Question

Perform the indicated operation. Write all results in lowest terms. $$\frac{5}{8}-\frac{3}{8}$$

Short answer

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

Step-by-step solution

Identify Common Denominator

Look at the denominators in the fractions. Both fractions have the same denominator of 8. This allows you to directly subtract the numerators.

Subtract the Numerators

Subtract the numerators of the two fractions: 5 - 3 = 2. The denominator remains the same, so the fraction becomes \( \frac{2}{8} \).

Simplify the Fraction

Find the greatest common divisor (GCD) of the numerator and denominator. For 2 and 8, the GCD is 2. Divide both the numerator and the denominator by the GCD: \( \frac{2}{2} = 1 \) and \( \frac{8}{2} = 4 \). The fraction simplifies to \( \frac{1}{4} \).

Key concepts

lowest terms

When we talk about writing fractions in lowest terms, we mean simplifying the fraction so that the numerator and the denominator have no common factors other than 1. This makes the fraction as simple as possible. Consider the fraction \( \frac{2}{8} \). To reduce this fraction to its lowest terms, we look for the greatest common divisor (GCD) of the numerator and the denominator. Once we find the GCD, we divide both the numerator and the denominator by this number. For our example, the GCD of 2 and 8 is 2. By dividing the top and bottom of the fraction by 2, we simplify it to \( \frac{1}{4} \). This is the simplest form, as no other number (except 1) can divide both the numerator and the denominator. Reducing to lowest terms is crucial since it provides a clearer, more easily understood view of what the fraction represents. This practice is common in mathematics to standardize answers.

common denominator

The idea of a common denominator is pivotal in fraction arithmetic, especially for addition and subtraction. A common denominator is a shared multiple of the denominators of two or more fractions. In fraction subtraction, just like in our exercise where \( \frac{5}{8} - \frac{3}{8} \) was solved, having a common denominator means you can directly subtract or add the numerators. Why Common Denominators Matter
- They simplify the arithmetic process.- They allow us to compare fractions easily.- They are necessary for performing operations like subtraction and addition.In our example, both fractions have 8 as the denominator, making it a straightforward operation to subtract: \( 5 - 3 = 2 \). The result is \( \frac{2}{8} \). This also shows that finding a common denominator upfront can make these types of problems less prone to mistakes and more manageable to solve.

greatest common divisor

The greatest common divisor (GCD) of two numbers is the largest number that divides both of them without leaving a remainder. It plays a crucial role in simplifying fractions. In our exercise, we arrive at the fraction \( \frac{2}{8} \) and need to simplify it. To do this, we find the GCD of 2 and 8. The GCD is found by identifying the common factors of both numbers and choosing the greatest one. How to Find the GCD
- List the factors of both numbers.- Identify the common factors.- Select the largest common factor.For 2, the factors are 1 and 2, while for 8, the factors are 1, 2, 4, and 8. The common factors are 1 and 2, so the GCD is 2. We then divide both numerator and denominator by this GCD to simplify, resulting in \( \frac{1}{4} \). Using the GCD allows us to ensure we are working with the simplest form of a fraction, which aids in making further arithmetic operations more efficient and comprehensible.