Problem 35 - Prealgebra

Question

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

Step-by-Step Solution

Answer

The expression \(z-x\) simplifies to \(\frac{1}{4}\).

Step 1: Identify values for x and z

The value for \(x\) is \(\frac{8}{12}\) and the value for \(z\) is \(\frac{11}{12}\).

Step 2: Perform subtraction z - x

Subtract the fraction \(x\) from \(z\). This gives: \[ z - x = \frac{11}{12} - \frac{8}{12} \] Since the denominators are the same, subtract the numerators directly: \(11 - 8 = 3\). Thus: \[ z - x = \frac{3}{12} \]

Step 3: Simplify the fraction

Simplify \(\frac{3}{12}\) by finding the greatest common divisor of 3 and 12, which is 3. Divide both the numerator and the denominator by 3: \[ \frac{3}{12} = \frac{3 \div 3}{12 \div 3} = \frac{1}{4} \]

Key Concepts

Simplifying Fractions

Simplifying fractions means reducing them to their simplest form. A fraction consists of two numbers: the numerator and the denominator. The numerator is the top part, while the denominator is the bottom part. When you simplify a fraction, you look for the greatest common divisor (GCD) of both the numerator and the denominator. The GCD is the largest number that can divide both the numerator and the denominator without leaving a remainder.

If a fraction is already expressed in terms of the smallest possible numbers that can maintain its value relative to one another, it is considered to be in its simplest form. By simplifying, you make fractions cleaner and easier to understand or work with.

For example, in the given exercise, once we performed the subtraction from the fractions, we were left with \( \frac{3}{12} \). The GCD of 3 and 12 is 3, so we divided both the numerator and the denominator by 3 to get the simplified fraction \( \frac{1}{4} \). This process helps in understanding the problem better and making calculations more straightforward.

Numerator and Denominator

Fractions are constructed with two essential parts: the numerator and the denominator. The numerator is the number above the fraction bar and it depicts how many parts of the whole are being considered. On the flip side, the denominator, located below the fraction bar, indicates the total number of equal parts that make up a whole.

Understanding the role of the numerator and the denominator is crucial when dealing with operations like addition, subtraction, and simplification of fractions. When the denominators are the same, as in the exercise, it allows direct operations on the numerators without adjusting the fractions further.

• Numerator: Tells us the number of parts taken.
• Denominator: Tells us the total number of equal parts.

These concepts ensure that mathematical operations on fractions are accurate and that their values are properly comprehended.

Subtraction of Fractions

Subtracting fractions involves a few simple steps, especially when they share the same denominator. Here’s a breakdown of how to subtract fractions like in our example exercise:

When subtracting fractions with the same denominator, you only need to subtract the numerators while keeping the denominator unchanged. This is one of the benefits of common denominators.
For example, with \( z = \frac{11}{12} \) and \( x = \frac{8}{12} \), both fractions have the denominator 12, which allows us to directly subtract the numerators:
\( 11 - 8 = 3 \). This operation results in \( \frac{3}{12} \).

When the denominators differ, it's necessary to find a common denominator before proceeding with subtraction, which often involves more steps such as finding the least common denominator. However, since our exercise had the same denominators, we skipped those additional steps, simplifying our work.