Fractions - Comparison and Ordering of Fractions

Like Fractions Comparison: Like fractions are fractions that have the same denominator. To compare them, simply look at the numerators; the fraction with the larger numerator is the greater fraction. For example, $\frac{7}{10} > \frac{3}{10}$ because $7 > 3$. Visually, if you have two identical circles both divided into $10$ equal slices, the circle with $7$ shaded slices clearly contains more area than the one with only $3$ shaded slices.

Unlike Fractions with Same Numerator: When comparing two fractions that have the same numerator but different denominators, the fraction with the smaller denominator is actually the larger value. For example, $\frac{2}{5} > \frac{2}{9}$. Visually, imagine two loaves of bread of the same size. If you cut one into $5$ thick slices and the other into $9$ thin slices, $2$ thick slices from the first loaf will be much larger than $2$ thin slices from the second loaf.

Unlike Fractions with Different Denominators (LCM Method): To compare fractions like $\frac{2}{3}$ and $\frac{3}{4}$, you must first make their denominators the same. Find the Least Common Multiple (LCM) of the denominators (for $3$ and $4$, the LCM is $12$). Convert both to equivalent fractions with the denominator $12$: $\frac{2 \times 4}{3 \times 4} = \frac{8}{12}$ and $\frac{3 \times 3}{4 \times 3} = \frac{9}{12}$. Since $9 > 8$, we find that $\frac{3}{4} > \frac{2}{3}$.

Cross-Multiplication Method: This is a quick technique to compare two fractions $\frac{a}{b}$ and $\frac{c}{d}$. You multiply the numerator of the first by the denominator of the second ($a \times d$) and the numerator of the second by the denominator of the first ($c \times b$). If the first product is larger, the first fraction is larger. Visually, this method checks the relative 'weight' of the numerators when scaled to a common base.

Ordering Fractions: Arranging fractions in Ascending Order means placing them from smallest to largest, while Descending Order means largest to smallest. To order multiple unlike fractions, convert all of them into like fractions using a common denominator (the LCM of all denominators) and then arrange them based on their new numerators.

Comparison with Whole Numbers: Any whole number $n$ can be written as a fraction $\frac{n}{1}$. When comparing a fraction like $\frac{5}{4}$ to $1$, remember that $1$ is equal to $\frac{4}{4}$. Since $5 > 4$, then $\frac{5}{4} > 1$. Visually, any proper fraction (where the numerator is smaller than the denominator) is always less than $1$ whole, while any improper fraction (where the numerator is equal to or larger than the denominator) is equal to or greater than $1$ whole.