Adding & subtracting fractions
The idea
Once two fractions are counted in the same size pieces, adding them is as easy as adding apples: 2 eighths plus 3 eighths is 5 eighths, just like 2 apples plus 3 apples is 5 apples. The piece size — eighths — is the thing being counted, and the top numbers are the counts. Subtracting works the same way: take 5/8 away from 8/8 and 3 eighths remain.
This explains the one rule that matters: the bottom number does not get added. Writing 2/8 + 3/8 = 5/16 is the classic mistake, and you can see why it fails — nobody cut the mile or the pizza into sixteenths, and sixteenths are smaller pieces anyway, so the total would shrink instead of grow. The denominator is a label for piece size, like the word apples; you would never add 2 apples and 3 apples and get 5 apple-apples. When the bottoms do not match, rename one or both fractions into equal pieces first, and only then count.
Worked example
Ana walks 2/8 of a mile to the park, rests, then walks 3/8 of a mile more to reach the library. How far has she walked in total, and how much less than a whole mile is that?
- Both distances use the same piece size — eighths of a mile — so you may simply count pieces: 2 pieces + 3 pieces = 5 pieces.
- That makes 2/8 + 3/8 = 5/8 of a mile. The bottom number stays 8 because the pieces are still eighths; only how many she walked changed.
- Adding the bottoms to get 5/16 would be wrong: sixteenths are smaller pieces that nobody cut, and a total should not come out smaller than its parts.
- A whole mile is 8/8. Subtract the part she walked: 8/8 − 5/8 = 3/8 of a mile still short of a full mile.
- Check that the parts rebuild the whole: 5/8 + 3/8 = 8/8, exactly one mile. The pieces fit back together perfectly.
Answer. Ana has walked 5/8 of a mile, which is 3/8 of a mile less than a whole mile.
Check your understanding
- Why does the bottom number stay the same when you add fractions that already have matching pieces?
- What would you do first to add 1/2 and 1/4, where the piece sizes do not match?
- How is adding 2 eighths and 3 eighths like adding 2 apples and 3 apples?
- How can drawing one bar split into eighths help you check a fraction sum or difference?