Problem 213 - Intermediate Algebra

Question

In the following exercises, simplify. $$ -\frac{7}{15}-\frac{y}{4} $$

Step-by-Step Solution

Answer

\[ \frac{-28 - 15y}{60} \]

Step 1: Find a Common Denominator

The common denominator between 15 and 4 can be found by calculating the least common multiple (LCM). The LCM of 15 and 4 is 60.

Step 2: Convert Fractions to have the Same Denominator

Convert each fraction so that both have a denominator of 60. \[ -\frac{7}{15} = -\frac{7 \times 4}{15 \times 4} = -\frac{28}{60} \ -\frac{y}{4} = -\frac{y \times 15}{4 \times 15} = -\frac{15y}{60} \ \text{So, the expression becomes:} \ -\frac{28}{60} -\frac{15y}{60} \]

Step 3: Combine the Fractions

Because the denominators of both fractions are now the same, we can combine the fractions into a single fraction: \[ -\frac{28}{60} -\frac{15y}{60} = \frac{-28 - 15y}{60} \]

Key Concepts

Least Common Multiple

When working with fractions that have different denominators, it is important to find a common denominator. The least common multiple (LCM) of two denominators will be the smallest number that both denominators can divide evenly into. This approach simplifies combining the fractions later. For example, to combine \(-\frac{7}{15}\) and \(-\frac{y}{4}\), we need an LCM for 15 and 4. Start listing out the multiples of each number:

• Multiples of 15: 15, 30, 45, 60, ...
• Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, ...

The smallest common multiple is 60. Thus, the LCM of 15 and 4 is 60.

Common Denominator

After determining the least common multiple (LCM), we convert the fractions to have a common denominator. This allows us to combine or perform operations on them easily. For example, we convert \(-\frac{7}{15}\) and \(-\frac{y}{4}\) so they both have a denominator of 60. Using the LCM we found earlier:

• For \(-\frac{7}{15}\), multiply both the numerator and denominator by 4: \- \frac{7 \times 4}{15 \times 4} = -\frac{28}{60}\
• For \(-\frac{y}{4}\), multiply both the numerator and denominator by 15: \- \frac{y \times 15}{4 \times 15} = -\frac{15y}{60}\

Now, both fractions have the same denominator of 60: \- \frac{28}{60} \text{ and } -\frac{15y}{60}\.

Combining Fractions

Once the fractions have a common denominator, it becomes straightforward to combine them. We simply add or subtract the numerators, keeping the common denominator intact. For our example, the fractions are \(-\frac{28}{60} \text{ and } -\frac{15y}{60}\). Since their denominators are already the same, we combine them:

• Combine the numerators: \(-28 - 15y\)
• Place this combined numerator over the common denominator of 60: \frac{-28 - 15y}{60}\

Thus, the simplified form of the expression \(-\frac{7}{15} - \frac{y}{4}\) is \frac{-28 - 15y}{60}\. This process highlights the importance of least common multiples and common denominators in simplifying and combining fractions.