Calculating Remaining Board Length A Carpenter's Cut

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In this article, we'll delve into a practical problem involving a carpenter and the lengths of wooden boards. This is a common scenario in woodworking and construction, where accurate measurements and calculations are crucial. We'll break down the problem step-by-step, ensuring a clear understanding of how to arrive at the correct solution. Specifically, we aim to determine the length of the remaining piece of board after a carpenter makes a cut.

Understanding the Problem

The problem states: A carpenter cuts a board 3383 \frac{3}{8} feet long off the end of a board that is 6126 \frac{1}{2} feet long. How long is the remaining piece of board? Our goal is to find the length of the board that remains after the cut is made. This involves subtracting the length of the cut piece from the original length of the board.

Initial Board Length

The initial length of the board is given as 6126 \frac{1}{2} feet. This is a mixed number, which means it has a whole number part (6) and a fractional part (12\frac{1}{2}). To work with this number mathematically, it's often easier to convert it into an improper fraction. An improper fraction has a numerator that is larger than its denominator.

To convert 6126 \frac{1}{2} to an improper fraction, we multiply the whole number (6) by the denominator (2) and then add the numerator (1). This gives us (6 * 2) + 1 = 13. We then place this result over the original denominator, giving us 132\frac{13}{2} feet. So, the initial length of the board is 132\frac{13}{2} feet.

Length of the Cut

The carpenter cuts off a piece that is 3383 \frac{3}{8} feet long. Like the initial length, this is also a mixed number. We need to convert this to an improper fraction as well.

To convert 3383 \frac{3}{8} to an improper fraction, we multiply the whole number (3) by the denominator (8) and then add the numerator (3). This gives us (3 * 8) + 3 = 27. We then place this result over the original denominator, giving us 278\frac{27}{8} feet. So, the length of the cut is 278\frac{27}{8} feet.

The Subtraction

Now that we have both lengths as improper fractions, we can subtract the length of the cut from the initial length to find the remaining length. The problem can be represented as:

132−278\frac{13}{2} - \frac{27}{8}

To subtract fractions, they must have a common denominator. In this case, the denominators are 2 and 8. The least common multiple (LCM) of 2 and 8 is 8. This means we need to convert 132\frac{13}{2} to an equivalent fraction with a denominator of 8.

To do this, we multiply both the numerator and the denominator of 132\frac{13}{2} by 4 (since 2 * 4 = 8). This gives us:

13∗42∗4=528\frac{13 * 4}{2 * 4} = \frac{52}{8}

Now we can rewrite the subtraction problem as:

528−278\frac{52}{8} - \frac{27}{8}

When subtracting fractions with a common denominator, we subtract the numerators and keep the denominator the same:

52−278=258\frac{52 - 27}{8} = \frac{25}{8}

Converting Back to a Mixed Number

The result, 258\frac{25}{8}, is an improper fraction. While this is a mathematically correct answer, it's often more practical to express the length as a mixed number, which is easier to visualize and understand in real-world contexts.

To convert 258\frac{25}{8} to a mixed number, we divide the numerator (25) by the denominator (8). 25 divided by 8 is 3 with a remainder of 1. The quotient (3) becomes the whole number part of the mixed number, and the remainder (1) becomes the numerator of the fractional part. The denominator remains the same (8).

Therefore, 258\frac{25}{8} is equal to 3183 \frac{1}{8}.

The Answer

The remaining piece of board is 3183 \frac{1}{8} feet long.

Step-by-Step Solution

Let's recap the steps we took to solve this problem:

  1. Convert mixed numbers to improper fractions:
    • 612=1326 \frac{1}{2} = \frac{13}{2}
    • 338=2783 \frac{3}{8} = \frac{27}{8}
  2. Find a common denominator:
    • The least common multiple of 2 and 8 is 8.
  3. Convert fractions to have the common denominator:
    • 132=528\frac{13}{2} = \frac{52}{8}
  4. Subtract the fractions:
    • 528−278=258\frac{52}{8} - \frac{27}{8} = \frac{25}{8}
  5. Convert the improper fraction back to a mixed number:
    • 258=318\frac{25}{8} = 3 \frac{1}{8}

Therefore, the remaining piece of board is 3183 \frac{1}{8} feet long.

