Comparing fractions with the same denominator or numerator
Two fractions share a bottom number, or share a top number. How can you tell which one is greater without measuring anything?
Where you are 3 in Comparing fractions
When you reach for it
Kittipong and Pranee are at a swim lesson. Kittipong swims \displaystyle \frac{5}{8} of a length of the pool before he stops to rest, and Pranee swims \displaystyle \frac{3}{8} of a length. Who swam farther? A cat eats \displaystyle \frac{2}{3} of a can of food one day and \displaystyle \frac{2}{6} of the same kind of can the next. On which day did it eat more? Each question asks you to compare two fractions: to say which one is greater, or that they are equal.
This lesson handles the two cases where the fractions share a number. When the denominators match, the pieces are the same size, and the count of pieces decides. When the numerators match, the counts are the same, and the size of the pieces decides. Both cases can be seen two ways: in an area picture, a whole cut into equal parts as in Area and length models of a fraction, and on a number line, where every fraction is a point, as in Locating a fraction on the number line.
A comparison is written with a symbol. The sentence \displaystyle \frac{5}{8} > \frac{3}{8} reads “five eighths is greater than three eighths”. The sentence \displaystyle \frac{3}{8} < \frac{5}{8} reads “three eighths is less than five eighths”. They say the same thing from opposite ends: the open side of the symbol faces the greater number and the point faces the smaller one. The sign = says the two fractions are the same number.
One condition sits under every comparison in this lesson: both fractions are fractions of the same whole. The same pool, the same size of can, the same hour. Part 4 shows what goes wrong without it.
The procedure
The lesson runs over four days. Part 1 compares fractions with the same denominator using area pictures. Part 2 does the same for fractions with the same numerator. Part 3 moves both cases onto the number line. Part 4 compares fractions greater than one, looks at when two fractions are equal, and shows why the whole must be the same.
Part 1: the same denominator, so the same size pieces
Take \displaystyle \frac{5}{8} and \displaystyle \frac{3}{8} of one length of the pool. The denominator is 8 in both, so in both the length is cut into 8 equal parts, and every piece is \displaystyle \frac{1}{8} of the length. The pieces are all the same size. The numerators count the pieces: 5 of them against 3 of them. Each piece adds the same amount, so 5 pieces cover more than 3 pieces, and
The reasoning never used the number 8, only the fact that the two denominators match. So for any two fractions of the same whole with the same denominator, the fraction with the greater numerator is the greater fraction. Comparing the fractions comes down to comparing two whole numbers, the numerators. If the numerators are equal too, the two fractions are the same fraction, and the sign between them is =.
Part 2: the same numerator, so the same number of pieces
Now take \displaystyle \frac{2}{3} and \displaystyle \frac{2}{6} of the same can. Both count 2 pieces, so the count cannot decide. The pieces are different sizes. Thirds come from cutting the can into 3 equal parts; sixths come from cutting the same can into 6 equal parts. Cutting each third in half makes sixths, so a sixth is half of a third, and \displaystyle \frac{1}{3} > \frac{1}{6}. Two of the bigger pieces cover more than two of the smaller ones:
The same holds for any denominators. Sharing one whole among more equal parts makes each part smaller: a can shared 8 ways gives smaller shares than the same can shared 4 ways. So among unit fractions of one whole, the larger denominator gives the smaller piece:
With the same numerator, both fractions take the same number of pieces, so the smaller denominator gives the greater fraction. This can look backward, since 3 is less than 6 and yet \displaystyle \frac{2}{3} is the greater fraction. The denominator is not an amount of anything. It tells how many parts the whole is cut into, and more parts means smaller parts.
Part 3: on the number line, farther right is greater
On a number line the numbers grow from left to right, so of two points the one farther right is the greater number, just as 7 sits to the right of 3. A fraction \displaystyle \frac{a}{b} is the point reached by a jumps of \displaystyle \frac{1}{b} from 0. Both rules from Parts 1 and 2 can be read off the jumps.
