Problem 16 - Prealgebra

Question

CARPENTRY How many boards, each 2 feet 8 inches long, can be cut from a board 16 feet long if there is no waste?

Step-by-Step Solution

Answer

You can cut 6 boards, each 2 feet 8 inches long, from a 16-foot board with no waste.

Step 1: Convert Measurements

First, we need to convert all measurements to the same unit. Convert 2 feet 8 inches to feet. Since there are 12 inches in a foot, 8 inches is equal to \(\frac{8}{12} = \frac{2}{3}\) feet. Thus, 2 feet 8 inches is equivalent to \(2 + \frac{2}{3} = \frac{8}{3}\) feet.

Step 2: Determine How Many Sections Fit

Next, divide the total length of the board, 16 feet, by the length of each piece, \(\frac{8}{3}\) feet, using the division of fractions. To divide by a fraction, multiply by the reciprocal:\[16 \div \frac{8}{3} = 16 \times \frac{3}{8} = \frac{48}{8} = 6\].

Step 3: Intepret the Result

The quotient \( 6 \) represents the number of full 2 feet 8 inch sections that can be cut from a 16-foot board without any waste.

Key Concepts

Measurement Conversion

When working with carpentry projects, or any kind of project that involves physical measurements, it is often necessary to convert between units. This process is known as measurement conversion.

In the original exercise, we needed to convert a length given in feet and inches entirely into feet. This consolidation into a single unit helps to simplify calculations and comparisons.

Here's how it was done:

• We started with 2 feet and 8 inches. Knowing that there are 12 inches in one foot, the inches were converted to feet by forming a fraction:
  • First, convert 8 inches into feet by dividing by 12. This gives \( \frac{8}{12} = \frac{2}{3} \) feet.
  • Then, add these converted inches to the 2 feet original measurement: \( 2 + \frac{2}{3} = \frac{8}{3} \) feet.

This type of conversion is crucial in ensuring that all measurements are compatible before starting any calculations.

Division of Fractions

Dividing fractions is a key skill in performing operations that involve fractional components. This concept was central to calculating how many boards could be cut from a main board in our problem.

The original problem required us to divide a fraction. To divide by a fraction, we use the multiplication by the reciprocal technique. Let's break it down:

• We needed to divide 16 feet by \( \frac{8}{3} \) feet.
• Instead of dividing directly, multiply 16 by the reciprocal of \( \frac{8}{3} \). The reciprocal of \( \frac{8}{3} \) is \( \frac{3}{8} \).
  • This changes the division into a multiplication: \( 16 \times \frac{3}{8} \).
  • Simplify the operation: \[16 \times \frac{3}{8} = \frac{48}{8} = 6\].

This calculation shows how many pieces of length \( \frac{8}{3} \) can fit into 16 feet, giving us straightforward and accurate results in practical applications.

Carpentry Math

Carpentry requires precise measurements and accurate calculations to ensure all pieces fit together correctly and efficiently. Such applications often involve converting measurements, as well as performing calculations with fractions and whole numbers.

In practical terms, this means:

• Understanding and using units like feet and inches effectively.
• Being adept at converting mixed units into a consistent single unit for easy calculations.
• Utilizing division of fractions to determine how materials should be cut or used.

In the carpentry problem provided, it was essential to determine how many smaller boards of a specific length could be cut from a longer board.

By combining measurement conversion and division of fractions, carpenters can maximize material usage, reduce waste, and ensure the accuracy of their construction projects. This makes carpentry math both a practical and necessary skill in the field.