Same Denominator: Order the Numerators

Order from least to greatest: $\frac{5}{9}$, $\frac{2}{9}$, $\frac{7}{9}$, $\frac{4}{9}$

Check the denominators: All four fractions are ninths = Same denominator: compare the numerators

Order the numerators: $2 < 4 < 5 < 7$ = $2, 4, 5, 7$

Write the fractions in that order: Keep the denominator $9$ = $\frac{2}{9} < \frac{4}{9} < \frac{5}{9} < \frac{7}{9}$

Answer: $\frac{2}{9}, \frac{4}{9}, \frac{5}{9}, \frac{7}{9}$

Same Numerator: Larger Denominator, Smaller Fraction

Order from greatest to least: $\frac{3}{5}$, $\frac{3}{8}$, $\frac{3}{4}$, $\frac{3}{10}$

Check the numerators: All four fractions have the numerator $3$ = Same numerator: compare the denominators

Remember the rule: More parts in the whole means smaller parts, so a larger denominator gives a smaller fraction = Smallest denominator = greatest fraction

Order the denominators from smallest to largest: $4 < 5 < 8 < 10$ = $\frac{3}{4} > \frac{3}{5} > \frac{3}{8} > \frac{3}{10}$

Answer: $\frac{3}{4}, \frac{3}{5}, \frac{3}{8}, \frac{3}{10}$

Different Denominators: Use the LCD

Order from least to greatest: $\frac{2}{3}$, $\frac{3}{4}$, $\frac{5}{8}$, $\frac{7}{12}$

Find the least common denominator of 3, 4, 8 and 12: Multiples of 12: 12, 24. Is 24 divisible by 3, 4 and 8? Yes = LCD $= 24$

Rewrite every fraction in twenty-fourths: $\frac{2}{3} = \frac{16}{24}$, $\frac{3}{4} = \frac{18}{24}$, $\frac{5}{8} = \frac{15}{24}$, $\frac{7}{12} = \frac{14}{24}$ = Numerators: $16, 18, 15, 14$

Order the new numerators: $14 < 15 < 16 < 18$ = $\frac{14}{24} < \frac{15}{24} < \frac{16}{24} < \frac{18}{24}$

Write the original fractions in that order: $\frac{14}{24} = \frac{7}{12}$, $\frac{15}{24} = \frac{5}{8}$, $\frac{16}{24} = \frac{2}{3}$, $\frac{18}{24} = \frac{3}{4}$ = $\frac{7}{12} < \frac{5}{8} < \frac{2}{3} < \frac{3}{4}$

Answer: $\frac{7}{12}, \frac{5}{8}, \frac{2}{3}, \frac{3}{4}$

Large LCD: Convert to Decimals

Order from least to greatest: $\frac{3}{8}$, $\frac{2}{5}$, $\frac{7}{20}$, $\frac{9}{25}$

Choose the method: The LCD of 8, 5, 20 and 25 is 200, which is large. These denominators all give short decimals = Convert to decimals

Divide for the first two fractions: $3 \div 8 = 0.375$ and $2 \div 5 = 0.4$ = $0.375$ and $0.4$

Divide for the other two fractions: $7 \div 20 = 0.35$ and $9 \div 25 = 0.36$ = $0.35$ and $0.36$

Order the decimals and go back to the fractions: $0.35 < 0.36 < 0.375 < 0.4$ = $\frac{7}{20} < \frac{9}{25} < \frac{3}{8} < \frac{2}{5}$

Answer: $\frac{7}{20}, \frac{9}{25}, \frac{3}{8}, \frac{2}{5}$

Benchmarks First, Then the Fastest Method

Order from greatest to least: $\frac{5}{8}$, $\frac{1}{10}$, $\frac{7}{8}$, $\frac{4}{9}$

Sort the fractions with the benchmarks 0, 1/2 and 1: $\frac{1}{10}$ is close to 0. Half of 9 is 4.5, so $\frac{4}{9} < \frac{1}{2}$. Half of 8 is 4, so $\frac{5}{8} > \frac{1}{2}$ and $\frac{7}{8} > \frac{1}{2}$ = Below half: $\frac{1}{10}, \frac{4}{9}$. Above half: $\frac{5}{8}, \frac{7}{8}$

