Problem 36 - Prealgebra

Question

Evaluate expression if \(x=\frac{8}{12}, y=2 \frac{1}{12},\) and \(z=\frac{11}{12} .\) Write in simplest form. \(y-x\)

Step-by-Step Solution

Answer

The simplified result of \(y-x\) is \(\frac{17}{12}\).

Step 1: Convert Mixed Number to Improper Fraction

The given value for \(y\) is a mixed number: \(y = 2\frac{1}{12}\). To convert it to an improper fraction, multiply the whole number part \(2\) by \(12\) and add the numerator of the fraction part \(1\): \(2 \times 12 + 1 = 24 + 1 = 25\). Therefore, \(y = \frac{25}{12}\).

Step 2: Ensure Common Denominator

Ensure that both fractions \(x = \frac{8}{12}\) and \(y = \frac{25}{12}\) have the same denominator, which they already do (\(12\)). This allows us to directly subtract the numerators.

Step 3: Subtract the Fractions

Subtract \(x\) from \(y\) by subtracting their numerators: \(\frac{25}{12} - \frac{8}{12} = \frac{25-8}{12} = \frac{17}{12}\).

Step 4: Simplify the Result

The fraction \(\frac{17}{12}\) is already in its simplest form since \(17\) and \(12\) share no common factors other than 1. Thus, the simplified result is \(\frac{17}{12}\).

Key Concepts

Mixed Numbers

Mixed numbers are a combination of a whole number and a fraction. They are often used to represent numbers greater than 1 in a more intuitive way. For example, the mixed number \(2 \frac{1}{12}\) consists of the whole number 2 and the fraction \(\frac{1}{12}\). Mixed numbers are useful because they can be easy to visualize. However, when performing certain operations like addition or subtraction with other fractions, it's helpful to convert mixed numbers into improper fractions first. This conversion helps in standardizing the format, making calculations simpler.

Improper Fractions

Improper fractions are fractions where the numerator is larger than or equal to the denominator, such as \(\frac{25}{12}\). Even if they may not look intuitive at first, they are very efficient for arithmetic operations. To convert a mixed number to an improper fraction:

• Multiply the whole number part by the denominator of the fraction part.
• Add the numerator of the fraction part to the result.
• The sum becomes the new numerator while the denominator remains the same.

Using mixed numbers in arithmetic processes can be limiting, so transitioning them to improper fractions facilitates easier calculations.

Simplifying Fractions

Simplifying fractions means reducing them to their simplest form, where the only common factor between the numerator and the denominator is 1. A fraction like \(\frac{17}{12}\) is in its simplest form because both 17 and 12 don't have other divisors in common except 1. Here’s a quick guide to simplify any fraction:

• Find the greatest common divisor (GCD) of the numerator and denominator.
• Divide both the numerator and denominator by their GCD.

In the case of \(\frac{17}{12}\), the GCD is 1, indicating it's already simplified. Simplifying ensures fractions are in their most digestible form, which is especially helpful in comparing, adding, subtracting, or thoroughly understanding them.