Solve the Fraction Equation: Finding the Factor in 5/7 × ? = 1/3

Q: Why do I multiply by the reciprocal instead of dividing?

Dividing by a fraction is the same as multiplying by its reciprocal! Since $ \frac{1}{3} ÷ \frac{5}{7} $ is hard to visualize, we flip $ \frac{5}{7} $ to get $ \frac{7}{5} $ and multiply instead.

Q: How do I find the reciprocal of a fraction?

Simply flip the numerator and denominator! The reciprocal of $ \frac{5}{7} $ is $ \frac{7}{5} $. When you multiply a fraction by its reciprocal, you always get 1.

Q: What if I get a different answer choice?

Double-check your multiplication: $ \frac{1}{3} \times \frac{7}{5} = \frac{1 \times 7}{3 \times 5} = \frac{7}{15} $. Make sure you're multiplying numerator × numerator and denominator × denominator.

Q: Can I simplify the answer further?

Check if the numerator and denominator share any common factors. Since 7 and 15 share no common factors (7 is prime, 15 = 3×5), $ \frac{7}{15} $ is already in simplest form.

Q: How do I verify my answer is correct?

Substitute back into the original equation: $ \frac{5}{7} \times \frac{7}{15} $. The 7's cancel out, leaving $ \frac{5}{15} = \frac{1}{3} $ ✓

Fraction Division with Reciprocal Multiplication

$\frac{5}{7}\times?=\frac{1}{3}$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find the unknown

00:03 Isolate the unknown

00:06 Write division as multiplication by the reciprocal, and calculate the products

00:09 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

$\frac{5}{7}\times?=\frac{1}{3}$

Step-by-step solution

To solve this problem, we'll proceed with the following steps:

Step 1: Set up the equation: $\frac{5}{7} \times x = \frac{1}{3}$ .
Step 2: Isolate $x$ by dividing both sides by $\frac{5}{7}$ , which is equivalent to multiplying both sides by the reciprocal, $\frac{7}{5}$ .
Step 3: Simplify the resulting expression to find $x$ .

Now, applying these steps:
Step 1: We have the equation $\frac{5}{7} \times x = \frac{1}{3}$ .
Step 2: Multiply both sides by $\frac{7}{5}$ to isolate $x$ :
$x = \frac{1}{3} \times \frac{7}{5}$ .
Step 3: Simplify the calculation:
$x = \frac{1 \times 7}{3 \times 5} = \frac{7}{15}$ .

Therefore, the solution to the problem is $x = \frac{7}{15}$ .

Final Answer

$\frac{7}{15}$

Key Points to Remember

Essential concepts to master this topic

Rule: To isolate unknown factor, divide by known factor
Technique: Division by $\frac{5}{7}$ equals multiplication by $\frac{7}{5}$
Check: Verify $\frac{5}{7} \times \frac{7}{15} = \frac{35}{105} = \frac{1}{3}$ ✓

Common Mistakes

Avoid these frequent errors

Adding fractions instead of using division
Don't try to add $\frac{5}{7} + ? = \frac{1}{3}$ by subtracting = wrong operation! This confuses addition with multiplication. Always use the reciprocal when dividing by a fraction to isolate the unknown factor.

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 by the reciprocal instead of dividing?

Dividing by a fraction is the same as multiplying by its reciprocal! Since $\frac{1}{3} ÷ \frac{5}{7}$ is hard to visualize, we flip $\frac{5}{7}$ to get $\frac{7}{5}$ and multiply instead.

How do I find the reciprocal of a fraction?

Simply flip the numerator and denominator! The reciprocal of $\frac{5}{7}$ is $\frac{7}{5}$ . When you multiply a fraction by its reciprocal, you always get 1.

What if I get a different answer choice?

Double-check your multiplication: $\frac{1}{3} \times \frac{7}{5} = \frac{1 \times 7}{3 \times 5} = \frac{7}{15}$ . Make sure you're multiplying numerator × numerator and denominator × denominator.

Can I simplify the answer further?

Check if the numerator and denominator share any common factors. Since 7 and 15 share no common factors (7 is prime, 15 = 3×5), $\frac{7}{15}$ is already in simplest form.

How do I verify my answer is correct?

Substitute back into the original equation: $\frac{5}{7} \times \frac{7}{15}$ . The 7's cancel out, leaving $\frac{5}{15} = \frac{1}{3}$ ✓

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