Verify the Fraction Simplification: (3·4)/(8·3) = 1/2

Q: Why can I cancel the 3 and 4 in this fraction?

You can cancel because the same factors appear in both numerator and denominator. The 3 appears once on top and once on bottom, same with the 4. This is like $ \frac{3 \times 4}{3 \times 4 \times 2} = \frac{12}{24} = \frac{1}{2} $

Q: What if the numbers don't obviously have common factors?

Look for ways to factor the numbers first! Like in this problem, we rewrote 8 as 2×4 to reveal the common factor of 4. Try breaking down larger numbers into their prime factors.

Q: Can I cancel across multiplication signs?

No! You can only cancel factors that are multiplied in both numerator and denominator. Never cancel across addition, subtraction, or between separate terms.

Q: How do I know when a fraction is fully simplified?

A fraction is fully simplified when the numerator and denominator share no common factors except 1. Check by finding the GCD (Greatest Common Divisor) - it should be 1.

Q: What's the difference between canceling and cross-multiplying?

Canceling removes common factors within one fraction, while cross-multiplying is used to solve equations with two fractions set equal. This problem uses canceling, not cross-multiplying!

Fraction Simplification with Common Factor Cancellation

Determine if the simplification below is correct:

$\frac{3\cdot4}{8\cdot3}=\frac{1}{2}$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Determine if the reduction is correct

00:06 Let's reduce what we can

00:21 Let's break down 8 into factors 4 and 2

00:24 Let's reduce what we can

00:30 Let's compare the expressions

00:33 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

Determine if the simplification below is correct:

$\frac{3\cdot4}{8\cdot3}=\frac{1}{2}$

Step-by-step solution

We simplify the expression on the left side of the approximate equality.

First let's consider the fact that the number 8 is a multiple of the number 4:

$8=2\cdot4$
Therefore, we will return to the problem in question and present the number 8 as a multiple of the number 4, then we will simplify the fraction:

$\frac{3\cdot4}{\underline{8}\cdot3}\stackrel{?}{= }\frac{1}{2}\\ \downarrow\\ \frac{3\cdot4}{\underline{2\cdot4}\cdot3}\stackrel{?}{= }\frac{1}{2}\\ \downarrow\\ \frac{\textcolor{blue}{\not{3}}\cdot\textcolor{red}{\not{4}}}{2\cdot\textcolor{red}{\not{4}}\cdot\textcolor{blue}{\not{3}}}\stackrel{?}{= }\frac{1}{2} \\ \downarrow\\ \frac{1}{2}\stackrel{!}{= }\frac{1}{2}$
Therefore, the described simplification is correct.

That is, the correct answer is A.

Final Answer

True

Key Points to Remember

Essential concepts to master this topic

Rule: Cancel common factors in numerator and denominator before multiplying
Technique: Rewrite 8 as 2×4, then cancel: $\frac{3 \cdot \cancel{4}}{2 \cdot \cancel{4} \cdot 3} = \frac{1}{2}$
Check: Verify by computing original: 3×4=12, 8×3=24, so 12/24 = 1/2 ✓

Common Mistakes

Avoid these frequent errors

Multiplying numbers before looking for common factors
Don't calculate 3×4=12 and 8×3=24 first = harder simplification! This creates larger numbers that are harder to factor. Always look for common factors like the 4 and 3 before multiplying anything.

Practice Quiz

Test your knowledge with interactive questions

Complete the corresponding expression for the denominator

$\frac{12ab}{?}=1$

FAQ

Everything you need to know about this question

Why can I cancel the 3 and 4 in this fraction?

You can cancel because the same factors appear in both numerator and denominator. The 3 appears once on top and once on bottom, same with the 4. This is like $\frac{3 \times 4}{3 \times 4 \times 2} = \frac{12}{24} = \frac{1}{2}$

What if the numbers don't obviously have common factors?

Look for ways to factor the numbers first! Like in this problem, we rewrote 8 as 2×4 to reveal the common factor of 4. Try breaking down larger numbers into their prime factors.

Can I cancel across multiplication signs?

No! You can only cancel factors that are multiplied in both numerator and denominator. Never cancel across addition, subtraction, or between separate terms.

How do I know when a fraction is fully simplified?

A fraction is fully simplified when the numerator and denominator share no common factors except 1. Check by finding the GCD (Greatest Common Divisor) - it should be 1.

What's the difference between canceling and cross-multiplying?

Canceling removes common factors within one fraction, while cross-multiplying is used to solve equations with two fractions set equal. This problem uses canceling, not cross-multiplying!

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