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Grade 4 · 4.NF.B.3C · 5.NF.B.3

Mixed numbers and improper fractions: rewriting and comparing

A baker has two whole loaves and three quarters of another, all cut into quarters. How many quarter pieces is that, and how do you go back from a pile of pieces to whole loaves?

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Mixed numbers and improper fractions: rewriting and comparing
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When you reach for it

Ozan, Levko and Jelena run a bakery stand. Some loaves sell whole. Others are cut into equal pieces first: quarters for the big loaves, fifths for the long thin ones. At the end of a morning the amounts left on the table are rarely whole numbers of loaves.

There are two ways to write such an amount. One counts whole loaves and then a part of a loaf: 2\tfrac{3}{4} loaves, a mixed number, which means 2 + \tfrac{3}{4} (Mixed numbers and improper fractions). The other counts only pieces: \tfrac{11}{4} loaves, eleven quarters, a fraction whose top is at least its bottom, called an improper fraction (Fractions greater than one (improper fractions)). Both name the same amount of bread.

You reach for this lesson whenever you need to switch between the two names. A count of pieces is easier to sell from: “I have 11 quarter pieces.” A count of wholes is easier to picture: “I have almost 3 loaves.” Later, adding, subtracting and multiplying mixed numbers all ask you to switch names in the middle of the work. And to say which of two amounts is bigger, it helps to write them the same way first.

The lesson has three parts, one for each day:

  • Part 1: a mixed number to an improper fraction, by cutting every whole into pieces.
  • Part 2: an improper fraction to a mixed number, by packing pieces into wholes, using division with a remainder.
  • Part 3: mixed numbers on the number line, and comparing mixed numbers and improper fractions.

The procedure

Part 1: cut every whole into pieces

Ozan has 2\tfrac{3}{4} loaves on his table. Every loaf is cut into quarters. How many quarter pieces does he have?

\tfrac{4}{4}
\tfrac{4}{4}
\tfrac{3}{4}
2\tfrac{3}{4} = \tfrac{11}{4}
Figure 1Each whole loaf is \tfrac{4}{4}. Two whole loaves and \tfrac{3}{4} more are 4 + 4 + 3 = 11 quarters, so 2\tfrac{3}{4} = \tfrac{11}{4}.

Look at one whole loaf. It is cut into 4 equal pieces, and all 4 are there, so the whole loaf is \tfrac{4}{4}. The same is true of the second whole loaf. The third loaf has only 3 of its quarters, \tfrac{3}{4}. So the mixed number can be written as a sum of quarters:

2\tfrac{3}{4} = 1 + 1 + \tfrac{3}{4} = \tfrac{4}{4} + \tfrac{4}{4} + \tfrac{3}{4}

Every number in that sum counts the same piece, a quarter of a loaf. So adding them is just counting quarters: 4 quarters, 4 more, and 3 more make 11 quarters. Breaking a fraction into its unit fractions like this is the work of Decomposing a fraction into unit fractions; here we run it the other way and gather the quarters up.

2\tfrac{3}{4} = \tfrac{4}{4} + \tfrac{4}{4} + \tfrac{3}{4} = \tfrac{11}{4}

Ozan has 11 quarter pieces. The denominator stays 4 the whole time, because the size of the pieces never changed. Only the count of pieces changed: it now includes the pieces hiding inside the whole loaves.

The quick way, and why it works. Writing \tfrac{4}{4} once for every whole loaf gets slow when there are many loaves. But each whole loaf adds the same number of pieces, 4. So the whole loaves are equal groups, and equal groups are counted by multiplying: 2 loaves of 4 quarters each are 2 \times 4 = 8 quarters. Then add the 3 quarters that were already there:

2\tfrac{3}{4} = \frac{2 \times 4 + 3}{4} = \frac{8 + 3}{4} = \tfrac{11}{4}

In words: multiply the whole number by the denominator, which counts the pieces in the wholes. Add the numerator, which counts the extra pieces. Keep the denominator, because the pieces are the same size. Each of those three moves is the picture, written with numbers.

A whole number by itself. A whole number of loaves can be written in pieces too. 3 loaves cut into fifths are 3 \times 5 = 15 fifths, so 3 = \tfrac{15}{5}. That is the quick way with no extra pieces to add.

Part 2: pack pieces into wholes

Jelena counts the pieces of the long loaves at the end of the day. She has 17 pieces, and each one is a fifth of a loaf. How many loaves is that?

