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Grade 4 · 4.NF.B.3 · 4.NF.B.3B

Adding and subtracting fractions with like denominators

At the class party, Basak eats some slices of a pizza and Taras eats some more slices of the same pizza. How much of the pizza did they eat together, written as one fraction?

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Decomposing a fraction into unit fractionsGrade 4Fractions greater than one (improper fractions)Grade 3
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Adding and subtracting mixed numbers with like denominatorsGrade 4Adding mixed numbersGrade 5
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Adding and subtracting fractions5
Adding and subtracting fractions with like denominators
Estimating with fractionsAdding tenths and hundredths as fractionsAdding and subtracting fractions with unlike denominatorsWord problems that add or subtract fractions

When you reach for it

The class party has pizza. Every pizza is the same size and is cut into equal slices. Basak eats some slices from one pizza, and Taras eats some more from the same pizza. How much of the pizza is gone? Later, Vuk carries part of a pizza away to another table. How much is left on the tray?

Questions like these join amounts or take them apart, and the amounts are fractions of one whole. When every fraction in the question has the same denominator, the pieces are all the same size. Then adding and subtracting is only counting pieces.

You already know the other direction. In Decomposing a fraction into unit fractions one fraction was broken into a sum of smaller parts. Adding reads that same kind of equation from the parts back to the total.

The procedure

This lesson takes three days, one part a day.

Part 1: adding pieces of the same size

A pizza at the party is cut into 8 equal slices. Each slice is \displaystyle \frac{1}{8} of the pizza, a unit fraction. Basak eats 3 slices, which is \displaystyle \frac{3}{8} of the pizza. Taras eats 4 slices, which is \displaystyle \frac{4}{8} of the pizza. Together they ate

\frac{3}{8} + \frac{4}{8}
B B B T T T T
Figure 1Basak’s 3 slices and Taras’s 4 slices are 7 of the 8 equal slices.

Look at the picture and count. Basak’s slices and Taras’s slices are 7 slices in all, and every one of them is an eighth of the pizza. So they ate \displaystyle \frac{7}{8} of the pizza:

\frac{3}{8} + \frac{4}{8} = \frac{7}{8}

Here is the same thing written with unit fractions. Each fraction is a count of eighths, so write out the eighths and put them together:

\begin{aligned} \frac{3}{8} + \frac{4}{8} &= \left(\frac{1}{8} + \frac{1}{8} + \frac{1}{8}\right) \\ &\quad + \left(\frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8}\right) \\ &= \frac{7}{8} \end{aligned}

Taking away the parentheses is allowed by the associative property of addition: how the numbers in a sum are grouped does not change the total. Once the parentheses are gone, there are seven copies of \displaystyle \frac{1}{8}, and seven copies of \displaystyle \frac{1}{8} is \displaystyle \frac{7}{8}.

A number line tells the same story with lengths. Cut the space from 0 to 1 into eighths. Start at 0 and make 3 hops of \displaystyle \frac{1}{8}. Then make 4 more hops. You land on \displaystyle \frac{7}{8} (counting by unit fractions: one eighth, two eighths, and so on up to seven eighths).

0
\tfrac{1}{8}
\tfrac{2}{8}
\tfrac{3}{8}
\tfrac{4}{8}
\tfrac{5}{8}
\tfrac{6}{8}
\tfrac{7}{8}
 1
\tfrac{3}{8}
\tfrac{4}{8}
Figure 2Three hops of \displaystyle \frac{1}{8}, then four more, land on \displaystyle \frac{7}{8}.

Why only the numerators add. The two numbers in a fraction do two different jobs (see What the numerator and the denominator each tell you). The denominator 8 names the size of the piece: eighths. The numerator counts how many of those pieces there are. When Basak’s slices and Taras’s slices are put on one plate, nothing is cut and nothing is glued together. Every slice on the plate is still one eighth of the pizza. So the size of the piece stays the same, and the denominator stays 8. The only thing that changes is how many slices there are, and that count is 3 + 4 = 7.