Same denominator. Every jump is the same length, \displaystyle \frac{1}{b}. More jumps from 0 carry you farther right, so the greater numerator lands on the greater point. The point \displaystyle \frac{5}{6} is one jump past \displaystyle \frac{4}{6}.
Same numerator. Draw two lines one above the other, with their 0 ticks lined up and their 1 ticks lined up, one cut into sixths and one into eighths. Eight jumps fit from 0 to 1 on the lower line where only six fit on the upper one, so a jump of \displaystyle \frac{1}{8} is shorter than a jump of \displaystyle \frac{1}{6}. The same number of shorter jumps stops farther left.
Lining up the 0 and 1 ticks is the number line’s way of keeping the whole the same. The unit is the stretch from 0 to 1, and both lines need the same unit, or “farther right” means nothing.
Part 4: past one, equal fractions, and the same whole
Past one. The jumps carry on past 1 exactly as before, so the rules do not change for fractions greater than one. \displaystyle \frac{7}{4} is two jumps of \displaystyle \frac{1}{4} past \displaystyle \frac{5}{4}. A whole number can join a comparison by taking a fraction name with the same denominator: the point 1 is \displaystyle \frac{4}{4}, four jumps of \displaystyle \frac{1}{4} (see Whole numbers written as fractions).
Equal fractions. Two fractions with the same denominator are equal only when their numerators are equal too, since different numbers of same-size jumps land on different points. Zero is a special case for the same-numerator rule. The fractions \displaystyle \frac{0}{3} and \displaystyle \frac{0}{8} both mean no pieces at all, no jumps from 0, so both are the point 0 and \displaystyle \frac{0}{3} = \frac{0}{8}. Fractions such as \displaystyle \frac{1}{2} and \displaystyle \frac{2}{4}, which share neither number and still name the same point, are the subject of Seeing equivalent fractions in pictures and on the number line.
The same whole. The rules compare fractions as numbers, all measured against one unit. A fraction of a thing becomes an amount only once the thing is fixed. One half of a small can and one third of a big can are pieces of two different wholes, and the fact that \displaystyle \frac{1}{2} > \frac{1}{3} says nothing about which is more food. Example 8 works it out.
Worked examples
Part 1: two swimmers in eighths of a length
At the swim lesson, Kittipong swims \displaystyle \frac{5}{8} of a length of the pool and Pranee swims \displaystyle \frac{3}{8} of a length of the same pool. Who swam farther, and by how much? Write the comparison with a symbol both ways.
Draw each swim as a bar for one length of the pool. Both lengths are the same pool, so the bars are the same length. Both fractions are eighths, so cut each bar into 8 equal parts, and every part is \displaystyle \frac{1}{8} of a length (Figure 1).
Shade 5 parts on Kittipong’s bar and 3 on Pranee’s. The pieces are the same size, so the bar with more shaded pieces shows the longer swim. 5 > 3, so Kittipong swam farther:
Kittipong’s shading runs 5 - 3 = 2 pieces past Pranee’s, so he swam \displaystyle \frac{2}{8} of a length farther.
Part 1: ordering three practice times
Wanchai practices piano for \displaystyle \frac{3}{4} of an hour on Monday, \displaystyle \frac{1}{4} of an hour on Tuesday and \displaystyle \frac{2}{4} of an hour on Wednesday. Put the three days in order from the shortest practice to the longest, and write the order with symbols.
SolutionAll three are fourths of the same hour. Picture the hour as a bar cut into 4 equal parts, each \displaystyle \frac{1}{4} of an hour. Each day’s practice is some number of those same-size parts: 3 parts, 1 part and 2 parts.
Same-size parts, so the order of the fractions is the order of the counts. The counts in order are 1 < 2 < 3, so
Tuesday was the shortest practice, then Wednesday, then Monday. A chain like this reads in both directions at once: \displaystyle \frac{2}{4} is greater than \displaystyle \frac{1}{4} and less than \displaystyle \frac{3}{4}. None of the three reaches the full hour, \displaystyle \frac{4}{4}, since 3 < 4.