\tfrac{5}{5}
\tfrac{5}{5}
\tfrac{5}{5}
\tfrac{2}{5}
\tfrac{17}{5} = 3\tfrac{2}{5}
Figure 2Seventeen fifths fill 3 loaves of 5 fifths each, with 2 fifths left over, so \tfrac{17}{5} = 3\tfrac{2}{5}.

Five fifths make one whole loaf. So pack the pieces into loaves, five at a time. The first 5 pieces make a loaf, the next 5 make a second, and the next 5 make a third. That uses 15 pieces. Only 2 are left, not enough for another loaf. So \tfrac{17}{5} is 3 whole loaves and \tfrac{2}{5} of a loaf:

\tfrac{17}{5} = \tfrac{15}{5} + \tfrac{2}{5} = 3 + \tfrac{2}{5} = 3\tfrac{2}{5}

Packing is dividing. “How many groups of 5 fit in 17, and how many are left?” is a division with a remainder:

17 \div 5 = 3 \text{ R } 2

The quotient 3 counts the full loaves, so it becomes the whole part. The remainder 2 counts the pieces left over, so it becomes the numerator of the fraction part. The divisor 5 is the number of pieces in a loaf, so it stays the denominator. In Remainders and what they mean a remainder is sometimes dropped and sometimes rounded up. Here it is kept and reported as it is, as pieces of a whole.

the divisionwhat it countsin the mixed number
quotient 3full loaveswhole part 3
remainder 2pieces left overnumerator 2
divisor 5pieces in one loafdenominator 5

Why the fraction part is always less than one. A remainder is always smaller than the divisor. If 5 or more pieces were left over, they would make one more loaf, and the packing would not be finished. So the leftover pieces never fill a whole, and the fraction part is always less than 1.

Checking by going back. Part 1 undoes Part 2. Cut Jelena’s 3 loaves back into fifths and add the 2 extra: 3 \times 5 + 2 = 17. That gives \tfrac{17}{5} again, so the answer is right. The check is the same sum as the one inside a division: quotient times divisor, plus remainder, gives back the number you divided.

When nothing is left over. If the remainder is 0, the pieces fill whole loaves exactly and the answer is a whole number. \tfrac{20}{5} is 20 \div 5 = 4 loaves, with no fraction part.

Part 3: the number line, and which is more

On the number line. Mark a number line with the whole numbers 0, 1, 2, 3. Then cut each space between whole numbers into 4 equal steps, so every step is \tfrac{1}{4} (Locating a fraction on the number line).

0
1
2
3
2\tfrac{3}{4} = \tfrac{11}{4}
0
1
2
3
4
3\tfrac{2}{5} = \tfrac{17}{5}
Figure 3A mixed number and its improper fraction are one point on the line. 2\tfrac{3}{4} is 3 quarter steps past 2, which is also 11 quarter steps from 0.

There are two ways to find 2\tfrac{3}{4} on this line, and they land on the same point. The mixed number says: go to 2, then take 3 more quarter steps. The improper fraction \tfrac{11}{4} says: start at 0 and take 11 quarter steps. Going to 2 already took 8 quarter steps, four for each whole, and 8 + 3 = 11. The two names are one point.

The same holds on the line cut into fifths: \tfrac{17}{5} is 17 fifth steps from 0, which is 3 wholes and 2 steps more, the point 3\tfrac{2}{5}. A mixed number also tells you at once which two whole numbers it sits between: 3\tfrac{2}{5} is between 3 and 4, because its fraction part is more than 0 and less than 1.

Comparing when the whole parts differ. Which is more, 3\tfrac{1}{5} loaves or 2\tfrac{4}{5} loaves? Look at the whole parts first. 2\tfrac{4}{5} is less than 3, because its fraction part is less than 1. And 3\tfrac{1}{5} is more than 3. So 2\tfrac{4}{5} < 3\tfrac{1}{5}, and the fraction parts did not matter. On the number line, 2\tfrac{4}{5} is in the space between 2 and 3, and 3\tfrac{1}{5} is in the space after it. Whenever the whole parts differ, the mixed number with the larger whole part is the larger number.

