Adding Fractions

Study Sheet

Adding Fractions

Same denominator, then different

Why Denominators Must Match

Concept
The Big Idea
= 3/5
Two fifths (heavy shading) and one more fifth (light shading) make three fifths.

You can only add pieces that are the same size. The denominator names the piece size, so it must match before you add the tops.

You can only add pieces that are the same size. Two fifths plus one fifth is three fifths — the pieces are fifths, so you count them. When the pieces are different sizes, like thirds and fourths, you first cut both into the same size (twelfths), and then the counting works again.

Example
Finding 28+38\dfrac{2}{8} + \dfrac{3}{8}
2 eighths (heavy shading) and 3 more eighths (light shading): 5 eighths.

The pieces are both eighths, so add the counts: 2+3=52 + 3 = 5 eighths, or 58\dfrac{5}{8}.

Example
Finding 45−15\dfrac45 - \dfrac15
0152535451back 1/5
One jump back of one fifth.

Start at 45\tfrac45 on a number line marked in fifths and jump back 11 fifth: you land on 35\tfrac35. The pieces are still fifths; only the count changed: 4−1=34 - 1 = 3.

Side note
Where it goes wrong

Adding the denominators: 13+13\tfrac13 + \tfrac13 is 23\tfrac23, not 26\tfrac26. The bottom number names the piece size, and adding pieces does not change their size.

Try it
Try it: same-size pieces

Find 37+27\dfrac37 + \dfrac27 and 69−29\dfrac69 - \dfrac29.

Show the answer

Sevenths: 3+2=53 + 2 = 5, so 57\mathbf{\dfrac57}. Ninths: 6−2=46 - 2 = 4, so 49\mathbf{\dfrac49}. The bottoms do not change, because the pieces do not change size.

In Plain Terms & More Examples

Concept
In plain terms

When the denominators already match, just add or subtract the top numbers — the piece size (the bottom) stays the same.

Example
Finding 310+410\dfrac{3}{10} + \dfrac{4}{10}
3 tenths and 4 more tenths: 7 tenths.

Same tenths, so add the tops: 3+4=73 + 4 = 7, giving 710\dfrac{7}{10}.

Example
Finding 56−26\dfrac{5}{6} - \dfrac{2}{6}

Same sixths, so subtract the tops: 5−2=35 - 2 = 3, giving 36=12\dfrac{3}{6} = \dfrac{1}{2}.

Side note
Where it goes wrong

Doing the same thing to the bottoms. Subtracting the denominators in 56−26\dfrac56 - \dfrac26 would give 30\dfrac{3}{0}, and a whole cut into 00 pieces makes no sense. The 66 tells you the size of the pieces; taking some pieces away leaves them sixths: 36\dfrac36.

Try it
Try it: twelfths

Find 212+712\dfrac{2}{12} + \dfrac{7}{12} and write the answer in lowest terms.

Show the answer

2+7=92 + 7 = 9 twelfths: 912\dfrac{9}{12}. The greatest common factor of 99 and 1212 is 33: 9÷312÷3=34\dfrac{9 \div 3}{12 \div 3} = \mathbf{\dfrac34}.

Mixed Numbers

Concept
Whole parts and fraction parts

Add the wholes and the fractions separately, then regroup if the fraction part is 11 or more. Or convert everything to improper fractions first — both work.

A mixed number is a whole-number part and a fraction part added together: 234=2+342\tfrac34 = 2 + \tfrac34. Adding wholes to wholes and fractions to fractions is allowed because you can add in any order. When the fraction parts make a whole or more, trade 44\tfrac44 for 11, the same regrouping as trading 1010 ones for a ten.

Example
Finding 234+1122\tfrac34 + 1\tfrac12
234112
Whole bars add to 3; the leftover pieces, 3/4 and 1/2, make one more whole and 1/4.

Wholes: 2+1=32 + 1 = 3. Fractions: 34+24=54=114\tfrac34 + \tfrac24 = \tfrac54 = 1\tfrac14. Total 3+114=4143 + 1\tfrac14 = 4\tfrac14.

Example
Regrouping to subtract: 414−1344\tfrac14 - 1\tfrac34
012345−134
Jumping back 1 3/4 from 4 1/4 lands on 2 1/2.

The fraction part 14\tfrac14 is smaller than 34\tfrac34, so trade one whole for 44\tfrac44: 414=3+44+14=3544\tfrac14 = 3 + \tfrac44 + \tfrac14 = 3\tfrac54. Now subtract. Wholes: 3−1=23 - 1 = 2. Fractions: 54−34=24\tfrac54 - \tfrac34 = \tfrac24. Answer: 224=2122\tfrac24 = 2\tfrac12.

Side note
Where it goes wrong

Taking the smaller fraction from the bigger one, whichever is on top. For 516−2565\tfrac16 - 2\tfrac56 it is tempting to do 56−16=46\tfrac56 - \tfrac16 = \tfrac46 and answer 3463\tfrac46. But the 56\tfrac56 has to be taken away from the 16\tfrac16, which needs a trade: 516=4765\tfrac16 = 4\tfrac76, and 476−256=226=2134\tfrac76 - 2\tfrac56 = 2\tfrac26 = 2\tfrac13.

Try it
Try it: subtract

Compute 315−1253\tfrac15 - 1\tfrac25.

