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Grade 4 · 3.NF.A.3 · 4.NF.A.2

Comparing against a benchmark

Two smoothie cups of the same size are filled to three eighths and to four sevenths. Which cup holds more, and can you tell without cutting either cup into new pieces?

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Benchmark fractionsGrade 4Comparing fractions with the same denominator or numeratorGrade 3
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Comparing fractions with the same denominator or numerator
Comparing against a benchmark
Comparing by finding a common denominator

When you reach for it

Ayse, Solomiya and Tijana run a smoothie stand. Every cup at the stand is the same size, and a cup is often only partly filled: one customer wants \tfrac{3}{8} of a cup to taste, another wants \tfrac{4}{7} of a cup. Which order is bigger?

One way is to cut both cups into matching pieces and count. That always works, and it is the subject of Common denominators and the least common denominator. But it takes work, and very often a quick look is enough. If one cup is less than half full and the other is more than half full, nothing else needs to be known. If both orders have three pieces, the order with the bigger pieces is bigger. If both cups are one piece short of full, the cup with the smaller gap is fuller.

Reach for these quick comparisons whenever two fractions need to be compared and you do not want to rename them. Each one rests on a fraction you already know well, a benchmark, such as \tfrac{1}{2} or 1, met in Benchmark fractions. When none of them decides, you still have the slow way.

The procedure

A fraction counts pieces of one whole. The bottom number, the denominator, says how many equal pieces make the whole. The top number, the numerator, says how many of those pieces you have. Every comparison below reads those two numbers and decides which fraction is larger without renaming either one.

Part 1: which side of one half?

Before comparing anything, check that both fractions are of the same whole.

\tfrac{1}{2}
\tfrac{1}{2}
 large cup small cup
Figure 1Both cups are \tfrac{1}{2} full, yet they do not hold the same amount. A comparison of fractions is only fair when both fractions are of the same whole.

Half of a large cup is more smoothie than half of a small cup, although both are \tfrac{1}{2}. So “which fraction is larger?” is a question about two fractions of one whole, such as two cups of the same size or two parts of the same number line. Every comparison in this lesson is about the same whole.

Now place each fraction against \tfrac{1}{2}. As Benchmark fractions showed, a fraction is more than half when its top number is more than half its bottom number. Doubling both numbers keeps which one is bigger, so it is the same to double the top and compare it with the bottom:

\begin{aligned} 2 \times \text{top} > \text{bottom} &\quad\text{more than half} \\ 2 \times \text{top} = \text{bottom} &\quad\text{exactly half} \\ 2 \times \text{top} < \text{bottom} &\quad\text{less than half} \end{aligned}

If one fraction is less than half and the other is more than half, the second is the larger one. The reason is a middle object, as in comparing with a middle object: the first fraction is smaller than \tfrac{1}{2}, and \tfrac{1}{2} is smaller than the second, so the first is smaller than the second. The two fractions are never compared with each other directly. Each is compared only with one half.

The benchmark 1 works the same way. A fraction whose top number is more than its bottom number has more pieces than the whole holds, so it is more than 1. A fraction whose top is less than its bottom is less than 1, and a fraction whose top equals its bottom is exactly 1. If 1 stands between two fractions, that settles them.

Part 2: the same top or the same bottom

When both fractions are on the same side of one half, one half cannot decide. Look at the numbers themselves.

The same bottom number. As in Comparing fractions with the same denominator or numerator, the pieces are the same size, so the fraction with more pieces is larger: \tfrac{2}{9} < \tfrac{7}{9}.

The same top number. Now both fractions have the same number of pieces, but the pieces are different sizes. Cutting a whole into fewer equal pieces makes each piece bigger: a cup shared four ways gives bigger shares than the same cup shared five ways, so \tfrac{1}{4} is more than \tfrac{1}{5}. Three bigger pieces make more than three smaller pieces. So with the same top number, the smaller bottom number gives the larger fraction. The bigger pieces win.

The two rules can be chained. If two fractions share no number, a third fraction that shares a top number with one and a bottom number with the other can act as a middle object, just as one half does in Part 1.

Part 3: how far from full?

