Adding & Subtracting Fractions
“Same bottom: just work the tops — the slice size stays.”
The formula
a/c ± b/c = (a ± b)/cHow to read it: When the bottoms are the same, add or subtract the tops and keep the bottom
- a, b
- — the numerators — the parts on top
- c
- — the denominator — the equal parts (must match)
The hook
One slice plus two slices makes three slices — adding fractions is just counting slices.
In plain words
When the slices are the same size (same bottom), just add or subtract how many slices (the tops).
The intuition
1/4 and 2/4 are both quarter-slices. Same size, so just 1+2 = 3 slices, i.e. 3/4. The bottom (slice size) never changes — only the count does.
How it's built
Same bottoms: work on the tops, keep the bottom. Different bottoms: first make them match (find a common bottom), then add. 1/2 becomes 3/6 and 1/3 becomes 2/6.
Example
Compute 1/2 + 1/3. Make both bottoms 6: 3/6 + 2/6 = 5/6.
Common mistake
Do not add the bottoms! 1/4 + 2/4 is 3/4, not 3/8. The slice size (the bottom) stays the same.
Where it's used
Combining recipe amounts, adding times (1/2 hour + 1/4 hour), finding what's left.
Where it came from
The rule of matching bottoms was refined over centuries by traders and scholars splitting grain and money.
Prerequisites
Quick check
What is 2/7 + 3/7?
- 5/14
- 5/7✓
- 6/7
- 1/7
Practice
Compute 1/6 + 4/6.
Answer: undefined
- Bottoms match (6).
- Add the tops: 1+4=5
- → 5/6
Takeaway: Same bottom, just add the tops.
Compute 5/8 − 2/8.
Answer: undefined
- Bottoms match (8).
- Subtract the tops: 5−2=3
- → 3/8
Takeaway: Subtraction works the tops too.
Compute 1/2 + 1/4. (Match the bottoms first.)
Answer: undefined
- Change 1/2 into 2/4.
- 2/4 + 1/4 = 3/4
Takeaway: Different bottoms? Make them match first.