Adding Fractions: Calculating 2/3 + 1/5 in Homework Completion

Q: Why can't I just add the numerators and denominators separately?

Fractions represent parts of a whole. Adding $ \frac{2}{3} + \frac{1}{5} $ as $ \frac{3}{8} $ is like adding 2 slices of a 3-piece pizza to 1 slice of a 5-piece pizza and claiming you have 3 slices of an 8-piece pizza - it doesn't make sense!

Q: How do I find the LCD of 3 and 5?

Since 3 and 5 are both prime numbers, their LCD is simply their product: 3 × 5 = 15. For other numbers, find the smallest number that both denominators divide into evenly.

Q: What if my answer seems too big compared to the original fractions?

Check if your answer makes sense! $ \frac{2}{3} $ is about 0.67 and $ \frac{1}{5} $ is 0.2, so their sum should be about 0.87. $ \frac{13}{15} $ ≈ 0.87, which is reasonable!

Q: Do I need to simplify my final answer?

Always check if your answer can be simplified! In this case, 13 and 15 share no common factors other than 1, so $ \frac{13}{15} $ is already in lowest terms.

Q: Can Sarah complete more than 100% of her homework?

Yes, mathematically! $ \frac{13}{15} $ is about 87%, which means Sarah still has $ \frac{2}{15} $ of her homework left to complete.

Fraction Addition with Unlike Denominators

Sarah is doing her homework.

In the first hour, she completes

$\frac{2}{3}$ of the work, while in the second hour she completes $\frac{1}{5}$ of her homework.

How much of her total homework has Sarah done?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Which part of the homework did Sarah do?

00:03 Let's connect the parts

00:06 Multiply each fraction by the second denominator to find the common denominator

00:09 Remember to multiply both numerator and denominator

00:24 Let's calculate the multiplications

00:32 Add under the common denominator

00:37 Calculate the numerator

00:42 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

Sarah is doing her homework.

In the first hour, she completes

$\frac{2}{3}$ of the work, while in the second hour she completes $\frac{1}{5}$ of her homework.

How much of her total homework has Sarah done?

Step-by-step solution

To solve this problem, we need to determine how much of her homework Sarah has completed by adding the fractions $\frac{2}{3}$ and $\frac{1}{5}$ .

First, find the least common denominator (LCD) for the fractions. The denominators are 3 and 5. The LCD of 3 and 5 is 15.

Next, convert each fraction to an equivalent fraction with the denominator of 15:

$\frac{2}{3}$ becomes $\frac{2 \times 5}{3 \times 5} = \frac{10}{15}$
$\frac{1}{5}$ becomes $\frac{1 \times 3}{5 \times 3} = \frac{3}{15}$

Now, add the two fractions:

$\frac{10}{15} + \frac{3}{15} = \frac{10 + 3}{15} = \frac{13}{15}$

Thus, Sarah has completed $\frac{13}{15}$ of her total homework.

The correct answer is therefore $\frac{13}{15}$ .

Final Answer

$\frac{13}{15}$

Key Points to Remember

Essential concepts to master this topic

Rule: Find LCD before adding fractions with different denominators
Technique: Convert $\frac{2}{3} = \frac{10}{15}$ and $\frac{1}{5} = \frac{3}{15}$
Check: Add numerators only: 10 + 3 = 13 over common denominator 15 ✓

Common Mistakes

Avoid these frequent errors

Adding denominators together
Don't add $\frac{2}{3} + \frac{1}{5} = \frac{3}{8}$ ! Adding denominators creates meaningless fractions that don't represent the actual sum. Always find the LCD first, convert both fractions, then add only the numerators.

Practice Quiz

Test your knowledge with interactive questions

$\frac{2}{4}+\frac{1}{4}=$ $$

FAQ

Everything you need to know about this question

Why can't I just add the numerators and denominators separately?

Fractions represent parts of a whole. Adding $\frac{2}{3} + \frac{1}{5}$ as $\frac{3}{8}$ is like adding 2 slices of a 3-piece pizza to 1 slice of a 5-piece pizza and claiming you have 3 slices of an 8-piece pizza - it doesn't make sense!

How do I find the LCD of 3 and 5?

Since 3 and 5 are both prime numbers, their LCD is simply their product: 3 × 5 = 15. For other numbers, find the smallest number that both denominators divide into evenly.

What if my answer seems too big compared to the original fractions?

Check if your answer makes sense! $\frac{2}{3}$ is about 0.67 and $\frac{1}{5}$ is 0.2, so their sum should be about 0.87. $\frac{13}{15}$ ≈ 0.87, which is reasonable!

Do I need to simplify my final answer?

Always check if your answer can be simplified! In this case, 13 and 15 share no common factors other than 1, so $\frac{13}{15}$ is already in lowest terms.

Can Sarah complete more than 100% of her homework?

Yes, mathematically! $\frac{13}{15}$ is about 87%, which means Sarah still has $\frac{2}{15}$ of her homework left to complete.

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