Expand (1/3a + b)(9b + 12): Binomial Multiplication with Fractions

Q: Why do I get 3ab instead of 3ab when multiplying the fractions?

When you multiply $ \frac{1}{3}a \times 9b $, think of it as $ \frac{1 \times 9 \times a \times b}{3} = \frac{9ab}{3} = 3ab $. The 9 and 3 cancel out perfectly!

Q: How do I know which terms to multiply together?

Use FOIL: First terms ($ \frac{1}{3}a \times 9b $), Outer terms ($ \frac{1}{3}a \times 12 $), Inner terms ($ b \times 9b $), Last terms ($ b \times 12 $).

Q: Do I need to combine like terms at the end?

In this problem, all terms are different ($ 9b^2, 3ab, 4a, 12b $), so no combining is needed. Just arrange them in descending order of powers.

Q: What if I get the order of terms wrong in my final answer?

The mathematical value is the same regardless of order, but it's good practice to write terms in descending order: highest degree terms first, like $ 9b^2 + 3ab + 4a + 12b $.

Q: How can I check if my expansion is correct?

Try factoring your answer back to the original form, or substitute simple values like $ a = 3, b = 1 $ into both the original expression and your answer to see if they match.

Distributive Property with Fractional Coefficients

$(\frac{1}{3}a+b)(9b+12)=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Solve

00:03 Open parentheses properly, multiply each factor by each factor

00:25 Calculate the products

00:49 Calculate the quotients

01:04 Arrange the expression

01:13 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

$(\frac{1}{3}a+b)(9b+12)=$

Step-by-step solution

The given expression is $(\frac{1}{3}a + b)(9b + 12)$ . We will expand this expression using the distributive property:

Step 1: Multiply the first terms of each binomial:

$\frac{1}{3}a \times 9b = 3ab$

Step 2: Multiply the outer terms:

$\frac{1}{3}a \times 12 = 4a$

Step 3: Multiply the inner terms:

$b \times 9b = 9b^2$

Step 4: Multiply the last terms:

$b \times 12 = 12b$

Combine all the resulting terms together:

$9b^2 + 3ab + 4a + 12b$

Therefore, the solution to the problem is $9b^2 + 3ab + 4a + 12b$ .

Final Answer

$9b^2+3ab+4a+12b$

Key Points to Remember

Essential concepts to master this topic

FOIL Method: First, Outer, Inner, Last terms multiply systematically
Technique: $\frac{1}{3}a \times 9b = 3ab$ by canceling common factors
Check: Expand backwards: factor out common terms to verify original form ✓

Common Mistakes

Avoid these frequent errors

Forgetting to multiply all four term combinations
Don't just multiply $\frac{1}{3}a \times 9b = 3ab$ and stop there = missing three terms! This gives incomplete expansions like $3ab + 12b$ . Always multiply each term in the first binomial by each term in the second binomial using FOIL.

Practice Quiz

Test your knowledge with interactive questions

It is possible to use the distributive property to simplify the expression below?

What is its simplified form?

$$ (ab)(c d) $$

$$

FAQ

Everything you need to know about this question

Why do I get 3ab instead of 3ab when multiplying the fractions?

When you multiply $\frac{1}{3}a \times 9b$ , think of it as $\frac{1 \times 9 \times a \times b}{3} = \frac{9ab}{3} = 3ab$ . The 9 and 3 cancel out perfectly!

How do I know which terms to multiply together?

Use FOIL: First terms ( $\frac{1}{3}a \times 9b$ ), Outer terms ( $\frac{1}{3}a \times 12$ ), Inner terms ( $b \times 9b$ ), Last terms ( $b \times 12$ ).

Do I need to combine like terms at the end?

In this problem, all terms are different ( $9b^2, 3ab, 4a, 12b$ ), so no combining is needed. Just arrange them in descending order of powers.

What if I get the order of terms wrong in my final answer?

The mathematical value is the same regardless of order, but it's good practice to write terms in descending order: highest degree terms first, like $9b^2 + 3ab + 4a + 12b$ .

How can I check if my expansion is correct?

Try factoring your answer back to the original form, or substitute simple values like $a = 3, b = 1$ into both the original expression and your answer to see if they match.

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