Problem 119 - Introductory Algebra for College Students

Question

Perform the indicated operations. Begin by performing operations in parentheses. $$\left(\frac{1}{2}-\frac{1}{3}\right) \div \frac{5}{8}$$

Step-by-Step Solution

Answer

\(\frac{4}{15}\)

Step 1: Subtract Fractions in Parentheses

The first task is to compute the subtraction in the parentheses: \(\frac{1}{2}-\frac{1}{3}\).\nTo subtract fractions, they need to have the same denominator. In this case, the least common denominator of 2 and 3 is 6. Start by transforming both fractions to equivalent fractions that have 6 as the denominator. This gives \(\frac{3}{6}-\frac{2}{6}\). Now subtract the numerators, leaving the denominator same. The result is \(\frac{1}{6}\).

Step 2: Division of Fractions

Next, divide the result from step 1 by \(\frac{5}{8}\). This equates to \(\frac{1}{6} \div \(\frac{5}{8}\)\). Recall that dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of \(\frac{5}{8}\) is \(\frac{8}{5}\). This changes the operation to multiplication: \(\frac{1}{6} * \(\frac{8}{5}\)\).

Step 3: Multiply Fractions

To multiply fractions, multiply the numerators together for the new numerator and the denominators together for the new denominator. This gives \(\frac{1*8}{6*5}\), which simplifies to \(\frac{8}{30}\). In the final step, simplify the fraction by dividing numerator and denominator by their greatest common divisor, which is 2. This results in \(\frac{4}{15}\).