It works just like adding with a word. Three apples and four apples are seven apples. Three eighths and four eighths are seven eighths. The word “eighths” names the kind of thing being counted, and it does not change when you count more of it.

The same reasoning works for any denominator. If two fractions have the same denominator, count the pieces in both and keep the size:

\frac{a}{n} + \frac{b}{n} = \frac{a + b}{n}

Here a and b are the two counts and n is the number of equal pieces in the whole. Three or more fractions with the same denominator add the same way: count all their pieces.

Why the denominators have to match. Suppose Vuk puts \displaystyle \frac{1}{2} of a pizza and \displaystyle \frac{1}{4} of another pizza of the same size on his plate. There are two pieces on the plate, but they are not two of the same piece. A half is bigger than a fourth. The count “two” would not say two of what, so the pieces cannot simply be counted together. The amount on the plate is real, but to name it with one fraction, the pieces must first be made the same size. Cut the half into two fourths, so \displaystyle \frac{1}{2} = \frac{2}{4}, and then \displaystyle \frac{2}{4} + \frac{1}{4} = \frac{3}{4}. Renaming fractions so their pieces match is the work of Adding and subtracting fractions with unlike denominators. In this lesson the pieces already match.

Part 2: subtracting pieces of the same size

A pizza on the tray is cut into 10 equal slices, and 9 slices are left. That is \displaystyle \frac{9}{10} of the pizza. Vuk takes 4 slices to another table, which is \displaystyle \frac{4}{10} of the pizza. How much is still on the tray?

\frac{9}{10} - \frac{4}{10}

Count what is left. There were 9 slices and 4 went away, so 9 - 4 = 5 slices are left, and each one is still a tenth of the pizza:

\frac{9}{10} - \frac{4}{10} = \frac{5}{10}
\tfrac{5}{10}
Figure 3Nine tenths on the tray, four tenths taken, five tenths left.

The denominator stays 10 for the same reason as before. Taking slices away does not change the size of the slices that stay behind. Each one is still one tenth. Only the count changes. In unit fractions: nine copies of \displaystyle \frac{1}{10}, take away four copies, and five copies remain. On a number line, start at \displaystyle \frac{9}{10} and hop back 4 tenths to land on \displaystyle \frac{5}{10}.

The rule for subtraction matches the rule for addition. When the denominators are the same, subtract the counts and keep the size:

\frac{a}{n} - \frac{b}{n} = \frac{a - b}{n}

Subtraction also answers “how much more?”. Taras has \displaystyle \frac{6}{10} of a pizza on his plate and Basak has \displaystyle \frac{2}{10} of a pizza of the same size. Nobody takes anything away. The question is how much more pizza Taras has than Basak.

Taras Basak
\tfrac{4}{10}
Figure 4Line the two amounts up. The extra part of Taras’s amount is \displaystyle \frac{4}{10}.

Line the two amounts up from the same starting edge. Basak’s 2 tenths match the first 2 of Taras’s 6 tenths. The part of Taras’s amount with nothing to match is the difference:

\frac{6}{10} - \frac{2}{10} = \frac{4}{10}

Taras has \displaystyle \frac{4}{10} of a pizza more than Basak. The same question can be asked as a missing part: what must be added to Basak’s \displaystyle \frac{2}{10} to make \displaystyle \frac{6}{10}? Since \displaystyle \frac{2}{10} + \frac{4}{10} = \frac{6}{10}, the answer is \displaystyle \frac{4}{10} again. That is why every subtraction can be checked by adding: the part that is left (or the difference) plus the part taken away must give back the starting amount (see Subtraction as an unknown addend).

Taking from a whole pizza. A whole pizza cut into 10 slices is 10 tenths of the pizza, so \displaystyle 1 = \frac{10}{10}. Before taking tenths away from 1, write the 1 as tenths:

1 - \frac{3}{10} = \frac{10}{10} - \frac{3}{10} = \frac{7}{10}

Now both amounts count the same pieces, and 10 - 3 = 7.

