Problem 30 - Beginning and Intermediate Algebra

Question

Find each sum. $$ 1 \frac{3}{8}+\left(-2 \frac{1}{4}\right) $$

Step-by-Step Solution

Answer

\( \frac{-3}{8} \)

Step 1: Convert Mixed Numbers to Improper Fractions

First, convert the mixed numbers to improper fractions. For \( 1 \frac{3}{8} \), multiply the whole number 1 by 8 and add 3 to get: \ \[ 1 \cdot 8 + 3 = 11 \] \ So, \( 1 \frac{3}{8} \) becomes \( \frac{11}{8} \). For \( -2 \frac{1}{4} \), multiply -2 by 4 and add 1 to get: \ \[ -2 \cdot 4 + 1 = -8 + 1 = -7 \] \ So, \( -2 \frac{1}{4} \) becomes \( -\frac{7}{4} \).

Step 2: Find a Common Denominator

Now, find a common denominator for the fractions \( \frac{11}{8} \) and \( -\frac{7}{4} \). The least common multiple of 8 and 4 is 8.

Step 3: Convert Fractions to Equivalent Fractions with Common Denominator

Convert \( -\frac{7}{4} \) to a fraction with a denominator of 8: \ \( -\frac{7}{4} \) is equivalent to \( -\frac{7 \cdot 2}{4 \cdot 2} = -\frac{14}{8} \). \ Now we have \( \frac{11}{8} \) and \( -\frac{14}{8} \).

Step 4: Add the Fractions

Add the fractions with the common denominator: \ \[ \frac{11}{8} + \left( -\frac{14}{8} \right) = \frac{11 - 14}{8} = \frac{-3}{8} \]

Step 5: Simplify the Answer

The fraction \( \frac{-3}{8} \) is already in its simplest form.

Key Concepts

improper fractions

Improper fractions are a key topic when dealing with mixed numbers. A mixed number consists of a whole number and a fraction. To convert it into an improper fraction, multiply the denominator by the whole number and then add the numerator. This sum becomes the new numerator, while the denominator remains the same.
For instance, to convert the mixed number '1 \(\frac{3}{8}\)', multiply 1 (the whole number) by 8 (the denominator), then add 3 (the numerator) to get: 1 \(\cdot 8 + 3 = 11\). Hence, '1 \(\frac{3}{8}\)' becomes \(\frac{11}{8}\). Similarly, for '-2 \(\frac{1}{4}\)', multiply -2 by 4, then add 1, resulting in: -2 \(\cdot 4 + 1 = -8 + 1 = -7\), making '-2 \(\frac{1}{4}\)' equal to \(-\frac{7}{4}\).
This method ensures that both parts of the mixed number are included, making subsequent operations simpler.

common denominator

Finding a common denominator is essential when adding or subtracting fractions. The common denominator should be a multiple of the denominators of the fractions involved. This allows the fractions to be added or subtracted easily.
For example, to add \(\frac{11}{8}\) and \(-\frac{7}{4}\), first identify their denominators: 8 and 4. The least common multiple of these is 8.
This means converting \(-\frac{7}{4}\) into a fraction with a denominator of 8. To do this, multiply both the numerator and the denominator by the same number to keep the value of the fraction unchanged: \(-\frac{7}{4} = -\frac{7 \cdot 2}{4 \cdot 2} = -\frac{14}{8}\). Now both fractions \(\frac{11}{8}\) and \(-\frac{14}{8}\) have the same denominator, making addition straightforward.

simplifying fractions

Once the fractions to be added or subtracted have a common denominator, the next step is simplification. Combining fractions with the same denominator involves adding or subtracting their numerators while retaining the common denominator. Afterwards, it's important to simplify the resulting fraction, if possible.
For instance, add \(\frac{11}{8}\) and \(-\frac{14}{8}\) by combining the numerators: \(11 - 14 = -3\). This results in \(\frac{-3}{8}\). Checking if the fraction can be simplified further involves identifying if the numerator and denominator share any common factors. Here, \(3\) and \(8\) have no common factors other than \(1\), so the fraction \(\frac{-3}{8}\) is already in its simplest form.
Simplifying fractions involves reducing them to their most basic form, making them easier to understand and work with in further calculations.