Solve: 1/2 + 3/4 + 2/5 - Adding Three Fractions with Different Denominators

Q: Why can't I just use any common denominator?

You can, but using the LCM keeps numbers smaller and easier to work with. Using 40 instead of 20 would give $ \frac{66}{40} $, which is harder to simplify!

Q: What if I forget to multiply both numerator and denominator?

You'll change the value of the fraction! Remember: whatever you do to the bottom, you must do to the top. $ \frac{1}{2} \times \frac{10}{10} = \frac{10}{20} $ keeps the same value.

Q: How do I know if my final answer needs simplifying?

Check if the numerator and denominator share any common factors. For $ \frac{33}{20} $, since 33 = 3 × 11 and 20 = 4 × 5, they share no common factors, so it's already simplified!

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Solve

00:03 We want to find the least common denominator

00:06 Therefore we'll multiply by 10, 5, and 4 respectively to find the common denominator

00:09 Remember to multiply both numerator and denominator

00:31 Let's calculate the multiplications

00:43 Let's add under the common denominator

00:51 Let's calculate the numerator

00:58 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

$\frac{1}{2}+\frac{3}{4}+\frac{2}{5}=$

2

Step-by-step solution

To solve this problem, we will add the fractions by finding a common denominator:

Step 1: Identify the denominators: 2, 4, and 5. Find the least common multiple (LCM) of these denominators.
Step 2: The LCM of 2, 4, and 5 is 20. Use this as the common denominator.
Step 3: Convert each fraction to have a denominator of 20:

Convert $\frac{1}{2}$ to $\frac{10}{20}$ by multiplying the numerator and denominator by 10.
Convert $\frac{3}{4}$ to $\frac{15}{20}$ by multiplying the numerator and denominator by 5.
Convert $\frac{2}{5}$ to $\frac{8}{20}$ by multiplying the numerator and denominator by 4.

Step 4: Add the three fractions: $\frac{10}{20} + \frac{15}{20} + \frac{8}{20}$ .
Step 5: Add the numerators: $10 + 15 + 8 = 33$ .
Step 6: Write the result as a single fraction: $\frac{33}{20}$ .
Step 7: Check if the fraction can be simplified. Since 33 and 20 have no common factors other than 1, $\frac{33}{20}$ is in its simplest form.
Step 8: Confirm this matches choice (4): $\frac{33}{20}$ .

Therefore, the solution to the problem is $\frac{33}{20}$ .

3

Final Answer

$\frac{33}{20}$

Key Points to Remember

Essential concepts to master this topic

Common Denominator: Find LCM of all denominators before adding
Conversion: $\frac{1}{2} = \frac{10}{20}$ by multiplying by $\frac{10}{10}$
Verification: Check if final answer can be simplified by finding GCD ✓

Common Mistakes

Avoid these frequent errors

Adding numerators and denominators separately
Don't add $\frac{1}{2} + \frac{3}{4} + \frac{2}{5}$ as $\frac{6}{11}$ ! This ignores the different denominators and gives completely wrong results. Always find a common denominator first, 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

How do I find the LCM of three different numbers?

+

List multiples of the largest number first! For 2, 4, and 5: multiples of 5 are 5, 10, 15, 20... Since 20 is divisible by both 2 and 4, the LCM is 20.

Why can't I just use any common denominator?

+

You can, but using the LCM keeps numbers smaller and easier to work with. Using 40 instead of 20 would give $\frac{66}{40}$ , which is harder to simplify!

What if I forget to multiply both numerator and denominator?

+

You'll change the value of the fraction! Remember: whatever you do to the bottom, you must do to the top. $\frac{1}{2} \times \frac{10}{10} = \frac{10}{20}$ keeps the same value.

How do I know if my final answer needs simplifying?

+

Check if the numerator and denominator share any common factors. For $\frac{33}{20}$ , since 33 = 3 × 11 and 20 = 4 × 5, they share no common factors, so it's already simplified!

Can I convert to decimals instead of finding common denominators?

+

You could, but you might lose precision with repeating decimals. The fraction method is exact and often required for homework. Plus, it builds important algebraic skills!

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Solve: 1/2 + 3/4 + 2/5 - Adding Three Fractions with Different Denominators

Fraction Addition with Different Denominators

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