Part 3: sums that pass one whole

Two pizzas of the same size are each cut into 6 equal slices. At the end of the party, Basak’s table has \displaystyle \frac{5}{6} of one pizza left, and Taras’s table has \displaystyle \frac{4}{6} of the other. They put all the leftover slices together. How much pizza is that?

The slices are all sixths of a pizza the same size, so count them: 5 + 4 = 9 sixths.

\frac{5}{6} + \frac{4}{6} = \frac{9}{6}

The numerator 9 is bigger than the denominator 6, so this is more than one whole pizza (see Fractions greater than one (improper fractions)). A whole pizza has only 6 slices, and there are 9.

\tfrac{6}{6} = 1
\tfrac{3}{6}
Figure 5Nine sixths of a pizza fill one whole pizza with three sixths of another left over.

Put 6 of the slices together and they fill one whole pizza. Since 9 = 6 + 3, group the sixths that way, using the associative property:

\begin{aligned} \frac{9}{6} &= \frac{6}{6} + \frac{3}{6} \\ &= 1 + \frac{3}{6} \\ &= 1\frac{3}{6} \end{aligned}

The leftovers are one whole pizza and \displaystyle \frac{3}{6} of another, which is the mixed number \displaystyle 1\frac{3}{6} (read “one and three sixths”). Both \displaystyle \frac{9}{6} and \displaystyle 1\frac{3}{6} name the same amount. The mixed number is often easier to picture: one full pizza and part of another. The quick way to go between the two forms is in Mixed numbers and improper fractions: rewriting and comparing.

0
\tfrac{1}{6}
\tfrac{2}{6}
\tfrac{3}{6}
\tfrac{4}{6}
\tfrac{5}{6}
 1
\tfrac{7}{6}
\tfrac{8}{6}
\tfrac{9}{6}
\tfrac{10}{6}
\tfrac{11}{6}
 2
\tfrac{5}{6}
\tfrac{4}{6}
Figure 6Five sixths and four sixths more pass 1 and land on \displaystyle \frac{9}{6}.

On the number line, the first 6 hops of \displaystyle \frac{1}{6} reach 1, and the last 3 hops go past it to \displaystyle \frac{9}{6}. The hops do not care where the whole number 1 sits. They just keep counting sixths.

Word problems with one whole. In a word problem, first decide what the whole is. At this party the whole is always one pizza, and every pizza is the same size. That is what lets Basak’s sixths and Taras’s sixths be counted together. Then read what happens to the amounts:

  • Pieces are put together, or the question asks for a total: add the counts.
  • Pieces are taken away, eaten or given away, and the question asks what is left: subtract.
  • Two amounts are compared, and the question asks how much more or how much less: subtract.

Answers more than one whole are written as a fraction greater than one or as a mixed number, whichever the question needs. Adding and subtracting amounts that already start as mixed numbers, such as \displaystyle 2\frac{1}{6} pizzas, is the next lesson, Adding and subtracting mixed numbers with like denominators.

Worked examples

Example 1

Two people share one pizza

A pizza is cut into 12 equal slices. Basak eats 2 slices and gives 7 slices to Vuk’s table. What fraction of the pizza did she eat or give away?

Solution

Each slice is \displaystyle \frac{1}{12} of the pizza, because the pizza is cut into 12 equal slices. She ate \displaystyle \frac{2}{12} and gave away \displaystyle \frac{7}{12}. The question asks for both together, so add:

\frac{2}{12} + \frac{7}{12} = \frac{9}{12}

All the slices are twelfths, so the denominator stays 12, and 2 + 7 = 9 counts them. She ate or gave away \displaystyle \frac{9}{12} of the pizza. Check by counting the slices left on her tray: 12 - 9 = 3.

Example 2

Three fractions that fill the whole

A pizza is cut into 8 equal slices. Basak takes \displaystyle \frac{2}{8} of it, Taras takes \displaystyle \frac{1}{8} and Vuk takes \displaystyle \frac{5}{8}. How much of the pizza did they take? Is any left?

Solution

All three amounts are eighths, so count all the eighths:

\frac{2}{8} + \frac{1}{8} + \frac{5}{8} = \frac{8}{8}

because 2 + 1 + 5 = 8. Eight eighths is every slice of the pizza, so \displaystyle \frac{8}{8} = 1. They took the whole pizza, and none is left.

The order of adding does not matter. By the commutative property (switching the order in a sum does not change the total), \displaystyle \frac{2}{8} + \frac{5}{8} + \frac{1}{8} gives the same 8 eighths.

Example 3

Taking slices from a whole pizza

A whole pizza is cut into 8 equal slices. The first table eats \displaystyle \frac{5}{8} of it. How much of the pizza is left?

Solution

The pizza starts whole, so the question is \displaystyle 1 - \frac{5}{8}. The 1 is not yet written in eighths, so write it that way first: a whole pizza is all 8 slices, so \displaystyle 1 = \frac{8}{8}. Now both amounts count eighths:

1 - \frac{5}{8} = \frac{8}{8} - \frac{5}{8} = \frac{3}{8}

because 8 - 5 = 3. The slices left are still eighths, so the denominator stays 8. There is \displaystyle \frac{3}{8} of the pizza left. Check by adding back: \displaystyle \frac{3}{8} + \frac{5}{8} = \frac{8}{8} = 1.

Example 4

How much more

Two pizzas of the same size are each cut into 12 equal slices. At the end of the party, Taras’s table has \displaystyle \frac{11}{12} of a pizza left and Vuk’s table has \displaystyle \frac{4}{12} of a pizza left. How much more pizza is left at Taras’s table?

Solution

Nothing is taken away here; two amounts are compared. Picture the two amounts lined up from the same edge. Vuk’s 4 twelfths match 4 of Taras’s 11 twelfths, and the twelfths with no match are the difference:

\frac{11}{12} - \frac{4}{12} = \frac{7}{12}

Taras’s table has \displaystyle \frac{7}{12} of a pizza more. Check with the missing part: Vuk’s amount plus the difference should give Taras’s amount, and \displaystyle \frac{4}{12} + \frac{7}{12} = \frac{11}{12}.

Example 5

Two people take from the same pizza

A whole pizza is cut into 10 equal slices. Taras takes \displaystyle \frac{2}{10} of it, and then Vuk takes \displaystyle \frac{5}{10}. How much of the pizza is left? Find it two ways.

Solution

First way, one step at a time. Write the whole pizza as \displaystyle \frac{10}{10}, then take away each person’s tenths in turn:

\begin{aligned} \frac{10}{10} - \frac{2}{10} &= \frac{8}{10} \\ \frac{8}{10} - \frac{5}{10} &= \frac{3}{10} \end{aligned}

Second way, add first. Together the two took \displaystyle \frac{2}{10} + \frac{5}{10} = \frac{7}{10} of the pizza. Take that away from the whole:

\frac{10}{10} - \frac{7}{10} = \frac{3}{10}

Both ways leave \displaystyle \frac{3}{10} of the pizza. They must agree, because the same 7 slices leave the tray whether they go one person at a time or both at once.

Example 6

A sum more than one whole

Two pizzas of the same size are each cut into 8 equal slices. Basak’s table eats \displaystyle \frac{5}{8} of one pizza, and Taras’s table eats \displaystyle \frac{7}{8} of the other. How much pizza did the two tables eat? Write the answer as a fraction and as a mixed number.

Solution

The slices are all eighths of the same size of pizza, so count them:

\frac{5}{8} + \frac{7}{8} = \frac{12}{8}

because 5 + 7 = 12. Twelve eighths is more than one pizza, since a pizza has only 8 slices. Split the count into one full pizza and the rest, 12 = 8 + 4:

\begin{aligned} \frac{12}{8} &= \frac{8}{8} + \frac{4}{8} \\ &= 1 + \frac{4}{8} = 1\frac{4}{8} \end{aligned}

The two tables ate \displaystyle \frac{12}{8} of a pizza, which is \displaystyle 1\frac{4}{8} pizzas: one whole pizza and four slices of another.

Example 7

A party with three pizzas

The class orders 3 pizzas of the same size, each cut into 6 equal slices. Basak’s table eats \displaystyle \frac{4}{6} of a pizza, Taras’s table eats \displaystyle \frac{5}{6} of a pizza and Vuk’s table eats \displaystyle \frac{4}{6} of a pizza. How much pizza was eaten? How much is left?

Solution

Every amount is a count of sixths of the same size of pizza, so add the counts:

\frac{4}{6} + \frac{5}{6} + \frac{4}{6} = \frac{13}{6}

because 4 + 5 + 4 = 13. Each whole pizza is 6 sixths, and 13 = 6 + 6 + 1. Group the sixths into full pizzas:

\begin{aligned} \frac{13}{6} &= \frac{6}{6} + \frac{6}{6} + \frac{1}{6} \\ &= 1 + 1 + \frac{1}{6} = 2\frac{1}{6} \end{aligned}

The class ate \displaystyle 2\frac{1}{6} pizzas. To find what is left, write the 3 pizzas as sixths too: 3 pizzas of 6 slices is 18 slices, so \displaystyle 3 = \frac{18}{6}. Then

\frac{18}{6} - \frac{13}{6} = \frac{5}{6}

There is \displaystyle \frac{5}{6} of a pizza left, five slices. Check: the eaten slices and the slices left should be all the slices, and 13 + 5 = 18.

Boundaries

  • The denominators must be the same. Counting pieces works only when every piece is the same size. A half and a fourth cannot be counted together as one kind of piece. They have to be renamed to pieces of one size first, which is Adding and subtracting fractions with unlike denominators.
  • The fractions must be of the same whole. An eighth of a small pizza and an eighth of a large pizza are different amounts of pizza, even though both are written \displaystyle \frac{1}{8}. Adding them as \displaystyle \frac{2}{8} would not be two eighths of any one pizza.
  • The pieces must be equal. A fraction like \displaystyle \frac{3}{8} means 3 of 8 equal pieces. If a pizza is cut into 8 slices of different sizes, one slice is not \displaystyle \frac{1}{8} of it.
  • You cannot take away more pieces than there are. With amounts of pizza, \displaystyle \frac{3}{8} - \frac{5}{8} has no answer: there are only 3 slices, and 5 cannot be taken from them. In a subtraction here, the first numerator is at least as big as the second.
  • The same amount has more than one name. A sum can be written as a fraction greater than one, \displaystyle \frac{9}{6}, or as a mixed number, \displaystyle 1\frac{3}{6}. Both are correct. Some answers can also be renamed in bigger pieces, such as \displaystyle \frac{4}{8} = \frac{1}{2}, which is the idea of Equivalent fractions. Renaming is never needed to add or subtract like fractions.
  • The denominator cannot be 0. A whole cut into 0 pieces has no pieces to count.

Where this goes next

Mixed numbers come next. With Adding and subtracting mixed numbers with like denominators you will add \displaystyle 1\frac{3}{6} pizzas and \displaystyle 2\frac{4}{6} pizzas by adding the wholes and the sixths, and sometimes trading a whole for 6 sixths to subtract.

Counting same-size pieces is also the reason tenths and hundredths can be added once tenths are written as hundredths, in Adding tenths and hundredths as fractions. Later, fractions with different denominators are renamed into shared pieces and then added exactly as here, in Adding and subtracting fractions with unlike denominators. Deciding whether a story joins, takes away or compares amounts, with any fractions, grows into Word problems that add or subtract fractions.

Exercises

Part 1: adding pieces of the same size

  1. Find \displaystyle \frac{2}{7} + \frac{3}{7}.
  2. Find \displaystyle \frac{4}{9} + \frac{4}{9}.
  3. Find \displaystyle \frac{3}{12} + \frac{8}{12}.
  4. Find \displaystyle \frac{1}{5} + \frac{2}{5} + \frac{1}{5}.
  5. A pizza is cut into 10 equal slices. Taras eats 3 slices and Vuk eats 6 slices. What fraction of the pizza did they eat together?
  6. Find \displaystyle \frac{3}{4} + \frac{1}{4}. What does the answer mean for a pizza cut into 4 slices?
  7. Find the missing numerator: \displaystyle \frac{3}{11} + \frac{?}{11} = \frac{10}{11}.
  8. A number line from 0 to 1 is marked in twelfths. Start at \displaystyle \frac{4}{12} and make 5 hops of \displaystyle \frac{1}{12}. Where do you land? Write the addition.

