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Grade 4 · 4.NF.B.3 · 4.NF.B.3B

Decomposing a fraction into unit fractions

Five of the six portions of cornbread are left in the pan. How can that one amount, five sixths of the pan, be broken into smaller amounts and still be the same cornbread?

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Decomposing a fraction into unit fractions

When you reach for it

Tolga bakes a pan of cornbread and cuts it into 6 equal portions. After lunch, 5 portions are left. That is \displaystyle \frac{5}{6} of the pan.

Now Kateryna wants some and Goran wants some. The \displaystyle \frac{5}{6} has to be split into smaller amounts. How much can each person get? What equation tells the story?

Questions like this need a fraction to come apart. To decompose a fraction is to write it as a sum of smaller fractions. In this lesson every part keeps the same denominator as the fraction you start with, so every part is made of the same size of piece.

Reach for it when an amount has to be shared, split or grouped. Reach for it, too, before you add or subtract fractions: a sum of fractions is a decomposition read the other way.

The procedure

This lesson takes three days, one part a day.

Part 1: a fraction is a count of unit fractions

A unit fraction is one piece when a whole is cut into equal parts, a fraction with numerator 1 (see Unit fractions). One portion of Tolga’s pan is \displaystyle \frac{1}{6} of the pan, because the pan is cut into 6 equal portions.

In \displaystyle \frac{5}{6}, the denominator 6 names the size of the piece, sixths. The numerator 5 counts how many of those pieces there are (see What the numerator and the denominator each tell you). So \displaystyle \frac{5}{6} means five pieces, each \displaystyle \frac{1}{6} of the pan. Write those five pieces as a sum:

\frac{5}{6} = \frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{1}{6}

That is the first way to decompose any fraction: break it all the way down into unit fractions, one for each piece the numerator counts.

\tfrac{1}{6}
\tfrac{1}{6}
\tfrac{1}{6}
\tfrac{1}{6}
\tfrac{1}{6}
\tfrac{5}{6}
Figure 1Five sixths is five pieces, each one sixth of the whole tape.

The tape is the pan stretched out in a row. It is cut into 6 equal pieces, and 5 are shaded. Each shaded piece is \displaystyle \frac{1}{6}, and together they are \displaystyle \frac{5}{6}.

A number line shows the same thing as lengths. Cut the space from 0 to 1 into 6 equal steps. Each step is \displaystyle \frac{1}{6} long. Start at 0 and hop one step at a time, counting “one sixth, two sixths, three sixths, four sixths, five sixths” (that is counting by unit fractions). After five hops you are at \displaystyle \frac{5}{6}.

0
\tfrac{1}{6}
\tfrac{2}{6}
\tfrac{3}{6}
\tfrac{4}{6}
\tfrac{5}{6}
 1
\tfrac{1}{6}
\tfrac{1}{6}
\tfrac{1}{6}
\tfrac{1}{6}
\tfrac{1}{6}
Figure 2Five hops of \displaystyle \frac{1}{6} from 0 land on \displaystyle \frac{5}{6}.

Why is the sum true? It is not a rule to remember. It is what the fraction means. The fraction \displaystyle \frac{5}{6} is defined as 5 pieces of size \displaystyle \frac{1}{6}, and a sum of five copies of \displaystyle \frac{1}{6} is 5 pieces of size \displaystyle \frac{1}{6}. The two sides name the same amount of cornbread.

The same works the other way. A sum of copies of one unit fraction is a single fraction: count the copies for the numerator and keep the size for the denominator. Three copies of \displaystyle \frac{1}{4} make \displaystyle \frac{3}{4}.

Part 2: many ways to break a fraction apart

Kateryna takes 2 of the 5 portions, and Goran takes the other 3. Kateryna has \displaystyle \frac{2}{6} of the pan and Goran has \displaystyle \frac{3}{6}. Nothing was eaten or added. The same five portions were only put on two plates. So

\frac{5}{6} = \frac{2}{6} + \frac{3}{6}

Here is why it is true, piece by piece. Start from the five unit fractions and group them:

\begin{aligned} \frac{5}{6} &= \frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{1}{6} \\ &= \left(\frac{1}{6} + \frac{1}{6}\right) \\ &\quad + \left(\frac{1}{6} + \frac{1}{6} + \frac{1}{6}\right) \\ &= \frac{2}{6} + \frac{3}{6} \end{aligned}

Putting in the parentheses is allowed by the associative property of addition: how you group the numbers in a sum does not change the total. The first group is two sixths and the second is three sixths.

The next day there are 5 portions left again. This time Kateryna takes 1 and Goran takes 4:

\frac{5}{6} = \frac{1}{6} + \frac{4}{6}
\tfrac{2}{6}
\tfrac{3}{6}
\tfrac{5}{6} = \tfrac{2}{6} + \tfrac{3}{6}
\tfrac{1}{6}
\tfrac{4}{6}
\tfrac{5}{6} = \tfrac{1}{6} + \tfrac{4}{6}
Figure 3The same five sixths, grouped two different ways. The pieces do not change, only how they are grouped.

Both equations are true, and so are others. The pattern in every one: the denominators all stay 6, and the numerators add up to 5. The denominator stays because grouping pieces never changes the size of a piece. A sixth on Kateryna’s plate is still a sixth of the pan. The numerators add to 5 because the plates hold all five pieces, each piece on exactly one plate.

So finding a decomposition of \displaystyle \frac{5}{6} is the same as breaking the whole number 5 into parts, then putting each part over 6. Since 5 = 2 + 3, we get \displaystyle \frac{2}{6} + \frac{3}{6}. Since 5 = 1 + 1 + 3, we get

\frac{5}{6} = \frac{1}{6} + \frac{1}{6} + \frac{3}{6}

Order does not matter either. Goran’s \displaystyle \frac{3}{6} could be listed first: \displaystyle \frac{3}{6} + \frac{2}{6} is the same total as \displaystyle \frac{2}{6} + \frac{3}{6}. That is the commutative property of addition: switching the order of the numbers in a sum does not change the total.

Record every decomposition as an equation, with the starting fraction on one side and the sum on the other. An equation says the two sides are the same amount, and that is exactly the claim a decomposition makes.

Part 3: fractions greater than one, and mixed numbers

At Goran’s party there are two pans of cornbread, the same size, each cut into 4 equal portions. At the end, 7 portions are left. Each portion is \displaystyle \frac{1}{4} of a pan, so the amount left is \displaystyle \frac{7}{4} of a pan. That is more than one whole pan, because a pan has only 4 portions. A fraction whose numerator is bigger than its denominator is more than 1 (see Fractions greater than one (improper fractions)).

Decompose it the same way as before. Seven pieces of size \displaystyle \frac{1}{4}:

\begin{aligned} \frac{7}{4} &= \frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4} \\ &\quad + \frac{1}{4} + \frac{1}{4} + \frac{1}{4} \end{aligned}

The first four portions fill one whole pan. Group them, since 7 = 4 + 3:

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

Four fourths of a pan is the whole pan, so \displaystyle \frac{4}{4} = 1. Put that in:

\frac{7}{4} = 1 + \frac{3}{4}
\tfrac{4}{4} = 1
\tfrac{3}{4}
Figure 4Seven fourths of a pan is one full pan and three fourths of another.

On a number line, the first four hops of \displaystyle \frac{1}{4} land on 1. Three more hops go past 1 and land on \displaystyle \frac{7}{4}.

0
\tfrac{1}{4}
\tfrac{2}{4}
\tfrac{3}{4}
 1
\tfrac{5}{4}
\tfrac{6}{4}
\tfrac{7}{4}
 2
Figure 5Seven hops of \displaystyle \frac{1}{4}. The first four reach 1; three more land on \displaystyle \frac{7}{4}.

The sum \displaystyle 1 + \frac{3}{4} is written \displaystyle 1\frac{3}{4} for short and read “one and three fourths”. A number written this way, a whole number with a fraction beside it, is a mixed number. The short form means the sum: \displaystyle 1\frac{3}{4} = 1 + \frac{3}{4}.

That meaning lets a mixed number come apart too. Kateryna has \displaystyle 2\frac{1}{3} pans of cornbread, with pans cut into 3 portions. It is \displaystyle 2 + \frac{1}{3}, and 2 is 1 + 1, so

2\frac{1}{3} = 1 + 1 + \frac{1}{3}

Each whole pan is 3 thirds, \displaystyle \frac{3}{3} = 1, so the same amount in thirds is

2\frac{1}{3} = \frac{3}{3} + \frac{3}{3} + \frac{1}{3}

Every portion is now a third of a pan, so the sum can be split right down to unit fractions if needed. Turning a mixed number into one single fraction in a quick way is its own skill, Mixed numbers and improper fractions: rewriting and comparing.

