

Set up the equation representing the total length of the board in terms of x x .
Given: - Length of the first piece: x x - Length of the second piece: 3x 3x - Length of the third piece: 4x 4x
The equation is: x+3x+4x=56 x + 3x + 4x = 56

STEP 4

Combine like terms in the equation from STEP_1.
8x=56 8x = 56

STEP 5

Solve for x x by dividing both sides of the equation by 8.
x=568 x = \frac{56}{8}
x=7 x = 7

STEP 6

Determine the lengths of the second and third pieces using the value of x x .
Length of the first piece: x=7inches x = 7 \, \text{inches}
Length of the second piece: 3x=3×7=21inches 3x = 3 \times 7 = 21 \, \text{inches}
Length of the third piece: 4x=4×7=28inches 4x = 4 \times 7 = 28 \, \text{inches}

STEP 7

Verify the solution by ensuring the sum of the three pieces equals the total length of the board.
Sum of the lengths of the three pieces: 7+21+28=56 7 + 21 + 28 = 56
Since the sum equals the total length of the board, the solution is verified correct.
Solution: The length of the first piece is 7 7 inches. The length of the second piece is 21 21 inches. The length of the third piece is 28 28 inches.

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