Adding Fractions with Different Denominators: 1/3 and 5/9

Q: Why can't I just add 1 + 5 and 3 + 9?

Because fractions represent parts of different-sized wholes! Adding $ \frac{1}{3} $ (thirds) to $ \frac{5}{9} $ (ninths) is like adding apples to oranges. You need the same denominator first.

Q: How do I know which denominator to use as the common one?

Use the least common multiple (LCM) of the denominators. Since 9 is a multiple of 3, the LCM of 3 and 9 is 9. Always choose the smallest number that both denominators divide into evenly.

Q: What if my answer doesn't match any of the choices?

Double-check your conversion! Make sure you multiplied both the numerator and denominator by the same number. If $ \frac{1}{3} \times \frac{3}{3} = \frac{3}{9} $, then $ \frac{3}{9} + \frac{5}{9} = \frac{8}{9} $.

Q: Should I always simplify my final answer?

Yes, when possible! However, $ \frac{8}{9} $ is already in simplest form since 8 and 9 share no common factors other than 1. Always check if you can reduce your fraction.

Q: Can I convert both fractions instead of just one?

Absolutely! You could convert both to eighteenths: $ \frac{1}{3} = \frac{6}{18} $ and $ \frac{5}{9} = \frac{10}{18} $, giving $ \frac{16}{18} = \frac{8}{9} $. However, using the smaller common denominator makes calculations easier.

Fraction Addition with Common Denominator Conversion

$\frac{1}{3}+\frac{5}{9}=$

❤️ Continue Your Math Journey!

We have hundreds of course questions with personalized recommendations + Account 100% premium

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Solve

00:03 Let's multiply the fraction by 6 and 2 respectively to find the common denominator

00:07 Make sure to multiply both numerator and denominator

00:19 Let's calculate the multiplications

00:26 Let's add with the common denominator

00:31 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

$\frac{1}{3}+\frac{5}{9}=$

Step-by-step solution

Let's solve the problem by finding the sum of $\frac{1}{3}$ and $\frac{5}{9}$ .

Step 1: Convert $\frac{1}{3}$ to have the same denominator as $\frac{5}{9}$ .
To convert $\frac{1}{3}$ to a fraction with a denominator of $9$ , multiply both the numerator and the denominator by $3$ :
$\frac{1}{3} = \frac{1 \times 3}{3 \times 3} = \frac{3}{9}$ .
Step 2: Add the numerators of the two fractions now that they have the same denominator:
$\frac{3}{9} + \frac{5}{9} = \frac{3 + 5}{9} = \frac{8}{9}$ .
Step 3: Simplify the result if possible. In this case, $\frac{8}{9}$ is already in its simplest form.
Step 4: Verify the result against the given choices. The correct choice is $\frac{8}{9}$ , which matches the given choice $2$ .

Therefore, the solution to the problem is $\frac{8}{9}$ .

Final Answer

$\frac{8}{9}$

Key Points to Remember

Essential concepts to master this topic

Common Denominator Rule: Fractions need same denominator before adding numerators
Conversion Technique: Multiply $\frac{1}{3}$ by $\frac{3}{3}$ to get $\frac{3}{9}$
Verification Check: Convert answer back: $\frac{8}{9} = \frac{3}{9} + \frac{5}{9}$ ✓

Common Mistakes

Avoid these frequent errors

Adding numerators and denominators separately
Don't add $\frac{1}{3} + \frac{5}{9}$ as $\frac{1+5}{3+9} = \frac{6}{12}$ ! This ignores the fundamental rule that fractions must have the same denominator before adding. Always convert to a common denominator first, then add only the numerators.

Practice Quiz

Test your knowledge with interactive questions

You have a pair of denominators, what is their least common multiple?

$\boxed 2~~~\boxed5$

FAQ

Everything you need to know about this question

Why can't I just add 1 + 5 and 3 + 9?

Because fractions represent parts of different-sized wholes! Adding $\frac{1}{3}$ (thirds) to $\frac{5}{9}$ (ninths) is like adding apples to oranges. You need the same denominator first.

How do I know which denominator to use as the common one?

Use the least common multiple (LCM) of the denominators. Since 9 is a multiple of 3, the LCM of 3 and 9 is 9. Always choose the smallest number that both denominators divide into evenly.

What if my answer doesn't match any of the choices?

Double-check your conversion! Make sure you multiplied both the numerator and denominator by the same number. If $\frac{1}{3} \times \frac{3}{3} = \frac{3}{9}$ , then $\frac{3}{9} + \frac{5}{9} = \frac{8}{9}$ .

Should I always simplify my final answer?

Yes, when possible! However, $\frac{8}{9}$ is already in simplest form since 8 and 9 share no common factors other than 1. Always check if you can reduce your fraction.

Can I convert both fractions instead of just one?

Absolutely! You could convert both to eighteenths: $\frac{1}{3} = \frac{6}{18}$ and $\frac{5}{9} = \frac{10}{18}$ , giving $\frac{16}{18} = \frac{8}{9}$ . However, using the smaller common denominator makes calculations easier.

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Simple Fractions questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime