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Grade 3 · 3.NF.A.3C

Fractions greater than one (improper fractions)

A pie is cut into four equal pieces, and you have five of those pieces. How much pie is that, and how can a fraction name more than one whole?

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Why this exists

In What the numerator and the denominator each tell you a fraction counted pieces of one whole: the denominator names the size of the piece, and the numerator counts how many of those pieces there are. In Counting by unit fractions the count went on past the whole: four fourths, then five fourths, six fourths. This lesson asks what an amount like five fourths is, and how to picture it.

Think of an apple pie cut into 4 equal pieces. Each piece is a fourth of the pie. You can have 1, 2, 3 or all 4 of them, and \displaystyle \frac{4}{4} is the whole pie. But a bakery has more than one pie. Five pieces, each a fourth of a pie, is a real amount of pie: one whole pie and one piece of another. The fraction \displaystyle \frac{5}{4} names it. A fraction can name an amount past one whole, as long as there is more than one whole to take pieces from.

This comes up every time there is more than one of something: more than one bagel’s worth of halves, more than one wheel’s worth of cheese wedges. It also gives a quick test. Comparing the numerator with the denominator tells at a glance whether a fraction is less than one whole, exactly one, or more.

The lesson runs over three days. Part 1 pictures an amount past one whole. Part 2 looks at fractions equal to one and sorts fractions into less than, equal to or more than one. Part 3 names an amount past one whole in words, as wholes and some more.

Definition

A fraction \displaystyle \frac{a}{b} names a copies of the unit fraction \displaystyle \frac{1}{b}, the piece you get by cutting one whole into b equal parts. Nothing in that reading says a must be smaller than b.

When the numerator is bigger than the denominator, the fraction names more pieces than one whole holds, so it is an amount greater than one. \displaystyle \frac{5}{4} is 5 copies of \displaystyle \frac{1}{4}, and one whole holds only 4 of them.

When the numerator equals the denominator, the fraction names exactly as many pieces as one whole holds, so it equals one: \displaystyle \frac{4}{4} = 1 and \displaystyle \frac{6}{6} = 1.

When the numerator is smaller than the denominator, some pieces of the whole are missing, so the fraction is less than one.

A fraction whose numerator is greater than or equal to its denominator is called an improper fraction, so \displaystyle \frac{5}{4} and \displaystyle \frac{4}{4} are both improper fractions. The name is old and does not mean anything is wrong: \displaystyle \frac{5}{4} is a number like any other. A fraction whose numerator is smaller than its denominator, like \displaystyle \frac{3}{4}, is a proper fraction.

The whole is still one thing. \displaystyle \frac{5}{4} of a pie is measured in pies, one pie at a time. Picturing it takes more than one whole, and every whole in the picture must be the same size and cut into the same number of equal pieces, so that every piece is the same \displaystyle \frac{1}{4} of a pie.

Intuition

Lay out wholes of the same size, all cut into the same number of equal pieces, and shade one piece for each count. The first whole fills up after b pieces. The next piece has no room in that whole, so it goes into a second whole, cut the same way. That is all a fraction greater than one is: one or more full wholes, and some pieces of the next.

From the other side, a whole with every piece shaded is one whole, however it was cut (Figure 1). A pie in 2 pieces with both shaded, \displaystyle \frac{2}{2}, and a pie of the same size in 8 pieces with all 8 shaded, \displaystyle \frac{8}{8}, are the same amount of pie. The pieces differ; the whole does not.

\displaystyle \frac{2}{2}
\displaystyle \frac{3}{3}
\displaystyle \frac{4}{4}
\displaystyle \frac{6}{6}
\displaystyle \frac{8}{8}
Figure 1Every piece shaded is one whole, however many pieces the whole was cut into.

So the numerator and the denominator can be read against each other. The denominator says how many pieces fill one whole. The numerator says how many pieces you have. Fewer than a whole’s worth is less than one, exactly a whole’s worth is one, and more than that is past one.

Results

Less than, equal to or more than one. For a fraction \displaystyle \frac{a}{b}:

The numerator isThe fraction isExample
smaller than the denominator, a < bless than one\displaystyle \frac{3}{4}
equal to the denominator, a = bexactly one\displaystyle \frac{4}{4} = 1
bigger than the denominator, a > bmore than one\displaystyle \frac{5}{4}

Wholes and some more. Every b pieces make one whole. A fraction greater than one is as many wholes as there are full groups of b pieces, and the pieces left over, fewer than b, are the part more. For example, \displaystyle \frac{5}{4} is one whole and \displaystyle \frac{1}{4} more, because 5 = 4 + 1.

