A very common answer to 1/2 + 1/3 is 2/5. It is wrong, and it is worth knowing why.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Check the size
2/5 is less than a half. But you started with a half and added more, so the answer must be bigger.
The reason
Halves and thirds are different things. Counting them together makes no sense.
Apples and elephants
2 apples plus 1 elephant is not 3 of anything. Halves plus thirds is the same problem.
Why the obvious answer is wrong
1/2 + 1/3 is not 2/5. Adding the tops and bottoms separately treats halves and thirds as if they were the same size, and they are not. Check it roughly: 1/2 alone is already larger than 2/5, so the answer cannot possibly be 2/5.
Make the units the same first
Cut both into sixths: 1/2 = 3/6 and 1/3 = 2/6. Now both count sixths, so 3/6 + 2/6 = 5/6. The rewriting is not an extra rule — it is what makes counting possible in the first place.
The same problem outside fractions
You cannot add 3 metres and 4 centimetres to get 7 of anything. You convert first, then add. Fractions with different denominators are exactly this situation: two measurements in different units, needing conversion before they can be combined.
Estimate to catch the error
Before adding any fractions, judge the rough size: 1/2 plus a bit more than a quarter should be somewhere near 3/4. That single estimate catches the add-the-bottoms mistake instantly, which is why it is worth doing every time.
Step 2: Try It Yourself
Set the bottom numbers differently and read what it says.
1/2
1/3
These are halves and thirds. They are different things, so they cannot be counted together yet.
Adding the denominators would be like adding apples to elephants.
Step 3: Watch an Example
One step at a time.
Watch Ana Catch the Mistake
Ana is shown the answer 1/2 + 1/3 = 2/5.
- Step 1
She checks the size: 2/5 is less than a half.
Step 4: Your Turn
Practice makes it stick.
The Size Test
Problem 1 of 2
Is 1/2 + 1/4 bigger or smaller than 1/2?
In Quarters
Problem 2 of 2
1/2 + 1/4, written in quarters. What is the top number?
Same Unit First
1 of 3
1/2 + 1/3, in sixths. What is the top number?
2 of 3
1/3 + 1/6, in sixths. What is the top number?
3 of 3
Can you add halves and thirds by counting them together?
Step 5: Quick Check
Show what you know.
Question 1 of 2
1/4 + 1/2, in quarters. What is the top number?
Question 2 of 2
How can you tell 2/5 is a wrong answer to 1/2 + 1/3?
What You Learned
- Adding the bottom numbers gives a wrong answer.
- Different units cannot be counted together.
- Rewrite both in one unit first, then count.