Ordering Fractions

Show your work for each problem.

1.
Which list shows $\frac{4}{7}$ , $\frac{1}{7}$ , $\frac{6}{7}$ , $\frac{3}{7}$ from least to greatest?
1. a) $\frac{4}{7}, \frac{1}{7}, \frac{6}{7}, \frac{3}{7}$
2. b) $\frac{1}{7}, \frac{4}{7}, \frac{3}{7}, \frac{6}{7}$
3. c) $\frac{6}{7}, \frac{4}{7}, \frac{3}{7}, \frac{1}{7}$
4. d) $\frac{1}{7}, \frac{3}{7}, \frac{4}{7}, \frac{6}{7}$
Answer: $\frac{1}{7}, \frac{3}{7}, \frac{4}{7}, \frac{6}{7}$
Same denominator: $1 < 3 < 4 < 6$ , so $\frac{1}{7} < \frac{3}{7} < \frac{4}{7} < \frac{6}{7}$ . The first list is in descending order.
2.
Order $\frac{5}{11}$ , $\frac{9}{11}$ , $\frac{2}{11}$ , $\frac{7}{11}$ from least to greatest. Which fraction comes first?
Answer: 2/11
All are elevenths and $2 < 5 < 7 < 9$ : $\frac{2}{11} < \frac{5}{11} < \frac{7}{11} < \frac{9}{11}$ . The first fraction is $\frac{2}{11}$ .
3.
Which list shows $\frac{2}{9}$ , $\frac{2}{3}$ , $\frac{2}{5}$ , $\frac{2}{7}$ from greatest to least?
1. a) $\frac{2}{3}, \frac{2}{7}, \frac{2}{5}, \frac{2}{9}$
2. b) $\frac{2}{9}, \frac{2}{7}, \frac{2}{5}, \frac{2}{3}$
3. c) $\frac{2}{3}, \frac{2}{5}, \frac{2}{7}, \frac{2}{9}$
4. d) $\frac{2}{5}, \frac{2}{3}, \frac{2}{9}, \frac{2}{7}$
Answer: $\frac{2}{3}, \frac{2}{5}, \frac{2}{7}, \frac{2}{9}$
All numerators are 2. Denominators from smallest to largest: $3 < 5 < 7 < 9$ , so $\frac{2}{3} > \frac{2}{5} > \frac{2}{7} > \frac{2}{9}$ . The first list is ascending, not descending.
4.
Order $\frac{4}{5}$ , $\frac{4}{9}$ , $\frac{4}{7}$ , $\frac{4}{11}$ from least to greatest. Which fraction comes first?
Answer: 4/11
All numerators are 4, so the order is reversed for the denominators: $\frac{4}{11} < \frac{4}{9} < \frac{4}{7} < \frac{4}{5}$ . The smallest is $\frac{4}{11}$ .
5.
Order $\frac{1}{8}$ , $\frac{9}{10}$ , $\frac{1}{2}$ from least to greatest using the benchmarks 0, 1/2 and 1.
Answer: 1/8 < 1/2 < 9/10
1. Which fraction is close to 0? 1/8
2. Which fraction is close to 1? 9/10
3. Which fraction comes second from least to greatest? 1/2
6.
Order $\frac{7}{12}$ , $\frac{11}{12}$ , $\frac{1}{12}$ , $\frac{5}{12}$ from greatest to least.
Answer: 11/12 > 7/12 > 5/12 > 1/12
1. All four fractions have the same denominator. What is it? 12
2. Which fraction is the greatest? 11/12
3. Which fraction comes third from greatest to least? 5/12
7.
What is the fastest way to order $\frac{5}{6}$ , $\frac{5}{9}$ , $\frac{5}{7}$ , $\frac{5}{12}$ ?