Part 2: two thirds or two sixths of a can
Pranee’s cat eats \displaystyle \frac{2}{3} of a can of cat food on day 1 and \displaystyle \frac{2}{6} of a can on day 2. The cans are all the same size. On which day did the cat eat more?
The numerators match: 2 pieces each day. So the count of pieces cannot decide, and the size of the pieces must.
Draw two same-size bars for the two cans (Figure 2). Cut the first into 3 equal parts and the second into 6 equal parts. Each third, cut in half, makes two sixths, so a sixth is half the size of a third. Shade 2 parts of each bar. The two thirds reach across most of the first can; the two sixths cover only as much as one third does.
Two bigger pieces are more than two smaller pieces of the same can, so
The cat ate more on day 1. In fact the picture shows twice as much, since each third matches two sixths.
Part 2: three cats, three fractions, the same numerator
Wanchai feeds three cats from cans that are all the same size. The orange cat eats \displaystyle \frac{3}{4} of a can, the gray cat eats \displaystyle \frac{3}{8} of a can and the black cat eats \displaystyle \frac{3}{6} of a can. Order the amounts from least to greatest.
SolutionEvery cat eats 3 pieces, so again the size of the pieces decides. One can cut into 8 equal pieces has smaller pieces than the same can cut into 6, and 6 pieces are smaller than 4, because the more parts one whole is shared among, the smaller each part. So \displaystyle \frac{1}{8} < \frac{1}{6} < \frac{1}{4}.
Three of each piece keeps that order: three of the smallest pieces is the least food, three of the largest is the most.
The gray cat ate the least, then the black cat, then the orange cat. With the same numerator the order of the denominators runs the other way: 8 is the largest denominator and gives the least.
Part 3: practice times on one number line
Kittipong practices piano for \displaystyle \frac{4}{6} of an hour and Wanchai practices for \displaystyle \frac{5}{6} of an hour. Use a number line to decide who practiced longer, and say how much longer.
Draw a line from 0 to 1 hour and cut it into 6 equal parts, so each jump is \displaystyle \frac{1}{6} of an hour (Figure 3). Kittipong’s time is 4 jumps from 0; Wanchai’s is 5 jumps from 0.
All jumps are the same length, so 5 jumps reach farther right than 4. The point farther right is the greater number:
Wanchai practiced longer. His point is 5 - 4 = 1 jump past Kittipong’s, so he practiced \displaystyle \frac{1}{6} of an hour longer. Both points are left of 1, so neither practiced a full hour.
Part 3: five sixths or five eighths on two matched lines
In the same pool, Pranee swims \displaystyle \frac{5}{6} of a length on her back and Kittipong swims \displaystyle \frac{5}{8} of a length on his back. Use two number lines to decide who swam farther.
The numerators match, so compare the jumps. Draw two lines, one above the other, for one length of the pool. The length is the same pool for both swimmers, so the 0 ticks line up and the 1 ticks line up (Figure 4). Cut the top line into 6 equal parts and the bottom line into 8 equal parts.
Eight jumps fill the stretch from 0 to 1 on the bottom line, and only six fill the same stretch on the top line, so each jump of \displaystyle \frac{1}{8} is shorter than a jump of \displaystyle \frac{1}{6}. Take 5 jumps on each line. The dashed line dropped from \displaystyle \frac{5}{6} lands to the right of \displaystyle \frac{5}{8}.
Pranee swam farther. Counting from the other end tells the same story: \displaystyle \frac{5}{6} is 6 - 5 = 1 jump of one sixth short of the wall, and \displaystyle \frac{5}{8} is 8 - 5 = 3 jumps of one eighth short.
Part 4: past one whole can
Over a weekend, Wanchai’s cat eats \displaystyle \frac{7}{4} cans of food and Kittipong’s cat eats \displaystyle \frac{5}{4} cans. The cans are all the same size. Which cat ate more? Did each cat eat more or less than one can, and more or less than two?