Order the group above half: Same denominator: $7 > 5$ = $\frac{7}{8} > \frac{5}{8}$

Order the group below half: $\frac{4}{9}$ is just below half, $\frac{1}{10}$ is close to 0 = $\frac{4}{9} > \frac{1}{10}$

Put the groups together, greatest first: Every fraction above half is greater than every fraction below half = $\frac{7}{8} > \frac{5}{8} > \frac{4}{9} > \frac{1}{10}$

Answer: $\frac{7}{8}, \frac{5}{8}, \frac{4}{9}, \frac{1}{10}$

Mistake: Ordering unit fractions by their denominators: writing $\frac{1}{3}, \frac{1}{4}, \frac{1}{6}$ as least to greatest because $3 < 4 < 6$

Why: With the same numerator, a larger denominator means the whole is cut into more, smaller pieces.

Correct: Least to greatest is $\frac{1}{6} < \frac{1}{4} < \frac{1}{3}$: the largest denominator gives the smallest fraction.

Mistake: Comparing only the numerators when the denominators differ, e.g. putting $\frac{5}{12}$ after $\frac{2}{3}$ because $5 > 2$

Why: Numerators can only be compared directly when the pieces have the same size.

Correct: Rewrite first: $\frac{2}{3} = \frac{8}{12}$, and $\frac{5}{12} < \frac{8}{12}$, so $\frac{5}{12} < \frac{2}{3}$.

Mistake: Changing only the denominator when rewriting: $\frac{3}{4} = \frac{3}{12}$

Why: Multiplying only the denominator makes the pieces smaller without adding more of them, so the value changes.

Correct: Multiply the numerator and the denominator by the same number: $\frac{3 \times 3}{4 \times 3} = \frac{9}{12}$.

Mistake: Mixing up ascending and descending order

Why: Students order correctly from least to greatest, but the question asked for greatest to least (or the other way round).

Correct: Underline the words in the question first. Ascending starts with the smallest fraction ($<$), descending starts with the greatest ($>$).

Recipes

Recipes often give ingredients as fractions of a cup, and you may want to know which ingredient you need the most or the least of.

A cake uses $\frac{3}{4}$ cup of flour, $\frac{1}{3}$ cup of sugar and $\frac{1}{2}$ cup of milk. With benchmarks: $\frac{1}{3} < \frac{1}{2} < \frac{3}{4}$, so flour is the largest amount.

Race Distances

Coaches compare training distances that are often given as fractions of a kilometer or a mile.

Three runners covered $\frac{3}{5}$ km, $\frac{3}{4}$ km and $\frac{3}{10}$ km. The numerators are the same, so $\frac{3}{10} < \frac{3}{5} < \frac{3}{4}$.

Survey Results

Survey results from groups of different sizes are often reported as fractions, and ranking them shows which group answered yes most often.

In a survey, $\frac{1}{4}$ of the students chose music, $\frac{1}{2}$ chose sports and $\frac{1}{8}$ chose art. Same numerators: $\frac{1}{8} < \frac{1}{4} < \frac{1}{2}$.

Q: How do I choose the fastest method?

A: Look at the whole list. Same denominators or same numerators? Use that rule. Do the benchmarks $0$, $\frac{1}{2}$ and $1$ split the fractions into clear groups? Sort first. Otherwise use the LCD when it is small (like 12 or 24), or decimals when the LCD is large or you have a calculator.

Q: What if two fractions in the list are equal?

A: Equivalent fractions such as $\frac{2}{4}$ and $\frac{3}{6}$ take the same place in the order. Write them next to each other with an equals sign: $\frac{1}{4} < \frac{2}{4} = \frac{3}{6} < \frac{5}{6}$.

Q: Do I have to use the least common denominator?

A: No, any common denominator works. Multiplying all the denominators together always gives one. The least common denominator just keeps the numbers small, which means fewer mistakes.

Q: How can I check my order?

A: Turn every fraction into a decimal with a calculator. In ascending order each decimal must be larger than the one before it, in descending order smaller.