Adding and Subtracting Fractions with Unlike Denominators

A fraction tells you both a piece size and a piece count. In $\frac{3}{8}$, the denominator, 8, means the whole has eight equal parts. The numerator, 3, means you have three of those parts.

Add the piece counts

Fractions with like denominators have the same bottom number. Their pieces match when their wholes are the same size. Think of two strips, each as long as one whole. Both are marked in sevenths.

Joining two sevenths and three sevenths gives five sevenths:

Add the numerators. Keep the denominator, because the pieces are still sevenths. Adding the bottom numbers would rename the pieces as fourteenths, even though you did not cut them smaller.

Take away matching pieces

For subtraction, count how many pieces remain. From seven eighths, take away three eighths:

The last step gives a simpler name for the same amount. Four eighths cover half the strip.

A number line shows the movement. Mark equal eighths between 0 and 1. To add two eighths to three eighths, start at $\frac{3}{8}$ and move two spaces right. You land at $\frac{5}{8}$. Subtracting moves left.

Reach or pass one whole

Two halves make one whole:

The answer cannot be $\frac{2}{4}$: that would still be only half a whole.

Sometimes you pass 1. Four fifths plus three fifths makes seven fifths:

A fraction whose top is at least as large as its bottom is an improper fraction. A mixed number has a whole-number part and a fraction part. Here, five fifths make the whole, leaving two fifths.

To subtract from 1, first name the whole in matching pieces:

You can also join three matching amounts at once:

A half and a third are different sizes. You cannot count them together as two matching pieces. Fractions with different bottom numbers have unlike denominators.

Cut smaller pieces without changing the amount

Try adding one half and one third. Cut each half into three equal pieces. Each new piece is a sixth, so one half becomes three sixths. Cut each third into two equal pieces. One third becomes two sixths.

These are equivalent fractions: different names for equal amounts. Now both amounts count sixths:

Renaming changes how you count the amount. It does not add any more.

Show it with strips or rectangles

Put a half-strip and a third-strip end to end. Compare their total length with a strip marked in sixths. The half covers three sixths; the third covers two more. Together they cover five sixths.

You can also draw two equal-size rectangles. Divide each into two equal columns and three equal rows. Each rectangle now has six equal cells.

• In the first rectangle, shade one column. It covers three cells: one half.
• In the second rectangle, shade one row. It covers two cells: one third.

Count the shaded cells across the two rectangles: five sixths. Keep the amounts in separate rectangles. Shading both on one rectangle would make one cell overlap.

Use a number line for subtraction

On a line marked in sixths, one half is at three sixths. One third is at two sixths. There is one sixth between them:

Check the whole

Half of a small pizza and half of a large pizza have different sizes. They do not join to make one pizza of either size. Fraction models need the same whole or equal-size wholes. Measurements also need matching units, such as cups with cups.

Adding the tops and bottoms of one half and one third gives two fifths. That amount is smaller than one half. Adding a positive amount to one half must make it larger, so two fifths cannot be the sum.

A multiple is a number you get by multiplying by 1, 2, 3, and so on. Multiples of 4 include 4, 8, 12, 16, 20, and 24. Multiples of 6 include 6, 12, 18, and 24. The numbers 12 and 24 appear in both lists.

Choose a shared piece size

A common denominator is a common multiple of the denominators. Both old denominators must divide into it evenly. For fourths and sixths, 12 works: each fourth becomes three twelfths, and each sixth becomes two twelfths.

The least common denominator, or LCD, is the smallest choice. It is the least common multiple, or LCM, of the denominators. Some books say "lowest common denominator."

Try the larger denominator first

When one denominator is a multiple of the other, use the larger one. For fifths and tenths, use tenths. Since $10÷5=2$, each fifth becomes two tenths. The tenths already have the right name.

But the larger number does not always work. For fourths and sixths, 6 is too small a choice: 4 does not divide evenly into 6.

List multiples until both fit

For denominators 4 and 6, start with multiples of the larger number:

Four does not divide evenly into 6, but it does into 12. The first shared multiple is 12, so that is the LCD.

For denominators 2, 3, and 4, check all three. Twelve is the smallest positive number that all three divide into evenly. You can use twelfths for all three fractions.