Both are fourths, so the jumps are the same size, \displaystyle \frac{1}{4} of a can. Draw a line from 0 to 2 with each unit cut into 4 equal parts (Figure 5). Seven jumps reach farther right than five:
Wanchai’s cat ate more, 7 - 5 = 2 jumps more, which is \displaystyle \frac{2}{4} of a can.
To compare with whole cans, give the whole numbers fourths names. One can is 4 jumps, the point \displaystyle \frac{4}{4}; two cans is 4 + 4 = 8 jumps, the point \displaystyle \frac{8}{4}. Both 5 and 7 are more than 4 and less than 8, so
Each cat ate more than one can and less than two.
Part 4: different cans, different wholes
Pranee’s cat eats \displaystyle \frac{1}{2} of a small can, which holds 6 ounces. Wanchai’s cat eats \displaystyle \frac{1}{3} of a big can, which holds 12 ounces. Since \displaystyle \frac{1}{2} > \frac{1}{3}, did Pranee’s cat eat more food?
The comparison \displaystyle \frac{1}{2} > \frac{1}{3} holds for halves and thirds of the same whole. These are two different wholes, so the fractions alone cannot answer. Find the amounts instead.
The small can’s 6 ounces cut into 2 equal parts gives 6 \div 2 = 3 ounces in each part, so Pranee’s cat ate 3 ounces. The big can’s 12 ounces cut into 3 equal parts gives 12 \div 3 = 4 ounces in each part, so Wanchai’s cat ate 4 ounces (Figure 6).
Since 4 > 3, Wanchai’s cat ate more, even though its fraction is the smaller one. A third of the big can is a bigger piece than a half of the small can because the big can is twice the size. A comparison of fractions answers a question about amounts only when both fractions are of the same whole.
Boundaries
The wholes must be the same. Every rule here compares fractions of one whole: one pool length, one size of can, one hour, one unit on the number line. Of two different wholes, the greater fraction can be the smaller amount (Example 8).
The parts must be equal. A denominator names equal parts. If a can is cut into 4 pieces of different sizes, “three of the four pieces” is not \displaystyle \frac{3}{4} and cannot be compared this way.
One number must be shared. These rules work when the two fractions have the same denominator or the same numerator. A pair like \displaystyle \frac{2}{3} and \displaystyle \frac{3}{4} shares neither, and neither rule applies directly; that needs Comparing against a benchmark or Comparing by finding a common denominator.
Zero pieces are equal. With the same numerator, the smaller denominator gives the greater fraction, except when the numerator is 0. Then both fractions are the number 0, so \displaystyle \frac{0}{2} = \frac{0}{6}.
The denominator is never zero. A whole cannot be cut into 0 parts, so there is no \displaystyle \frac{1}{0} to compare.
Two lines need the same unit. Comparing on two number lines works only when the stretch from 0 to 1 is the same length on both.
Where this goes next
Later in this unit, Seeing equivalent fractions in pictures and on the number line shows how two fractions that share neither number, like \displaystyle \frac{1}{2} and \displaystyle \frac{2}{4}, can still be equal. Once every fraction can be renamed, any two fractions can be compared: Comparing against a benchmark sets each one beside \displaystyle \frac{1}{2} or 1, and Comparing by finding a common denominator renames both into same-size pieces so that Part 1 of this lesson finishes the job. Fractions past one, such as \displaystyle \frac{7}{4}, are compared again in Mixed numbers and improper fractions: rewriting and comparing, written as a whole number and a fraction.
Exercises
Part 1: the same denominator
- Write >, < or = between \displaystyle \frac{2}{6} and \displaystyle \frac{5}{6}. Explain with pieces.
- Write >, < or = between \displaystyle \frac{7}{8} and \displaystyle \frac{4}{8}.
- At the swim lesson, Wanchai swims \displaystyle \frac{6}{8} of a length holding a kickboard and Pranee swims \displaystyle \frac{7}{8} of a length of the same pool. Who swims farther, and by how much?
- Put \displaystyle \frac{3}{6}, \displaystyle \frac{1}{6}, \displaystyle \frac{5}{6} and \displaystyle \frac{2}{6} in order from least to greatest.