Two fractions just under 1 are hard to compare by their pieces, but easy to compare by what is missing. A cup filled to \tfrac{5}{6} is one sixth short of full. A cup filled to \tfrac{9}{10} is one tenth short. In general a fraction less than 1 is short of the whole by

\text{bottom} - \text{top} \quad\text{pieces}

of the same size. The fuller cup is the one with the smaller gap. And the gaps are fractions too, so they can be compared with Part 2: if both gaps have the same number of pieces, the gap made of smaller pieces is smaller.

To record a comparison, write the symbol between the two fractions. The symbol < is read “is less than” and > is read “is greater than”; the open side of the symbol always faces the larger number. The symbol = says the two fractions are equal, which happens when they name the same point, as \tfrac{2}{4} = \tfrac{1}{2} does in Equivalent fractions. The same fact can be written either way round: \tfrac{1}{4} > \tfrac{1}{5} and \tfrac{1}{5} < \tfrac{1}{4} say the same thing.

Sometimes none of these quick looks decides: both fractions are on the same side of one half, they share no top or bottom number, and their gaps do not match either. Then rename both fractions in matching pieces, as in Common denominators and the least common denominator, and count.

Worked examples

The lesson is taught in three parts. Examples 1 and 2 belong to the first: comparing with one half and with one. Examples 3 and 4 belong to the second: the same top or the same bottom number. Examples 5 to 7 belong to the third: comparing by the missing piece, writing the result with a symbol, and seeing when a benchmark cannot decide. In every example, all cups are the same size.

Example 1

Opposite sides of one half

Ayse fills a tasting cup to \tfrac{3}{8} full. Solomiya fills another cup of the same size to \tfrac{4}{7} full. Whose cup holds more smoothie?

Solution

The denominators differ, so the pieces are different sizes and cannot be counted against each other. Place each fraction against one half instead.

  • \tfrac{3}{8}: double the top, 2 \times 3 = 6. That is less than the bottom, 8, so \tfrac{3}{8} is less than half. In pieces, half of 8 is 4, and Ayse has 3.
  • \tfrac{4}{7}: 2 \times 4 = 8, which is more than 7, so \tfrac{4}{7} is more than half. Half of 7 is 3\tfrac{1}{2}, and Solomiya has 4.
0
\tfrac{1}{2}
 1
\tfrac{3}{8}
\tfrac{4}{7}
 less than half more than half
Figure 2One half stands between \tfrac{3}{8} and \tfrac{4}{7}, so it settles which one is larger.

One half sits between the two cups’ levels:

\tfrac{3}{8} < \tfrac{1}{2} < \tfrac{4}{7}

So \tfrac{3}{8} < \tfrac{4}{7}. Solomiya’s cup holds more. Notice what was never needed: the size of an eighth compared with a seventh, or how far apart the two fractions are.

Example 2

Exactly half, and more than one whole

(a) Tijana’s cup is \tfrac{4}{8} full and Ayse’s is \tfrac{3}{10} full. Which holds more? (b) One order needs \tfrac{7}{6} cups of smoothie, which takes one full cup and part of a second cup. Another needs \tfrac{9}{10} of a cup. Which order is bigger?

Solution

(a) For \tfrac{4}{8}, 2 \times 4 = 8, exactly the bottom number, so \tfrac{4}{8} is exactly one half. For \tfrac{3}{10}, 2 \times 3 = 6, which is less than 10, so \tfrac{3}{10} is less than half. Anything less than half is less than exactly half:

\tfrac{3}{10} < \tfrac{4}{8}

Tijana’s cup holds more. Here one of the two fractions is the benchmark itself.

(b) In \tfrac{7}{6} the top is more than the bottom: there are 7 sixths and only 6 sixths fit in one whole, so \tfrac{7}{6} is more than 1. In \tfrac{9}{10} the top is less than the bottom, so it is less than 1. The benchmark 1 stands between them:

\tfrac{9}{10} < 1 < \tfrac{7}{6}

So \tfrac{7}{6} > \tfrac{9}{10}. The first order is bigger. Both fractions are more than half, so one half could not have decided this one; one whole did.

Example 3

The same number of pieces

Solomiya fills one cup to \tfrac{3}{4} and Tijana fills another to \tfrac{3}{5}. Which cup is fuller?

