Problem 34 - Prealgebra

Question

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

Step-by-Step Solution

Answer

The simplified expression is 3.

Step 1: Convert Mixed Numbers to Improper Fractions

First, convert the mixed number \(y = 2 \frac{1}{12}\) to an improper fraction. To do this, multiply the whole number 2 by the denominator 12, then add the numerator 1: \(2 \times 12 + 1 = 25\). So, \(y = \frac{25}{12}\).

Step 2: Add Fractions

Add the fractions \(z = \frac{11}{12}\) and \(y = \frac{25}{12}\). Since both fractions have the same denominator, add only the numerators: \(11 + 25 = 36\). So, the sum is \(\frac{36}{12}\).

Step 3: Simplify the Fraction

Simplify \(\frac{36}{12}\). The greatest common divisor of 36 and 12 is 12. Divide the numerator and denominator by 12: \(\frac{36}{12} = 3\). Therefore, the simplified form of the expression is 3.

Key Concepts

Understanding Improper Fractions

Fractions come in various types, the improper fraction being a common one. An improper fraction is simply where the numerator (top number) is greater than or equal to the denominator (bottom number). This often occurs when converting a mixed number to a fraction. For instance, the mixed number \(2 \frac{1}{12}\) converts to the improper fraction \(\frac{25}{12}\).

• The process involves multiplying the whole number by the denominator.
• Add this product to the numerator.
• The result becomes your new numerator, with the denominator remaining unchanged.

Knowing how to handle improper fractions is crucial for calculations because it transforms mixed numbers into a format that's easy to add or subtract.

The Simplification of Fractions

Simplifying fractions involves reducing them to their simplest form. This means the numerator and denominator have no common divisors other than 1. For example, the fraction \(\frac{36}{12}\) simplifies because both 36 and 12 share the greatest common divisor (GCD) of 12.

• First, identify the GCD of the numerator and the denominator. In this case, it's 12.
• Then, divide both the numerator and the denominator by this GCD.
• This results in a simplified fraction. Here, \(\frac{36}{12}\) simplifies to 3.

Simplifying fractions makes them easier to understand and work with, especially when further operations are needed.

Adding Fractions with Like Denominators

Adding fractions is a straightforward process when they share the same denominator. This allows you to simply add the numerators while keeping the denominator constant. In the exercise, the fractions \(\frac{11}{12}\) and \(\frac{25}{12}\) share the same denominator of 12.

• When adding, combine the numerators: \(11 + 25 = 36\).
• The denominator remains the same, resulting in \(\frac{36}{12}\).
• The final step is to simplify the resulting fraction, if possible.

Remember, fractions need the same denominator to be directly added. This maintains the integrity of the fractions' size in relation to the whole.