Different Whole Parts

Order from least to greatest: $3\frac{1}{2}$, $1\frac{4}{5}$, $2\frac{1}{3}$

Look at the whole parts: $3\frac{1}{2}$ has 3, $1\frac{4}{5}$ has 1, $2\frac{1}{3}$ has 2 = Whole parts: 3, 1, 2

Check for equal whole parts: All three whole parts are different, so the fraction parts do not matter = Sort by whole parts only

Sort the whole parts from least to greatest: $1 < 2 < 3$ = $1\frac{4}{5} < 2\frac{1}{3} < 3\frac{1}{2}$

Answer: $1\frac{4}{5} < 2\frac{1}{3} < 3\frac{1}{2}$. Even though $\frac{4}{5}$ is the biggest fraction part, $1\frac{4}{5}$ is the smallest number.

Mixed Numbers and Improper Fractions Together

Order from least to greatest: $\frac{7}{3}$, $2\frac{1}{4}$, $\frac{5}{2}$, $1\frac{7}{8}$

Change the improper fractions into mixed numbers: $7 \div 3 = 2$ remainder $1$, so $\frac{7}{3} = 2\frac{1}{3}$. $5 \div 2 = 2$ remainder $1$, so $\frac{5}{2} = 2\frac{1}{2}$ = $2\frac{1}{3}$, $2\frac{1}{4}$, $2\frac{1}{2}$, $1\frac{7}{8}$

Sort by whole parts: Only $1\frac{7}{8}$ has whole part 1, the other three have whole part 2 = $1\frac{7}{8}$ is the smallest

Compare the fraction parts of the 2-group with a common denominator: The LCD of 3, 4 and 2 is 12: $\frac{1}{3} = \frac{4}{12}$, $\frac{1}{4} = \frac{3}{12}$, $\frac{1}{2} = \frac{6}{12}$ = $\frac{3}{12} < \frac{4}{12} < \frac{6}{12}$

Write the order with the original numbers: $2\frac{1}{4}$, then $2\frac{1}{3} = \frac{7}{3}$, then $2\frac{1}{2} = \frac{5}{2}$ = $1\frac{7}{8} < 2\frac{1}{4} < \frac{7}{3} < \frac{5}{2}$

Answer: $1\frac{7}{8} < 2\frac{1}{4} < \frac{7}{3} < \frac{5}{2}$

Planks from Longest to Shortest (Benchmarks)

Four planks are $2\frac{5}{8}$ m, $\frac{17}{6}$ m, $2\frac{3}{7}$ m and $3\frac{1}{10}$ m long. Order them from longest to shortest.

Change the improper fraction into a mixed number: $17 = 2 \times 6 + 5$, so $\frac{17}{6} = 2\frac{5}{6}$ = $2\frac{5}{8}$, $2\frac{5}{6}$, $2\frac{3}{7}$, $3\frac{1}{10}$

Find the biggest whole part: Only $3\frac{1}{10}$ has whole part 3 = $3\frac{1}{10}$ m is the longest

Compare the fraction parts with the benchmark $\frac{1}{2}$: $\frac{3}{7} < \frac{1}{2}$ (3 is less than half of 7), while $\frac{5}{8} > \frac{1}{2}$ and $\frac{5}{6} > \frac{1}{2}$ = $2\frac{3}{7}$ m is the shortest

Compare the two fractions above one half: $\frac{5}{6}$ and $\frac{5}{8}$ have the same numerator, and sixths are bigger pieces than eighths = $\frac{5}{6} > \frac{5}{8}$

Answer: $3\frac{1}{10} > \frac{17}{6} > 2\frac{5}{8} > 2\frac{3}{7}$ (in meters, longest to shortest)

Race Times with Decimals

Four cyclists finish a race in $\frac{11}{4}$ h, $2\frac{2}{3}$ h, $2\frac{7}{10}$ h and $\frac{23}{8}$ h. Order the times from fastest to slowest.

Decide the direction: The fastest cyclist has the smallest time = Order from least to greatest

Change every time into a decimal: $\frac{11}{4} = 2.75$, $2\frac{2}{3} \approx 2.667$, $2\frac{7}{10} = 2.7$, $\frac{23}{8} = 2.875$ = $2.75$, $2.667$, $2.7$, $2.875$

Order the decimals: $2.667 < 2.7 < 2.75 < 2.875$ = Smallest time: $2\frac{2}{3}$ h

Write the order with the original numbers: Replace each decimal by its time = $2\frac{2}{3} < 2\frac{7}{10} < \frac{11}{4} < \frac{23}{8}$

Answer: From fastest to slowest: $2\frac{2}{3}$ h, $2\frac{7}{10}$ h, $\frac{11}{4}$ h, $\frac{23}{8}$ h

Five Recipe Amounts

A recipe uses $1\frac{3}{4}$ cups of flour, $\frac{5}{3}$ cups of milk, $1\frac{5}{8}$ cups of oats, $\frac{13}{6}$ cups of water and $1\frac{1}{2}$ cups of sugar. Order the amounts from least to greatest. Which amount is the second largest?