Know when the product helps

Multiplying the denominators always gives a common denominator. For 4 and 6, the product is 24. It works, although 12 keeps the numbers smaller.

When the denominators share no factor except 1, their product is the LCD. Three and five share only the factor 1, so their LCD is $3\times 5=15$.

Here is the same sum using two choices:

Adding the denominators gives 10, which fails the test: neither 4 nor 6 divides into 10 evenly.

Once you choose a common denominator, each fraction needs that bottom number. The top must change with it so that you keep the same amount.

Find the multiplier

To write three fourths in twelfths, ask how many fourths fit into twelve:

Multiply both the numerator and denominator by 3:

Picture each fourth cut into three equal pieces. Your three fourths now contain nine smaller pieces. Nothing was added or taken away.

The multiplier keeps the value because $\frac{3}{3}=1$. Multiplying an amount by 1 leaves it unchanged.

Rename each fraction separately

For $\frac{3}{4}+\frac{5}{6}$, choose twelfths. The fourths need a multiplier of 3. The sixths need a multiplier of 2, because $12÷6=2$.

The multipliers differ because the original pieces differ. Once renamed, both amounts count twelfths. You can also compare them: nine twelfths is less than ten twelfths, so three fourths is less than five sixths.

Leave a matching denominator alone

For $\frac{2}{5}+\frac{3}{10}$, choose tenths. Rename two fifths as four tenths. Three tenths already has the chosen denominator, so it stays three tenths.

Check the new name

Simplify the renamed fraction to check it:

You get the starting fraction back.

Changing only the bottom number would give three twelfths. That equals one fourth, so it loses part of the starting amount. Adding the same number to top and bottom also changes the value: adding 8 to both parts of three fourths gives eleven twelfths, which is too large. Use the same multiplication on both parts.

Keep four steps in order: choose a common denominator, rename, calculate, and simplify. The first two steps make the pieces match. Then you use the same counting rule as for like denominators.

Add two fractions

Work out $\frac{2}{3}+\frac{3}{4}$. The LCD is 12.

1. Rename two thirds: $\frac{2}{3}=\frac{8}{12}$.
2. Rename three fourths: $\frac{3}{4}=\frac{9}{12}$.
3. Add the numerators: $\frac{8}{12}+\frac{9}{12}=\frac{17}{12}$.
4. Write the amount as a mixed number: $\frac{17}{12}=1\frac{5}{12}$.

You have seventeen twelfths, including one whole group of twelve. Keep the denominator 12 when adding. Seventeen twenty-fourths would use pieces of a different size.

Subtract in the given order

For $\frac{5}{6}-\frac{1}{4}$, use twelfths again:

Check by adding back the amount removed:

Addition lets you swap the two amounts. Subtraction does not: the starting amount must stay first.

A denominator may already match. In $\frac{7}{8}-\frac{1}{2}$, only the half needs a new name. It becomes four eighths, leaving three eighths.

Improper fractions use the same steps:

Work with three fractions

Give all three fractions one common denominator. Then add or subtract from left to right:

Before calculating, check every fraction has been renamed. An unchanged third among twelfths would still have the wrong piece size.

Find a missing amount

For $\frac{1}{3}+\mathrm{\square }=\frac{3}{4}$, take the known part from the total:

For $\mathrm{\square }-\frac{1}{6}=\frac{1}{2}$, rebuild the starting amount by adding:

An answer can have the right value and still need simplifying. Simplest form, also called lowest terms, means the numerator and denominator share no factor except 1.

Divide both parts by a common factor

A factor divides a number evenly. For eighteen twenty-fourths, both 18 and 24 divide evenly by 6:

The amount stays the same. You are counting fewer, larger pieces.

The greatest common factor, or GCF, is the largest factor both numbers share. List their factors to find it:

• Factors of 18: 1, 2, 3, 6, 9, 18.
• Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24.

The largest shared factor is 6. Dividing by it finishes the simplifying in one step. Smaller steps work too: eighteen twenty-fourths becomes nine twelfths after dividing by 2, then three fourths after dividing by 3.

Divide both parts. Dividing only the numerator changes the amount. Subtracting the same number from both parts also changes it.

