Variables and Algebraic Expressions: Using fractions

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 need to follow a systematic approach:

• Step 1: Convert all mixed numbers into improper fractions.
• Step 2: Carry out the multiplication between the terms $4\frac{1}{3}x \cdot 7.3x$.
• Step 3: Combine the terms by adding the coefficients of $x$ and $x^2$.

Let's proceed with these steps:
Step 1: Convert mixed numbers to improper fractions.
$6\frac{4}{5}x = \left(\frac{34}{5}\right)x$ $7\frac{1}{2}x = \left(\frac{15}{2}\right)x$ $4\frac{1}{3} = \frac{13}{3}$ Step 2: Compute the multiplication and product.
$\left(\frac{13}{3}\right)x \cdot 7.3x = \left(\frac{13}{3} \cdot 7.3\right)x^2 = \left(\frac{94.9}{3}\right)x^2$ Approximate $\frac{94.9}{3} = 31.633...$ or convert to a fraction $31\frac{19}{30}$.
Step 3: Combine all like terms by their variables.
$\left(\frac{34}{5}\right)x + \left(\frac{15}{2}\right)x + 31\frac{19}{30}x^2$ Converting to decimal form: $6.8x + 7.5x = 14.3x$ Thus, the final expression combines neatly as:
$14.3x + 31\frac{19}{30}x^2$

After reviewing the steps and calculations, the solution to the expression is $14.3x + 31\frac{19}{30}x^2$.

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.

To solve this algebraic expression problem, we proceed with the following steps:

• Step 1: Identify the Like Terms
• Step 2: Combine the Like Terms
• Step 3: Present the Final Simplified Expression

Step 1: Identify the Like Terms:
The expression given is $\frac{2}{3}a + \frac{1}{4}b + \frac{1}{8}c + \frac{1}{4}a$. Notice that the terms $\frac{2}{3}a$ and $\frac{1}{4}a$ are like terms involving $a$.

Step 2: Combine the Like Terms:
To combine $\frac{2}{3}a$ and $\frac{1}{4}a$, we need a common denominator. The least common denominator of 3 and 4 is 12.
Rewriting these fractions with a denominator of 12 gives: $\frac{2}{3}a = \frac{8}{12}a$, and $\frac{1}{4}a = \frac{3}{12}a$.
Adding these gives: $\frac{8}{12}a + \frac{3}{12}a = \frac{11}{12}a$.

Step 3: Present the Final Simplified Expression:
Now, substitute back the simplified terms involving $a$ into the expression: $\frac{11}{12}a + \frac{1}{4}b + \frac{1}{8}c$.
Therefore, the simplified expression is $\frac{11}{12}a + \frac{1}{4}b + \frac{1}{8}c$.

This matches with choice 3 in the provided options.

The final solution is: $\frac{11}{12}a + \frac{1}{4}b + \frac{1}{8}c$.

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

• Step 1: Identify the given expression and separate like terms.
• Step 2: Find a common denominator for terms with same variables.
• Step 3: Add the fractions for each group of like terms.
• Step 4: Simplify the expression to find the final result.

Now, let's work through each step:
Step 1: The expression is $\frac{2}{5}x + \frac{4}{3}y + \frac{7}{9}x + \frac{3}{4}y$. Group like terms together:
$(\frac{2}{5}x + \frac{7}{9}x) + (\frac{4}{3}y + \frac{3}{4}y)$.

Step 2: For $(\frac{2}{5}x + \frac{7}{9}x)$, find a common denominator for the fractions $2/5$ and $7/9$, which is 45.
Convert $\frac{2}{5}$ to $\frac{18}{45}$ and $\frac{7}{9}$ to $\frac{35}{45}$.

Step 3: Add the fractions for $x$:
$\frac{18}{45}x + \frac{35}{45}x = \frac{53}{45}x$.

For $(\frac{4}{3}y + \frac{3}{4}y)$, find a common denominator, which is 12.
Convert $\frac{4}{3}$ to $\frac{16}{12}$ and $\frac{3}{4}$ to $\frac{9}{12}$.

Add the fractions for $y$:
$\frac{16}{12}y + \frac{9}{12}y = \frac{25}{12}y$.

Step 4: Combine results to express the simplified form:
$\frac{53}{45}x + \frac{25}{12}y$.

As mixed numbers, the solution becomes:
$1\frac{8}{45}x + 2\frac{1}{12}y$.

Therefore, the solution to the problem is $1\frac{8}{45}x + 2\frac{1}{12}y$.

To solve this problem, we'll proceed as follows:

• Step 1: Group the like terms involving $x$ and $y$ separately.
• Step 2: Find a common denominator for the terms involving $x$.
• Step 3: Add the fractions to simplify the $x$ terms.
• Step 4: Find a common denominator for the terms involving $y$.
• Step 5: Add the fractions to simplify the $y$ terms.
• Step 6: Combine the simplified terms for $x$ and $y$.

Now, let's perform these steps in detail:
Step 1: Identify and group the terms:
$\left(\frac{4}{7}x + \frac{3}{4}x\right)$ and $\left(\frac{5}{7}y + \frac{8}{9}y\right)$.

Step 2: Find a common denominator for the $x$-terms:
The denominators are 7 and 4. The least common denominator (LCD) is 28.

Step 3: Add the $x$-terms:
$\frac{4}{7}x = \frac{4 \cdot 4}{28}x = \frac{16}{28}x$
$\frac{3}{4}x = \frac{3 \cdot 7}{28}x = \frac{21}{28}x$
Adding them gives $\frac{16}{28}x + \frac{21}{28}x = \frac{37}{28}x = 1\frac{9}{28}x$.

Step 4: Find a common denominator for the $y$-terms:
The denominators are 7 and 9. The LCD is 63.