1. a)Cross multiply every pair of fractions
2. b)Same numerator: order the denominators
3. c)Convert every fraction to a decimal
4. d)Rewrite all of them with the LCD 252
Answer: Same numerator: order the denominators
All numerators are 5, so the rule for the same numerator orders them at once: $\frac{5}{12} < \frac{5}{9} < \frac{5}{7} < \frac{5}{6}$ . The LCD would be 252 and decimals or six cross multiplications take much longer.
8.
Order $\frac{1}{2}$ , $\frac{2}{5}$ , $\frac{3}{10}$ , $\frac{7}{20}$ from least to greatest. Which fraction comes second?
Answer: 7/20
LCD 20: $\frac{1}{2} = \frac{10}{20}, \frac{2}{5} = \frac{8}{20}, \frac{3}{10} = \frac{6}{20}, \frac{7}{20} = \frac{7}{20}$ . Then $6 < 7 < 8 < 10$ , so $\frac{3}{10} < \frac{7}{20} < \frac{2}{5} < \frac{1}{2}$ . Second is $\frac{7}{20}$ .
9.
Order $\frac{3}{4}$ , $\frac{5}{6}$ , $\frac{2}{3}$ , $\frac{7}{12}$ from least to greatest.
Answer: 7/12 < 2/3 < 3/4 < 5/6
1. What is the least common denominator of 4, 6, 3 and 12? 12
2. Write $\frac{3}{4}$ in twelfths. What is the numerator? 9
3. Write $\frac{2}{3}$ in twelfths. What is the numerator? 8
4. Write $\frac{5}{6}$ in twelfths. What is the numerator? 10
5. Which fraction comes third from least to greatest? 3/4
10.
Order $\frac{7}{8}$ , $\frac{4}{5}$ , $\frac{17}{20}$ , $\frac{21}{25}$ from greatest to least by converting to decimals.
Answer: 7/8 > 17/20 > 21/25 > 4/5
1. Write $\frac{7}{8}$ as a decimal. 0.875
2. Write $\frac{17}{20}$ as a decimal. 0.85
3. Write $\frac{21}{25}$ as a decimal. 0.84
4. Which fraction comes second from greatest to least? 17/20
11.
Four runners ran $\frac{5}{9}$ km, $\frac{2}{3}$ km, $\frac{1}{2}$ km, $\frac{7}{18}$ km. Which list shows the distances from shortest to longest?
1. a) $\frac{1}{2}, \frac{7}{18}, \frac{2}{3}, \frac{5}{9}$
2. b) $\frac{7}{18}, \frac{1}{2}, \frac{5}{9}, \frac{2}{3}$
3. c) $\frac{7}{18}, \frac{5}{9}, \frac{1}{2}, \frac{2}{3}$
4. d) $\frac{2}{3}, \frac{5}{9}, \frac{1}{2}, \frac{7}{18}$
Answer: $\frac{7}{18}, \frac{1}{2}, \frac{5}{9}, \frac{2}{3}$
In eighteenths: $\frac{5}{9} = \frac{10}{18}, \frac{2}{3} = \frac{12}{18}, \frac{1}{2} = \frac{9}{18}$ , and $\frac{7}{18}$ . $7 < 9 < 10 < 12$ , so $\frac{7}{18} < \frac{1}{2} < \frac{5}{9} < \frac{2}{3}$ .
12.
A recipe uses $\frac{3}{8}$ cup of oil, $\frac{1}{4}$ cup of honey, $\frac{1}{3}$ cup of sugar and $\frac{5}{12}$ cup of milk. Which amount is the second largest? Write it as a fraction.
Answer: 3/8
In twenty-fourths: $\frac{3}{8} = \frac{9}{24}$ , $\frac{1}{4} = \frac{6}{24}$ , $\frac{1}{3} = \frac{8}{24}$ , $\frac{5}{12} = \frac{10}{24}$ . Order: $\frac{1}{4} < \frac{1}{3} < \frac{3}{8} < \frac{5}{12}$ , so the second largest is $\frac{3}{8}$ (oil).
13.