Solution

Try one half first. 2 \times 3 = 6 is more than 4 and more than 5, so both fractions are more than half. One half does not stand between them, so it cannot decide.

Look at the numbers instead. Both tops are 3: each cup holds three pieces. The pieces are different sizes. A cup cut into 4 equal levels has bigger levels than a cup of the same size cut into 5, because the same whole is shared among fewer pieces. So one fourth is more than one fifth, and three fourths is more than three fifths.

\tfrac{3}{4}
\tfrac{3}{5}
Figure 3Three pieces in each cup. Fourths are bigger than fifths, so three fourths fills the cup higher than three fifths.
\tfrac{3}{4} > \tfrac{3}{5}

Solomiya’s cup is fuller. The larger denominator, 5, belongs to the smaller fraction: with the same top number, the bigger pieces win.

Example 4

A middle fraction

Ayse has three cups filled to \tfrac{7}{10}, \tfrac{2}{12} and \tfrac{7}{12}. Put them in order from least full to most full.

Solution

Look for pairs that share a number.

\tfrac{2}{12} and \tfrac{7}{12} have the same bottom number, so the pieces are all twelfths and 2 twelfths is less than 7 twelfths: \tfrac{2}{12} < \tfrac{7}{12}.

\tfrac{7}{12} and \tfrac{7}{10} have the same top number: seven pieces each. Tenths are bigger than twelfths, since the whole is shared among fewer pieces, so \tfrac{7}{12} < \tfrac{7}{10}.

Now \tfrac{7}{12} sits in the middle of both comparisons:

\tfrac{2}{12} < \tfrac{7}{12} < \tfrac{7}{10}

The cups in order are \tfrac{2}{12}, \tfrac{7}{12}, \tfrac{7}{10}. The fractions \tfrac{2}{12} and \tfrac{7}{10} share no number at all, but they never had to be compared directly; \tfrac{7}{12} was the middle object. As a check with one half, 2 \times 2 = 4 is less than 12, so \tfrac{2}{12} is less than half while the other two are more, which agrees.

Example 5

One piece short of full

Tijana pours one cup to \tfrac{5}{6} full and another to \tfrac{7}{8} full. Which cup is fuller?

Solution

Both fractions are more than half (10 > 6 and 14 > 8), and they share neither a top nor a bottom number. So look at what is missing.

  • The cup at \tfrac{5}{6} has 6 - 5 = 1 piece missing: it is \tfrac{1}{6} short of full.
  • The cup at \tfrac{7}{8} has 8 - 7 = 1 piece missing: it is \tfrac{1}{8} short of full.
\tfrac{5}{6}
\tfrac{7}{8}
\text{missing }\tfrac{1}{6}
\text{missing }\tfrac{1}{8}
Figure 4Each cup is one piece short of full. The eighth is the smaller gap, so \tfrac{7}{8} is the fuller cup.

Both gaps are one piece, and an eighth is a smaller piece than a sixth, because the cup is shared among more pieces. So the \tfrac{7}{8} cup has the smaller gap. The cup with less missing has more in it:

\tfrac{5}{6} < \tfrac{7}{8}

The \tfrac{7}{8} cup is fuller. It is a little surprising at first that the fraction with the bigger pieces lost this time. It did not lose because of its pieces: it lost because its missing piece is bigger.

Example 6

Two pieces short

Solomiya has a cup at \tfrac{8}{10} and Ayse has one at \tfrac{4}{6}. Whose cup is fuller?

Solution

Both are more than half (16 > 10 and 8 > 6), and no number is shared. Count the gaps.

  • \tfrac{8}{10} is short of full by 10 - 8 = 2 pieces: \tfrac{2}{10}.
  • \tfrac{4}{6} is short of full by 6 - 4 = 2 pieces: \tfrac{2}{6}.

The two gaps have the same top number, 2, so Part 2 compares them: tenths are smaller pieces than sixths, so two tenths is a smaller gap than two sixths, \tfrac{2}{10} < \tfrac{2}{6}. The smaller gap belongs to the fuller cup:

\tfrac{8}{10} > \tfrac{4}{6}

Solomiya’s cup is fuller. As a check, \tfrac{8}{10} names the same amount as \tfrac{4}{5} (each fifth is two tenths), and \tfrac{4}{5} against \tfrac{4}{6} has the same top number with bigger pieces in fifths. The same answer, by another road.

