A carpenter is working with a wooden board that is exactly 80 inches long. He needs to cut… — Armed Services Vocational Aptitude Battery (ASVAB)

A carpenter is working with a wooden board that is exactly $80$ inches long. He needs to cut three separate pieces from this board with the following lengths: $14 \frac{7}{8}$ inches, $21 \frac{5}{16}$ inches, and $19 \frac{1}{2}$ inches. To extract these three pieces, he makes exactly $3$ cuts. Each cut removes exactly $\frac{1}{8}$ of an inch of wood as sawdust (kerf).

What is the exact length of the remaining portion of the board?
(Enter your answer as a simplified mixed number with a space between the whole number and the fraction, like 5 1/2)

Answer:The remaining portion of the board measures 【23 15/16】 inches.

Answer

23 15/16

The correct answer accurately accounts for both the sum of the three pieces and the material lost from the three saw cuts. By converting all fractions to a common denominator of 16, adding the pieces together, adding the total kerf waste, and then subtracting that combined sum from the initial 80 inches, we find the exact remaining length.

Step-by-Step Solution

Find a common denominator for all fractions to make addition straightforward.

The common denominator is 16. The pieces convert to

14 \frac{14}{16}

21 \frac{5}{16}

, and

19 \frac{8}{16}

. The kerf waste per cut converts to

\frac{2}{16}

Fractions must have the same denominator before they can be added or subtracted.

Calculate the total length of the three cut pieces.

14 \frac{14}{16} + 21 \frac{5}{16} + 19 \frac{8}{16} = 54 \frac{27}{16}

, which simplifies to

55 \frac{11}{16}

inches.

This represents the total usable material extracted from the board.

Calculate the total amount of wood lost to sawdust (kerf) and add it to the pieces' length.

3 cuts

\times \frac{2}{16}

inch/cut =

\frac{6}{16}

inch lost. Total wood consumed is

55 \frac{11}{16} + \frac{6}{16} = 55 \frac{17}{16}

, which simplifies to

56 \frac{1}{16}

inches.

Each cut removes material that is turned into sawdust, so this length must be accounted for as part of the board that is no longer available.

Subtract the total wood consumed from the original length of the board.

80 - 56 \frac{1}{16} = 79 \frac{16}{16} - 56 \frac{1}{16} = 23 \frac{15}{16}

inches.

Subtracting the total used (pieces plus waste) from the starting length gives the exact remaining length.

Key Concept

Adding and subtracting mixed numbers with unlike denominators while accounting for systemic deductions (like kerf waste) in real-world contexts.

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