- Write >, < or =: (a) \displaystyle \frac{3}{4} and \displaystyle \frac{3}{4}; (b) \displaystyle \frac{1}{2} and \displaystyle \frac{2}{2}.
- Which whole numbers can go in the box to make \displaystyle \frac{\square}{8} < \frac{3}{8} true?
Part 2: the same numerator
- Write >, < or = between \displaystyle \frac{1}{3} and \displaystyle \frac{1}{6}. Explain with pieces.
- Write >, < or = between \displaystyle \frac{4}{6} and \displaystyle \frac{4}{8}.
- Wanchai’s cat eats \displaystyle \frac{1}{4} of a can and Kittipong’s cat eats \displaystyle \frac{1}{2} of a can. The cans are the same size. Whose cat eats more?
- Put \displaystyle \frac{2}{8}, \displaystyle \frac{2}{2} and \displaystyle \frac{2}{4} in order from least to greatest.
- The denominator can be 2, 3, 4, 6 or 8. Which of these make \displaystyle \frac{3}{\square} > \frac{3}{4} true?
Part 3: on the number line
- During a kickboard drill, Kittipong stops at \displaystyle \frac{6}{8} of a length and Wanchai stops at \displaystyle \frac{3}{8} of a length of the same pool. Using a line cut into eighths, say who went farther and how far apart they stopped.
- On a line from 0 to 1 cut into thirds, compare \displaystyle \frac{2}{3} and \displaystyle \frac{1}{3}, and say how far apart the two points are.
- Two lines from 0 to 2 are drawn with their whole numbers lined up. The top is cut into halves and the bottom into fourths. Compare \displaystyle \frac{3}{2} and \displaystyle \frac{3}{4}, and say which side of 1 each is on.
- Pranee practices piano for \displaystyle \frac{5}{6} of an hour and Wanchai for \displaystyle \frac{2}{6} of an hour. Write the comparison both ways, with > and with <.
Part 4: past one, equal, and the same whole
- Compare \displaystyle \frac{5}{3} and \displaystyle \frac{4}{3} on a line from 0 to 2.
- Compare \displaystyle \frac{9}{8} and \displaystyle \frac{7}{8}. Which one is more than 1?
- Compare \displaystyle \frac{3}{2} and \displaystyle \frac{3}{3}. What whole number is \displaystyle \frac{3}{3}?
- Kittipong’s cat eats \displaystyle \frac{1}{4} of a big can of 12 ounces. Pranee’s cat eats \displaystyle \frac{1}{4} of a small can of 8 ounces. Did the two cats eat the same amount?
- Write >, < or = between \displaystyle \frac{0}{6} and \displaystyle \frac{0}{2}.
Reasoning
- Explain why, when two fractions of the same whole have the same numerator, the one with the larger denominator is the smaller fraction.
- Is \displaystyle \frac{5}{8} of something always more than \displaystyle \frac{3}{8} of something? Say what must be true for the answer to be yes.
- Two lines are drawn one above the other, both starting at the same 0. The top line is cut into thirds; the bottom line is cut into sixths, but its stretch from 0 to 1 is twice as long as the top line’s. Where do the dots for \displaystyle \frac{2}{3} on the top line and \displaystyle \frac{2}{6} on the bottom line fall? Can these lines compare the two fractions?
- Can the rules of this lesson compare \displaystyle \frac{2}{3} and \displaystyle \frac{3}{4}? Say why or why not.
- Using a picture of one can, explain why \displaystyle \frac{1}{3} of the can is more than \displaystyle \frac{1}{4} of the same can.
Answers
- \displaystyle \frac{2}{6} < \frac{5}{6}. Both are sixths, so the pieces are the same size, and 2 pieces are fewer than 5.
- \displaystyle \frac{7}{8} > \frac{4}{8}, because 7 > 4 pieces of the same size.
- Pranee, by 7 - 6 = 1 piece, so \displaystyle \frac{1}{8} of a length farther: \displaystyle \frac{7}{8} > \frac{6}{8}.
