In plain words
Is 1/5 + 2/5 equal to 3/10?
No. 1/5 + 2/5 = 3/5. Joining one fifth and two fifths gives three fifths: the number of pieces changes, but each piece is still a fifth. 3/10 is only half as much as 3/5.
- The big idea
- Adding like parts changes how many parts there are, not their size: 1/5 + 2/5 = 3/5.
- The common confusion
- When fractions have the same denominator, add the denominators too, so 1/5 + 2/5 = 3/10. It often sounds like this: “(one fifth) + (two fifths) is (three tenths), because five plus five is ten.”
Why it works
The denominator names the size of the pieces. In 1/5 and 2/5, every piece is one fifth of the same whole.
Adding joins the pieces: one fifth and two more fifths make three fifths. The number of pieces changes, but their size does not, so 1/5 + 2/5 = 3/5.
Adding the denominators as well gives 3/10. But tenths are half the size of fifths, so 3/10 is only half of 3/5. Joining pieces can’t make each piece smaller.
It helps to read the denominator as a unit, like a word: 1 apple + 2 apples = 3 apples, and 1 fifth + 2 fifths = 3 fifths.
The same counting works past one whole: 4/5 + 3/5 = 7/5, which is one whole and two fifths more.
A worked example
A chocolate bar is made of 5 equal squares, so each square is 1/5 of the bar. Eat 1 square, then 2 more, and you have eaten 3 squares: 3/5 of the bar. The squares didn’t shrink into tenths along the way.
Why this confusion is so common
Children who have learned to add whole numbers often add every number they can see, so 1/5 + 2/5 becomes 3/10. It usually means the denominator is being read as a count of things rather than as the name of a unit. Saying the units aloud helps: one fifth plus two fifths is three fifths, just as one apple plus two apples is three apples.
Try this away from the screen
Make a whole from 5 identical blocks or tiles, so each one is 1/5. Hand over 1 block, then 2 more, counting aloud in fifths: one fifth, two fifths, three fifths. Ask: did any block change size when we joined them? Would 3/10 mean the blocks got smaller?
Helpful things to say
| Instead of | Try |
|---|---|
| “Add the tops, but leave the bottoms alone.” | “What size are these pieces? When we join them, how many of those pieces do we have?” |
| “The bottom number stays the same because that’s the rule.” | “Say it in words: one fifth plus two fifths. What do you get?” |
Practice questions
For grown-ups helping out: the answers are tucked away so learners can try first.
Mia eats 2/7 of a chocolate bar and Leo eats 3/7 of it. How much of the bar do they eat altogether?
Show the answer
Answer: 5/7. They eat 2 pieces and 3 pieces of the same 7-piece bar: 5 pieces. The pieces are still sevenths, so 2/7 + 3/7 = 5/7.
What is 3/4 + 3/4? Use the number line to help.
Show the answer
Answer: 6/4. Fourths stay fourths: 3 fourths + 3 fourths = 6 fourths. Six fourths from 0 is one whole and two fourths more, so 3/4 + 3/4 = 6/4.
What is 1/6 + 4/6?
Show the answer
Answer: 5/6. One slice and four more slices of the same 6-slice pizza make five slices. The slices are still sixths, so 1/6 + 4/6 = 5/6.
Ana says 3/8 + 3/8 = 6/16. Is she right?
Show the answer
Answer: No. Eighths stay eighths, so 3/8 + 3/8 = 6/8. Ana’s 6/16 is the same as 3/8, which would mean adding 3/8 made no difference.
Use sixth pieces to build 2/6 + 4/6.
Show the answer
Answer: use pieces of 1/6 to make 6/6. Two sixths and four more sixths make six sixths, which fill the whole: 2/6 + 4/6 = 6/6 = 1. Adding the bottom numbers would give 6/12, which is only one half.
Questions people ask
Why does the denominator stay the same when you add fractions with the same denominator?
Because the denominator names the size of the pieces. One fifth plus two fifths is three pieces that are each one fifth: 3/5. Joining pieces doesn’t make them smaller, so the denominator stays the same.
What happens when the answer is more than one whole?
Keep counting the same pieces. 5/6 + 4/6 = 9/6, which is one whole (6/6) and three sixths more. You can also write it as 1 3/6, or 1 1/2.
How do I explain this to my child?
Say the unit out loud. “One fifth plus two fifths” works like “one apple plus two apples”: the answer is three of the same thing, three fifths. Then check it with paper strips or a drawn bar.
Words to know
- Denominator
- The bottom number of a fraction. It tells how many equal parts make one whole.
- Numerator
- The top number of a fraction. It counts how many of the equal parts we have.
- Unit fraction
- A fraction with 1 on top, such as 1/2, 1/3 or 1/8. It is one equal part of the whole.
- Improper fraction
- A fraction whose top number is equal to or bigger than its bottom number, such as 6/4 or 5/5. It is worth one whole or more.
Curriculum connections
We don’t assume any particular curriculum. For teachers in the United States, here’s how this slice lines up with the Common Core State Standards:
- 4.NF.B.3
Understand a fraction a/b with a > 1 as a sum of fractions 1/b.
- 4.NF.B.3.a
Understand addition and subtraction of fractions as joining and separating parts referring to the same whole.
- 4.NF.B.3.d
Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problem.
Related slices
Adding Different Pieces
Is (one half) + (one fourth) equal to (two sixths)?
Why 1/2 + 1/4 is 3/4, not 2/6: make the pieces the same size before you count them. · About 3 minutes
The Top Number
What does the top number of a fraction mean?
What the top number of a fraction counts: how many of the equal parts you have. · About 3 minutes
The Bottom Number
What does the bottom number of a fraction mean?
What the bottom number of a fraction counts: all the equal parts that make one whole. · About 3 minutes
More Than One Whole
Can a fraction be bigger than 1?
Why 6/4 makes sense: a fraction can equal one whole or go beyond it. · About 3 minutes
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