Adding and subtracting fractions with unlike denominators

When you reach for it

Aiyana, Dohyun and Lanre take a ballet class. They keep a log of their practice time, in fractions of an hour, and they learn their routine a piece at a time, so they talk about fractions of a routine too. After class they make paper banners for the spring show, and each banner gets painted in parts.

When two amounts are counted in the same pieces, adding them is easy. Two tenths and three tenths are five tenths, because tenths can be counted like apples (Adding and subtracting fractions with like denominators). But Aiyana’s log says \tfrac{1}{2} hour at the barre and \tfrac{3}{10} hour of jumps. Halves and tenths are different sizes. You reach for this lesson whenever two fractions with different denominators must be joined or one taken from the other. It has four parts, one for each day:

• Part 1: why different pieces cannot be counted together, and two pictures that fix it: fraction strips and area models.
• Part 2: adding when one denominator is a multiple of the other, and simplifying the answer.
• Part 3: adding when neither denominator is a multiple of the other, using the product of the denominators or the least common denominator.
• Part 4: subtracting, sums greater than 1, and adding three fractions.

The procedure

Part 1: the pieces must match

A fraction counts pieces. \tfrac{1}{2} is 1 piece of size one half. \tfrac{3}{10} is 3 pieces of size one tenth. Adding means putting the pieces together and saying how many there are. But 1 big piece and 3 small pieces are not “4” of anything. There is no single piece size to name.

Try adding the tops and the bottoms anyway, and you get \tfrac{4}{12}. That cannot be right. \tfrac{4}{12} is the same as \tfrac{1}{3}, which is less than \tfrac{1}{2}. Aiyana’s total practice cannot be less than her time at the barre alone. So the bottom numbers do not add. The denominator is not a count. It names the size of the pieces.

Fraction strips. The fix is to cut the pieces again until both fractions use one size. A half is the same as 5 tenths, because cutting a half into 5 equal parts makes pieces that are each one tenth of the whole strip.

Now both amounts are in tenths, so count them: 5 tenths and 3 tenths make 8 tenths.

Aiyana practiced \tfrac{8}{10} of an hour. Notice that only the halves were cut again. The tenths were already small enough.

Area models. Sometimes neither strip fits inside the other. Thirds cannot be cut into quarters, and quarters cannot be cut into thirds. A square solves this. Cut it into 3 rows to show thirds. Cut a square of the same size into 4 columns to show quarters. Then draw the other set of lines on each square too, as dashed lines.

Each square is now 3 rows by 4 columns, which is 3 \times 4 = 12 equal cells, so every cell is one twelfth. Two rows of the first square cover 2 \times 4 = 8 cells. One column of the second square covers 3 cells. So

The grid trick always works. Rows for one denominator and columns for the other cut the whole into pieces that both fractions can be counted in. Parts 2 and 3 do the same cutting with numbers instead of pictures.

Part 2: one denominator is a multiple of the other

Cutting each half into 5 parts is the same as multiplying the top and the bottom of \tfrac{1}{2} by 5. There are 5 times as many pieces, and each one is 5 times smaller, so the amount does not change (Equivalent fractions). Multiplying the top and bottom by 5 is the same as multiplying by \tfrac{5}{5}, which is 1.

How do you know to use 5? Because 10 is a multiple of 2: 2 \times 5 = 10. When the larger denominator is a multiple of the smaller one, the larger denominator is the shared piece size, and only one fraction has to change. To test it, divide the larger denominator by the smaller. If the answer is a whole number, that number is what the other fraction gets multiplied by.

With the pieces matched, add the numerators and keep the denominator. The denominator stays because the pieces are still tenths; you have only counted more of them. In general, for fractions with the same denominator d,

Simplifying the answer. The sum \tfrac{8}{10} is correct, but it can be written with fewer, bigger pieces. Both 8 and 10 can be divided by 2. Joining the tenths in pairs makes fifths, since 2 tenths is 1 fifth, and 8 tenths make 4 such pairs.

A fraction is in simplest form when no whole number bigger than 1 divides both its top and its bottom. Both \tfrac{8}{10} and \tfrac{4}{5} are right, and they name the same amount, but the simplest form is the one people usually write, because it is easiest to picture. So Aiyana’s practice was \tfrac{4}{5} of an hour, which is 48 minutes.