Check even after using the LCD

The smallest common denominator keeps the calculation small. It does not promise a simplified result:

You can also simplify before calculating. Three sixths plus one fourth becomes one half plus one fourth, which is three fourths.

Turn extra pieces into wholes

For an improper fraction, divide the numerator by the denominator. The whole-number answer counts complete wholes. The remainder counts pieces left over.

For nineteen twelfths, $19÷12$ gives 1 remainder 7:

You can simplify before or after finding the wholes:

Or first write two and six eighths, then simplify the fraction part. Keep the two wholes: $2\frac{6}{8}=2\frac{3}{4}$.

Sometimes no pieces remain. Sixteen eighths makes exactly two wholes, so write $\frac{16}{8}=2$.

Before doing the exact work, think about the answer's size. An estimate uses nearby, easy numbers. For fractions, useful benchmarks are 0, one half, and 1.

Find a nearby benchmark

One ninth is near 0. Four ninths is near one half. Eight ninths is near 1. Compare the numerator with the denominator to picture how full the whole is.

To check which side of one half a fraction lies on, double its numerator. For three eighths, $2\times 3=6$. Since 6 is less than 8, three eighths is less than one half. For four sevenths, the double is 8, which is greater than 7, so four sevenths is more than one half.

Estimate before calculating

For $\frac{9}{10}+\frac{5}{8}$, use 1 plus one half. Expect about one and one half.

The exact work gives:

That is close to the estimate. An answer below one half would be much too small.

For mixed numbers, round each amount to its nearest whole number. Three and seven eighths plus two and one sixth is about $4+2=6$. The exact sum is six and one twenty-fourth.

Use size rules too

A sum of positive amounts is larger than each amount. If you add one half and three eighths, an answer of four tenths is too small: it is below the half you started with.

Two fractions each below one half have a sum below 1. Two each above one half have a sum above 1. Three fifths and four sevenths are both above one half, so their sum must pass 1.

When you take away a positive amount, the result is smaller than your starting amount. For eleven twelfths minus one tenth, the answer should be a little below 1. The exact result is forty-nine sixtieths.

An estimate gives a rough check. Two different answers can both be close to it. For an exact subtraction check, add the removed amount to your difference. You should get the starting amount back.

The mixed number $2\frac{3}{4}$ means two wholes plus three fourths. When you add mixed numbers, keep track of both kinds of parts.

Add the wholes and the fractions

For $2\frac{1}{3}+1\frac{1}{4}$, add the wholes to get 3. Rename the fraction parts in twelfths:

Put the parts together: $3\frac{7}{12}$. The fraction total is below 1, so no extra whole is needed.

Carry a new whole

Now add $3\frac{2}{3}+1\frac{3}{4}$.

1. Add the wholes: $3+1=4$.
2. Add the renamed fractions: $\frac{8}{12}+\frac{9}{12}=\frac{17}{12}$.
3. Make a whole from twelve of those twelfths: $\frac{17}{12}=1\frac{5}{12}$.
4. Add that whole to the four wholes: $4+1\frac{5}{12}=5\frac{5}{12}$.

This is called carrying a whole. Writing four and five twelfths would lose the whole made by the fractions. A finished mixed number has a fraction part smaller than 1.

A mixed number plus a fraction can need carrying too:

Try improper fractions

You can count all the pieces together first. To turn a mixed number into an improper fraction, multiply the whole part by the denominator, then add the numerator.

Three and two thirds has nine thirds in its wholes, plus two more:

One and three fourths is seven fourths. Add these improper fractions:

Both methods give the same result. Keeping wholes separate often uses smaller numbers.

Simplify the final fraction

One and one half plus two and one sixth gives three and four sixths. Divide the fraction's top and bottom by 2:

Keep the three wholes. For a rough check, round the starting mixed numbers to whole numbers and add those.

Start mixed-number subtraction by giving the fraction parts a common denominator. Compare those parts before subtracting.

Subtract when the fraction pieces are enough

For $5\frac{3}{4}-2\frac{1}{6}$, rename in twelfths. Nine twelfths is enough to take away two twelfths:

You subtract the wholes and matching pieces separately.