Part 2: subtracting pieces of the same size

  1. Find \displaystyle \frac{6}{7} - \frac{2}{7}.
  2. Find \displaystyle \frac{11}{12} - \frac{2}{12}.
  3. Find \displaystyle 1 - \frac{2}{5}.
  4. Find \displaystyle \frac{5}{9} - \frac{5}{9}. What does the answer mean?
  5. A pizza is cut into 8 equal slices, and \displaystyle \frac{7}{8} of it is on the tray. Vuk takes 4 slices. How much of the pizza is still on the tray?
  6. Basak has \displaystyle \frac{9}{12} of a pizza and Taras has \displaystyle \frac{4}{12} of a pizza of the same size. How much more pizza does Basak have?
  7. Find the missing numerator: \displaystyle \frac{?}{10} - \frac{3}{10} = \frac{6}{10}.
  8. Find \displaystyle \frac{7}{8} - \frac{2}{8} - \frac{1}{8}.

Part 3: sums that pass one whole

  1. Find \displaystyle \frac{2}{3} + \frac{2}{3}. Write the answer as a fraction and as a mixed number.
  2. Find \displaystyle \frac{7}{10} + \frac{6}{10}. Write the answer as a fraction and as a mixed number.
  3. Find \displaystyle \frac{4}{5} + \frac{4}{5} + \frac{3}{5}. Write the answer as a mixed number.
  4. Two pizzas of the same size are each cut into 12 equal slices. Taras’s table eats \displaystyle \frac{9}{12} of one and Vuk’s table eats \displaystyle \frac{7}{12} of the other. How much pizza did they eat? Write the answer as a mixed number.
  5. The class has 2 pizzas of the same size, each cut into 8 equal slices. By the end of the party, \displaystyle \frac{11}{8} of a pizza has been eaten. How much pizza is left?
  6. Which sum is more than one whole, \displaystyle \frac{3}{7} + \frac{5}{7} or \displaystyle \frac{2}{9} + \frac{6}{9}? Explain without drawing.

