Problem 90 - Beginning and Intermediate Algebra

Question

Perform each indicated operation. $$ \left(-\frac{3}{4}-\frac{5}{2}\right)-\left(-\frac{1}{8}-1\right) $$

Step-by-Step Solution

Answer

-\(\frac{17}{8}\)

Step 1: Simplify the first parentheses

First, simplify \(-\frac{3}{4}-\frac{5}{2}\). To subtract these fractions, find a common denominator. The common denominator for 4 and 2 is 4. Convert \(-\frac{5}{2}\) to a fraction with denominator 4: \(-\frac{5 \cdot 2}{2 \cdot 2} = -\frac{10}{4}\). Now add the fractions: \(-\frac{3}{4}-\frac{10}{4} = -\frac{13}{4}\).

Step 2: Simplify the second parentheses

Next, simplify \(-\frac{1}{8}-1\). Convert -1 to a fraction with the same denominator as 8: \(-1 = -\frac{8}{8}\). Now add the fractions: \(-\frac{1}{8}-\frac{8}{8} = -\frac{1+8}{8} = -\frac{9}{8}\).

Step 3: Subtract the second result from the first

Subtract the second simplified expression from the first: \(-\frac{13}{4} - (-\frac{9}{8})\). Subtracting a negative is the same as adding a positive, so this becomes \(-\frac{13}{4} + \frac{9}{8}\). Find a common denominator for the fractions. The common denominator for 4 and 8 is 8. Convert \(-\frac{13}{4}\) to have a denominator of 8: \(-\frac{13 \cdot 2}{4 \cdot 2} = -\frac{26}{8}\).

Step 4: Add the fractions

Add the two fractions: \(-\frac{26}{8} + \frac{9}{8} = -\frac{26-9}{8} = -\frac{17}{8}\). Thus, the result of the operation is \-\frac{17}{8}\.

Key Concepts

Common Denominator

When we work with fractions, we need to ensure that they have the same denominator. This is important because it allows us to easily apply addition or subtraction operations to the fractions. The denominator is the bottom number of the fraction and represents the total number of equal parts the whole is divided into. To find a common denominator, we typically look for the least common multiple (LCM) of the denominators involved.

Subtracting Fractions

Once fractions have the same denominator, subtracting them becomes simple. We subtract the numerators (the top numbers) while keeping the common denominator. For example, to subtract \(\frac{3}{4}\) from \(\frac{5}{4}\), we perform the operation \(\frac{5 - 3}{4} = \frac{2}{4}\). If the fractions have different denominators, like \(\frac{1}{2}\) and \(\frac{1}{3}\), we first convert them to equivalent fractions with a common denominator before subtracting. For example, the common denominator for 2 and 3 is 6, so we convert these fractions to \(\frac{3}{6}\) and \(\frac{2}{6}\), then subtract: \(\frac{3 - 2}{6} = \frac{1}{6}\).

Simplifying Fractions

After performing operations on fractions, the resulting fraction may not be in its simplest form. Simplifying fractions means reducing them to their lowest terms. This is done by finding the greatest common divisor (GCD) of the numerator and the denominator and dividing both by it. For example, to simplify \(\frac{10}{15}\), we find the GCD of 10 and 15, which is 5, and divide both the numerator and the denominator by 5, resulting in \(\frac{2}{3}\).

Negative Fractions

Negative fractions are handled similarly to positive fractions, but they include a negative sign. When subtracting negative fractions, remember that subtracting a negative is equivalent to adding a positive. For example, \(-\frac{3}{4} - (-\frac{1}{2})\) becomes \(-\frac{3}{4} + \frac{1}{2} \). Convert \(\frac{1}{2}\) to have the same denominator as \(\frac{3}{4}\), which is \(\frac{2}{4}\), then perform the operation: \(-\frac{3}{4} + \frac{2}{4} = -\frac{1}{4}\).