Regroup when you need more pieces

For $5\frac{1}{4}-2\frac{2}{3}$, the fraction parts become three twelfths and eight twelfths. The first part needs more pieces.

Break one of the five wholes into twelve twelfths. Add those pieces to the three twelfths:

This trade is regrouping, also called renaming or borrowing. The total value stays the same. There is one fewer whole and twelve more fraction pieces.

Now subtract:

Keep the original subtraction order. Swapping three twelfths and eight twelfths would change the question. Also remember to reduce the whole part when you trade it. Keeping all five wholes would count one twice.

One whole is worth the number of pieces named by the denominator. With twelfths, add twelve pieces, not ten.

Subtract from a whole number

Four wholes can be written as three wholes and eight eighths:

Taking a fraction from a mixed number works the same way:

Choose another method or check

Improper fractions let you subtract without trading a separate whole:

You can also count up between the amounts on a number line. From two and two thirds to 3 is one third. From 3 to 5 is 2. From 5 to five and one fourth is one fourth. The gap is two and seven twelfths.

Check by adding back:

Read what you know and what you need to find. The action in a story helps you choose between adding and subtracting.

Join parts or take an amount away

"Maya walked three fourths of a mile to school, then two thirds of a mile to the library. How far did she walk?"

The two distances join, so add:

Maya walked $1\frac{5}{12}$ miles.

"A jug has seven eighths of a gallon of juice. Kai pours out one third of a gallon. How much remains?"

An amount is removed, so subtract:

There is $\frac{13}{24}$ gallon left.

Find a gap or a missing part

"Ben's ribbon is five sixths of a yard long. Ana's is three fourths of a yard. How much longer is Ben's?"

Subtract the shorter length from the longer one:

Ben's ribbon is $\frac{1}{12}$ yard longer.

For a fraction of one book, the whole is 1. If Lena has read two fifths of her book, the unread part is $1-\frac{2}{5}=\frac{3}{5}$.

A missing-part story can also be written with a box. A recipe needs two and one fourth cups of flour. Sam has one and two thirds cups:

Subtract what Sam has from what is needed. The gap is seven twelfths of a cup.

Draw the parts

A tape diagram is a bar that shows amounts. Draw one bar for the total. Mark known parts and put a question mark on the unknown part. Add when both parts are known and you need the total. Subtract when the total and one part are known.

Do not let one word choose for you. "Jo used one fourth of a gallon of paint and has one half gallon left. How much was there at first?" Here, both parts are known. Add the used paint and the paint left to get three fourths of a gallon.

Keep wholes and units clear

Fractions of different-size pizzas cannot be added as though each pizza were the same whole. Measurements must also share a unit. Keep "mile," "cup," or "hour" with your answer so the number has a clear meaning.

A two-step story has a useful result along the way. Label that result so you know whether it answers the question.

Find the remaining part

"One third of a garden's beds have tomatoes, one fourth have beans, and the rest have flowers. Each bed has just one crop. What fraction have flowers?"

First find the fraction used by tomatoes and beans:

Seven twelfths describes the vegetable beds. The question asks about flowers, so there is another step:

Five twelfths of the beds have flowers. Check by adding the three parts: four twelfths, three twelfths, and five twelfths make twelve twelfths, or the whole garden.

Track lengths used and left

"Rosa has a board 6 feet long. She cuts off one and three fourths feet, then two and one half feet. How much remains?"

Add the lengths cut off:

Then subtract from the starting length:

Rosa has $1\frac{3}{4}$ feet left. Giving four and one fourth feet would answer how much she cut off.

Make a decision with the total

A bag holds at most 3 pounds. Will one and one half pounds of apples and one and two thirds pounds of pears fit?

The fruit is one sixth of a pound over the limit, so it will not fit.

Change units before adding

A shelf is two and one fourth feet long. An extension adds 6 inches. Since 12 inches make 1 foot, 6 inches is one half foot.

The new shelf is $2\frac{3}{4}$ feet long. Adding 6 directly would mix inches and feet.

Check the answer against the story

Estimate before calculating, then compare the result with what happened. A leftover must be smaller than the starting amount. Add the used amounts and the leftover to rebuild the whole. Finally, read the question again and write a sentence naming the amount it asks for.