Problem 63 - Introductory Algebra for College Students

Question

Simplify each series of additions and subtractions. $$1-\frac{2}{3}-\left(-\frac{5}{6}\right)$$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{7}{6}\)

Step 1: Interpret the negative before the bracket

The bracket contains a negative fraction. Remember that a negative sign in front of a bracket negates all the elements inside. Change \(-\left(-\frac{5}{6}\right)\) to \(+\frac{5}{6}\) thus rewriting the expression as: \(1-\frac{2}{3}+\frac{5}{6}\)

Step 2: Finding a common denominator

To add or subtract fractions, a common denominator is required. Here, \(3\) and \(6\) appear in the denominators. The least common multiple (LCM) of \(3\) and \(6\) is \(6\), therefore, convert all fractions to have \(6\) as the denominator. The expression becomes: \(1-\frac{4}{6}+\frac{5}{6}\)

Step 3: Simplification

Now, perform the addition and subtraction operations: \(1-\frac{4}{6}+\frac{5}{6} = 1 + \frac{1}{6}\)

Step 4: Convert the mixed number

Convert \(1 + \frac{1}{6}\) into an improper fraction. Multiply the denominator of the fraction by the whole number, add the numerator, and put over the original denominator. This gives \(\frac{7}{6}\)