Solve the Fraction Equation: Finding the Denominator in 10a/? = 6a/3

Q: Why don't I include the variable 'a' in my answer?

Great question! The variable 'a' appears in both numerators and will cancel out when the fractions are equal. You only need the numerical denominator that makes the equation balance.

Q: How do I know which fraction to simplify first?

Always start with the side that has obvious common factors. Here, $ \frac{6a}{3} $ simplifies easily since 6 and 3 share a factor of 3.

Q: Can I solve this without cross-multiplying?

Yes! You can also think: "What number times 2 gives me 10?" Since 10 ÷ 2 = 5, the missing denominator is 5.

Q: What if the equation had different variables in numerators?

If the numerators had different variables (like 10x and 6y), you'd need to keep those variables in your solution. But here, both have 'a', so they cancel out.

Q: Is there a pattern for these fraction equations?

Yes! When fractions are equal, their cross products are equal. So if $ \frac{10a}{5} = \frac{6a}{3} $, then 10a × 3 = 6a × 5 = 30a.

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Complete the appropriate denominator

00:06 Let's factor 6 into factors 3 and 2

00:14 Let's reduce what we can

00:21 We want to isolate the denominator, so we'll multiply by the denominator

00:30 Let's isolate the denominator

00:38 Let's reduce what we can

00:46 Let's calculate the quotient

00:50 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

Complete the corresponding expression for the denominator

$\frac{10a}{?}=\frac{6a}{3}$

2

Step-by-step solution

After examining the problem, note that the fraction on the right side can be reduced:

$\frac{10a}{?}=\frac{\not{6}a}{\not{3}} \\ \downarrow\\ \frac{10a}{?}=\frac{2a}{1}$

Using the following factorisation:

$6=2\cdot3$

Remember the process of reducing a fraction:

In both the numerator of the expression on the right side as well as in the numerator of the expression on the left side the expression $a$ is present. Therefore in the expression we are looking for there are no variables (given that we don't want to reduce them from the expression in the numerator on the left side),

Next, we ask which number is placed in the denominator of the expression on the left side so that when reduced with the number 10 it will yield the number 2. The answer to this is of course - the number 5,

Due to the fact that:

$10=2\cdot 5$

Verify that this choice gives us the expression on the right side:

$\frac{10a}{?}=\frac{2a}{1} \\ \downarrow\\ \frac{\not{10}a}{\textcolor{red}{\not{5}}}\stackrel{?}{= }\frac{2a}{1} \\ \downarrow\\ \boxed{\frac{2a}{1}\stackrel{!}{= }\frac{2a}{1}}$

Therefore this choice is indeed correct.

In other words - the correct answer is answer C.

3

Final Answer

$5$

Key Points to Remember

Essential concepts to master this topic

Cross-Multiplication Rule: For fractions $\frac{a}{b} = \frac{c}{d}$ , multiply diagonally
Simplification Technique: Reduce $\frac{6a}{3} = \frac{2a}{1}$ by dividing by 3
Verification Check: Substitute back: $\frac{10a}{5} = 2a$ equals $\frac{6a}{3} = 2a$ ✓

Common Mistakes

Avoid these frequent errors

Keeping variables in the unknown denominator
Don't include 'a' in the missing denominator like 5a = wrong answer! This creates an unbalanced equation where variables don't cancel properly. Always solve for just the numerical value - the variable 'a' appears in both numerators and cancels out.

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 don't I include the variable 'a' in my answer?

+

Great question! The variable 'a' appears in both numerators and will cancel out when the fractions are equal. You only need the numerical denominator that makes the equation balance.

How do I know which fraction to simplify first?

+

Always start with the side that has obvious common factors. Here, $\frac{6a}{3}$ simplifies easily since 6 and 3 share a factor of 3.

Can I solve this without cross-multiplying?

+

Yes! You can also think: "What number times 2 gives me 10?" Since 10 ÷ 2 = 5, the missing denominator is 5.

What if the equation had different variables in numerators?

+

If the numerators had different variables (like 10x and 6y), you'd need to keep those variables in your solution. But here, both have 'a', so they cancel out.

How can I check my work without substituting?

+

Cross-multiply your final answer: 10a × 1 should equal 2a × 5. Since 10a = 10a, your answer is correct!

Is there a pattern for these fraction equations?

+

Yes! When fractions are equal, their cross products are equal. So if $\frac{10a}{5} = \frac{6a}{3}$ , then 10a × 3 = 6a × 5 = 30a.

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Solve the Fraction Equation: Finding the Denominator in 10a/? = 6a/3

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