Change the improper fractions into mixed numbers: $\frac{5}{3} = 1\frac{2}{3}$ and $\frac{13}{6} = 2\frac{1}{6}$ = $1\frac{3}{4}$, $1\frac{2}{3}$, $1\frac{5}{8}$, $2\frac{1}{6}$, $1\frac{1}{2}$

Sort by whole parts: Only $2\frac{1}{6}$ has whole part 2 = $\frac{13}{6}$ cups of water is the largest

Use a common denominator for the four fraction parts with whole part 1: The LCD of 4, 3, 8 and 2 is 24: $\frac{3}{4} = \frac{18}{24}$, $\frac{2}{3} = \frac{16}{24}$, $\frac{5}{8} = \frac{15}{24}$, $\frac{1}{2} = \frac{12}{24}$ = $12 < 15 < 16 < 18$

Write the full order: Use the original numbers = $1\frac{1}{2} < 1\frac{5}{8} < \frac{5}{3} < 1\frac{3}{4} < \frac{13}{6}$

Answer: $1\frac{1}{2} < 1\frac{5}{8} < \frac{5}{3} < 1\frac{3}{4} < \frac{13}{6}$. The second largest amount is $1\frac{3}{4}$ cups of flour.

Mistake: Thinking an improper fraction like $\frac{13}{6}$ is small because it looks like an ordinary fraction

Why: A fraction whose numerator is bigger than its denominator is more than 1. $\frac{13}{6}$ is more than 2.

Correct: Change it first: $\frac{13}{6} = 2\frac{1}{6}$, which is greater than $1\frac{3}{4}$.

Mistake: Ordering by the fraction parts and ignoring the whole parts, for example $2\frac{1}{4} < 1\frac{7}{8}$

Why: The fraction part is always less than 1, so it can never make up for a smaller whole part.

Correct: Sort by whole parts first: $1\frac{7}{8} < 2\frac{1}{4}$ because $1 < 2$.

Mistake: Thinking a bigger denominator means a bigger number, for example $2\frac{1}{8} > 2\frac{1}{5}$

Why: With the same numerator, a bigger denominator means smaller pieces.

Correct: $\frac{1}{8} < \frac{1}{5}$, so $2\frac{1}{8} < 2\frac{1}{5}$.

Mistake: Writing the order in the wrong direction

Why: The question asked for greatest to least (or for the fastest time first), but the list was written from least to greatest out of habit.

Correct: Read the question again before writing the answer: greatest to least uses $>$, least to greatest uses $<$. The fastest time is the smallest number.

Woodwork and Building

Carpenters sort boards by length before cutting, so the longest boards are used for the longest parts.

Boards of $2\frac{1}{2}$ m, $\frac{9}{4}$ m and $2\frac{3}{5}$ m: since $\frac{9}{4} = 2\frac{1}{4}$, the order from shortest to longest is $\frac{9}{4} < 2\frac{1}{2} < 2\frac{3}{5}$.

Race Results

Race organizers rank finishing times from smallest to largest to find the winner and the podium.

Times of $1\frac{1}{2}$ h, $\frac{7}{5}$ h and $1\frac{3}{4}$ h: since $\frac{7}{5} = 1.4$, $1\frac{1}{2} = 1.5$ and $1\frac{3}{4} = 1.75$, the winner is the runner with $\frac{7}{5}$ h.

Cooking and Baking

Cooks compare the amounts in a recipe to see which ingredients they need most of and what to buy first.

A bread recipe uses $\frac{7}{2}$ cups of flour, $1\frac{1}{4}$ cups of water and $\frac{3}{4}$ cup of seeds: $\frac{3}{4} < 1\frac{1}{4} < \frac{7}{2}$, so flour is the biggest amount.

Q: Do I have to change improper fractions into mixed numbers?

A: It is the quickest way to see the whole part. You can also change every mixed number into an improper fraction (for example $2\frac{1}{3} = \frac{7}{3}$) and compare with a common denominator, but the numbers get bigger.

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

A: Then they share the same place in the order. For example $\frac{9}{4} = 2\frac{1}{4}$, so you write $2\frac{1}{4} = \frac{9}{4}$ next to each other.

Q: When are decimals better than a common denominator?

A: Decimals are handy when the denominators share no small common multiple, like 7, 8 and 10, or when you have a calculator. A common denominator is better for halves, thirds, quarters and sixths, because it gives exact answers.

Q: Where does a whole number like 3 go?

A: A whole number is a mixed number with fraction part 0. So $2\frac{9}{10} < 3 < 3\frac{1}{9}$.