Why it is true

The denominator b means one whole is cut into b equal parts, and each part is \displaystyle \frac{1}{b}. So b copies of \displaystyle \frac{1}{b} cover every part of the whole, with none missing and none extra: \displaystyle \frac{b}{b} = 1.

With fewer than b copies, at least one part of the whole is not covered, so the amount is less than one whole. With more than b copies, all b parts of the first whole are covered and there are still copies left over, so the amount is more than one whole. Those extra copies are pieces of a second whole of the same size, which is why the picture needs more than one whole.

The same reasoning counts the wholes. Every b copies fill one whole. Setting aside b copies for each full whole leaves fewer than b copies, and those are the part more.

Worked examples

Example 1

Part 1: five fourths of an apple pie

Nahia bakes two apple pies of the same size and cuts each one into 4 equal pieces. After a bake sale she has 5 pieces left: all 4 pieces of one pie and 1 piece of the other (Figure 2). How much apple pie is left, as a fraction of one pie?

\displaystyle \frac{4}{4}
 one whole pie
\displaystyle \frac{1}{4}
 one fourth more
Figure 2Nahia’s leftover pie. Five fourths fill one pie and spill one piece into a second pie of the same size.
Solution

The whole is one apple pie. Each pie is cut into 4 equal pieces, so each piece is \displaystyle \frac{1}{4} of a pie.

Nahia has 5 of those pieces, 5 copies of \displaystyle \frac{1}{4}. That is \displaystyle \frac{5}{4} of a pie, five fourths.

One pie holds only 4 fourths, so 5 fourths cannot all come from one pie. Four of them make the full pie, \displaystyle \frac{4}{4} = 1, and the fifth is a piece of the second pie. So \displaystyle \frac{5}{4} of a pie is one whole pie and one fourth of a pie more, which is more than one pie.

The two pies have to be the same size and cut the same way. Then the piece from the second pie is the same size as a piece of the first, and calling it “one fourth of a pie” is true.

The whole matters too. If the whole were both pies together, it would have 8 equal pieces, and 5 of them would be \displaystyle \frac{5}{8} of the two pies, which is less than one. The pieces on the table are the same; the fraction changes because the whole changed. The question asks about one pie, so the answer is \displaystyle \frac{5}{4} of a pie.

Example 2

Part 1: seven sixths of a cheese wheel

Txomin sells cheese in wedges. He cuts every cheese wheel into 6 equal wedges, and all his wheels are the same size. A restaurant orders \displaystyle \frac{7}{6} of a wheel. How many wedges is that? Can Txomin fill the order from one wheel? Describe what he hands over.

\displaystyle \frac{6}{6}
 one whole wheel
\displaystyle \frac{1}{6}
 one sixth more
Figure 3The order of \displaystyle \frac{7}{6} of a wheel. One wheel holds only 6 wedges, so the seventh comes from a second wheel.
Solution

The denominator 6 says each wedge is \displaystyle \frac{1}{6} of a wheel. The numerator 7 says the order is 7 of those wedges.

One wheel is cut into only 6 wedges, \displaystyle \frac{6}{6} of a wheel, so one wheel is 1 wedge short. Txomin cuts a second wheel into 6 wedges the same way and takes 1 wedge from it.

He hands over all 6 wedges of the first wheel and 1 wedge more, 6 + 1 = 7 wedges (Figure 3). In words, \displaystyle \frac{7}{6} of a wheel is one whole wheel and one sixth of a wheel more. The second wheel still has 6 - 1 = 5 wedges for other customers.

The quick test agrees: 7 is bigger than 6, so the order is more than one wheel.

Example 3

Part 1: three halves of a bagel

Jonne makes open-face sandwiches for a picnic. He slices each bagel through the middle into a top half and a bottom half, and each sandwich sits on one half. All his bagels are the same size. He makes 3 sandwiches. What fraction of a bagel does he use? How many bagels must he slice, and how much is left over?

\displaystyle \frac{2}{2}
 one whole bagel
\displaystyle \frac{1}{2}
 one half more
Figure 4Three halves of a bagel take both halves of one bagel and one half of another.
Solution

Each bagel is cut into 2 equal parts, so each half is \displaystyle \frac{1}{2} of a bagel. Three sandwiches use 3 halves, 3 copies of \displaystyle \frac{1}{2}, which is \displaystyle \frac{3}{2} of a bagel, three halves.

