Problem 78 - Algebra 2

Question

Evaluate each expression. \(\frac{5}{8}-\frac{1}{4}\)

Step-by-Step Solution

Answer

The expression evaluates to \(\frac{3}{8}\).

Step 1: Identify Common Denominator

Both fractions have different denominators, 8 and 4. To subtract them, we need a common denominator. The least common multiple of 8 and 4 is 8, which will be our common denominator.

Step 2: Convert to Equivalent Fractions

We already have \(\frac{5}{8}\) with the denominator 8. For \(\frac{1}{4}\), we multiply both the numerator and the denominator by 2 to get an equivalent fraction: \(\frac{1 \times 2}{4 \times 2} = \frac{2}{8}\).

Step 3: Subtract the Fractions

Now that both fractions have the same denominator, we can subtract the numerators: \(\frac{5}{8} - \frac{2}{8} = \frac{5 - 2}{8} = \frac{3}{8}\).

Step 4: Simplify the Result

The resulting fraction \(\frac{3}{8}\) is already in its simplest form, as 3 and 8 have no common factors other than 1.

Key Concepts

Common Denominator

When subtracting fractions, having a common denominator is crucial. Think of a common denominator as a shared foundation or base that allows us to directly compare and combine fractions. In our exercise, we wanted to subtract \( \frac{5}{8} \) and \( \frac{1}{4} \). The denominators (8 and 4) are different, meaning we can't subtract them straight away.

So, what's the solution? We look for the lowest shared or common multiple of these denominators. In this instance, 8 is the least common multiple of 8 and 4 because both numbers can fit neatly into it without any leftovers. Now, with a common denominator, we can proceed to the next steps of adjusting fractions before subtracting them.

Equivalent Fractions

Once we've established a common denominator, it's time to adjust each fraction accordingly so that they're equivalent. This is like converting different currencies into the same type before doing any calculations. Why is this important? Because only then can we easily add or subtract fractions.

In our exercise, \( \frac{5}{8} \) doesn't need any adjustment because it's already expressed with 8. However, \( \frac{1}{4} \) must be restructured to an equivalent form with a denominator of 8. By multiplying the numerator (1) and the denominator (4) by 2, we get the fraction \( \frac{2}{8} \). This is what makes \( \frac{1}{4} \) equivalent to \( \frac{2}{8} \), just expressed differently.

This transforming process is the key step that aligns all fractions on the same measurement threshold before further arithmetic operations are performed on them.

Simplifying Fractions

After performing arithmetic operations, like subtraction, simplification is often the final step. Simplifying fractions means reducing them to their simplest form, where the numerator and denominator share no common factors other than 1.

When we subtracted \( \frac{5}{8} - \frac{2}{8} \) in our example, we obtained \( \frac{3}{8} \). To see if \( \frac{3}{8} \) can be simplified, we check the greatest common factor (GCF) of 3 and 8. Since these two numbers don't share any factors except 1, \( \frac{3}{8} \) is already in its simplest form.

Always remember this step as it brings fractions to their neatest form, which is typically the desired result in any fraction operation.