Example 7

Equal, and too close to call

Tijana writes two comparisons on the board at the stand.

(a) \tfrac{6}{8} \;?\; \tfrac{9}{12}

(b) \tfrac{3}{5} \;?\; \tfrac{5}{8}

Put <, > or = in place of each question mark.

Solution

(a) Both fractions are landmarks from Benchmark fractions. Eight pieces in four equal groups is 2 in each group, so three fourths of 8 is 6, and \tfrac{6}{8} is \tfrac{3}{4}. Twelve pieces in four groups is 3 in each group, so three fourths of 12 is 9, and \tfrac{9}{12} is \tfrac{3}{4} too. They name the same point:

\tfrac{6}{8} = \tfrac{9}{12}

(b) Try every quick look in turn. Both are more than half, since 6 > 5 and 10 > 8. Both are less than \tfrac{3}{4}: three fourths of 5 would be 3\tfrac{3}{4} pieces, more than 3, and three fourths of 8 is 6, more than 5. So both sit between \tfrac{1}{2} and \tfrac{3}{4}. They share no top or bottom number. Their gaps are \tfrac{2}{5} and \tfrac{3}{8}, which share nothing either. No benchmark decides.

So rename both in matching pieces. Fortieths work for both, since 5 \times 8 = 40:

\begin{aligned} \tfrac{3}{5} &= \tfrac{3 \times 8}{5 \times 8} = \tfrac{24}{40} \\ \tfrac{5}{8} &= \tfrac{5 \times 5}{8 \times 5} = \tfrac{25}{40} \end{aligned}

Now the pieces match and 24 < 25, so

\tfrac{3}{5} < \tfrac{5}{8}

The two fractions differ by only \tfrac{1}{40}, which is why no quick look could see it. Read aloud, the answer is “three fifths is less than five eighths”, or the other way round, “five eighths is greater than three fifths”.

Boundaries

  • Every comparison is for fractions of the same whole. Half of a large cup is more than half of a small cup. A statement such as \tfrac{3}{5} < \tfrac{3}{4} is about two parts of one whole (or of two equal wholes), never about two different-sized things.
  • The pieces of each whole must be equal. The test “double the top and compare with the bottom” and the rule “bigger pieces win” both count equal pieces.
  • One half decides only when the two fractions are on opposite sides of it, or one of them is exactly one half. Two fractions on the same side need another look.
  • The same-top rule needs the same top number, and the missing-piece rule needs the gaps to be comparable, which in this lesson means the same number of missing pieces. Otherwise use the common-denominator way.
  • The same-top rule needs a top number above 0. With a top of 0 there are no pieces at all, so \tfrac{0}{4} = \tfrac{0}{5}, and neither is larger.
  • The missing-piece look is for fractions less than 1. A fraction more than 1 has nothing missing from its first whole, and is compared with 1 as in Example 2.
  • A quick look says which fraction is larger, not by how much. To find the difference, rename both in matching pieces.

Where this goes next

When no benchmark decides, two fractions are renamed in matching pieces, as in Common denominators and the least common denominator. The full comparison by a common denominator, including fractions against decimals and a shortcut by multiplying across, comes in Comparing by finding a common denominator. The habit of placing a fraction near 0, \tfrac{1}{2} or 1 returns when estimating sums before working them out, in Estimating with fractions.

Exercises

Exercises 1 to 8 go with the first part of the lesson (Examples 1 and 2), Exercises 9 to 16 with the second (Examples 3 and 4), Exercises 17 to 22 with the third (Examples 5 to 7), and the reasoning group draws on all three. All cups are the same size.

