Lesson 5 of 10 · Grades 3 to 5
Add fractions with the same denominator
When two fractions have the same denominator, their parts are the same size. Adding them is as easy as counting: two fifths plus one fifth is three fifths. The denominator stays the same because the size of each part does not change.
Counting parts of the same size
When two fractions have the same denominator, their parts are all the same size. That makes adding simple: count the parts. Two fifths plus one fifth is three fifths, just as two apples plus one apple is three apples. The size of the parts never changes, so the denominator never changes either.
The rule is short: add the numerators and keep the denominator. For 2/5 + 1/5, add 2 and 1 to get 3, and keep the 5: the answer is 3/5. If you wrote 3/10, you would be adding the names of the parts as well as the counts, which changes the size of the pieces.
Why the denominator stays
Think of the denominator as a label, like the word "metres". If you walk 2 metres and then 1 more metre, you have walked 3 metres, not 3 something else. The "metres" stays. In the same way, fifths plus fifths stay fifths; only the count of fifths changes. The denominator names the unit of the parts, and adding does not change the unit.
Answers that fill the whole
Sometimes the count reaches the denominator, and the answer is one whole. Imagine 3/8 of a cake on a plate and a second slice that is 5/8 of the cake. Together they are 8/8, which is the whole cake. Whenever the numerators add up to the denominator, the answer is 1. If they add up to more than the denominator, the answer is more than one whole, and you can write it as a mixed number. You will practise that in a later lesson.
You can also simplify an answer when the numerator and denominator share a factor. 2/8 + 2/8 is 4/8, and 4/8 simplifies to 1/2. Both 4/8 and 1/2 are correct, but the simplest form is tidy. Simplifying does not change the amount; it just names it with the fewest pieces.
Worked example
Add 5/12 and 4/12. Step 1: the denominators match, so the parts are both twelfths. Step 2: add the numerators: 5 + 4 = 9. Step 3: keep the denominator: the sum is 9/12. Step 4: simplify if you can. Both 9 and 12 divide by 3, so 9/12 becomes 3/4. The answer is 3/4.
Try a word problem. Sam runs 3/10 of a kilometre in the morning and 4/10 of a kilometre in the afternoon. How far did he run in total? The parts are both tenths, so add 3 + 4 = 7 and keep 10. Sam ran 7/10 of a kilometre. Check that the answer makes sense: 7/10 is less than a whole kilometre, and the two trips together were less than one kilometre.
Watch out for this
The most tempting mistake is to add the denominators too. 1/2 + 1/2 is 2/2, which is 1, not 2/4. Adding the denominators would mean cutting the whole into more pieces, and that would change the value of both fractions. Keep the denominator; touch only the numerators.
Remember
- Same denominator: add the numerators and keep the denominator.
- Never add the denominators.
- An answer can equal 1 when the parts add up to the whole.
- Simplify the answer when both numbers share a factor.
Try it
Answer the five questions below. Work them out first, then open each answer to check.
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1/6 + 2/6 = ?
Show answer
3/6, which simplifies to 1/2.
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2/7 + 3/7 = ?
Show answer
5/7.
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3/10 + 4/10 = ?
Show answer
7/10.
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A jug has 2/5 of a litre of juice in it. You pour in another 2/5 of a litre. How much juice is in the jug now?
Show answer
4/5 of a litre.
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1/8 + 3/8 + 2/8 = ?
Show answer
6/8, which simplifies to 3/4.
Puzzle · Fraction pyramid
Puzzle time
In this pyramid, each brick is the sum of the two bricks under it. The top brick is 9/9. What fraction goes in the empty brick?
Show a hint
Guess a fraction for the empty brick. Fill in the middle row, then add to the top. Did you get 9/9? If not, change your guess.
Show the answer
Answer: 3/9. The middle row is 1/9 + 3/9 = 4/9 and 3/9 + 2/9 = 5/9. The top is 4/9 + 5/9 = 9/9, which is 1 whole. The middle brick is counted twice, because it sits under both middle bricks.