Comparing when the whole parts are the same. Which is more, 4\tfrac{1}{4} or 4\tfrac{3}{4}? Both have 4 wholes, so the comparison comes down to the fraction parts. They have the same denominator, so the pieces are the same size, and 3 quarters are more than 1 quarter (Comparing fractions with the same denominator or numerator). So 4\tfrac{1}{4} < 4\tfrac{3}{4}.

Comparing a mixed number with an improper fraction. To compare 3\tfrac{2}{5} with \tfrac{16}{5}, first give them the same kind of name. Either name works:

  • Rewrite the mixed number: 3\tfrac{2}{5} = \tfrac{3 \times 5 + 2}{5} = \tfrac{17}{5}. Now both count fifths, and 17 fifths are more than 16 fifths.
  • Or rewrite the improper fraction: 16 \div 5 = 3 \text{ R } 1, so \tfrac{16}{5} = 3\tfrac{1}{5}. Now the whole parts are both 3, and \tfrac{2}{5} is more than \tfrac{1}{5}.

Both routes say \tfrac{16}{5} < 3\tfrac{2}{5}. Turning everything into pieces is handy when you put a long list in order: once every number counts the same piece, the order is the order of the numerators.

Worked examples

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

Example 1

A mixed number of fifths

Levko has 3\tfrac{1}{5} long loaves, all cut into fifths. Write the amount as an improper fraction.

Solution

Each whole loaf is 5 fifths, so the 3 whole loaves are 3 equal groups of 5 fifths: 3 \times 5 = 15 fifths. The fraction part adds 1 more fifth.

3\tfrac{1}{5} = \frac{3 \times 5 + 1}{5} = \frac{15 + 1}{5} = \tfrac{16}{5}

The pieces are still fifths, so the denominator stays 5. Check by counting the long way: \tfrac{5}{5} + \tfrac{5}{5} + \tfrac{5}{5} + \tfrac{1}{5} is 5 + 5 + 5 + 1 = 16 fifths.

Example 2

How many pieces to sell

Ozan sells bread by the quarter loaf. He has 5\tfrac{3}{4} loaves, cut into quarters. How many quarter pieces can he sell?

Solution

The question asks for a count of pieces, which is the numerator of the improper fraction. The 5 whole loaves hold 5 \times 4 = 20 quarters, because each loaf holds 4. The part loaf adds 3 more.

5\tfrac{3}{4} = \frac{5 \times 4 + 3}{4} = \frac{20 + 3}{4} = \tfrac{23}{4}

Ozan can sell 23 quarter pieces. A size check: 5\tfrac{3}{4} is a little less than 6 loaves, and 6 loaves would be 24 quarters, so 23 is just right, one quarter short of 24.

Example 3

Packing fifths into loaves

At closing, Jelena has 38 pieces of long loaf, each a fifth of a loaf. How many loaves is that? Write the answer as a mixed number.

Solution

The amount is \tfrac{38}{5} loaves. Pack the fifths into whole loaves of 5 each, which is dividing by 5:

38 \div 5 = 7 \text{ R } 3

since 7 \times 5 = 35 and 38 - 35 = 3. Seven full loaves, with 3 fifths left over, and 3 is less than 5, so no more loaves can be made.

\tfrac{38}{5} = 7\tfrac{3}{5}

Jelena has 7\tfrac{3}{5} loaves. Check by going back: 7 \times 5 + 3 = 38.

Example 4

Pieces short of a full loaf

The seeded loaves are cut into eighths. Levko has 45 slices, each an eighth of a loaf. How many loaves is that? How many more slices would he need to make a whole number of loaves?

Solution

The amount is \tfrac{45}{8} loaves. Pack the eighths, 8 to a loaf:

45 \div 8 = 5 \text{ R } 5

since 5 \times 8 = 40 and 45 - 40 = 5. So \tfrac{45}{8} = 5\tfrac{5}{8} loaves: five full loaves and 5 slices of a sixth loaf.

The sixth loaf needs 8 slices and has 5, so it is 8 - 5 = 3 slices short. With 3 more slices Levko would have 48 slices, and 48 \div 8 = 6 loaves exactly, remainder 0.

Example 5

Placing improper fractions on the number line

Between which two whole numbers does \tfrac{13}{4} lie? Where is \tfrac{7}{4}? Describe each point on a number line marked in quarters.

Solution

Rewrite each as a mixed number, because the whole part names the whole number just before the point. 13 \div 4 = 3 \text{ R } 1, so \tfrac{13}{4} = 3\tfrac{1}{4}. The point lies between 3 and 4, one quarter step past 3.