Step 5: Add the $y$-terms:
$\frac{5}{7}y = \frac{5 \cdot 9}{63}y = \frac{45}{63}y$
$\frac{8}{9}y = \frac{8 \cdot 7}{63}y = \frac{56}{63}y$
Adding them gives $\frac{45}{63}y + \frac{56}{63}y = \frac{101}{63}y = 1\frac{38}{63}y$.

Step 6: Combine the simplified terms:
The final expression is $1\frac{9}{28}x + 1\frac{38}{63}y$.

Therefore, the solution to the problem is $1\frac{9}{28}x + 1\frac{38}{63}y$.

To solve this problem, we'll simplify the algebraic expression and combine like terms:

• Convert mixed numbers to improper fractions and simplify.
• Simplify each part of the expression.
• Combine like terms by variable type.

Let's go through each step:

First, convert and simplify $3\frac{b}{a} \cdot 1\frac{3}{8}a$: - Change $1\frac{3}{8}$ to an improper fraction: $1\frac{3}{8} = \frac{11}{8}$.

Thus, multiply: $3\frac{b}{a} \cdot \frac{11}{8}a = \frac{33b}{8}$.

Then, look at each term:

• The expression now consists of: $\frac{33b}{8} + \frac{5}{8}b + \frac{9}{10}a + \frac{4}{18}m + \frac{2}{3}m$.
• Simplify each component: - $\frac{4}{18}m = \frac{2}{9}m$ (reducing the fraction).
• Now, add like terms: - Combine terms involving $b$: $\frac{33b}{8} + \frac{5b}{8} = \frac{33b + 5b}{8} = \frac{38b}{8} = 4\frac{3}{4}b$.
• There are no like terms with $a$, so $\frac{9}{10}a$ remains unchanged.
• Combine terms involving $m$: $\frac{2}{9}m + \frac{2}{3}m = \frac{2m}{9} + \frac{6m}{9} = \frac{8m}{9}$.

After the simplification, the expression becomes: $4\frac{3}{4}b + \frac{9}{10}a + \frac{8}{9}m$.

Therefore, the solution to the problem is $4\frac{3}{4}b + \frac{9}{10}a + \frac{8}{9}m$.

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

• Identify and combine like terms involving $z$.

• Simplify the term involving both $z$ and $m$.

• Combine and simplify fractions.

Now, let us solve the expression step-by-step:

Given expression: $\frac{3z}{4} + \frac{1}{4}m + \frac{12}{13}z \cdot \frac{4}{5}m + \frac{1}{7}z$

Step 1: Combine like terms for $z$.

Terms involving $z$ are: $\frac{3z}{4}$, $\frac{1}{7}z$.

To combine these, we need a common denominator. The least common multiple of 4 and 7 is 28:

$\frac{3z}{4} = \frac{21z}{28}$ and $\frac{1z}{7} = \frac{4z}{28}$.

Add these: $\frac{21z}{28} + \frac{4z}{28} = \frac{25z}{28}$.

Step 2: Simplify the term involving both $z$ and $m$.

$\frac{12}{13}z \cdot \frac{4}{5}m = \frac{48}{65}zm$.

This expression is already in its simplest form.

Step 3: Write the whole expression in simplified form:

$\frac{25z}{28} + \frac{1}{4}m + \frac{48}{65}zm$.

Therefore, the simplified expression is: $\frac{25}{28}z + \frac{1}{4}m + \frac{48}{65}zm$.

To solve this problem, we'll simplify the expression step by step using the distributive law.

Step 1: Apply the distributive property to the first part of the expression: $(\frac{3}{4} + 2a)(8a + 9ba)$.

• Distribute $\frac{3}{4}$ to $8a$ and $9ba$: $\frac{3}{4} \times 8a = 6a$ and $\frac{3}{4} \times 9ba = \frac{27}{4}ab$.
• Distribute $2a$ to $8a$ and $9ba$: $2a \times 8a = 16a^2$ and $2a \times 9ba = 18a^2b$.

The first part expands to: $6a + \frac{27}{4}ab + 16a^2 + 18a^2b$.

Step 2: Apply the distributive property to the second part of the expression: $(5 + a)(\frac{3}{2}a + b)$.

• Distribute $5$ to $\frac{3}{2}a$ and $b$: $5 \times \frac{3}{2}a = \frac{15}{2}a$ and $5 \times b = 5b$.
• Distribute $a$ to $\frac{3}{2}a$ and $b$: $a \times \frac{3}{2}a = \frac{3}{2}a^2$ and $a \times b = ab$.

The second part expands to: $\frac{15}{2}a + 5b + \frac{3}{2}a^2 + ab$.

Step 3: Simplify the expression by subtracting the second part from the first:

• Combine like terms: $6a - \frac{15}{2}a$ and $\frac{27}{4}ab - ab$.
• Subtract constants and like terms: - $6a - \frac{15}{2}a = -\frac{3}{2}a$. - $\frac{27}{4}ab - ab = \frac{27}{4}ab - \frac{4}{4}ab = \frac{23}{4}ab$. - $(16a^2 - \frac{3}{2}a^2) + (18a^2b - 5b)$.

The full simplified expression is: $-\frac{3}{2}a + \frac{23}{4}ab + \left(16a^2 - \frac{3}{2}a^2\right) + (18a^2b - 5b)$.

Recognize that $16a^2 - \frac{3}{2}a^2 = \frac{32}{2}a^2 - \frac{3}{2}a^2 = \frac{29}{2}a^2$, the final answer is:

The simplified expression is: $-\frac{3}{2}a + 5\frac{3}{4}ab + 14\frac{1}{2}a^2 + (18a^2-5)b$.