Solve: Product of 1/2 × 2/3 × 1/2 Using Substitutive Property

Q: What is the substitutive property in fraction multiplication?

The substitutive property allows us to group and rearrange factors without changing the result. In $ \frac{1}{2} \times \frac{2}{3} \times \frac{1}{2} $, we can substitute and regroup as needed while following multiplication rules.

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

Unlike addition, multiplication of fractions is much simpler! You multiply numerator × numerator and denominator × denominator. No need for common denominators like in addition.

Q: How do I know when to simplify the answer?

Always simplify when possible! Find the GCD of numerator and denominator. In our case, $ \frac{2}{12} $ simplifies to $ \frac{1}{6} $ because both 2 and 12 divide by 2.

Q: Can I cancel numbers before multiplying?

Yes! You can cancel common factors diagonally. For example, the 2 in the numerator of $ \frac{2}{3} $ cancels with one 2 in a denominator, making calculation easier.

Fraction Multiplication with Substitutive Property

Solve the exercise using the substitutive property:

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

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Solve using the distributive property

00:03 We will use the distributive property and arrange the exercise in a way that's convenient to solve

00:12 Make sure to multiply numerator by numerator and denominator by denominator

00:20 Let's calculate the multiplications

00:31 Reduce the fraction as much as possible

00:34 Make sure to divide both numerator and denominator

00:37 And this is the solution to the problem

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Solve the exercise using the substitutive property:

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

Step-by-step solution

To solve the problem $\frac{1}{2} \times \frac{2}{3} \times \frac{1}{2}$ , follow these steps:

Step 1: Multiply the numerators

We multiply the numerators of the fractions together: $1 \times 2 \times 1 = 2$ .

Step 2: Multiply the denominators

Similarly, multiply the denominators of the fractions: $2 \times 3 \times 2 = 12$ .

Step 3: Form the resulting fraction

After multiplying the numerators and denominators, we form the fraction: $\frac{2}{12}$ .

Step 4: Simplify the fraction

We simplify $\frac{2}{12}$ by finding the greatest common divisor (GCD) of 2 and 12, which is 2. Dividing both the numerator and denominator by 2, we get:

$\frac{2}{12} = \frac{2 \div 2}{12 \div 2} = \frac{1}{6}$ .

Thus, the correct solution is $\frac{1}{6}$ .

Final Answer

$\frac{1}{6}$

Key Points to Remember

Essential concepts to master this topic

Rule: Multiply numerators together, then denominators together for fractions
Technique: Calculate $1 \times 2 \times 1 = 2$ and $2 \times 3 \times 2 = 12$
Check: Simplify $\frac{2}{12}$ by dividing both by GCD of 2 to get $\frac{1}{6}$ ✓

Common Mistakes

Avoid these frequent errors

Adding fractions instead of multiplying
Don't add numerators and denominators like $\frac{1+2+1}{2+3+2} = \frac{4}{7}$ = wrong answer! This confuses addition and multiplication rules. Always multiply numerators together and denominators together separately.

Practice Quiz

Test your knowledge with interactive questions

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

FAQ

Everything you need to know about this question

What is the substitutive property in fraction multiplication?

The substitutive property allows us to group and rearrange factors without changing the result. In $\frac{1}{2} \times \frac{2}{3} \times \frac{1}{2}$ , we can substitute and regroup as needed while following multiplication rules.

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

Unlike addition, multiplication of fractions is much simpler! You multiply numerator × numerator and denominator × denominator. No need for common denominators like in addition.

How do I know when to simplify the answer?

Always simplify when possible! Find the GCD of numerator and denominator. In our case, $\frac{2}{12}$ simplifies to $\frac{1}{6}$ because both 2 and 12 divide by 2.

Can I cancel numbers before multiplying?

Yes! You can cancel common factors diagonally. For example, the 2 in the numerator of $\frac{2}{3}$ cancels with one 2 in a denominator, making calculation easier.

What if I get a whole number instead of a fraction?

Sometimes fraction multiplication gives whole numbers! This happens when the numerator is divisible by the denominator. Always double-check your work by ensuring you followed all multiplication steps correctly.

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