Problem 331 - Intermediate Algebra

Question

In the following exercises, simplify. $$ \frac{1}{2}+\frac{7}{8}+\left(-\frac{1}{2}\right) $$

Step-by-Step Solution

Answer

\(\frac{7}{8}\)

Step 1 - Combine like terms

Group the fractions that have the same denominator. In this case, combine \(\frac{1}{2}\) and \(-\frac{1}{2}\).

Step 2 - Simplify combined terms

Simplify the combined fractions \(\frac{1}{2}\) and \(-\frac{1}{2}\). This results in 0: \(\frac{1}{2} + (-\frac{1}{2}) = 0\).

Step 3 - Add remaining term

Add the result from Step 2 to the remaining fraction, \(\frac{7}{8}\). Since 0 adds no value, the simplified expression is \(\frac{7}{8}\).

Key Concepts

combining fractions

Combining fractions can seem tricky, but it's all about finding common terms. Start by identifying fractions with the same denominator. For instance, in the exercise \(\frac{1}{2}+\frac{7}{8}+\bigg(-\frac{1}{2}\bigg)\), we see the two fractions \(\frac{1}{2}\) and \(-\frac{1}{2}\) can be grouped together because they share a denominator of 2. Combining like terms helps simplify the problem early on. This makes further calculations easier. Group fractions with the same denominator and perform the arithmetic operation.

simplifying fractions

Simplifying fractions involves reducing them to their simplest form. This means making the numerator and the denominator as small as possible while still keeping the fraction's value the same. In our exercise, after combining \( \frac{1}{2} \) and \( -\frac{1}{2} \), they simplify to 0. When you add a fraction and its negative counterpart, they cancel each other out: \( \frac{1}{2} + \bigg(-\frac{1}{2}\bigg) = 0 \). Simplification helps make fractions easier to work with and can reveal hidden simplicity in complex problems.

adding fractions

Adding fractions requires a common denominator. If the denominators differ, convert them to a common base. In the exercise, after simplifying \(\frac{1}{2} + \big(-\frac{1}{2}\big) = 0\), we're left with \( \frac{7}{8} \). Adding 0 doesn't change the value, so the final answer remains \( \frac{7}{8} \). For fractions with different denominators, find the least common denominator (LCD). For example, to add \( \frac{1}{2} \) and \( \frac{1}{3} \), the LCD is 6. Convert and add: \( \frac{3}{6} + \frac{2}{6} = \frac{5}{6} \).