Reasoning

  1. Explain, using the size of the slices, why \displaystyle \frac{3}{5} + \frac{1}{5} has denominator 5, and why its numerator is 4.
  2. Vuk eats \displaystyle \frac{1}{2} of a small pizza and \displaystyle \frac{1}{2} of a large pizza. He says he ate one pizza in all. Explain why “one pizza” does not describe what he ate.
  3. Taras says that \displaystyle \frac{11}{12} - \frac{5}{12} and the missing part in \displaystyle \frac{5}{12} + \frac{?}{12} = \frac{11}{12} always have the same answer. Explain why, and give the answer.
Answers
  1. \displaystyle \frac{5}{7}. Both are sevenths, so count them: 2 + 3 = 5 sevenths.
  2. \displaystyle \frac{8}{9}, since 4 + 4 = 8 ninths.
  3. \displaystyle \frac{11}{12}, since 3 + 8 = 11 twelfths.
  4. \displaystyle \frac{4}{5}, since 1 + 2 + 1 = 4 fifths.
  5. \displaystyle \frac{9}{10} of the pizza. Each slice is \displaystyle \frac{1}{10}, and \displaystyle \frac{3}{10} + \frac{6}{10} = \frac{9}{10}, since 3 + 6 = 9.
  6. \displaystyle \frac{4}{4} = 1. Three slices and one more slice are all 4 slices, the whole pizza.
  7. 7, so the missing part is \displaystyle \frac{7}{11}, since 3 + 7 = 10.
  8. You land on \displaystyle \frac{9}{12}, and \displaystyle \frac{4}{12} + \frac{5}{12} = \frac{9}{12}, since each hop is one twelfth and 4 + 5 = 9.
  9. \displaystyle \frac{4}{7}, since 6 - 2 = 4 sevenths.
  10. \displaystyle \frac{9}{12}, since 11 - 2 = 9 twelfths.
  11. \displaystyle \frac{3}{5}. Write \displaystyle 1 = \frac{5}{5}, then \displaystyle \frac{5}{5} - \frac{2}{5} = \frac{3}{5}.
  12. \displaystyle \frac{0}{9} = 0. All five ninths are taken away, so nothing is left.
  13. \displaystyle \frac{3}{8} of the pizza. The 4 slices are \displaystyle \frac{4}{8}, and \displaystyle \frac{7}{8} - \frac{4}{8} = \frac{3}{8}.
  14. \displaystyle \frac{5}{12} of a pizza more, since \displaystyle \frac{9}{12} - \frac{4}{12} = \frac{5}{12}. Check: \displaystyle \frac{4}{12} + \frac{5}{12} = \frac{9}{12}.
  15. 9, so the starting amount is \displaystyle \frac{9}{10}. Add back what was taken: \displaystyle \frac{6}{10} + \frac{3}{10} = \frac{9}{10}.
  16. \displaystyle \frac{4}{8}. One step at a time: \displaystyle \frac{7}{8} - \frac{2}{8} = \frac{5}{8}, then \displaystyle \frac{5}{8} - \frac{1}{8} = \frac{4}{8}. Or take away \displaystyle \frac{3}{8} at once, since 2 + 1 = 3.
  17. \displaystyle \frac{4}{3} = 1\frac{1}{3}. Count: 2 + 2 = 4 thirds, and \displaystyle \frac{4}{3} = \frac{3}{3} + \frac{1}{3} = 1 + \frac{1}{3}.
  18. \displaystyle \frac{13}{10} = 1\frac{3}{10}. Count: 7 + 6 = 13 tenths, and 13 = 10 + 3, so \displaystyle \frac{13}{10} = \frac{10}{10} + \frac{3}{10}.
  19. \displaystyle 2\frac{1}{5}. Count: 4 + 4 + 3 = 11 fifths, and 11 = 5 + 5 + 1, so \displaystyle \frac{11}{5} = \frac{5}{5} + \frac{5}{5} + \frac{1}{5} = 2\frac{1}{5}.
  20. \displaystyle 1\frac{4}{12} pizzas. Count: \displaystyle \frac{9}{12} + \frac{7}{12} = \frac{16}{12}, and 16 = 12 + 4, so \displaystyle \frac{16}{12} = 1 + \frac{4}{12}.
  21. \displaystyle \frac{5}{8} of a pizza. Two pizzas are 16 slices, so \displaystyle 2 = \frac{16}{8}, and \displaystyle \frac{16}{8} - \frac{11}{8} = \frac{5}{8}.
  22. \displaystyle \frac{3}{7} + \frac{5}{7}. It is \displaystyle \frac{8}{7}, and 8 sevenths is more than the 7 sevenths in a whole. The other sum is \displaystyle \frac{8}{9}, less than the 9 ninths in a whole.
  23. Each fraction counts fifths, pieces of a whole cut into 5 equal parts. Putting 3 fifths and 1 fifth together does not cut or join any piece, so every piece is still a fifth and the denominator stays 5. The numerator counts the pieces: 3 + 1 = 4. So the sum is \displaystyle \frac{4}{5}.
  24. A half of the small pizza and a half of the large pizza are not the same size of piece, because the wholes are different. Two halves make one pizza only when both are halves of the same pizza. Vuk ate more than one small pizza and less than one large pizza, so “one pizza” names no single whole that matches what he ate.
  25. Subtracting asks what is left when 5 twelfths are taken from 11 twelfths. The missing part asks what must be put with 5 twelfths to make 11 twelfths. Both look for the part that, together with the 5 twelfths, makes the 11 twelfths, so they are the same number. The answer is \displaystyle \frac{6}{12}, since 11 - 5 = 6 and 5 + 6 = 11.
kg1128 · updated 2026-10-20 · CC-BY-SA-4.0