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Comparing and Ordering Fractions

Find equivalent representations of rational numbers and represent rational numbers on a number line.

You cannot compare fractions until the pieces are the same size. That is the whole method — make the bottoms match, then just look at the tops.

Builds onYr 7 · Equivalent Fractions and Simplifying

Method

Same bottom, compare the tops
37<57\frac{3}{7} < \frac{5}{7}
If the pieces are the same size, more pieces means more.
Same top, the bigger bottom is smaller
18<13\frac{1}{8} < \frac{1}{3}
More pieces in the whole means each piece is smaller.
Otherwise use the lowest common denominator
LCD=LCM of the bottomsLCD = \text{LCM of the bottoms}
Rewrite both fractions over it, then compare the tops.
Compare with a half as a quick check
Is the top more or less than half the bottom? That often settles it in one look.

Worked example

e.g. Put 23, 34 and 58 in order, smallest first.\text{Put } \frac{2}{3},\ \frac{3}{4} \text{ and } \frac{5}{8} \text{ in order, smallest first.}
  1. Look at the bottoms.3, 4, 83,\ 4,\ 8
  2. Find the lowest common multiple of them.LCM(3,4,8)=24LCM(3, 4, 8) = 24
  3. Rewrite each fraction over 24.23=162434=182458=1524\frac{2}{3} = \frac{16}{24} \qquad \frac{3}{4} = \frac{18}{24} \qquad \frac{5}{8} = \frac{15}{24}
  4. Now the pieces are the same size, so compare the tops.15<16<1815 < 16 < 18
  5. Write the answer using the original fractions.58, 23, 34\frac{5}{8},\ \frac{2}{3},\ \frac{3}{4}
    The three fractions marked in order on a number line from zero to one015/82/33/4

Practice

Rewrite over a common denominator, then compare.

1
Which is larger: 35 or 59?\text{Which is larger: } \frac{3}{5} \text{ or } \frac{5}{9}?
Answer2745>2545∴35\frac{27}{45} > \frac{25}{45} \quad\therefore\quad \frac{3}{5}
2
Which is larger: 16 or 14?\text{Which is larger: } \frac{1}{6} \text{ or } \frac{1}{4}?
Answer14 — bigger bottom means smaller pieces\frac{1}{4} \text{ — bigger bottom means smaller pieces}
3
Order smallest first: 12, 25, 310\text{Order smallest first: } \frac{1}{2},\ \frac{2}{5},\ \frac{3}{10}
Answer310, 25, 12\frac{3}{10},\ \frac{2}{5},\ \frac{1}{2}
4
Order smallest first: 56, 712, 34\text{Order smallest first: } \frac{5}{6},\ \frac{7}{12},\ \frac{3}{4}
Answer712, 34, 56\frac{7}{12},\ \frac{3}{4},\ \frac{5}{6}
5
Is 715 more or less than a half?\text{Is } \frac{7}{15} \text{ more or less than a half?}
AnswerLess — half of 15 is 7,5\text{Less — half of } 15 \text{ is } 7{,}5
6
Put a fraction between 14 and 12\text{Put a fraction between } \frac{1}{4} \text{ and } \frac{1}{2}
Answer38 (any value between works)\frac{3}{8} \text{ (any value between works)}
Next step
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