Problem 33 - Prealgebra

Question

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

Step-by-Step Solution

Answer

The simplified form of the expression is \(\frac{11}{4}\).

Step 1: Convert Mixed Numbers To Improper Fractions

Let's convert the mixed number for \(y\) to an improper fraction. Given \(y = 2 \frac{1}{12}\), it can be written as \(y = \frac{2 \times 12 + 1}{12} = \frac{25}{12}\).

Step 2: Write The Given Expression

Now, let's write the expression \(x + y\) using fractions. We have \(x = \frac{8}{12}\) and \(y = \frac{25}{12}\). So, the expression becomes \(\frac{8}{12} + \frac{25}{12}\).

Step 3: Add The Fractions

Since the fractions have the same denominator, we can add them directly. \(\frac{8}{12} + \frac{25}{12} = \frac{8+25}{12} = \frac{33}{12}\).

Step 4: Simplify The Resulting Fraction

To simplify \(\frac{33}{12}\), we find the greatest common divisor (GCD) of 33 and 12, which is 3. Divide both the numerator and denominator by 3: \(\frac{33 \div 3}{12 \div 3} = \frac{11}{4}\). This is the simplest form of the fraction.

Key Concepts

Understanding Fractions

Fractions are one of the building blocks of mathematics and they represent a part of a whole. A fraction consists of two numbers - a numerator and a denominator. The numerator is the top number, showing how many parts we have. The denominator is the bottom number, indicating into how many parts the whole is divided. For example, in the fraction \(\frac{8}{12}\), 8 is the numerator and 12 is the denominator.

Fractions are used in various real-world situations like cooking, measuring, and even dividing pizza among friends! They can also be used to express ratios and proportions. Fractions with the same denominator are known as like fractions and can be added or subtracted directly. When dealing with fractions that have different denominators, a common denominator must be found to perform addition or subtraction. This ensures that the fractions are compared or combined consistently.

Converting Mixed Numbers to Improper Fractions

Improper fractions are fractions where the numerator is equal to or greater than the denominator. They often arise from mixed numbers, which are numbers containing both an integer and a fraction. A common task is converting mixed numbers to improper fractions for easier calculations.

To convert a mixed number like \(2 \frac{1}{12}\) to an improper fraction, follow these steps:

• Multiply the whole number by the fraction's denominator: \(2 \times 12 = 24\).
• Add the numerator of the fraction to this product: \(24 + 1 = 25\).
• Write the result over the original denominator: \(\frac{25}{12}\).

Now, the mixed number is expressed as an improper fraction, which makes it easier to perform additional operations, like addition or subtraction, especially when combined with other fractions.

Techniques for Simplifying Fractions

Simplifying fractions means reducing them to their simplest form, where the numerator and denominator have no common divisors other than 1. This process not only makes fractions easier to understand but also more helpful in solving math problems.

Here's a simple way to simplify fractions:

• Identify the greatest common divisor (GCD) of the numerator and denominator.
• Divide both the numerator and the denominator by this GCD.
• The resulting fraction is the simplest form.

For instance, in the fraction \(\frac{33}{12}\), the GCD of 33 and 12 is 3. Divide both numbers by 3 to get \(\frac{11}{4}\). Thus, \(\frac{11}{4}\) is the simplest form of \(\frac{33}{12}\).

Remember, simplifying fractions makes it easier to compare them, perform operations like addition or subtraction, and solve equations consistently.