Part 3: neither denominator is a multiple of the other

For \tfrac{2}{3} + \tfrac{1}{4}, neither 3 nor 4 divides the other. Both fractions must change. The area model used 3 \times 4 = 12 cells, and the numbers do the same thing. Multiply the top and bottom of \tfrac{2}{3} by 4, which is cutting each third into 4 parts. Multiply the top and bottom of \tfrac{1}{4} by 3, which is cutting each quarter into 3 parts.

The product of the denominators always works as a common denominator. Each fraction is multiplied, top and bottom, by the other fraction’s denominator, and both bottoms become the same product. It needs no searching.

The least common denominator is the smallest number other than 0 that both denominators divide into evenly. Renaming two fractions this way has its own lesson, Common denominators and the least common denominator, and the search uses lists of multiples: list multiples of the larger denominator until one is also a multiple of the smaller. For 3 and 4 the list is 4, 8, 12, and 12 is the first that 3 divides. Here the least common denominator is the product. That happens whenever the two denominators share no factor except 1.

When the denominators do share a factor, the least common denominator is smaller than the product. For sixths and eighths, the product is 6 \times 8 = 48, but the multiples of 8 go 8, 16, 24, and 24 is already a multiple of 6. Either choice gives the same final answer, because forty-eighths are just twenty-fourths cut in half. The smaller choice keeps the numbers small and often means less simplifying at the end. The product is a safe fallback when the least common denominator does not come to mind.

Part 4: subtracting, sums greater than one, three fractions

Subtracting. Taking away has the same problem as adding: you cannot take 3 eighths from 5 sixths until both are counted in one size. The move is the same. For \tfrac{5}{6} - \tfrac{3}{8}, the least common denominator is 24, as Part 3 found. Cut each sixth into 4 and each eighth into 3:

Once the pieces match, subtract the numerators and keep the denominator, for the same reason as before: the pieces are still twenty-fourths.

On the number line, the difference is the distance from \tfrac{3}{8} up to \tfrac{5}{6}. In twenty-fourths that is 20 - 9 = 11 marks. In this lesson the first fraction is always the larger one, so the difference is never below zero. Subtracting a larger fraction from a smaller one gives a negative number, which Operations with rational numbers handles.

Sums greater than one. Two fractions less than 1 can add up to more than 1. Dohyun practices \tfrac{4}{5} hour on Monday and \tfrac{1}{2} hour on Tuesday:

The top is bigger than the bottom, so this is more than one whole hour. Ten of the thirteen tenths make 1 whole, and 3 tenths are left over, so \tfrac{13}{10} = 1\tfrac{3}{10}. The mixed number 1\tfrac{3}{10} reads “one and three tenths”: one whole hour and three tenths of another. To turn a fraction into a mixed number, divide the top by the bottom. The quotient is the whole number and the remainder is the new top: 13 \div 10 = 1 remainder 3. Simplify the fraction part if it can be simplified.

Three fractions. Adding three fractions works the same way. Find one denominator that all three denominators divide into, rename each fraction, and add the three numerators. Example 8 does this. Addition can be grouped and ordered in any way (the associative and commutative properties), so you may also add two first and then the third, but one shared denominator for all three is usually fewer steps.

Checking. Before trusting an answer, ask whether its size makes sense. \tfrac{4}{5} is close to 1 and \tfrac{1}{2} is a half, so their sum should be a bit less than 1\tfrac{1}{2}, and 1\tfrac{3}{10} is. Rounding each fraction to 0, \tfrac{1}{2} or 1 this way is the subject of Estimating with fractions.

Worked examples

Examples 1 and 2 belong to Part 1, Examples 3 and 4 to Part 2, Examples 5 and 6 to Part 3, and Examples 7 and 8 to Part 4.