7 \div 4 = 1 \text{ R } 3, so \tfrac{7}{4} = 1\tfrac{3}{4}. The point lies between 1 and 2, three quarter steps past 1, which is one quarter step before 2.

Counting from 0 agrees: \tfrac{13}{4} is 13 quarter steps, and the first 12 of them reach 3; \tfrac{7}{4} is 7 quarter steps, and the first 4 reach 1.

Example 6

Who has more bread

Ozan has 3\tfrac{1}{4} big loaves left. Levko has \tfrac{14}{4} big loaves left, in quarter pieces. Who has more, and by how much?

Solution

The two amounts are written in different ways, so write them the same way first. Rewrite Levko’s: 14 \div 4 = 3 \text{ R } 2, so \tfrac{14}{4} = 3\tfrac{2}{4}. Both have 3 whole loaves, and \tfrac{2}{4} is more than \tfrac{1}{4}, since the pieces are the same size and 2 of them are more than 1. So Levko has more.

The other route agrees. Rewrite Ozan’s: 3\tfrac{1}{4} = \tfrac{3 \times 4 + 1}{4} = \tfrac{13}{4}, and 13 quarters are fewer than 14 quarters.

In pieces the difference is plain: Levko has 14 quarters and Ozan has 13, so Levko has \tfrac{1}{4} of a loaf more.

Example 7

Putting five amounts in order

Jelena writes down how much of each kind of long loaf is left, in fifths of a loaf:

breadamount left
wheat2\tfrac{4}{5}
rye\tfrac{13}{5}
oat3
seeded\tfrac{19}{5}
raisin3\tfrac{3}{5}

Put the amounts in order from least to greatest.

Solution

Three different kinds of name are mixed together: mixed numbers, improper fractions and a whole number. Turn every one into fifths, so that each counts the same piece.

\begin{aligned} 2\tfrac{4}{5} &= \tfrac{2 \times 5 + 4}{5} = \tfrac{14}{5} \\ 3 &= \tfrac{3 \times 5}{5} = \tfrac{15}{5} \\ 3\tfrac{3}{5} &= \tfrac{3 \times 5 + 3}{5} = \tfrac{18}{5} \end{aligned}

Rye is \tfrac{13}{5} and seeded is \tfrac{19}{5} already. Now every amount counts fifths, so the order is the order of the numerators: 13 < 14 < 15 < 18 < 19.

\tfrac{13}{5} < 2\tfrac{4}{5} < 3 < 3\tfrac{3}{5} < \tfrac{19}{5}

From least to greatest: rye, wheat, oat, raisin, seeded. A check with whole parts: rye is \tfrac{13}{5} = 2\tfrac{3}{5} and wheat is 2\tfrac{4}{5}, both between 2 and 3, and seeded is \tfrac{19}{5} = 3\tfrac{4}{5}, which is just past raisin’s 3\tfrac{3}{5}.

Boundaries

  • The pieces must be equal, and of the same whole. \tfrac{4}{4} is one loaf only when the 4 quarters are equal pieces of that one loaf. Four quarters of a big loaf are not one small loaf. Every amount in a comparison must be measured in the same whole.
  • No zero denominator. The denominator says how many equal pieces fill a whole, so it is never 0. Packing pieces into groups of 0 makes no sense, just as dividing by 0 makes no sense.
  • The fraction part of a mixed number is less than one. That is what lets a mixed number name the whole number just before it on the line, and what lets you compare by whole parts first. A form such as 2\tfrac{5}{4} names an amount (2 + \tfrac{5}{4} = 3\tfrac{1}{4}), but it is not finished: its fraction part still holds a whole.
  • Comparing fraction parts piece by piece needs the same denominator. 4\tfrac{1}{4} and 4\tfrac{3}{4} compare by their numerators because quarters match quarters. To compare 2\tfrac{1}{3} with 2\tfrac{2}{5}, the parts must first be renamed in the same pieces (Comparing by finding a common denominator).
  • Not always in simplest form. 3\tfrac{2}{4} and 3\tfrac{1}{2} are the same amount. This lesson keeps the pieces the problem gives, quarters as quarters, and does not ask you to simplify.
  • No negative amounts. Every amount here is 0 or more.

