Why is 1/2 + 1/3 not 2/5?
Updated
The short answer
To add fractions, give them the same denominator first and then add the numerators: 1/2 + 1/3 = 3/6 + 2/6 = 5/6. Adding the tops and the bottoms separately gives 2/5, which is smaller than 1/2 on its own.
What it looks like
| Wrong | Right |
|---|---|
| 1/2 + 1/3 = 2/5 | 1/2 + 1/3 = 3/6 + 2/6 = 5/6 |
| x/3 + x/4 = 2x/7 | x/3 + x/4 = 4x/12 + 3x/12 = 7x/12 |
The wrong answer fails a simple size test. 1/2 + 1/3 has to be bigger than 1/2, because you're adding something positive to it. 2/5 is smaller than 1/2, so it can't be right. The slip is the 5 on the bottom, which came from adding the 2 and the 3.
With letters, the same slip turns 1/x + 1/y into 2/(x + y). The right sum is (x + y)/(xy), which you get by rewriting both fractions over xy: y/(xy) + x/(xy).
In alai, an iPad app that reads your handwriting as you work, the tutor, Ally, asks about the first line that doesn't follow from the one before.
Why it happens
Multiplying fractions really does work top with top and bottom with bottom: 1/2 × 1/3 = 1/6. So adding them the same way feels consistent. Eric Schechter's page of common errors in college math notes that many college students have trouble adding fractions, and ties a lot of their mistakes to what he calls the belief that everything is additive.
The denominator tells you the size of the pieces, and the numerator counts how many pieces you have. You can add counts only when the pieces are the same size. Halves and thirds are different sizes, so first turn both into sixths: 1/2 is 3 sixths and 1/3 is 2 sixths. Now you can add the counts, 3 + 2 = 5, and the pieces are still sixths. The denominator doesn't get added, because putting pieces together doesn't change how big each piece is.
How to catch it yourself
Before you add, look at the two denominators. If they're different, the fractions aren't ready to add yet. Then point at the bottom of your answer and ask yourself:
Where did the 5 on the bottom come from?
After you add, run the size test. The sum of two positive fractions has to be bigger than each of them.
Any common denominator works. Multiplying the two denominators always gives one, and the smallest one keeps the numbers small. For 1/6 + 1/9, using 54 gives 9/54 + 6/54 = 15/54, which reduces to 5/18, the same answer that 18 gives with smaller numbers.
Try these
- 1/4 + 1/6
- 2/5 + 1/3
- x/2 + x/5
Answers
- 5/12, because 1/4 + 1/6 = 3/12 + 2/12
- 11/15, because 2/5 + 1/3 = 6/15 + 5/15
- 7x/10, because x/2 + x/5 = 5x/10 + 2x/10
How alai helps
In Homework help, when you pause after writing a step, Ally reads it and asks about the first line that doesn't follow from the one before. If you're not sure why the bottoms don't add, you can hold to talk and ask, and Ally answers out loud.