Problem 67 - Introductory and Intermediate Algebra for College Students 4th

Question

Simplify each series of additions and subtractions. $$-\frac{3}{4}-\frac{1}{4}-\left(-\frac{5}{8}\right)$$

Step-by-Step Solution

Answer

The simplified answer is -\(\frac{3}{8}\).

Step 1: Handle the negative sign inside the parentheses

The negative sign before the parentheses changes the sign of the fraction inside the parentheses. Thus, \(-(-\frac{5}{8})\) becomes \(+\frac{5}{8}\). So the series of operations becomes -\(\frac{3}{4}\) - \(\frac{1}{4}\) + \(\frac{5}{8}\).

Step 2: Convert fractions to have common denominator

In order to add or subtract fractions, fractions need to have a common denominator. As 4 and 8 have a common multiple of 8, convert the fractions to have 8 as the denominator. Thus, -\(\frac{3}{4}\) becomes -\(\frac{6}{8}\) and -\(\frac{1}{4}\) becomes -\(\frac{2}{8}\). Now the expressions becomes -\(\frac{6}{8}\) - \(\frac{2}{8}\) + \(\frac{5}{8}\). By simplifying this, you get -\(\frac{3}{8}\).

Step 3: Summarize the expressions

Now simply add and subtract the fractions as indicated: -\(\frac{6}{8}\) - \(\frac{2}{8}\) + \(\frac{5}{8}\) = -\(\frac{3}{8}\).

Key Concepts

Simplification of Fractions

Simplification of fractions is an essential skill in math that allows you to express fractions in the simplest form. When we simplify a fraction, we reduce it to its smallest possible terms without changing its value. This involves finding the greatest common divisor (GCD) of the numerator and the denominator and dividing both by this number. For example, if you have the fraction \( \frac{6}{8} \), you would divide both 6 and 8 by their GCD, which is 2:

• \( 6 \div 2 = 3 \)
• \( 8 \div 2 = 4 \)

Thus, \( \frac{6}{8} \) simplifies to \( \frac{3}{4} \). Simplifying fractions not only makes them easier to work with but also helps in making correct calculations when combining with other fractions. Always remember, the key to simplifying fractions is to look for that GCD.

Common Denominator

When you're adding or subtracting fractions, having a common denominator is a must. A common denominator is a shared multiple of the denominators of two or more fractions. Having the same denominator is crucial because it allows you to combine the fractions directly.To find a common denominator, you usually take the least common multiple (LCM) of the denominators. For instance, with denominators 4 and 8, the common denominator is 8, as it is the smallest number both can divide into evenly. This enables us to convert fractions into equivalent fractions with a common denominator:

• Convert \( -\frac{3}{4} \) to \( -\frac{6}{8} \)
• Convert \( -\frac{1}{4} \) to \( -\frac{2}{8} \)

Having a common denominator aligns the fractions and makes addition or subtraction straightforward.

Addition and Subtraction of Fractions

Once fractions have a common denominator, adding or subtracting them becomes simple. You keep the denominator the same and either add or subtract the numerators. Let's look at an example:Given the expression \( -\frac{6}{8} - \frac{2}{8} + \frac{5}{8} \), you can perform the operations step-by-step:

• \( -6 - 2 = -8 \)
• Add the next numerator: \( -8 + 5 = -3 \)

Therefore, the result is \( \frac{-3}{8} \).Remember, the denominator does not change throughout the addition or subtraction process, so ensure it's the same across all fractions you are working with. This keeps calculations simple and accurate.