Problem 37 - Prealgebra

Question

The problems below will allow you to review subtraction of fractions and mixed numbers. $$\frac{5}{8}-\frac{1}{4}$$

Step-by-Step Solution

Answer

\(\frac{5}{8} - \frac{1}{4} = \frac{3}{8}\)

Step 1: Find a Common Denominator

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

Step 2: Convert Fractions to the Common Denominator

Convert \(\frac{1}{4}\) to a fraction with a denominator of 8. Multiply the numerator and the denominator by 2: \(\frac{1\times2}{4\times2} = \frac{2}{8}\). Now the fractions are \(\frac{5}{8}\) and \(\frac{2}{8}\).

Step 3: Subtract the Numerators

Keep the denominator the same and subtract the numerators: \(\frac{5}{8} - \frac{2}{8} = \frac{3}{8}\).

Step 4: Simplify the Result

The fraction \(\frac{3}{8}\) is already in its simplest form, so no further simplification is needed.

Key Concepts

Understanding Common Denominators

When working with fractions, especially in subtraction or addition, it's crucial that the fractions have the same denominator. This shared denominator is known as the common denominator.

Why do we need a common denominator? The answer lies in the basic nature of fractions. Denominators tell us the size of the parts we are dealing with. Therefore, to accurately compare or subtract these parts, they must be of the same size.

• The lowest number that both original denominators divide into evenly is called the least common denominator (LCD).
• The LCD for \(\frac{5}{8}\) and \(\frac{1}{4}\) is 8 because both 8 and 4 divide into 8.

After identifying the LCD, adjust each fraction to have this common denominator. This might require multiplying the numerator and denominator of one or more of the fractions by the same number.

Working With Mixed Numbers

Mixed numbers are those that contain both a whole number and a fraction part, such as 2\(\frac{3}{4}\). They can make subtraction seem tricky, but breaking the process into clear steps helps.

When subtracting mixed numbers:

• First, ensure that the fractions have common denominators, just as you would with simple fractions.
• If necessary, convert mixed numbers to improper fractions (where the numerator is larger than the denominator).
• Perform the subtraction, addressing whole numbers and fractions separately if they appear together.

Mixing numbers can add a layer of challenge, but consistency in following these steps ensures clarity and accuracy.

Simplifying Fractions After Subtraction

Once you've subtracted fractions and found your answer, simplifying might be necessary. Simplifying a fraction means reducing it to its simplest form.

For example, after subtracting to get a result like \(\frac{3}{8}\), here's how you check if simplification is needed:

• Identify the greatest common divisor (GCD) of the numerator and the denominator.
• If the GCD is greater than 1, divide both the numerator and denominator by this number.
• In this case, \(\frac{3}{8}\) is already simplified because the GCD of 3 and 8 is 1.

Simplifying fractions helps ensure your final answers in math appear clean, neat, and easiest to understand.