Solve 1/2 × 1/3 × 1/2: Applying the Substitutive Property

Q: Why do I multiply the denominators instead of adding them?

In fraction multiplication, we're finding parts of parts. $ \frac{1}{2} \times \frac{1}{3} $ means "one-third of one-half," which creates smaller pieces that need a larger denominator!

Q: Can I simplify before multiplying?

Yes! Look for common factors you can cancel. Here, both fractions have $ \frac{1}{2} $, so you could write it as $ \left(\frac{1}{2}\right)^2 \times \frac{1}{3} $, but the final answer stays the same.

Q: How do I know if my fraction answer is in simplest form?

Check if the numerator and denominator share any common factors. Since 1 only has itself as a factor, and 12 = 2² × 3, they share no common factors, so $ \frac{1}{12} $ is fully simplified.

Q: What's the substitutive property mentioned in the problem?

The substitutive property means you can substitute equivalent expressions. Here, it likely refers to recognizing that $ \frac{1}{2} \times \frac{1}{3} \times \frac{1}{2} $ can be rewritten and solved systematically.

Q: Why is my answer so much smaller than the original fractions?

When multiplying fractions less than 1, the result gets smaller! You're taking a fraction of a fraction, which creates even tinier pieces. $ \frac{1}{12} $ is much smaller than $ \frac{1}{2} $ or $ \frac{1}{3} $.

Fraction Multiplication with Three Factors

Solve the exercise using the substitutive property:

$\frac{1}{2}\times\frac{1}{3}\times\frac{1}{2}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Solve using the distributive law

00:03 We'll use the distributive law and arrange the exercise in a convenient way to solve

00:09 Make sure to multiply numerator by numerator and denominator by denominator

00:18 Let's calculate the multiplications

00:29 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

Solve the exercise using the substitutive property:

$\frac{1}{2}\times\frac{1}{3}\times\frac{1}{2}=$

Step-by-step solution

Let's solve this problem step-by-step:

Step 1: Multiply the numerators of the fractions. We have $1 \times 1 \times 1 = 1$ .
Step 2: Multiply the denominators of the fractions. We have $2 \times 3 \times 2 = 12$ .
Step 3: Combine the results to form a fraction: $\frac{1}{12}$ .
Step 4: Simplify the fraction if possible. Since the fraction $\frac{1}{12}$ is already in its simplest form, no further simplification is needed.

Thus, the product of $\frac{1}{2} \times \frac{1}{3} \times \frac{1}{2}$ is $\frac{1}{12}$ .

The correct answer choice is $\frac{1}{12}$ , matching choice 4.

Therefore, the solution to the problem is $\frac{1}{12}$ .

Final Answer

$\frac{1}{12}$

Key Points to Remember

Essential concepts to master this topic

Rule: Multiply all numerators together, then all denominators together
Technique: $\frac{1}{2} \times \frac{1}{3} \times \frac{1}{2} = \frac{1 \times 1 \times 1}{2 \times 3 \times 2}$
Check: Verify $\frac{1}{12}$ cannot be simplified further since 1 and 12 share no common factors ✓

Common Mistakes

Avoid these frequent errors

Adding denominators instead of multiplying them
Don't add denominators like 2 + 3 + 2 = 7 to get $\frac{1}{7}$ ! This gives a completely wrong result because fraction multiplication requires multiplying denominators. Always multiply: 2 × 3 × 2 = 12.

Practice Quiz

Test your knowledge with interactive questions

$\frac{2}{3}\times\frac{5}{7}=$

FAQ

Everything you need to know about this question

Why do I multiply the denominators instead of adding them?

In fraction multiplication, we're finding parts of parts. $\frac{1}{2} \times \frac{1}{3}$ means "one-third of one-half," which creates smaller pieces that need a larger denominator!

Can I simplify before multiplying?

Yes! Look for common factors you can cancel. Here, both fractions have $\frac{1}{2}$ , so you could write it as $\left(\frac{1}{2}\right)^2 \times \frac{1}{3}$ , but the final answer stays the same.

How do I know if my fraction answer is in simplest form?

Check if the numerator and denominator share any common factors. Since 1 only has itself as a factor, and 12 = 2² × 3, they share no common factors, so $\frac{1}{12}$ is fully simplified.

What's the substitutive property mentioned in the problem?

The substitutive property means you can substitute equivalent expressions. Here, it likely refers to recognizing that $\frac{1}{2} \times \frac{1}{3} \times \frac{1}{2}$ can be rewritten and solved systematically.

Why is my answer so much smaller than the original fractions?

When multiplying fractions less than 1, the result gets smaller! You're taking a fraction of a fraction, which creates even tinier pieces. $\frac{1}{12}$ is much smaller than $\frac{1}{2}$ or $\frac{1}{3}$ .

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