Problem 82 - Prealgebra

Question

The following problems review addition and subtraction with fractions and mixed numbers. $$\frac{5}{8}-\frac{1}{4}$$

Step-by-Step Solution

Answer

The result is \(\frac{3}{8}\).

Step 1: Identify the Least Common Denominator

To subtract fractions, they must have the same denominator. Identify the least common denominator (LCD) for the fractions \(\frac{5}{8}\) and \(\frac{1}{4}\). The denominators are 8 and 4, and the LCD is 8.

Step 2: Convert Fractions to Have Common Denominator

Convert \(\frac{1}{4}\) to an equivalent fraction with a denominator of 8. Multiply the numerator and the denominator of \(\frac{1}{4}\) by 2 to achieve this: \(\frac{1}{4} = \frac{1 \times 2}{4 \times 2} = \frac{2}{8}\). Now both fractions are \(\frac{5}{8}\) and \(\frac{2}{8}\).

Step 3: Subtract the Fractions

With both fractions having the same denominator (8), you can subtract the numerators directly: \(\frac{5}{8} - \frac{2}{8} = \frac{5-2}{8} = \frac{3}{8}\).

Step 4: Simplify the Fraction (if necessary)

Check if the fraction \(\frac{3}{8}\) can be simplified. Since 3 and 8 have no common factors other than 1, \(\frac{3}{8}\) is already in its simplest form.

Key Concepts

Understanding the Least Common Denominator

When working with fraction operations, particularly addition and subtraction, it's crucial to ensure that the fractions involved share the same denominator. That's where the Least Common Denominator (LCD) comes in. The LCD is the smallest number that is evenly divisible by the denominators of the fractions involved. It helps in aligning the fractions to have a uniform base, allowing for straightforward arithmetic operations.

For instance, if you have fractions with denominators of 8 and 4, like in our example with \(\frac{5}{8}\) and \(\frac{1}{4}\), you need to find a common multiple. In this case, 8 is the smallest number that 4 and 8 can both divide into evenly. Thus, the LCD is 8.

Finding the LCD involves:

• Listing the multiples of each denominator.
• Identifying the smallest one that appears in both lists.

Once you have the LCD, you can convert each fraction to have this common denominator and proceed with your calculations smoothly.

Exploring Equivalent Fractions

Equivalent fractions represent the same value or proportion, even though they might look different. By adjusting the numerators and denominators of fractions through multiplication or division, you can make one fraction equivalent to another.

For instance, to convert \(\frac{1}{4}\) into a fraction with a denominator of 8, you multiply both the numerator and denominator by the same number, in this case, 2. Doing so, you get \(\frac{1 \times 2}{4 \times 2} = \frac{2}{8}\). Now, \(\frac{1}{4}\) and \(\frac{2}{8}\) are equivalent because they represent the same portion of a whole.

Steps to find equivalent fractions:

• Select a number to multiply both the numerator and denominator by. It should result in a desired denominator.
• Perform the multiplication.
• Check if the new fraction represents the same proportion as the original.

This process is essential for ensuring that fractions can be easily added or subtracted.

Mastering Simplifying Fractions

Simplifying fractions, or reducing them, involves converting a fraction to its simplest form. This means adjusting a fraction so that its numerator and denominator are as small as possible, while still maintaining the same value. The process of simplifying a fraction is about finding common factors and dividing them out.

Consider the fraction \(\frac{3}{8}\). To check if it can be simplified, identify any common factors between the numerator and the denominator. Here, 3 and 8 have no common factors other than 1. Thus, \(\frac{3}{8}\) is already in simplest form.

Steps to simplify fractions include:

• Identifying common factors between the numerator and denominator.
• Dividing both by their greatest common factor (GCF).
• Ensuring that the fraction cannot be reduced further.

Simplifying fractions makes them easier to work with in calculations and can help in identifying and comparing their sizes more readily.