Variables and Algebraic Expressions: Common denominators with variables

To solve this problem, we'll follow these steps:

• Step 1: Combine like terms.
• Step 2: Simplify the expression.

Now, let's work through each step:
Step 1: Start by identifying and combining like terms in the expression $\frac{1}{4}a + \frac{1}{3}x + \frac{2}{4}a + \frac{1}{8} + \frac{3}{8}$. Recognize that $\frac{1}{4}a$ and $\frac{2}{4}a$ are like terms since they both involve the variable $a$.

Combine these terms:

$\frac{1}{4}a + \frac{2}{4}a = \frac{3}{4}a$

Step 2: Look at the constant terms $\frac{1}{8} + \frac{3}{8}$. Since these fractions have a common denominator, add them directly:

$\frac{1}{8} + \frac{3}{8} = \frac{4}{8} = \frac{1}{2}$

Combine all the terms together to form the simplified expression:

$\frac{3}{4}a + \frac{1}{3}x + \frac{1}{2}$

Therefore, the solution to the problem is $\frac{3}{4}a + \frac{1}{3}x + \frac{1}{2}$.

To solve this problem, we'll follow these steps:

• Step 1: Simplify the expression by combining like terms and simplifying fractions.
• Step 2: Analyze coefficients of like terms and reduce them to simplest form.
• Step 3: Choose the correct multiple-choice option as the final answer.

Now, let's work through each step:
Step 1: The expression is $\frac{3}{4}+\frac{1}{8}m+\frac{2}{8}n+\frac{17}{8}m$. We identify like terms and consider the fraction denominators.

Step 2: Combine the terms involving $m$:

$\frac{1}{8}m + \frac{17}{8}m = \frac{1+17}{8}m = \frac{18}{8}m$.

We simplify $\frac{18}{8}$ to $\frac{9}{4}$. Therefore, this becomes $\frac{9}{4}m$.

Step 3: Include the other terms that cannot be further simplified as they are alone. Thus, the entire expression becomes:

$\frac{3}{4} + \frac{9}{4}m + \frac{1}{4}n$.

This can be rearranged to $\frac{3}{4} + 2\frac{1}{4}m + \frac{1}{4}n$, where each part is shown in terms of mixed numbers for clearer expression matching.

Therefore, the solution to the problem is $\frac{3}{4}+2\frac{1}{4}m+\frac{1}{4}n$.

To solve this problem, we'll follow these steps:

• Step 1: Make sure all terms have a common denominator.
• Step 2: Combine like terms.
• Step 3: Simplify the algebraic expression.

Now, let's work through each step:
Step 1: Convert all terms to have a common denominator. The common denominator for 4 and 8 is 8. Thus:
- $\frac{x}{4}$ becomes $\frac{2x}{8}$ when converted to have the denominator 8,
- $\frac{1}{4}x$ also becomes $\frac{2x}{8}$ when converted to 8.

Step 2: Combine the like terms.
- Combine the $x$ terms: $\frac{x}{8} + \frac{2x}{8} + \frac{2x}{8} = \frac{5x}{8}$.
- The constant term: $\frac{31}{8}$.
- The term with $y$: $\frac{y}{8}$.

Step 3: Write the final expression:

The simplified expression is $\frac{31}{8} + \frac{5x}{8} + \frac{y}{8}$.

Therefore, combining it we have $3\frac{7}{8}+\frac{5}{8}x+\frac{y}{8}$. This matches the given choice $3\frac{7}{8}+\frac{5}{8}x+\frac{y}{8}$.

Hence, the solution to the problem is $3\frac{7}{8}+\frac{5}{8}x+\frac{y}{8}$.

To solve this problem, we'll follow these steps:

• Step 1: Group and simplify terms with the same variable.
• Step 2: Convert any mixed numbers to improper fractions.
• Step 3: Find a common denominator to combine fractions.
• Step 4: Simplify the expression.

Let's work through the steps:
Step 1: Start by grouping like terms. The expression is:
$\frac{3}{8}a + \frac{6}{8}a + \frac{14}{9}b + 1\frac{1}{9}b$.
Step 2: Convert the mixed number to an improper fraction. For $1\frac{1}{9}b$: $1\frac{1}{9}b = \frac{10}{9}b$.
Rewrite the expression: $\frac{3}{8}a + \frac{6}{8}a + \frac{14}{9}b + \frac{10}{9}b$.
Step 3: Combine the $a$-terms and $b$-terms separately:
The $a$-terms: $\frac{3}{8}a + \frac{6}{8}a = \left(\frac{3}{8} + \frac{6}{8}\right)a = \frac{9}{8}a$.
For the $b$-terms: $\frac{14}{9}b + \frac{10}{9}b = \left(\frac{14}{9} + \frac{10}{9}\right)b = \frac{24}{9}b$.
Simplify $\frac{24}{9}$: $\frac{24}{9} = \frac{8}{3}$ after dividing by the greatest common divisor 3.
Step 4: Combine simplified terms: $\frac{9}{8}a + \frac{8}{3}b$.
Convert $\frac{9}{8}a$ to a mixed number: $\frac{9}{8}a = 1\frac{1}{8}a$.
Convert $\frac{8}{3}b$ to a mixed number: $\frac{8}{3}b = 2\frac{2}{3}b$.
Thus, the simplified expression is: $1\frac{1}{8}a + 2\frac{2}{3}b$.

Therefore, the solution to the problem is $1\frac{1}{8}a + 2\frac{2}{3}b$.

To solve this problem, we'll follow these steps:

• Step 1: Simplify and combine fractions with $a$.
• Step 2: Simplify and combine terms with $x$.
• Step 3: Express the final simplified algebraic expression.

Now, let's work through each step:
Step 1: Look at the terms with $a$: $\frac{1}{4a} + \frac{1}{2a}$.
Find a common denominator, which is $4$ in this case:

$\frac{1}{4a} + \frac{2}{4a} = \frac{1 + 2}{4a} = \frac{3}{4a}$.

Step 2: Simplify and combine terms with $x$:

$\frac{x}{3} + \frac{2}{3}x = \frac{1x}{3} + \frac{2x}{3} = \frac{3x}{3} = x$.

Step 3: Combine all terms together in the expression. Keep the $b$-related term as it is and combine:

$\frac{3}{4a} + x + \frac{4}{3}b$.

Therefore, the solution to the problem is $\frac{3}{4a} + x + \frac{4}{3}b$, which matches choice 2.