Worked examples

Example 1

A fraction as unit fractions

Kateryna cuts a pan of cornbread into 8 equal portions and wraps 3 of them for a neighbor. What fraction of the pan did she wrap? Write it as a sum of unit fractions.

Solution

Each portion is \displaystyle \frac{1}{8} of the pan, because the pan is cut into 8 equal portions. She wrapped 3 portions, so she wrapped \displaystyle \frac{3}{8} of the pan. The numerator 3 counts the pieces, so the sum has three copies of \displaystyle \frac{1}{8}:

\frac{3}{8} = \frac{1}{8} + \frac{1}{8} + \frac{1}{8}

Check on a number line in eighths: three hops of \displaystyle \frac{1}{8} from 0 land on \displaystyle \frac{3}{8}.

Example 2

Unit fractions that fill the whole

Goran cuts a small pan into 3 equal portions. He eats one, Tolga eats one, and Kateryna eats one. Write the amount eaten as a sum of unit fractions and as a single fraction. How much of the pan is left?

Solution

Each portion is \displaystyle \frac{1}{3} of the pan. Three people each ate one portion, three copies of \displaystyle \frac{1}{3}:

\frac{1}{3} + \frac{1}{3} + \frac{1}{3} = \frac{3}{3}

The count of pieces, 3, is the numerator, and the size, thirds, is the denominator. But 3 portions out of a pan of 3 portions is the whole pan, so \displaystyle \frac{3}{3} = 1, and nothing is left. On a number line in thirds, three hops land exactly on 1.

Any fraction with the same numerator and denominator is 1 in the same way: \displaystyle \frac{2}{2}, \displaystyle \frac{9}{9} and \displaystyle \frac{10}{10} are each a whole.

Example 3

Every way to split into two parts

Tolga has \displaystyle \frac{6}{8} of a pan left and puts it on two plates, at least one portion on each. Write an equation for every way he could do it.

Solution

The plates hold all 6 pieces, each an eighth, so the numerators must add to 6 and both denominators stay 8. List the ways to break 6 into two whole numbers, each at least 1, by counting up the first plate:

\begin{aligned} \frac{6}{8} &= \frac{1}{8} + \frac{5}{8} \\ \frac{6}{8} &= \frac{2}{8} + \frac{4}{8} \\ \frac{6}{8} &= \frac{3}{8} + \frac{3}{8} \\ \frac{6}{8} &= \frac{4}{8} + \frac{2}{8} \\ \frac{6}{8} &= \frac{5}{8} + \frac{1}{8} \end{aligned}

The first plate can hold 1 to 5 portions, so there are 5 equations. The commutative property pairs them up: \displaystyle \frac{1}{8} + \frac{5}{8} and \displaystyle \frac{5}{8} + \frac{1}{8} are the same sum in a different order. If order does not count, there are 3 different splits: 1 and 5, 2 and 4, 3 and 3. The last one gives each plate the same amount.

Example 4

Finding a missing part

Kateryna has \displaystyle \frac{7}{10} of a pan of cornbread. She gives \displaystyle \frac{3}{10} of the pan to Goran and keeps the rest. Write an equation with a missing part, and find what she keeps.

Solution

The two parts together are all she had, so

\frac{7}{10} = \frac{3}{10} + \frac{?}{10}

The missing part is also in tenths, because the pieces do not change size when they move to Goran’s plate. The numerators must add to 7, and 3 + 4 = 7, so the missing numerator is 4:

\frac{7}{10} = \frac{3}{10} + \frac{4}{10}

Kateryna keeps \displaystyle \frac{4}{10} of the pan. Check by counting pieces: 3 tenths and 4 tenths are 7 tenths.

Example 5

More than one whole

Pans at a bake sale are cut into 5 equal portions. At closing time, 9 portions are left. Write the amount as a fraction of a pan, then as a whole number of pans and a fraction.

Solution

Each portion is \displaystyle \frac{1}{5} of a pan, and 9 are left, so the amount is \displaystyle \frac{9}{5} of a pan. That is more than one pan, because 9 is more than 5.

One full pan is 5 portions, \displaystyle \frac{5}{5} = 1. Split the 9 portions into one full pan and the rest: 9 = 5 + 4. So, grouping the unit fractions by the associative property,

\begin{aligned} \frac{9}{5} &= \frac{5}{5} + \frac{4}{5} \\ &= 1 + \frac{4}{5} \end{aligned}

There is one full pan and \displaystyle \frac{4}{5} of another, which is \displaystyle 1\frac{4}{5} pans.