- \displaystyle \frac{1}{6} < \frac{2}{6} < \frac{3}{6} < \frac{5}{6}, the order of the numerators 1 < 2 < 3 < 5.
- (a) \displaystyle \frac{3}{4} = \frac{3}{4}, the same fraction. (b) \displaystyle \frac{1}{2} < \frac{2}{2}: same denominator, 1 < 2.
- 0, 1 and 2. Eighths are all the same size, so the numerator must be less than 3.
- \displaystyle \frac{1}{3} > \frac{1}{6}. A can cut into 3 equal parts has bigger parts than the same can cut into 6; each third is two sixths.
- \displaystyle \frac{4}{6} > \frac{4}{8}. Four pieces each, and sixths are bigger than eighths.
- Kittipong’s cat: \displaystyle \frac{1}{2} > \frac{1}{4}, since a half is bigger than a fourth of the same can.
- \displaystyle \frac{2}{8} < \frac{2}{4} < \frac{2}{2}. Same numerator, so the order of the denominators runs the other way.
- 2 and 3. With 3 pieces each, the fraction is greater than \displaystyle \frac{3}{4} when its pieces are bigger than fourths, so the denominator must be less than 4.
- Kittipong went farther: \displaystyle \frac{6}{8} > \frac{3}{8}. They stopped 6 - 3 = 3 jumps apart, \displaystyle \frac{3}{8} of a length.
- \displaystyle \frac{2}{3} > \frac{1}{3}; it is one jump farther right. They are 2 - 1 = 1 jump apart, \displaystyle \frac{1}{3}.
- \displaystyle \frac{3}{2} > \frac{3}{4}. Three jumps of \displaystyle \frac{1}{2} pass 1, which is \displaystyle \frac{2}{2}, so \displaystyle \frac{3}{2} is between 1 and 2. Three jumps of \displaystyle \frac{1}{4} stop short of 1, which is \displaystyle \frac{4}{4}.
- \displaystyle \frac{5}{6} > \frac{2}{6} and \displaystyle \frac{2}{6} < \frac{5}{6}. Pranee practiced longer.
- \displaystyle \frac{5}{3} > \frac{4}{3}. Both are thirds, and 5 jumps reach one jump past 4.
- \displaystyle \frac{9}{8} > \frac{7}{8}. The point 1 is \displaystyle \frac{8}{8}, so \displaystyle \frac{9}{8} is more than 1 and \displaystyle \frac{7}{8} is less.
- \displaystyle \frac{3}{2} > \frac{3}{3}, because halves are bigger than thirds. \displaystyle \frac{3}{3} = 1, three thirds filling one whole.
- No. A fourth of 12 ounces is 12 \div 4 = 3 ounces, and a fourth of 8 ounces is 8 \div 4 = 2 ounces. The same fraction of different wholes is a different amount.
- \displaystyle \frac{0}{6} = \frac{0}{2}. No pieces at all is 0 either way.
- Both fractions take the same number of pieces. The larger denominator cuts the same whole into more equal parts, so each part is smaller, and the same number of smaller parts covers less.
- No. It is yes when both are of the same whole, or of two wholes of the same size. Of different wholes, \displaystyle \frac{3}{8} of a big one can be more than \displaystyle \frac{5}{8} of a small one.
- At the same place. The bottom line’s unit is twice as long, so its \displaystyle \frac{2}{6} lies two sixths of a long unit from 0, which is as far as two thirds of the short unit. The lines cannot compare the fractions, because their units differ; with the same unit, \displaystyle \frac{2}{3} > \frac{2}{6}.
- No. They share neither the numerator nor the denominator, so neither the count of pieces nor the size of pieces is the same. A benchmark or a common denominator is needed.
- Draw one can as a bar, cut it into 3 equal parts, and draw the same bar cut into 4 equal parts. Sharing the can among more parts makes each part smaller, so one third is longer than one fourth: \displaystyle \frac{1}{3} > \frac{1}{4}.