Problem 96 - Algebra for College Students

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 length after trimming is 9 feet.

Step 1: Convert Mixed Numbers to Improper Fractions

First, we convert the mixed numbers to improper fractions. For \(11 \frac{3}{8}\), multiply 11 by 8 and add 3 to get \(\frac{91}{8}\). For \(1 \frac{5}{8}\), multiply 1 by 8 and add 5 to get \(\frac{13}{8}\).

Step 2: Subtract the First Trim Length

Subtract the length of the first trim, \(\frac{13}{8}\), from the original piece. The equation is \(\frac{91}{8} - \frac{13}{8}\), resulting in \(\frac{78}{8}\).

Step 3: Subtract the Second Trim Length

Now, subtract the length of the second trim, \(\frac{3}{4}\), from the current length. Convert \(\frac{3}{4}\) to a fraction with a common denominator of 8, which is \(\frac{6}{8}\). Use \(\frac{78}{8} - \frac{6}{8}\) to result in \(\frac{72}{8}\).

Step 4: Simplify the Final Fraction

Simplify \(\frac{72}{8}\) by dividing both the numerator and denominator by 8, resulting in \(9\).

Key Concepts

Mixed Numbers

Mixed numbers are a combination of a whole number and a fraction. They are quite common in everyday measurements and arithmetic, such as when you might have a length of wood that is measured in feet and inches. For example, in our exercise, the original piece of the molding was 11 \(\frac{3}{8}\) feet long. To work with mixed numbers in calculations, it's often easier to convert them to improper fractions. This allows you to handle the fraction part of the number more easily in arithmetic operations. For instance, converting \(11 \frac{3}{8}\) requires these steps:

• Multiply the whole number 11 by the denominator 8.
• Add the numerator 3 to get 91.

The improper fraction you get is \(\frac{91}{8}\). This method helps in aligning mixed numbers into a single fraction that is easier to calculate.

Improper Fractions

Improper fractions have numerators larger than or equal to their denominators. They represent quantities greater than one and are useful in various calculations. In our example, we converted the mixed numbers to improper fractions before performing arithmetic operations. To change \(11 \frac{3}{8}\) into an improper fraction involves recalculating the entire value into a single fraction form, \(\frac{91}{8}\). Similarly, converting \(1 \frac{5}{8}\) to an improper fraction results in \(\frac{13}{8}\). Working with improper fractions allows us to manage more complex operations such as addition, subtraction, multiplication, and division on fractions without additional conversion until the final step. This simplifies problem-solving, especially in subtraction or addition of fractions which require a common denominator.

Arithmetical Operations

Arithmetical operations on fractions include addition, subtraction, multiplication, and division. In this exercise, we focused on subtraction as we trimmed two sections off the piece of molding. The key was converting all parts into fractions with a common denominator of 8.First, we subtracted the length \(\frac{13}{8}\) from the improper fraction \(\frac{91}{8}\) obtained from the original length, giving us \(\frac{78}{8}\). The next step involved another subtraction: \(\frac{6}{8}\) (since \(\frac{3}{4} = \frac{6}{8}\)), which resulted in a final fraction of \(\frac{72}{8}\). These operations emphasize the importance of a common denominator, simplifying arithmetic across fractions.

Simplifying Fractions

Simplifying fractions is the process of reducing the fraction to its simplest form. This is done by dividing the numerator and the denominator by their greatest common divisor (GCD). A simpler fraction is often more intuitive to understand and easier to work with.In our exercise, after performing the subtraction operations, we obtained \(\frac{72}{8}\). We simplified this by dividing both 72 and 8 by their GCD, which is 8:

• Divide the numerator 72 by 8 to get 9.
• Divide the denominator 8 by 8 to result in 1.

The simplified form is thus the whole number 9. Simplifying fractions is a crucial final step for providing a clear, easily interpretable outcome of fraction operations.