Problem 42 - Prealgebra

Question

The problems below will allow you to review subtraction of fractions and mixed numbers. $$1-\frac{1}{2}$$

Step-by-Step Solution

Answer

\(\frac{1}{2}\)

Step 1: Understand the Problem

We are given a subtraction problem involving fractions - a whole number minus a fraction: \(1 - \frac{1}{2}\). Our goal is to find the difference.

Step 2: Convert the Whole Number to a Fraction

Convert the whole number 1 into a fraction with the same denominator as \(\frac{1}{2}\). Since \(\frac{1}{2}\) has a denominator of 2, express 1 as \(\frac{2}{2}\).

Step 3: Subtract the Fractions

Now, take \(\frac{2}{2}\) and subtract \(\frac{1}{2}\) from it. Use the formula for subtracting fractions with the same denominator: \(\frac{a}{c} - \frac{b}{c} = \frac{a-b}{c}\). In this case, \(2 - 1 = 1\), so the result is \(\frac{1}{2}\).

Step 4: Final Result

The solution to the subtraction problem \(1 - \frac{1}{2}\) is \(\frac{1}{2}\).

Key Concepts

Fractions

Fractions represent a part of a whole. A fraction consists of two parts: a numerator and a denominator. The numerator is the top number, representing how many parts you have. The denominator is the bottom number, showing the total number of equal parts into which the whole is divided. For example, in the fraction \( \frac{1}{2} \), 1 is the numerator and 2 is the denominator. This means one part out of two equal parts. It's important to remember that the larger the denominator, the smaller each individual part becomes, as the whole is divided into more pieces.

Fractions can be less than, equal to, or greater than 1. When the numerator is less than the denominator, the fraction is less than 1. When the numerator equals the denominator, the fraction equals 1. When the numerator is greater than the denominator, the fraction is greater than 1 and is often referred to as an improper fraction. One can convert improper fractions to mixed numbers and vice versa, which is a handy skill when performing operations like subtraction.

Mixed Numbers

Mixed numbers combine whole numbers and fractions, offering a more intuitive way of representing amounts greater than 1. For instance, the mixed number \(1 \frac{1}{2}\) consists of the whole number 1 and the fraction \( \frac{1}{2} \). Hence, it equates to 1 plus half, summing up to one and a half or 1.5.

Mixed numbers can be converted to improper fractions for calculation purposes, which is often crucial in operations like subtraction or addition. To convert a mixed number to an improper fraction, multiply the whole number by the denominator of the fraction and add the result to the numerator. This sum becomes the new numerator, while the denominator remains unchanged. For example, converting \(1 \frac{1}{2}\) to an improper fraction involves multiplying 1 by 2 and adding 1, which results in \(\frac{3}{2}\).

• They visually convey both the whole and fractional parts of a number, which can be especially useful in visualizing arithmetic operations.
• They are commonly used in everyday contexts such as cooking or carpentry, where measurements often comprise both whole and partial units.

Subtraction Steps

Subtracting fractions with like denominators is straightforward, but what about unlike denominators or mixed numbers? Here's a simple guide to help you out!

1. **Same Denominator:** If the fractions have the same denominator, simply subtract the numerators. Stick with the denominator the same. For instance, \(\frac{3}{4} - \frac{1}{4} = \frac{2}{4}\), which simplifies to \(\frac{1}{2}\).

2. **Different Denominators:** When denominators differ, find a common denominator, ideally the least common denominator (LCD). Convert each fraction to this common denominator, then subtract the numerators.

3. **Mixed Numbers:** For subtracting mixed numbers, either convert them to improper fractions first or break them down. For example, subtract \(1 \frac{1}{2} - \frac{3}{4}\) by converting \(1 \frac{1}{2}\) to \(\frac{3}{2}\), then find a common denominator.

• Convert mixed numbers to improper fractions if needed to simplify the process.
• Ensure the fractions are simplified at the end by dividing by the greatest common divisor (GCD) if possible.

Understanding these steps ensures a smoother calculation process and minimizes errors. It's all about laying a solid foundation for any problem involving fractions and mixed numbers.