Example 6

Taking apart a mixed number

Goran has \displaystyle 2\frac{3}{8} pans of cornbread, and every pan is cut into 8 equal portions. Write it as a sum of wholes and a fraction, then as a sum of eighths only.

Solution

A mixed number means a sum, so \displaystyle 2\frac{3}{8} = 2 + \frac{3}{8}. Two whole pans is 1 + 1:

2\frac{3}{8} = 1 + 1 + \frac{3}{8}

Each whole pan is 8 eighths, \displaystyle \frac{8}{8} = 1. Put that in for each 1:

2\frac{3}{8} = \frac{8}{8} + \frac{8}{8} + \frac{3}{8}

Now every part is in eighths, the same size of piece. The last part could go all the way down to unit fractions too: \displaystyle \frac{3}{8} = \frac{1}{8} + \frac{1}{8} + \frac{1}{8}.

Example 7

Two decompositions, one amount

Goran has \displaystyle \frac{13}{8} of a pan of cornbread. Kateryna writes \displaystyle \frac{13}{8} = 1 + \frac{5}{8}. Tolga writes \displaystyle \frac{13}{8} = \frac{4}{8} + \frac{4}{8} + \frac{5}{8}. Are both right? Show how one turns into the other.

Solution

Check Tolga first. Every part is in eighths and 4 + 4 + 5 = 13, so his parts hold exactly 13 eighths. He is right.

Check Kateryna. One whole pan is \displaystyle \frac{8}{8}, so \displaystyle 1 + \frac{5}{8} = \frac{8}{8} + \frac{5}{8}, and 8 + 5 = 13. She is right too.

To get from Tolga’s sum to Kateryna’s, group his first two parts, using the associative property:

\begin{aligned} &\left(\frac{4}{8} + \frac{4}{8}\right) + \frac{5}{8} \\ &= \frac{8}{8} + \frac{5}{8} \\ &= 1 + \frac{5}{8} \end{aligned}

Tolga’s two plates of \displaystyle \frac{4}{8} together make a whole pan. The two decompositions are the same 13 portions, grouped two ways.

Boundaries

  • The parts must be fractions of the same whole. A sixth of Tolga’s pan and a sixth of a bigger pan are different amounts of cornbread. A decomposition only makes sense when every part is a piece of the one whole the fraction is about.
  • The pieces must be equal. A unit fraction is one of several equal parts. If a pan is cut into 6 pieces of different sizes, one piece is not \displaystyle \frac{1}{6} of the pan.
  • In this lesson, every part has the starting denominator. Only the numerator is split; the denominator, the size of the piece, is never added or split. Cutting the pieces smaller, sixths into twelfths for example, is a different job that changes the denominator.
  • A unit fraction cannot come apart this way. \displaystyle \frac{1}{6} is a single piece. There is no way to write it as two or more parts, each at least one sixth.
  • Parts of zero add nothing. \displaystyle \frac{5}{6} = \frac{5}{6} + \frac{0}{6} is true, but the decompositions here use parts of at least one piece.
  • The denominator cannot be 0. A whole cut into 0 parts has no pieces at all, so there is no unit fraction to count.

Where this goes next

Read a decomposition from right to left and it is an addition: \displaystyle \frac{2}{6} + \frac{3}{6} is two sixths and three sixths, five sixths in all. Adding and subtracting fractions with the same denominator is counting pieces of one size, which is Adding and subtracting fractions with like denominators.

A fraction as a count of one unit fraction also says \displaystyle \frac{5}{6} is 5 copies of \displaystyle \frac{1}{6}, that is \displaystyle 5 \times \frac{1}{6}. Taking copies of a fraction is A fraction times a whole number.

Exercises

Part 1: a fraction is a count of unit fractions

  1. Write \displaystyle \frac{4}{7} as a sum of unit fractions.
  2. Write \displaystyle \frac{2}{9} as a sum of unit fractions.
  3. Write \displaystyle \frac{1}{12} + \frac{1}{12} + \frac{1}{12} + \frac{1}{12} + \frac{1}{12} as a single fraction.
  4. Tolga cuts a pan of cornbread into 10 equal portions, and 6 are left. Write the fraction of the pan that is left as a sum of unit fractions.
  5. How many copies of \displaystyle \frac{1}{8} make \displaystyle \frac{7}{8}?
  6. A number line from 0 to 1 is cut into fifths. Start at 0 and make 3 hops of \displaystyle \frac{1}{5}. Where do you land? Write the hops as a sum.
  7. How many copies of \displaystyle \frac{1}{7} make one whole? Write the sum.
  8. Goran writes \displaystyle \frac{3}{5} = \frac{1}{5} + \frac{1}{5} + \frac{1}{5} + \frac{1}{5}. Is that right? Explain.