Part 1: one half and one whole

  1. Compare \tfrac{2}{7} and \tfrac{5}{9}. Write <, > or = between them.
  2. Compare \tfrac{7}{12} and \tfrac{4}{10}.
  3. Compare \tfrac{5}{10} and \tfrac{3}{7}.
  4. Solomiya’s cup is \tfrac{4}{9} full and Ayse’s cup is \tfrac{6}{11} full. Whose cup holds more?
  5. Compare \tfrac{8}{7} and \tfrac{11}{12}.
  6. Put \tfrac{9}{8}, \tfrac{3}{10} and \tfrac{4}{6} in order from least to greatest.
  7. Compare \tfrac{9}{16} and \tfrac{7}{15}.
  8. Tijana is choosing a top number from 1 to 12 to make \tfrac{\square}{12}. Which top numbers make \tfrac{\square}{12} at least one half? Explain why each of them makes \tfrac{\square}{12} > \tfrac{4}{9}.

Part 2: the same top or the same bottom

  1. Compare \tfrac{3}{10} and \tfrac{7}{10}.
  2. Compare \tfrac{4}{5} and \tfrac{4}{9}.
  3. Compare \tfrac{2}{12} and \tfrac{2}{8}.
  4. Solomiya’s cup is \tfrac{6}{10} full and Tijana’s is \tfrac{6}{8} full. Whose cup is fuller?
  5. Put \tfrac{3}{7}, \tfrac{3}{4} and \tfrac{1}{7} in order from least to greatest.
  6. Put \tfrac{1}{10}, \tfrac{3}{4} and \tfrac{1}{4} in order from least to greatest.
  7. Which is greater, \tfrac{9}{12} or \tfrac{9}{10}?
  8. Which whole-number denominators make \tfrac{5}{\square} > \tfrac{5}{8} true?

Part 3: missing pieces and symbols

  1. Compare \tfrac{4}{5} and \tfrac{9}{10} by their missing pieces.
  2. Compare \tfrac{11}{12} and \tfrac{3}{4} by their missing pieces.
  3. Compare \tfrac{10}{12} and \tfrac{6}{8}.
  4. Solomiya’s cup is \tfrac{9}{12} full and Tijana’s is \tfrac{5}{8} full. Whose cup is fuller? Write the comparison with a symbol.
  5. Write <, > or =: (a) \tfrac{4}{8} \;?\; \tfrac{6}{12}, (b) \tfrac{2}{9} \;?\; \tfrac{2}{5}, (c) \tfrac{12}{10} \;?\; \tfrac{10}{12}.
  6. Explain why one half, the same-top rule and the missing pieces all fail to decide \tfrac{2}{3} against \tfrac{7}{10}. Then rename both in thirtieths and compare.

