Problem 62 - Algebra 2

Question

ACT/SAT What is the sum of \(\frac{x-y}{5}\) and \(\frac{x+y}{4} ?\) $$ \begin{array}{l}{\text { A } \frac{x+9 y}{20}} \\ {\text { B } \frac{9 x+y}{20}} \\ {\text { C } \frac{9 x-y}{20}} \\ {\text { D } \frac{x-9 y}{20}}\end{array} $$

Step-by-Step Solution

Answer

The sum is \( \frac{9x+y}{20} \), corresponding to option B.

Step 1: Express the Sum

To find the sum of \( \frac{x-y}{5} \) and \( \frac{x+y}{4} \), we write it as a single expression: \( \frac{x-y}{5} + \frac{x+y}{4} \).

Step 2: Find the Common Denominator

The denominators are 5 and 4. The least common multiple of 5 and 4 is 20. We need to express both fractions with the common denominator of 20.

Step 3: Convert to Common Denominator

Multiply \( \frac{x-y}{5} \) by \( \frac{4}{4} \) and \( \frac{x+y}{4} \) by \( \frac{5}{5} \) to get the common denominator:\[ \frac{4(x-y)}{20} + \frac{5(x+y)}{20} \]

Step 4: Simplify the Expression

Now that both fractions have the same denominator, combine them into a single fraction:\[ \frac{4(x-y) + 5(x+y)}{20} \]

Step 5: Distribute Terms

Distribute the 4 and the 5 in the numerator:\[ \frac{4x - 4y + 5x + 5y}{20} \]

Step 6: Combine Like Terms

Combine the \( x \) and \( y \) terms:\[ \frac{(4x + 5x) + (-4y + 5y)}{20} = \frac{9x + y}{20} \]

Step 7: Select the Correct Answer

The simplified sum is \( \frac{9x+y}{20} \), which corresponds to option B.

Key Concepts

Common Denominator

When working with fractions that have different denominators, like \( \frac{x-y}{5} \) and \( \frac{x+y}{4} \), you first need to find a common denominator to add them together. A denominator is the bottom number in a fraction, and the common denominator is a shared multiple of these denominators. In this case, the denominators are 5 and 4.
To find a common denominator, look for the least common multiple (LCM) of the two numbers. The LCM is the smallest number that is a multiple of both denominators. Here:

• The number 5 can be counted as 5, 10, 15, 20, 25, and so on.
• The number 4 can be counted as 4, 8, 12, 16, 20, and so on.

The smallest shared multiple is 20. Therefore, 20 is the common denominator that we will use, which allows us to rewrite our fractions with this common base.

Fraction Addition

After finding the common denominator, the next step is to rewrite and add the fractions. Let's take the fractions \( \frac{x-y}{5} \) and \( \frac{x+y}{4} \) with the common denominator of 20.
To convert each fraction:

• Multiply \( \frac{x-y}{5} \) by \( \frac{4}{4} \) to turn it into \( \frac{4(x-y)}{20} \).
• Multiply \( \frac{x+y}{4} \) by \( \frac{5}{5} \) to turn it into \( \frac{5(x+y)}{20} \).

Now, since both fractions have the same denominator, you can add them directly by combining the numerators:\[\frac{4(x-y) + 5(x+y)}{20}\]This step keeps the denominator the same while adding together the expressions in the numerators, an essential part of adding fractions.

Simplifying Expressions

Once the fractions have been set with a common denominator and combined, it’s time to simplify the expression. Simplifying reduces the expression to its simplest form by performing all possible arithmetic operations.

In our example, we ended up with the fraction:\[\frac{4(x-y) + 5(x+y)}{20}\]Start by distributing the numbers outside the parentheses in the numerator:

• Distribute the 4 in \(4(x-y)\) to get \(4x - 4y\).
• Distribute the 5 in \(5(x+y)\) to get \(5x + 5y\).

Combine these results:\[\frac{4x - 4y + 5x + 5y}{20}\]Finally, combine like terms in the numerator:

• Combine \(4x + 5x\) to get \(9x\).
• Combine \(-4y + 5y\) to get \(y\).

Thus, the simplified form is:\[\frac{9x + y}{20}\]Simplifying expressions like this helps express the result in an easier-to-read and often more useful form.