Part 2: many ways to break a fraction apart

  1. Write \displaystyle \frac{4}{5} as a sum of two fractions with denominator 5 in two different ways.
  2. Find the missing part: \displaystyle \frac{7}{9} = \frac{2}{9} + \frac{?}{9}.
  3. Find the missing part: \displaystyle \frac{8}{12} = \frac{?}{12} + \frac{3}{12}.
  4. Find every way to write \displaystyle \frac{5}{8} as a sum of two fractions with denominator 8, each at least \displaystyle \frac{1}{8}.
  5. Write \displaystyle \frac{8}{10} as a sum of three fractions with denominator 10 in two different ways.
  6. Kateryna has \displaystyle \frac{9}{12} of a pan. She puts \displaystyle \frac{4}{12} of the pan on a plate for Goran and the rest on a plate for Tolga. Write an equation, and say how much Tolga gets.
  7. Is \displaystyle \frac{3}{7} + \frac{2}{7} a decomposition of \displaystyle \frac{5}{7}? Is \displaystyle \frac{2}{7} + \frac{2}{7} + \frac{2}{7}?
  8. Tolga writes \displaystyle \frac{6}{9} = \frac{2}{9} + \frac{2}{9} + \frac{2}{9}. Goran writes \displaystyle \frac{6}{9} = \frac{3}{9} + \frac{3}{9}. Who is right?

Part 3: fractions greater than one, and mixed numbers

  1. Write \displaystyle \frac{5}{3} as a whole number plus a fraction.
  2. Write \displaystyle \frac{9}{4} as a sum of wholes and a fraction, then as a whole number plus a fraction.
  3. Write \displaystyle 1\frac{2}{5} as a sum of fifths.
  4. Write \displaystyle 3\frac{1}{2} as a sum of ones and a half, then as a sum of halves.
  5. Goran has \displaystyle \frac{11}{8} of a pan, with pans cut into 8 portions. How many full pans is that, and how many extra portions?
  6. Kateryna has \displaystyle \frac{14}{5} pans of cornbread. Write it as a sum of fifths that shows the full pans, then as a whole number plus a fraction.