Where this goes next

Adding and subtracting mixed numbers with the same denominator comes next (Adding and subtracting mixed numbers with like denominators). There the two names trade back and forth in the middle of a problem: a sum of fraction parts such as \tfrac{3}{5} + \tfrac{4}{5} = \tfrac{7}{5} gets packed into 1\tfrac{2}{5}, and taking away more pieces than you have means cutting a whole into pieces first. Both moves are the two parts of this lesson.

Further on, multiplying a mixed number (Multiplying mixed numbers) starts by rewriting it as an improper fraction, because the quick multiplication rule works on one fraction, not on a whole and a part side by side. And the idea of Part 2, that a division a \div b with quotient q and remainder r means a = q \times b + r, comes back whenever a fraction is turned into a decimal.

Exercises

Part 1: mixed numbers to improper fractions

Write each mixed number or whole number as an improper fraction.

  1. 1\tfrac{1}{4}
  2. 4\tfrac{1}{5}
  3. 5\tfrac{1}{3}
  4. 6\tfrac{5}{8}
  5. 4, as thirds
  6. 10\tfrac{1}{2}
  7. Ozan has 3\tfrac{3}{4} big loaves, cut into quarters. How many quarter pieces does he have?
  8. Levko writes 4\tfrac{3}{8} = \tfrac{7}{8}. Explain how you can tell this is wrong without working it out, and then find the correct improper fraction.

Part 2: improper fractions to mixed numbers

Write each improper fraction as a mixed number or a whole number.

  1. \tfrac{9}{4}
  2. \tfrac{26}{5}
  3. \tfrac{25}{6}
  4. \tfrac{30}{5}
  5. \tfrac{47}{8}
  6. \tfrac{50}{3}
  7. Jelena has 29 pieces of big loaf, each a quarter of a loaf. How many loaves is that? How many more pieces would make a whole number of loaves?
  8. Ozan sells 24 pieces of long loaf in a morning, each a fifth of a loaf. How many loaves did he sell, as a mixed number? He cuts a loaf only when he needs a piece from it. How many loaves did he cut?

Part 3: the number line and comparing

  1. Between which two whole numbers does \tfrac{19}{4} lie? How many quarter steps past the smaller whole number is it?
  2. Which is greater, 3\tfrac{4}{5} or 4\tfrac{1}{5}?
  3. Which is greater, 5\tfrac{2}{6} or 5\tfrac{5}{6}?
  4. Which is greater, \tfrac{23}{5} or 4\tfrac{2}{5}?
  5. Levko has 2\tfrac{5}{8} seeded loaves and Jelena has \tfrac{22}{8} seeded loaves. Who has more?
  6. Put in order from least to greatest: 1\tfrac{5}{8}, \tfrac{11}{8}, 2, \tfrac{19}{8}, 2\tfrac{1}{8}.

