Solve: 1/4a + 1/2a + x/3 + 4b/3 + 2x/3 - Combining Mixed Variable Fractions

Q: Why can't I combine terms with different variables like a and x?

Variables represent different unknown values. You can only combine terms that have exactly the same variables with the same exponents. Think of it like combining apples with apples, not apples with oranges!

Q: How do I know when fractions have the same denominator?

Look carefully at what's in the denominator. $ \frac{1}{4a} $ and $ \frac{2}{4a} $ both have 4a in the bottom, so they can be combined. But $ \frac{1}{4a} $ and $ \frac{x}{3} $ have different denominators.

Q: What's the difference between 4b/3 and 4/3 × b?

They're exactly the same! $ \frac{4b}{3} $ and $ \frac{4}{3}b $ are just different ways to write multiplication. You can write the variable before or after the fraction.

Q: How do I combine x/3 + 2x/3?

Since both terms have the same denominator (3), add the numerators: $ \frac{x + 2x}{3} = \frac{3x}{3} = x $. Remember that x means 1x!

Algebraic Simplification with Multiple Variable Terms

$\frac{1}{4a}+\frac{1}{2a}+\frac{x}{3}+\frac{4}{3}b+\frac{2}{3}x=\text{?}$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the expression

00:03 Mark the appropriate variables

00:07 Use the substitution law and arrange the appropriate variables together

00:22 Expand the fraction to find the common denominator

00:25 Make sure to multiply both numerator and denominator

00:31 Collect terms, multiply by the common denominator

01:03 Reduce what is possible

01:11 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

$\frac{1}{4a}+\frac{1}{2a}+\frac{x}{3}+\frac{4}{3}b+\frac{2}{3}x=\text{?}$

Step-by-step solution

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.

Final Answer

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

Key Points to Remember

Essential concepts to master this topic

Like Terms Rule: Combine terms with same variables and denominators
Technique: Find common denominators: $\frac{1}{4a} + \frac{2}{4a} = \frac{3}{4a}$
Check: Group similar variables separately and verify no terms missed ✓

Common Mistakes

Avoid these frequent errors

Incorrectly combining terms with different variables
Don't add terms like $\frac{3}{4a} + x$ together = impossible combination! These have different variables and cannot be simplified further. Always keep terms with different variables separate in your final answer.

Practice Quiz

Test your knowledge with interactive questions

Are the expressions the same or not?

$$ 3+3+3+3 $$

$3\times4$

FAQ

Everything you need to know about this question

Why can't I combine terms with different variables like a and x?

Variables represent different unknown values. You can only combine terms that have exactly the same variables with the same exponents. Think of it like combining apples with apples, not apples with oranges!

How do I know when fractions have the same denominator?

Look carefully at what's in the denominator. $\frac{1}{4a}$ and $\frac{2}{4a}$ both have 4a in the bottom, so they can be combined. But $\frac{1}{4a}$ and $\frac{x}{3}$ have different denominators.

What's the difference between 4b/3 and 4/3 × b?

They're exactly the same! $\frac{4b}{3}$ and $\frac{4}{3}b$ are just different ways to write multiplication. You can write the variable before or after the fraction.

Do I always write my final answer in a specific order?

While order doesn't change the mathematical meaning, it's good practice to write terms in alphabetical order by variable. So write $\frac{3}{4a} + \frac{4}{3}b + x$ rather than mixing them up.

How do I combine x/3 + 2x/3?

Since both terms have the same denominator (3), add the numerators: $\frac{x + 2x}{3} = \frac{3x}{3} = x$ . Remember that x means 1x!

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