One bagel gives only 2 halves, \displaystyle \frac{2}{2}, one whole bagel. The third sandwich needs a half from a second bagel. So Jonne slices 2 bagels and uses both halves of the first and one half of the second (Figure 4). He has 1 half left over, \displaystyle \frac{1}{2} of a bagel.

In words, he uses one whole bagel and one half of a bagel more. The numerator 3 is bigger than the denominator 2, so \displaystyle \frac{3}{2} is more than one bagel, as the picture shows.

The halves count as halves of “a bagel” only because the bagels are the same size. Half of a much smaller bagel is not \displaystyle \frac{1}{2} of the bagels Jonne is using.

Example 4

Part 2: two wheels, sold out

On Monday, Txomin cuts one cheese wheel into 8 equal wedges and sells all 8. On Tuesday he cuts a wheel of the same size into 6 equal wedges and sells all 6. Write each day’s sale as a fraction of a wheel. On which day did he sell more cheese?

Solution

On Monday each wedge is \displaystyle \frac{1}{8} of a wheel, and he sells 8 of them: \displaystyle \frac{8}{8} of a wheel. All 8 parts of the wheel are gone, so \displaystyle \frac{8}{8} = 1, one whole wheel.

On Tuesday each wedge is \displaystyle \frac{1}{6} of a wheel, and he sells 6 of them: \displaystyle \frac{6}{6} of a wheel. Again every part is gone, so \displaystyle \frac{6}{6} = 1.

He sold the same amount of cheese on both days, one whole wheel. Monday’s wedges were smaller and there were more of them; Tuesday’s were bigger and there were fewer. Whenever the numerator equals the denominator, you have as many pieces as the whole was cut into, which is the whole and nothing more. The same is true of \displaystyle \frac{2}{2}, \displaystyle \frac{3}{3} and \displaystyle \frac{4}{4} in Figure 1.

Example 5

Part 2: less than, equal to or more than one

Txomin labels each cheese package with how much of a wheel is inside. Without drawing, sort these labels into less than one wheel, exactly one wheel and more than one wheel:

\frac{2}{3}, \quad \frac{3}{3}, \quad \frac{9}{8}, \quad \frac{3}{4}, \quad \frac{5}{3}, \quad \frac{1}{2}.

How far from one whole wheel are the packages labeled \displaystyle \frac{3}{4} and \displaystyle \frac{9}{8}?

Solution

The denominator says how many pieces make one wheel, and the numerator says how many pieces are in the package. Compare the two.

For \displaystyle \frac{2}{3}, 2 is smaller than 3, so a third of the wheel is missing: less than one. For \displaystyle \frac{3}{3}, 3 equals 3, a whole wheel’s worth: exactly one. For \displaystyle \frac{9}{8}, 9 is bigger than 8, more pieces than a wheel holds: more than one. For \displaystyle \frac{3}{4}, 3 is smaller than 4: less than one. For \displaystyle \frac{5}{3}, 5 is bigger than 3: more than one. For \displaystyle \frac{1}{2}, 1 is smaller than 2: less than one.

Less than oneExactly oneMore than one
\displaystyle \frac{2}{3}, \displaystyle \frac{3}{4}, \displaystyle \frac{1}{2}\displaystyle \frac{3}{3}\displaystyle \frac{9}{8}, \displaystyle \frac{5}{3}

For the distances, count pieces against a whole wheel. A wheel is 4 fourths and the package labeled \displaystyle \frac{3}{4} has 3, so it is 4 - 3 = 1 fourth short: \displaystyle \frac{1}{4} of a wheel less than one wheel. A wheel is 8 eighths and the package labeled \displaystyle \frac{9}{8} has 9, so it is 9 - 8 = 1 eighth past: \displaystyle \frac{1}{8} of a wheel more than one wheel.

Only the numerator against its own denominator decides. The numbers in \displaystyle \frac{9}{8} are bigger than the numbers in \displaystyle \frac{5}{3}, and both are more than one; the numbers in \displaystyle \frac{3}{3} are bigger than those in \displaystyle \frac{1}{2}, and only one of them is a whole.