Reasoning

  1. Kateryna splits \displaystyle \frac{9}{10} of a pan onto three plates. Explain why every part in her equation has denominator 10.
  2. Explain why \displaystyle \frac{1}{4} cannot be written as a sum of two fractions with denominator 4, each at least \displaystyle \frac{1}{4}, but \displaystyle \frac{2}{4} can.
  3. Explain why \displaystyle \frac{6}{6} = 1, and use it to explain why \displaystyle \frac{8}{6} = 1 + \frac{2}{6}.
Answers
  1. \displaystyle \frac{4}{7} = \frac{1}{7} + \frac{1}{7} + \frac{1}{7} + \frac{1}{7}. Four sevenths is four pieces, each \displaystyle \frac{1}{7}.
  2. \displaystyle \frac{2}{9} = \frac{1}{9} + \frac{1}{9}.
  3. \displaystyle \frac{5}{12}. There are five copies of \displaystyle \frac{1}{12}, so the numerator is 5 and the denominator stays 12.
  4. Each portion is \displaystyle \frac{1}{10}, and 6 are left: \displaystyle \frac{6}{10} = \frac{1}{10} + \frac{1}{10} + \frac{1}{10} + \frac{1}{10} + \frac{1}{10} + \frac{1}{10}.
  5. 7 copies. The numerator 7 counts the eighths.
  6. \displaystyle \frac{3}{5}, and \displaystyle \frac{1}{5} + \frac{1}{5} + \frac{1}{5} = \frac{3}{5}.
  7. 7 copies: \displaystyle \frac{1}{7} + \frac{1}{7} + \frac{1}{7} + \frac{1}{7} + \frac{1}{7} + \frac{1}{7} + \frac{1}{7} = \frac{7}{7} = 1, since seven sevenths fill the whole.
  8. No. Four copies of \displaystyle \frac{1}{5} make \displaystyle \frac{4}{5}. Three fifths is three pieces: \displaystyle \frac{3}{5} = \frac{1}{5} + \frac{1}{5} + \frac{1}{5}.
  9. \displaystyle \frac{4}{5} = \frac{1}{5} + \frac{3}{5} and \displaystyle \frac{4}{5} = \frac{2}{5} + \frac{2}{5}, since 4 = 1 + 3 = 2 + 2. (\displaystyle \frac{3}{5} + \frac{1}{5} is the first sum in the other order.)
  10. \displaystyle \frac{5}{9}, since 2 + 5 = 7.
  11. \displaystyle \frac{5}{12}, since 5 + 3 = 8.
  12. \displaystyle \frac{5}{8} = \frac{1}{8} + \frac{4}{8} and \displaystyle \frac{5}{8} = \frac{2}{8} + \frac{3}{8}, from 5 = 1 + 4 = 2 + 3. Counting order, there are also \displaystyle \frac{4}{8} + \frac{1}{8} and \displaystyle \frac{3}{8} + \frac{2}{8}, four in all.
  13. For example, \displaystyle \frac{8}{10} = \frac{2}{10} + \frac{3}{10} + \frac{3}{10} and \displaystyle \frac{8}{10} = \frac{1}{10} + \frac{3}{10} + \frac{4}{10}. Any three whole numbers of at least 1 that add to 8 work, such as 1 + 1 + 6 or 2 + 2 + 4.
  14. \displaystyle \frac{9}{12} = \frac{4}{12} + \frac{5}{12}. Tolga gets \displaystyle \frac{5}{12} of the pan, since 4 + 5 = 9.
  15. Yes for the first: 3 + 2 = 5, all in sevenths. No for the second: \displaystyle \frac{2}{7} + \frac{2}{7} + \frac{2}{7} is 6 sevenths, \displaystyle \frac{6}{7}.
  16. Both. All parts are ninths, and 2 + 2 + 2 = 6 and 3 + 3 = 6.
  17. \displaystyle \frac{5}{3} = \frac{3}{3} + \frac{2}{3} = 1 + \frac{2}{3}, since 5 = 3 + 2 and \displaystyle \frac{3}{3} = 1.
  18. \displaystyle \frac{9}{4} = \frac{4}{4} + \frac{4}{4} + \frac{1}{4} = 1 + 1 + \frac{1}{4}, so \displaystyle \frac{9}{4} = 2 + \frac{1}{4}. Here 9 = 4 + 4 + 1.
  19. \displaystyle 1\frac{2}{5} = 1 + \frac{2}{5} = \frac{5}{5} + \frac{2}{5}, seven copies of \displaystyle \frac{1}{5} in all.
  20. \displaystyle 3\frac{1}{2} = 1 + 1 + 1 + \frac{1}{2} = \frac{2}{2} + \frac{2}{2} + \frac{2}{2} + \frac{1}{2}, since each whole is two halves.
  21. One full pan and 3 extra portions: \displaystyle \frac{11}{8} = \frac{8}{8} + \frac{3}{8} = 1 + \frac{3}{8}, since 11 = 8 + 3.
  22. \displaystyle \frac{14}{5} = \frac{5}{5} + \frac{5}{5} + \frac{4}{5} = 2 + \frac{4}{5}, since 14 = 5 + 5 + 4. That is two full pans and \displaystyle \frac{4}{5} of another.
  23. The \displaystyle \frac{9}{10} is nine pieces, each a tenth of the pan. Putting pieces on plates does not cut or join them, so every piece on every plate is still a tenth. Each plate’s amount counts tenths, so each part has denominator 10, and the numerators add to 9.
  24. \displaystyle \frac{1}{4} is one piece. Two parts of at least one piece each need at least two pieces, and there is only one. \displaystyle \frac{2}{4} is two pieces, so \displaystyle \frac{2}{4} = \frac{1}{4} + \frac{1}{4}.
  25. \displaystyle \frac{6}{6} is 6 pieces of a whole cut into 6 equal pieces, which is every piece, the whole thing: \displaystyle \frac{6}{6} = 1. Then 8 = 6 + 2, so \displaystyle \frac{8}{6} = \frac{6}{6} + \frac{2}{6} = 1 + \frac{2}{6}.
kg1128 · updated 2026-10-20 · CC-BY-SA-4.0