Reasoning

  1. Explain why, when two fractions of the same whole have the same top number, the one with the smaller bottom number is larger.
  2. Ayse says: “My large cup is \tfrac{1}{2} full and Tijana’s small cup is \tfrac{1}{2} full, so we have the same amount of smoothie.” Is she right? Explain.
  3. Tijana says: “If one fraction is less than half and the other is more than half, I do not need to know anything else to compare them.” Explain why she is right.
Answers
  1. \tfrac{2}{7} < \tfrac{5}{9}. 2 \times 2 = 4 is less than 7, so \tfrac{2}{7} is less than half; 2 \times 5 = 10 is more than 9, so \tfrac{5}{9} is more than half.
  2. \tfrac{7}{12} > \tfrac{4}{10}. 2 \times 7 = 14 is more than 12; 2 \times 4 = 8 is less than 10.
  3. \tfrac{5}{10} > \tfrac{3}{7}. 2 \times 5 = 10, so \tfrac{5}{10} is exactly half; 2 \times 3 = 6 is less than 7, so \tfrac{3}{7} is less than half.
  4. Ayse’s. 2 \times 4 = 8 is less than 9, so \tfrac{4}{9} is less than half; 2 \times 6 = 12 is more than 11, so \tfrac{6}{11} is more than half. So \tfrac{6}{11} > \tfrac{4}{9}.
  5. \tfrac{8}{7} > \tfrac{11}{12}. The top of \tfrac{8}{7} is more than its bottom, so it is more than 1; \tfrac{11}{12} is less than 1.
  6. \tfrac{3}{10} < \tfrac{4}{6} < \tfrac{9}{8}. \tfrac{3}{10} is less than half (6 < 10); \tfrac{4}{6} is more than half (8 > 6) but less than 1 (4 < 6); \tfrac{9}{8} is more than 1 (9 > 8).
  7. \tfrac{9}{16} > \tfrac{7}{15}. 2 \times 9 = 18 is more than 16; 2 \times 7 = 14 is less than 15.
  8. The top numbers from 6 to 12. 2 \times 6 = 12, so \tfrac{6}{12} is exactly half, and every larger top is more than half. Since 2 \times 4 = 8 is less than 9, \tfrac{4}{9} is less than half, so each of these is greater than \tfrac{4}{9}.
  9. \tfrac{3}{10} < \tfrac{7}{10}. Same pieces, and 3 tenths is fewer than 7 tenths.
  10. \tfrac{4}{5} > \tfrac{4}{9}. Four pieces each; fifths are bigger than ninths.
  11. \tfrac{2}{12} < \tfrac{2}{8}. Two pieces each; twelfths are smaller than eighths.
  12. Tijana’s: \tfrac{6}{8} > \tfrac{6}{10}. Both are more than half, so one half cannot decide, but both have 6 pieces and eighths are bigger than tenths.
  13. \tfrac{1}{7} < \tfrac{3}{7} < \tfrac{3}{4}. Same sevenths, 1 < 3; then three pieces each, and fourths are bigger than sevenths.
  14. \tfrac{1}{10} < \tfrac{1}{4} < \tfrac{3}{4}. One piece each, and tenths are smaller than fourths; then the same fourths, 1 < 3.
  15. \tfrac{9}{10}. Nine pieces each, and tenths are bigger than twelfths, so \tfrac{9}{10} > \tfrac{9}{12}.
  16. Every denominator from 1 to 7. With the same top 5, the fraction is larger exactly when the pieces are bigger, that is when the bottom number is less than 8.
  17. \tfrac{4}{5} < \tfrac{9}{10}. \tfrac{4}{5} is \tfrac{1}{5} short of full and \tfrac{9}{10} is \tfrac{1}{10} short; a tenth is the smaller gap.
  18. \tfrac{11}{12} > \tfrac{3}{4}. The gaps are \tfrac{1}{12} and \tfrac{1}{4}; a twelfth is the smaller gap.
  19. \tfrac{10}{12} > \tfrac{6}{8}. The gaps are \tfrac{2}{12} and \tfrac{2}{8}, two pieces each, and twelfths are smaller, so \tfrac{10}{12} is missing less.
  20. Solomiya’s: \tfrac{9}{12} > \tfrac{5}{8}. The gaps are \tfrac{3}{12} and \tfrac{3}{8}, three pieces each, and twelfths are smaller than eighths, so Solomiya’s cup is missing less.
  21. (a) \tfrac{4}{8} = \tfrac{6}{12}, both exactly half (2 \times 4 = 8, 2 \times 6 = 12). (b) \tfrac{2}{9} < \tfrac{2}{5}, two pieces each and fifths are bigger. (c) \tfrac{12}{10} > \tfrac{10}{12}, since \tfrac{12}{10} is more than 1 and \tfrac{10}{12} is less than 1.
  22. \tfrac{2}{3} < \tfrac{7}{10}. Both are more than half (4 > 3, 14 > 10), they share no top or bottom number, and the gaps \tfrac{1}{3} and \tfrac{3}{10} have different numbers of pieces. Renamed, \tfrac{2}{3} = \tfrac{20}{30} and \tfrac{7}{10} = \tfrac{21}{30}, and 20 < 21.
  23. Both fractions have the same number of pieces. The smaller bottom number cuts the same whole into fewer equal pieces, so each of its pieces is bigger. The same number of bigger pieces makes a larger amount.
  24. No. Each cup is half of its own whole, and the wholes are different sizes. Half of the large cup is more smoothie than half of the small cup. Two fractions can only be compared when they are fractions of the same whole.
  25. The first fraction is less than \tfrac{1}{2} and \tfrac{1}{2} is less than the second, so the first is less than the second. One half is a middle object between them, and the size of the pieces or the gap between the fractions makes no difference.
kg1128 · updated 2026-10-20 · CC-BY-SA-4.0