Example 6

Part 3: eleven fourths, in words

Nahia brings three apple pies of the same size to a party, each cut into 4 equal pieces. At the end, 11 pieces are left (Figure 5). Name the leftover pie as a fraction of one pie, and then say it in words as whole pies and some more.

\displaystyle \frac{4}{4}
\displaystyle \frac{4}{4}
\displaystyle \frac{3}{4}
Figure 5Eleven fourths of a pie, grouped into whole pies. Two pies are full and three fourths are left over.
Solution

Each piece is \displaystyle \frac{1}{4} of a pie, and there are 11 of them, so the leftover pie is \displaystyle \frac{11}{4} of a pie, eleven fourths. Since 11 is bigger than 4, it is more than one pie.

To say how much more, group the pieces into whole pies. Every 4 pieces make one whole pie. From 11 pieces, 4 make one pie and leave 11 - 4 = 7. Another 4 make a second pie and leave 7 - 4 = 3. Three pieces are fewer than 4, so they cannot make another whole pie.

11 = 4 + 4 + 3

So \displaystyle \frac{11}{4} of a pie is two whole pies and three fourths of a pie more. Figure 5 shows the same thing: two pies full, and the third with 3 of its 4 pieces left.

Check by building it back up: two whole pies are 4 + 4 = 8 fourths, and 3 fourths more make 8 + 3 = 11 fourths.

Example 7

Part 3: from words to a fraction

A family asks Txomin for one whole cheese wheel and five sixths of a wheel more. His wheels are cut into 6 equal wedges. Write the order as one fraction of a wheel. He cuts two fresh wheels to fill it. How much cheese is left over?

Solution

Count everything in sixths, since every wedge is \displaystyle \frac{1}{6} of a wheel. One whole wheel is 6 sixths, and five sixths more is 5 more wedges. The order is 6 + 5 = 11 wedges, which is \displaystyle \frac{11}{6} of a wheel.

Two fresh wheels hold 6 + 6 = 12 wedges. The order takes 11 of them, so 12 - 11 = 1 wedge is left over: \displaystyle \frac{1}{6} of a wheel.

The quick test fits: 11 is bigger than 6, so \displaystyle \frac{11}{6} is more than one wheel, and it is 1 wedge short of two whole wheels.

Boundaries

The wholes must match. A picture of \displaystyle \frac{5}{4} of a pie needs every pie the same size and cut into 4 equal pieces. A piece of a smaller pie, or of a pie cut into 6, is not \displaystyle \frac{1}{4} of the pie being measured, and counting it in names no fraction of that pie.

The whole is one thing, even when the picture shows several. \displaystyle \frac{5}{4} of a pie is measured against one pie. If the whole is taken to be both pies together, the same five pieces are \displaystyle \frac{5}{8} of the two pies (Example 1). Say what the whole is before naming the fraction.

“Out of” stops working past one. \displaystyle \frac{3}{4} can be read as 3 out of the 4 pieces of one pie. \displaystyle \frac{5}{4} cannot be 5 out of 4, since there are not 5 pieces among 4. The reading that works for every fraction is copies: \displaystyle \frac{5}{4} is 5 copies of \displaystyle \frac{1}{4}.

Equal to one is not past one. \displaystyle \frac{b}{b} is exactly one whole, not more. It is called an improper fraction because its numerator is not smaller than its denominator, but it is not greater than one.

The denominator is never zero. A whole cannot be cut into 0 equal parts, so a fraction with denominator 0 names no piece and no amount.

Where this goes next

The words “two whole pies and three fourths more” have a short written form, the mixed number, a whole number written beside a fraction. Changing a fraction like \displaystyle \frac{11}{4} into that form and back is Mixed numbers and improper fractions: rewriting and comparing, which comes later.

Fractions greater than one also get a place on the number line, where \displaystyle \frac{5}{4} sits one fourth past 1. Finding that place is Locating a fraction on the number line. The fact that \displaystyle \frac{b}{b} is exactly one comes back in Whole numbers written as fractions, where other whole numbers are written as fractions too.

Putting same-size pieces together, as four fourths and one fourth more make five fourths, is the start of Adding and subtracting fractions with like denominators, and taking several copies of one fraction is A fraction times a whole number. The next lesson, A fraction of a set, uses a fraction for part of a group of separate things rather than pieces of one whole.