Show the answer

Convert: 165−75=95=145\tfrac{16}{5} - \tfrac{7}{5} = \tfrac{9}{5} = \mathbf{1\tfrac45}. (Subtracting fraction parts first would need borrowing: 15<25\tfrac15 < \tfrac25.)

Word Problems

Concept
Which operation?

In all, altogether and total put amounts together: add. Left, how much more and how much farther take away or compare: subtract. Then make the pieces the same size before adding or subtracting.

A tape diagram helps you decide. Draw one bar for each amount, to scale. If the question asks for both bars end to end, add; if it asks for the difference between them, subtract. The picture tells you what to do before any arithmetic starts.

Example
Ribbon
78used 3/8left: ?
The ribbon is 7 eighths long; 3 eighths are used.

Maya has 78\tfrac{7}{8} m of ribbon and uses 38\tfrac{3}{8} m. Left: 78−38=48=12\tfrac{7}{8} - \tfrac{3}{8} = \tfrac{4}{8} = \tfrac12 m.

Example
How much farther?
SamMia
The difference is the 3 sixths where Sam’s bar is longer.

Sam ran 56\tfrac56 km and Mia ran 13\tfrac13 km. How much farther did Sam run? "How much farther" compares, so subtract. Make the pieces match: 13=1×23×2=26\tfrac13 = \tfrac{1 \times 2}{3 \times 2} = \tfrac26. Then 56−26=36=12\tfrac56 - \tfrac26 = \tfrac36 = \tfrac12 km.

Side note
Where it goes wrong

Adding tops and adding bottoms. For Leo's walk, 23+56\tfrac23 + \tfrac56 is not 2+53+6=79\tfrac{2+5}{3+6} = \tfrac79. A quick check shows why: 79\tfrac79 is less than 56\tfrac56, yet Leo walked 56\tfrac56 km after lunch alone. Adding more walking cannot give a smaller total.

Try it
Try it: the walk

Leo walked 23\tfrac23 km before lunch and 56\tfrac56 km after. How far did he walk in all?

Show the answer

Sixths: 46+56=96=112\tfrac46 + \tfrac56 = \tfrac96 = \mathbf{1\tfrac12} km.

Unlike Denominators

Concept
Cut the pieces to one size

To add 13+14\tfrac13 + \tfrac14, find a number both denominators divide into, here 1212, and rewrite both fractions as twelfths. Then the pieces match and you add the tops.

Thirds and fourths are different-sized pieces, so "one third plus one fourth" is like "one foot plus one inch": you cannot just count them. Cutting every third into 44 and every fourth into 33 turns both into twelfths, a size that fits both. Equivalent fractions make the change without altering either amount.

Example
Finding 12+14\tfrac12 + \tfrac14
1214sum
Half is two quarters; one more quarter makes three.

Fourths fit both: 12=24\tfrac12 = \tfrac24. Now the pieces match: 24+14=34\tfrac24 + \tfrac14 = \tfrac34.

Example
Finding 13+14\tfrac13 + \tfrac14
1314sum
4 twelfths (heavy shading) and 3 twelfths (light shading) make 7 twelfths.

Twelfths fit both. 13=1×43×4=412\tfrac13 = \tfrac{1 \times 4}{3 \times 4} = \tfrac{4}{12} and 14=1×34×3=312\tfrac14 = \tfrac{1 \times 3}{4 \times 3} = \tfrac{3}{12}. Add: 412+312=712\tfrac{4}{12} + \tfrac{3}{12} = \tfrac{7}{12}.

Example
Finding 34−23\tfrac34 - \tfrac23

Twelfths again: 34=912\tfrac34 = \tfrac{9}{12} and 23=812\tfrac23 = \tfrac{8}{12}. Subtract: 912−812=112\tfrac{9}{12} - \tfrac{8}{12} = \tfrac{1}{12}. The two amounts are very close, just one twelfth apart.

Side note
Where it goes wrong

Changing the bottom but not the top. Writing 13+14\tfrac13 + \tfrac14 as 112+112=212\tfrac{1}{12} + \tfrac{1}{12} = \tfrac{2}{12} changes the amounts: 112\tfrac{1}{12} is much less than 13\tfrac13. Whatever you multiply the bottom by, multiply the top by the same number.

Try it
Try it: fifths and tenths

Find 25+310\dfrac25 + \dfrac{3}{10}.

Show the answer

Tenths fit both: 25=2×25×2=410\dfrac25 = \dfrac{2 \times 2}{5 \times 2} = \dfrac{4}{10}. Then 410+310=710\dfrac{4}{10} + \dfrac{3}{10} = \mathbf{\dfrac{7}{10}}.

Why It Works

A denominator is a unit, exactly like "inches" or "apples". Three apples plus two apples is five apples, and 35+25\tfrac35 + \tfrac25 is five fifths for the same reason: the unit "fifth" is unchanged and only the count changes. When units differ we convert to a common unit first — inches to feet, or thirds and fourths to twelfths (13=412\tfrac13 = \tfrac4{12}, 14=312\tfrac14 = \tfrac{3}{12}) — and then 412+312=712\tfrac{4}{12} + \tfrac{3}{12} = \tfrac{7}{12} is just counting twelfths.

Tip
Reading a proof

Do not read a proof the way you read a rule. Cover the last line, ask what the previous line forces, and only then uncover it. If a step feels like magic, that is the step to work by hand with a concrete number.