Problem 100 - Intermediate Algebra

Question

Write a numerical statement to represent the problem. Then simplify the numerical expression to answer the question. Max bought a piece of trim molding that measured \(11 \frac{3}{8}\) feet in length. Because of defects in the wood, he had to trim \(1 \frac{5}{8}\) feet off one end, and he also had to remove \(\frac{3}{4}\) of a foot off the other end. How long was the piece of molding after he trimmed the ends?

Step-by-Step Solution

Answer

The molding is 9 feet long after trimming.

Step 1: Represent the Problem Numerically

First, convert the mixed numbers to improper fractions. Max bought a piece of trim molding that measured \(11 \frac{3}{8}\) feet in length. This can be written as \(\frac{91}{8}\). He trimmed off \(1 \frac{5}{8}\) feet, which is \(\frac{13}{8}\), and \(\frac{3}{4}\) of a foot. To simplify calculations, convert \(\frac{3}{4}\) to a fraction with a common denominator of 8, \(\frac{6}{8}\).

Step 2: Setup the Numerical Expression

Max trimmed \(\frac{13}{8}\) feet and \(\frac{6}{8}\) feet off the original length. The expression representing the process would be: \(\frac{91}{8} - \frac{13}{8} - \frac{6}{8}\).

Step 3: Simplify the Expression

Combine the fractions: \(\frac{91}{8} - \frac{13}{8} = \frac{78}{8}\). Then subtract the result by \(\frac{6}{8}\): \(\frac{78}{8} - \frac{6}{8} = \frac{72}{8}\).

Step 4: Convert Improper Fraction to Mixed Number

Simplify \(\frac{72}{8}\) to a whole number since \(72\div8=9\).

Step 5: Draw the Conclusion

After trimming both ends, the piece of molding is 9 feet long.

Key Concepts

Fraction Operations

Fractions are a fundamental concept in algebra and mathematics as a whole. Understanding operations with fractions is crucial for solving many types of problems. A fraction represents a part of a whole and consists of a numerator (top number) and a denominator (bottom number).

Common operations with fractions include:

• Addition and Subtraction: Require a common denominator, which is a shared multiple of the denominators of the fractions involved. This allows fractions to be combined or subtracted directly.
• Multiplication: Involves multiplying the numerators together and the denominators together. No common denominator is needed.
• Division: Similar to multiplication, but involves flipping the second fraction (taking the reciprocal) and then multiplying.

In the exercise, we focus on subtraction. The first step involved converting all parts into fractions with the same denominator, which was 8 for ease of calculation. This allowed for straightforward subtraction by simply adjusting the numerators while maintaining the denominator.

Mixed Numbers

Mixed numbers are numbers composed of a whole number and a fraction. They are commonly used to express quantities that are not whole numbers without using decimal points.

Operations with mixed numbers often require converting them to improper fractions first.

• Converting Mixed Numbers: Multiply the whole number by the denominator of the fraction and add the numerator. Use this sum as the new numerator over the original denominator. For example, to convert \(11 \frac{3}{8}\) to an improper fraction, multiply 11 by 8, then add 3 to get \(\frac{91}{8}\).

By performing such conversions, we simplify the process of carrying out fraction operations, especially when subtracting multiple values as seen in the exercise.

Numerical Expression Simplification

Simplifying numerical expressions involves reducing them to their simplest form to make calculations easier and more intuitive. This often requires combining like terms and following the order of operations (PEMDAS: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

In our scenario, simplification occurred by carefully conducting subtraction operations after ensuring the fractions had a common denominator. Here’s the process:

• Represent the whole expression with improper fractions to align all terms.
• Perform the subtraction step-by-step: First, subtract \(\frac{13}{8}\) from \(\frac{91}{8}\), and then subtract \(\frac{6}{8}\).
• The result was \(\frac{72}{8}\), which simplifies to 9 when the fraction is converted back to a whole number as \(72 \div 8 = 9\).

This final simplification step is crucial in ensuring clarity in the solution and confirming the length of the molding after trimming the defective parts.