Exercises

Part 1: picturing an amount past one whole

  1. Nahia cuts each of her apple pies into 6 equal slices. After a picnic she has one full pie and 3 slices of a second pie of the same size. What fraction of a pie does she have?
  2. Txomin has 5 bagel halves, all cut from bagels of the same size. What fraction of a bagel is that? What is the fewest bagels he could have sliced?
  3. A cheese wheel is cut into 8 equal wedges. A customer buys 10 wedges. What fraction of a wheel is that? How many wheels must be cut, and how many wedges come from each?
  4. Describe a picture of \displaystyle \frac{4}{3} of an apple pie: how many pies, how many equal pieces in each, and how many pieces shaded.
  5. Jonne wants \displaystyle \frac{6}{4} of an apple pie for his class. Is one pie enough? How many fourths does he need from a second pie of the same size?
  6. A picture shows two circles of the same size, each cut into 3 equal parts, with 4 parts shaded. What fraction does it show when the whole is one circle? When the whole is both circles together?

Part 2: one whole, and less or more than one

  1. Write the fraction that names one whole bagel cut into halves, and the fraction that names one whole apple pie cut into fourths.
  2. Sort into less than one, exactly one and more than one: \displaystyle \frac{5}{8}, \frac{2}{2}, \frac{7}{3}, \frac{1}{4}, \frac{11}{8}, \frac{5}{6}.
  3. For each fraction, say whether it is short of one whole or past it, and by how much: \displaystyle \frac{5}{6}, \displaystyle \frac{10}{8}, \displaystyle \frac{1}{3}.
  4. Fill in each box. (a) \displaystyle \frac{\square}{6} is exactly one. (b) The smallest numerator that makes \displaystyle \frac{\square}{8} more than one. (c) The largest numerator that makes \displaystyle \frac{\square}{4} less than one.
  5. Txomin has \displaystyle \frac{5}{8} of a cheese wheel. How many more eighths would make one whole wheel?
  6. Which of these is equal to one: \displaystyle \frac{3}{3}, \displaystyle \frac{3}{6}, \displaystyle \frac{6}{3}? Say whether each of the others is less or more than one.

Part 3: wholes and some more, in words

  1. Say each in words, as wholes and some more: (a) \displaystyle \frac{10}{6} of a cheese wheel, (b) \displaystyle \frac{7}{3} of a pie, (c) \displaystyle \frac{11}{8} of a wheel.
  2. Write each as one fraction: (a) one whole wheel and five eighths more, (b) two whole pies and one fourth more, (c) one whole pie and one third more.
  3. Txomin’s case holds 14 wedges, each \displaystyle \frac{1}{6} of a wheel. Name the amount as a fraction and in words.
  4. Jonne needs \displaystyle \frac{9}{2} bagels for open-face sandwiches, one half for each sandwich. How many bagels must he slice, and how many halves are left over? Say \displaystyle \frac{9}{2} in words.
  5. Which is more, \displaystyle \frac{7}{4} of a pie or one whole pie and two fourths more? By how much?
  6. Nahia has two apple pies of the same size, each cut into 8 equal slices. She serves 4 slices. What fraction of a pie is left? Say it in words.
  7. A fraction has denominator 3, and its numerator is one of 1, 2, 3, 4, 5. Which of these five fractions are more than one? Say each of them in words.