In a class survey about favorite sports, $\frac{3}{8}$ chose football, $\frac{1}{4}$ basketball, $\frac{1}{6}$ tennis and $\frac{5}{24}$ swimming. Order the fractions from least to greatest.
Answer: 1/6 < 5/24 < 1/4 < 3/8
1. What is the least common denominator of 8, 4, 6 and 24? 24
2. Write $\frac{1}{6}$ in twenty-fourths. What is the numerator? 4
3. Write $\frac{1}{4}$ in twenty-fourths. What is the numerator? 6
4. Which fraction comes second from least to greatest? 5/24
14.
Order $\frac{5}{7}$ , $\frac{2}{3}$ , $\frac{3}{5}$ , $\frac{7}{10}$ from least to greatest. Which fraction comes third?
Answer: 7/10
LCD 210: $\frac{5}{7} = \frac{150}{210}$ , $\frac{2}{3} = \frac{140}{210}$ , $\frac{3}{5} = \frac{126}{210}$ , $\frac{7}{10} = \frac{147}{210}$ . So $\frac{3}{5} < \frac{2}{3} < \frac{7}{10} < \frac{5}{7}$ and the third is $\frac{7}{10}$ .
15.
Which list shows $\frac{4}{9}$ , $\frac{7}{12}$ , $\frac{5}{11}$ , $\frac{3}{5}$ from greatest to least?
1. a) $\frac{7}{12}, \frac{3}{5}, \frac{4}{9}, \frac{5}{11}$
2. b) $\frac{3}{5}, \frac{7}{12}, \frac{5}{11}, \frac{4}{9}$
3. c) $\frac{3}{5}, \frac{7}{12}, \frac{4}{9}, \frac{5}{11}$
4. d) $\frac{4}{9}, \frac{5}{11}, \frac{7}{12}, \frac{3}{5}$
Answer: $\frac{3}{5}, \frac{7}{12}, \frac{5}{11}, \frac{4}{9}$
Above half: $\frac{3}{5} = 0.6$ and $\frac{7}{12} \approx 0.583$ . Below half: $\frac{5}{11} \approx 0.455$ and $\frac{4}{9} \approx 0.444$ . So $\frac{3}{5} > \frac{7}{12} > \frac{5}{11} > \frac{4}{9}$ .
16.
Order $\frac{2}{11}$ , $\frac{6}{7}$ , $\frac{5}{9}$ , $\frac{9}{10}$ from greatest to least without a common denominator.
Answer: 9/10 > 6/7 > 5/9 > 2/11
1. Which fraction is close to 0? 2/11
2. Which fraction is just above one half? 5/9
3. $\frac{6}{7}$ is $\frac{1}{7}$ away from 1. How far is $\frac{9}{10}$ from 1? 1/10
4. Which fraction is the greatest? 9/10
17.
In a race, Ana ran $\frac{5}{6}$ km, Ben $\frac{7}{9}$ km, Cleo $\frac{3}{4}$ km and Dan $\frac{13}{18}$ km. Order the distances from shortest to longest.
Answer: 13/18 < 3/4 < 7/9 < 5/6
1. What is the least common denominator of 6, 9, 4 and 18? 36
2. Write $\frac{7}{9}$ in thirty-sixths. What is the numerator? 28
3. Write $\frac{3}{4}$ in thirty-sixths. What is the numerator? 27
4. Write $\frac{13}{18}$ in thirty-sixths. What is the numerator? 26
5. Which distance is the second longest? Write it as a fraction. 7/9
18.
Four groups answered yes to a survey question: group A $\frac{7}{16}$ , group B $\frac{3}{8}$ , group C $\frac{9}{20}$ , group D $\frac{2}{5}$ . Which fraction is the largest?
Answer: 9/20
1. Write $\frac{7}{16}$ as a decimal. 0.4375
2. Write $\frac{9}{20}$ as a decimal. 0.45
3. Write $\frac{2}{5}$ as a decimal. 0.4
4. Which fraction is the largest? 9/20