Problem 92 - Elementary Algebra

Question

For exercises 81-96, evaluate. $$ -\frac{13}{16}+\frac{3}{4} $$

Step-by-Step Solution

Answer

\(\frac{-1}{16}\)

Step 1: Find a common denominator

To add or subtract fractions, find a common denominator. The denominators are 16 and 4. The least common multiple (LCM) of 16 and 4 is 16.

Step 2: Convert fractions to have the same denominator

Convert \(\frac{3}{4}\) to a fraction with denominator 16 by multiplying both the numerator and the denominator by 4. This gives: \(\frac{3 \times 4}{4 \times 4} = \frac{12}{16}\).

Step 3: Perform the subtraction

Subtract the fractions: \(-\frac{13}{16} + \frac{12}{16}\). Since the denominators are the same, subtract the numerators: \(\frac{-13 + 12}{16} = \frac{-1}{16}\).

Key Concepts

Common Denominator

When adding or subtracting fractions, it's important to have a common denominator. In simple terms, a common denominator is a shared multiple of the denominators of the fractions. This means the bottom numbers of the fractions must be the same before you can combine them.
Finding a common denominator helps to align the fractions so the numerators (the top numbers) can be easily added or subtracted. Let's look at why and how to find a common denominator:
Imagine you're adding \(-\frac{13}{16}+\frac{3}{4}\). Since the denominators are different (16 and 4), we need to convert them to a common denominator. This makes calculations straightforward and prevents errors. Always choose the least common multiple (LCM) as the common denominator to simplify your work.

Least Common Multiple

Determining the least common multiple (LCM) is crucial in finding a common denominator. The LCM is the smallest number that is a multiple of both denominators. In our example, we have denominators 16 and 4. Let's determine the LCM:
Start by listing the multiples of both numbers:

• Multiples of 16: 16, 32, 48, …
• Multiples of 4: 4, 8, 12, 16, ...

The first common multiple shared by both lists is 16, making it the LCM.
So, 16 is the LCM and hence our common denominator.

Converting Fractions

Once the common denominator is found, the next step is to convert the fractions to equivalent fractions with this denominator. This process is called converting fractions. Let's use our example:
We have \(-\frac{13}{16}+\frac{3}{4}\). The first fraction, -\frac{13}{16}, already has the common denominator. Now, we need to convert \frac{3}{4}\} to have the same denominator as 16.
We do this by multiplying both the numerator and the denominator of \frac{3}{4}\} by 4:

• \frac{3 \times 4}{4 \times 4} = \frac{12}{16}\

Now, both fractions have the same denominator (16). From here, you can proceed to add or subtract the numerators as required. In our case, \frac{-13}{16} + \frac{12}{16}\ becomes:
\frac{-13 + 12}{16}\} = \frac{-1}{16}\
Converting fractions with a common denominator ensures accurate and straightforward addition or subtraction.