Reasoning

  1. Explain, using loaves cut into pieces, why 4\tfrac{2}{3} can be rewritten by multiplying 4 by 3 and adding 2. Why does the denominator stay 3?
  2. When an improper fraction is rewritten by dividing its numerator by its denominator, explain why the fraction part of the answer is always less than 1.
  3. Ozan says: “If one mixed number has a bigger whole part than another, it is the bigger number, whatever the fraction parts are.” Is he right? Explain why.
Answers
  1. \tfrac{5}{4}. One whole is 4 quarters, and 1 \times 4 + 1 = 5.
  2. \tfrac{21}{5}. The 4 wholes are 4 \times 5 = 20 fifths, and 20 + 1 = 21.
  3. \tfrac{16}{3}. The 5 wholes are 5 \times 3 = 15 thirds, and 15 + 1 = 16.
  4. \tfrac{53}{8}. The 6 wholes are 6 \times 8 = 48 eighths, and 48 + 5 = 53.
  5. \tfrac{12}{3}. Each whole is 3 thirds, so 4 wholes are 4 \times 3 = 12 thirds.
  6. \tfrac{21}{2}. The 10 wholes are 10 \times 2 = 20 halves, and 20 + 1 = 21.
  7. 15 pieces. 3\tfrac{3}{4} = \tfrac{3 \times 4 + 3}{4} = \tfrac{15}{4}, which is 15 quarters.
  8. \tfrac{35}{8}. The amount 4\tfrac{3}{8} is more than 4, but \tfrac{7}{8} is less than 1, so they cannot be equal. (Levko added 4 + 3.) The 4 wholes are 4 \times 8 = 32 eighths, and 32 + 3 = 35.
  9. 2\tfrac{1}{4}. 9 \div 4 = 2 \text{ R } 1. Check: 2 \times 4 + 1 = 9.
  10. 5\tfrac{1}{5}. 26 \div 5 = 5 \text{ R } 1. Check: 5 \times 5 + 1 = 26.
  11. 4\tfrac{1}{6}. 25 \div 6 = 4 \text{ R } 1. Check: 4 \times 6 + 1 = 25.
  12. 6. 30 \div 5 = 6 \text{ R } 0, so the pieces fill 6 wholes exactly.
  13. 5\tfrac{7}{8}. 47 \div 8 = 5 \text{ R } 7. Check: 5 \times 8 + 7 = 47.
  14. 16\tfrac{2}{3}. 50 \div 3 = 16 \text{ R } 2. Check: 16 \times 3 + 2 = 50.
  15. 7\tfrac{1}{4} loaves, and 3 more pieces. 29 \div 4 = 7 \text{ R } 1, so \tfrac{29}{4} = 7\tfrac{1}{4}. The eighth loaf has 1 of its 4 quarters, so it needs 4 - 1 = 3 more; then 32 \div 4 = 8 loaves exactly.
  16. 4\tfrac{4}{5} loaves sold, and 5 loaves cut. 24 \div 5 = 4 \text{ R } 4, so \tfrac{24}{5} = 4\tfrac{4}{5}. The 4 leftover pieces came from a fifth loaf, so he cut 5 loaves, and 5 \times 5 - 24 = 1 fifth of a loaf is still unsold. Here the remainder means one more loaf had to be cut.
  17. Between 4 and 5, three quarter steps past 4. 19 \div 4 = 4 \text{ R } 3, so \tfrac{19}{4} = 4\tfrac{3}{4}.
  18. 4\tfrac{1}{5}. Its whole part is bigger: 3\tfrac{4}{5} is less than 4, and 4\tfrac{1}{5} is more than 4.
  19. 5\tfrac{5}{6}. The whole parts are both 5, and 5 sixths are more than 2 sixths.
  20. \tfrac{23}{5}. 23 \div 5 = 4 \text{ R } 3, so \tfrac{23}{5} = 4\tfrac{3}{5}, and \tfrac{3}{5} is more than \tfrac{2}{5}. Or in fifths: 4\tfrac{2}{5} = \tfrac{22}{5}, and 23 > 22.
  21. Jelena. 2\tfrac{5}{8} = \tfrac{2 \times 8 + 5}{8} = \tfrac{21}{8}, and 21 eighths are fewer than 22 eighths. Jelena has \tfrac{1}{8} of a loaf more.
  22. \tfrac{11}{8} < 1\tfrac{5}{8} < 2 < 2\tfrac{1}{8} < \tfrac{19}{8}. In eighths the five numbers are \tfrac{13}{8}, \tfrac{11}{8}, \tfrac{16}{8}, \tfrac{19}{8} and \tfrac{17}{8}, and 11 < 13 < 16 < 17 < 19.
  23. 4\tfrac{2}{3} is 4 whole loaves and 2 thirds of another, with every loaf cut into thirds. Each whole loaf is 3 thirds, so the 4 whole loaves are 4 equal groups of 3 thirds, which is 4 \times 3 = 12 thirds. The 2 extra thirds make 12 + 2 = 14 thirds, so 4\tfrac{2}{3} = \tfrac{14}{3}. The denominator stays 3 because cutting the whole loaves into thirds makes more pieces but does not change their size: every piece is still a third.
  24. Dividing packs the pieces into wholes, as many as fit. The remainder is the count of pieces left after that, and a remainder is always less than the divisor. The divisor is the denominator, the number of pieces that fill one whole. So the leftover pieces are fewer than a whole needs, and the fraction they make is less than 1. If they were not fewer, they would fill another whole and the packing would not be finished.
  25. Yes. The fraction part of a mixed number is less than 1. So a mixed number with whole part 3, say, is at least 3 and less than 4. A mixed number with a bigger whole part, 4 or more, is at least 4. Every number in the first group is less than every number in the second, whatever the fraction parts are. This needs the fraction parts to be less than 1: an unfinished form such as 3\tfrac{7}{4} is 4\tfrac{3}{4}, more than 4\tfrac{1}{4}.
kg1128 · updated 2026-10-20 · CC-BY-SA-4.0