Lesson 4: Find common units or number of units to compare two fractions.

Property

A common denominator for two fractions, $\frac{a}{b}$ and $\frac{c}{d}$, can be found visually using two identical area models.
By partitioning the first model (representing $\frac{a}{b}$) with $d$ horizontal lines and the second model (representing $\frac{c}{d}$) with $b$ vertical lines, both models are decomposed into $b \times d$ equal parts.

Examples

Property

To compare improper fractions, decompose each fraction into a whole number and a proper fraction.
If the whole numbers are different, the fraction with the larger whole number is greater.
If the whole numbers are the same, compare the remaining proper fractions.

where $c$ is a whole number.

Examples

• To compare $\frac{7}{4}$ and $\frac{5}{3}$:

Decompose each fraction: $\frac{7}{4} = 1 + \frac{3}{4}$ and $\frac{5}{3} = 1 + \frac{2}{3}$.
Since the whole numbers are both $1$, compare the parts: $\frac{3}{4}$ and $\frac{2}{3}$.
Find a common denominator: $\frac{3}{4} = \frac{9}{12}$ and $\frac{2}{3} = \frac{8}{12}$.
Since $\frac{9}{12} > \frac{8}{12}$, then $\frac{7}{4} > \frac{5}{3}$.

• To compare $\frac{11}{5}$ and $\frac{9}{4}$:

Decompose each fraction: $\frac{11}{5} = 2 + \frac{1}{5}$ and $\frac{9}{4} = 2 + \frac{1}{4}$.
Since the whole numbers are both $2$, compare the parts: $\frac{1}{5}$ and $\frac{1}{4}$.
Because fourths are larger than fifths, $\frac{1}{4} > \frac{1}{5}$.
Therefore, $\frac{9}{4} > \frac{11}{5}$.

Explanation

Using number bonds is a strategy to compare improper fractions. First, decompose each improper fraction into a mixed number, which is a whole number and a proper fraction. If one whole number is larger than the other, its original improper fraction is greater. If the whole numbers are equal, you then compare the fractional parts by finding a common denominator, just as you would with proper fractions.