Solve: 2/4 × 2/3 × 2/4 Using the Substitutive Property

Q: Can I simplify the fractions before multiplying?

Yes! This makes the problem much easier. Since $ \frac{2}{4} = \frac{1}{2} $, you can solve $ \frac{1}{2} \times \frac{2}{3} \times \frac{1}{2} $ instead.

Q: Why do we multiply straight across instead of finding common denominators?

Common denominators are only needed for adding or subtracting fractions. When multiplying, we always multiply numerators together and denominators together - no common denominators needed!

Q: How do I find the GCD to simplify my answer?

List the factors of both numbers. For $ \frac{8}{48} $: factors of 8 are {1,2,4,8} and factors of 48 include {1,2,4,8...}. The greatest common factor is 8.

Q: What does the substitutive property have to do with this problem?

The substitutive property allows us to replace $ \frac{2}{4} $ with $ \frac{1}{2} $ since they're equal. This substitution makes calculations easier while keeping the same answer!

Q: Is there a shortcut when the same fraction appears multiple times?

Yes! Since $ \frac{2}{4} $ appears twice, you can write it as $ \left(\frac{2}{4}\right)^2 \times \frac{2}{3} $. This gives $ \frac{4}{16} \times \frac{2}{3} = \frac{1}{4} \times \frac{2}{3} = \frac{1}{6} $.

Fraction Multiplication with Repeated Factors

Solve the exercise using the substitutive property:

$\frac{2}{4}\times\frac{2}{3}\times\frac{2}{4}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:08 Let's solve this using the distributive law.

00:12 Arrange the exercise using the distributive law, so it's easy to solve.

00:19 Remember to multiply the numerators together, and the denominators together.

00:26 Now, go ahead and calculate the multiplications.

00:38 Try to reduce the fraction as much as possible.

00:43 Make sure you divide both the numerator and the denominator.

00:49 And that's how we find the solution to this 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:

$\frac{2}{4}\times\frac{2}{3}\times\frac{2}{4}=$

Step-by-step solution

To solve this problem, we need to multiply the three given fractions:
$\frac{2}{4} \times \frac{2}{3} \times \frac{2}{4}$

The steps involved in multiplying these fractions are as follows:

Step 1: Multiply the numerators. The numerators are 2, 2, and 2. Multiplying these together gives:
$2 \times 2 \times 2 = 8$ .
Step 2: Multiply the denominators. The denominators are 4, 3, and 4. Multiplying these together gives:
$4 \times 3 \times 4 = 48$ .
Step 3: Combine the results to form a new fraction:
$\frac{8}{48}$ .
Step 4: Simplify the fraction. To simplify $\frac{8}{48}$ , we find the greatest common divisor (GCD) of 8 and 48, which is 8. Divide both the numerator and denominator by 8:
$\frac{8 \div 8}{48 \div 8} = \frac{1}{6}$ .

Therefore, the solution to the given exercise is $\frac{1}{6}$ .

Final Answer

$\frac{1}{6}$

Key Points to Remember

Essential concepts to master this topic

Rule: Multiply numerators together and denominators together when multiplying fractions
Technique: Simplify before multiplying: $\frac{2}{4} = \frac{1}{2}$ to make calculations easier
Check: Final answer should be in lowest terms: $\frac{8}{48} = \frac{1}{6}$ by dividing by GCD ✓

Common Mistakes

Avoid these frequent errors

Adding fractions instead of multiplying
Don't add denominators and numerators like $\frac{2+2+2}{4+3+4} = \frac{6}{11}$ ! This mixes addition and multiplication rules, giving a completely wrong result. Always multiply straight across: numerators together and denominators together.

Practice Quiz

Test your knowledge with interactive questions

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

FAQ

Everything you need to know about this question

Can I simplify the fractions before multiplying?

Yes! This makes the problem much easier. Since $\frac{2}{4} = \frac{1}{2}$ , you can solve $\frac{1}{2} \times \frac{2}{3} \times \frac{1}{2}$ instead.

Why do we multiply straight across instead of finding common denominators?

Common denominators are only needed for adding or subtracting fractions. When multiplying, we always multiply numerators together and denominators together - no common denominators needed!

How do I find the GCD to simplify my answer?

List the factors of both numbers. For $\frac{8}{48}$ : factors of 8 are {1,2,4,8} and factors of 48 include {1,2,4,8...}. The greatest common factor is 8.

What does the substitutive property have to do with this problem?

The substitutive property allows us to replace $\frac{2}{4}$ with $\frac{1}{2}$ since they're equal. This substitution makes calculations easier while keeping the same answer!

Is there a shortcut when the same fraction appears multiple times?

Yes! Since $\frac{2}{4}$ appears twice, you can write it as $\left(\frac{2}{4}\right)^2 \times \frac{2}{3}$ . This gives $\frac{4}{16} \times \frac{2}{3} = \frac{1}{4} \times \frac{2}{3} = \frac{1}{6}$ .

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