Reasoning

  1. To picture \displaystyle \frac{7}{3} of a pie, why must every pie in the picture be the same size and cut into 3 equal pieces? How many pies does the picture need?
  2. Explain why a fraction whose numerator equals its denominator is always exactly one whole, whatever the denominator is.
  3. Without drawing, explain why \displaystyle \frac{4}{6} is less than one and \displaystyle \frac{9}{6} is more than one.
  4. Two fractions both have denominator 4. One is less than one and the other is more than one. What can you say about each numerator, and why?
Answers
  1. \displaystyle \frac{9}{6} of a pie. Each slice is \displaystyle \frac{1}{6} of a pie; one full pie is 6 slices and 3 more make 6 + 3 = 9 slices.
  2. \displaystyle \frac{5}{2} of a bagel. Two bagels give only 2 + 2 = 4 halves, so he sliced at least 3 bagels.
  3. \displaystyle \frac{10}{8} of a wheel. One wheel has only 8 wedges, so 2 wheels are cut: all 8 wedges of one and 10 - 8 = 2 of the second.
  4. Two pies of the same size, each cut into 3 equal pieces. All 3 pieces of the first pie are shaded and 1 piece of the second, 3 + 1 = 4 pieces in all.
  5. No. One pie is only \displaystyle \frac{4}{4}, and 6 is bigger than 4. He needs 6 - 4 = 2 fourths from a second pie.
  6. With one circle as the whole, each part is \displaystyle \frac{1}{3}, so it shows \displaystyle \frac{4}{3}. With both circles as the whole, there are 3 + 3 = 6 equal parts, so it shows \displaystyle \frac{4}{6}.
  7. \displaystyle \frac{2}{2} for the bagel and \displaystyle \frac{4}{4} for the pie. Each names every piece of the whole.
  8. Less than one: \displaystyle \frac{5}{8}, \frac{1}{4}, \frac{5}{6}. Exactly one: \displaystyle \frac{2}{2}. More than one: \displaystyle \frac{7}{3}, \frac{11}{8}. Each time, compare the numerator with the denominator.
  9. \displaystyle \frac{5}{6} is 6 - 5 = 1 sixth short, \displaystyle \frac{1}{6} less than one. \displaystyle \frac{10}{8} is 10 - 8 = 2 eighths past, \displaystyle \frac{2}{8} more than one. \displaystyle \frac{1}{3} is 3 - 1 = 2 thirds short, \displaystyle \frac{2}{3} less than one.
  10. (a) 6, since \displaystyle \frac{6}{6} = 1. (b) 9: \displaystyle \frac{8}{8} is exactly one, and one more eighth is past it. (c) 3: \displaystyle \frac{4}{4} is already one whole.
  11. 3 more eighths, since a whole wheel is 8 eighths and 8 - 5 = 3.
  12. Only \displaystyle \frac{3}{3} is one: 3 pieces, and 3 make the whole. \displaystyle \frac{3}{6} is less than one, since 3 is smaller than 6. \displaystyle \frac{6}{3} is more than one, since 6 is bigger than 3.
  13. (a) One whole wheel and four sixths more, since 10 = 6 + 4. (b) Two whole pies and one third more, since 7 = 3 + 3 + 1. (c) One whole wheel and three eighths more, since 11 = 8 + 3.
  14. (a) \displaystyle \frac{13}{8}, since 8 + 5 = 13. (b) \displaystyle \frac{9}{4}, since 4 + 4 + 1 = 9. (c) \displaystyle \frac{4}{3}, since 3 + 1 = 4.
  15. \displaystyle \frac{14}{6} of a wheel. Since 14 = 6 + 6 + 2, it is two whole wheels and two sixths more.
  16. He needs 9 halves. Four bagels give 4 + 4 = 8 halves, one short, so he slices 5 bagels, which give 10 halves, and 10 - 9 = 1 half is left over. In words, \displaystyle \frac{9}{2} is four whole bagels and one half more.
  17. One whole pie and two fourths more is 4 + 2 = 6 fourths, \displaystyle \frac{6}{4}. So \displaystyle \frac{7}{4} is more, by 7 - 6 = 1 fourth, \displaystyle \frac{1}{4} of a pie.
  18. \displaystyle \frac{12}{8} of a pie. The pies had 8 + 8 = 16 slices and 16 - 4 = 12 are left. Since 12 = 8 + 4, it is one whole pie and four eighths more.
  19. \displaystyle \frac{4}{3} and \displaystyle \frac{5}{3}, the two whose numerators are bigger than 3. \displaystyle \frac{4}{3} is one whole and one third more; \displaystyle \frac{5}{3} is one whole and two thirds more.
  20. Each piece has to be the same \displaystyle \frac{1}{3} of a pie, so that the seven pieces are 7 copies of one unit fraction; a piece from a different size of pie, or cut another way, is a different size. Every pie holds 3 pieces and 7 = 3 + 3 + 1, so the picture needs 3 pies.
  21. The denominator b says the whole is cut into b equal pieces. A numerator of b means you have b of them: every piece of the whole, none missing and none extra. That is one whole.
  22. One whole is 6 sixths. \displaystyle \frac{4}{6} has only 4, so 2 sixths of the whole are missing: less than one. \displaystyle \frac{9}{6} has 9, which is 3 more than a whole holds: more than one.
  23. One whole is 4 fourths. The fraction less than one has a numerator smaller than 4; the fraction more than one has a numerator bigger than 4. A numerator of exactly 4 would be one whole.
kg1128 · updated 2026-10-21 · CC-BY-SA-4.0