Boundaries

• The same whole. Adding \tfrac{1}{2} and \tfrac{3}{10} gives \tfrac{4}{5} only when both are fractions of the same whole, such as one hour or one banner. Half of a small banner and three tenths of a large one are not \tfrac{4}{5} of either.
• Only the numerators add. The denominator names the size of the pieces. Once the pieces match, the count changes and the size stays. Adding the denominators would change the size of the pieces, which gives a wrong amount.
• No zero denominators. A denominator says how many equal parts the whole is cut into, so it is never 0. Every fraction in this lesson has a denominator of at least 2.
• Larger minus smaller. In this lesson every difference takes a smaller fraction from a larger one. The other order gives a number below zero, which needs negative numbers (Operations with rational numbers).
• Proper fractions only. Every fraction added or subtracted here is less than 1 (once, in Exercise 23, a part is taken from the whole, 1). Adding or subtracting mixed numbers, such as 2\tfrac{1}{3} + 1\tfrac{3}{4}, is the work of Adding mixed numbers and Subtracting mixed numbers.
• Adding and subtracting only. A common denominator is needed to add or subtract. It is not needed to multiply fractions (Multiplying fractions), where tops multiply and bottoms multiply.

Where this goes next

Mixed numbers come next. To find 1\tfrac{2}{3} + 2\tfrac{1}{4}, add the whole numbers and add the fraction parts, and the fraction parts are exactly the sums of this lesson (Adding mixed numbers). Subtracting mixed numbers sometimes needs a whole number broken into pieces first (Subtracting mixed numbers). Longer story problems, where you must read the situation to decide whether amounts are joined or taken away, are in Word problems that add or subtract fractions.

Later, the same renaming works with negative fractions (Operations with rational numbers), and later still with fractions that have letters in them. The idea never changes: rename to one piece size, then count.

Exercises

Part 1: strips and area models

Draw a picture for each sum, then write the sum as a fraction.

1. Use fraction strips to find \tfrac{1}{2} + \tfrac{3}{8}.
2. Use an area model to find \tfrac{1}{2} + \tfrac{1}{5}.
3. Use an area model to find \tfrac{2}{3} + \tfrac{1}{5}.

Part 2: one denominator a multiple of the other

Add. Write each answer in simplest form.

4. \tfrac{1}{3} + \tfrac{5}{9}
5. \tfrac{2}{5} + \tfrac{1}{10}
6. \tfrac{1}{4} + \tfrac{5}{12}
7. \tfrac{3}{7} + \tfrac{5}{14}
8. \tfrac{1}{6} + \tfrac{7}{18}
9. Dohyun spends \tfrac{1}{6} hour warming up and \tfrac{2}{3} hour learning a new dance. How long is that altogether?

Part 3: neither denominator a multiple of the other

Add. Write each answer in simplest form.

10. \tfrac{3}{4} + \tfrac{1}{5}
11. \tfrac{2}{7} + \tfrac{1}{3}
12. \tfrac{2}{9} + \tfrac{1}{6}
13. \tfrac{3}{10} + \tfrac{2}{15}
14. \tfrac{5}{12} + \tfrac{1}{8}
15. Aiyana covers \tfrac{1}{6} of a banner with stars and \tfrac{3}{10} of it with stripes. What fraction of the banner is covered?
16. Find \tfrac{3}{10} + \tfrac{1}{4} twice: once with the product of the denominators and once with the least common denominator. Show that the answers agree.

Part 4: subtracting, sums over one, three fractions

Write each answer in simplest form. Write a sum greater than 1 as a mixed number.

17. \tfrac{3}{4} - \tfrac{1}{3}
18. \tfrac{7}{10} - \tfrac{1}{2}
19. \tfrac{5}{6} - \tfrac{2}{9}
20. \tfrac{2}{3} + \tfrac{3}{5}
21. \tfrac{7}{8} + \tfrac{5}{12}
22. \tfrac{1}{2} + \tfrac{2}{3} + \tfrac{1}{6}
23. Lanre learned \tfrac{3}{8} of his routine on Monday and \tfrac{1}{3} of it on Wednesday. What fraction of the routine does he still have to learn?

Reasoning

24. Explain why, when one denominator is a multiple of the other, the larger denominator is always the least common denominator.
25. A proper fraction is less than 1. Explain why the sum of two proper fractions is always less than 2, and why the difference of two proper fractions (larger minus smaller) is always less than 1.