Importance of Understanding Fractions in Practical Applications

This problem highlights the importance of understanding fractions in real-world applications. Carpenters, engineers, chefs, and many other professionals regularly use fractions in their work. Being able to confidently work with fractions ensures accuracy and efficiency in these fields. For example, if a carpenter miscalculates a cut, it could lead to wasted materials and time. Similarly, in cooking, accurate measurements of ingredients are crucial for achieving the desired outcome.

Further Applications

Beyond the specific example of carpentry, understanding fractions is essential in various other contexts, including:

  • Measurement and conversion: Converting between different units of measurement often involves fractions. For instance, converting inches to feet or ounces to pounds requires working with fractional relationships.
  • Scaling recipes: When adjusting the serving size of a recipe, you need to multiply or divide the quantities of ingredients, which often involves fractions.
  • Financial calculations: Calculating interest rates, discounts, and taxes often requires working with fractions and percentages.
  • Construction and engineering: Architects and engineers rely heavily on fractions and decimals for accurate measurements and calculations in building design and construction.

Common Mistakes and How to Avoid Them

When working with fractions, there are some common mistakes that students and even professionals sometimes make. Being aware of these mistakes can help you avoid them and ensure accuracy in your calculations.

Not Finding a Common Denominator

One of the most common mistakes is trying to add or subtract fractions without first finding a common denominator. As we saw in the solution above, fractions must have the same denominator before they can be added or subtracted. If you try to add or subtract fractions with different denominators, you will get an incorrect result. Always remember to find the least common multiple (LCM) of the denominators and convert the fractions accordingly.

Incorrectly Converting Mixed Numbers

Another common mistake is incorrectly converting mixed numbers to improper fractions or vice versa. It's crucial to follow the correct procedure: To convert a mixed number to an improper fraction, multiply the whole number by the denominator, add the numerator, and then place the result over the original denominator. To convert an improper fraction to a mixed number, divide the numerator by the denominator; the quotient is the whole number, the remainder is the numerator, and the denominator remains the same.

Simplifying Fractions Incorrectly

Simplifying fractions is an important step in many problems involving fractions. However, it's crucial to do it correctly. Remember that you can only simplify a fraction by dividing both the numerator and the denominator by the same non-zero number. If you divide only the numerator or only the denominator, you will change the value of the fraction.

Misunderstanding Fraction Operations

It's also important to have a clear understanding of the different operations involving fractions. For example, adding and subtracting fractions require a common denominator, while multiplying fractions involves multiplying the numerators and the denominators directly. Dividing fractions involves multiplying the first fraction by the reciprocal of the second fraction. Make sure you understand the rules for each operation and apply them correctly.

Practice Problems

To solidify your understanding of working with fractions, try solving these practice problems:

  1. A baker has 5145 \frac{1}{4} cups of flour. She uses 2232 \frac{2}{3} cups for a cake. How much flour does she have left?
  2. A runner runs 101210 \frac{1}{2} miles in a week. If she runs the same distance each day, how many miles does she run per day?
  3. A carpenter needs to cut a board into pieces that are 1341 \frac{3}{4} feet long. If the board is 14 feet long, how many pieces can he cut?

Working through practice problems like these will help you build your confidence and skills in working with fractions. Remember to break down each problem into smaller steps, and carefully apply the rules for fraction operations.

Conclusion

In conclusion, solving practical problems like the carpenter's board cutting scenario requires a solid understanding of fractions and their operations. By converting mixed numbers to improper fractions, finding common denominators, performing the necessary calculations, and converting back to mixed numbers, we can accurately determine the remaining length of the board. This example highlights the importance of fractions in real-world applications and underscores the need for a strong foundation in mathematics. Remember to practice regularly and pay attention to common mistakes to enhance your problem-solving skills. With consistent effort, you can master the art of working with fractions and apply them effectively in various contexts.