Ordering of proper fractions is nothing but an arrangement of these fractions from the least to the greatest or from the greatest to the least.

This can be done by comparing it first and then set the order for each participating fractions.

**Steps of Ordering**:

**Step 1: **Find the LCM of the denominators

**Step 2:** multiplying numerator and denominator of each fraction by a number to make the LCM

**Step 3:** Ordering the fractions with like denominators

Let us do one comparing and ordering of fractions.

$\frac{2}{5}$, $\frac{4}{7}$, $\frac{5}{8}$

The LCM for the denominators is 280

Multiply the numerator and the denominator of each fraction by the fractor to make LCM

$\frac{2 \times 56}{5 \times 56}$ = $\frac{112}{280}$ ; $\frac{4 \times 40}{7 \times 40}$ = $\frac{160}{280}$ ; $\frac{5 \times 35}{8 \times 35}$ = $\frac{175}{280}$

From the above values the ordering in least to greatest would be

$\frac{2}{5}$ < $\frac{4}{7}$ < $\frac{5}{8}$

**Examples on Ordering Proper Fractions with the Same Denominators:**

1) Write the order of the given fractions:

$\frac{5}{8}$,

$\frac{7}{8}$,

$\frac{1}{8}$,

$\frac{3}{8}$?

**Solution:****Step 1:** Compare the numerators, 1 < 3 < 5 < 7

**Step 2:** So,

$\frac{1}{8}$ <

$\frac{3}{8}$ <

$\frac{5}{8}$ <

$\frac{7}{8}$

2) Write the order of the given fractions: $\frac{15}{11}$, $\frac{17}{11}$, $\frac{12}{11}$, $\frac{13}{11}$?

**Solution:**

**Step 1:** Compare the numerators $\Rightarrow$ 12 < 13 < 15 < 17

**Step 2:** So, $\frac{12}{11}$ < $\frac{13}{11}$ < $\frac{15}{11}$ < $\frac{17}{11}$

**Examples on Ordering Proper Fractions with Different Denominators:**

1) Write the order of the given fractions:

$\frac{2}{7}$,

$\frac{4}{9}$,

$\frac{1}{6}$,

$\frac{3}{14}$?

**Solution:****Step 1:** Least common multiple of the denominators (LCD) 7, 9, 6 and 14 = 126

**Step 2:** Make the fractions as equivalent fractions with the denominator as the LCD.

$\frac{2}{7}$ =

$\frac{(2 \times 18)}{(7 \times 18)}$ =

$\frac{36}{126}$$\frac{4}{9}$ =

$\frac{(4 \times 14)}{(9 \times 14)}$ =

$\frac{56}{126}$ $\frac{1}{6}$ =

$\frac{(1 \times 21)}{(6 \times 21)}$ =

$\frac{21}{126}$ $\frac{3}{14}$ =

$\frac{(3 \times 9)}{(14 \times 9)}$ =

$\frac{27}{126}$**Step 3:** Compare the numerators

$\Rightarrow$ 21< 27 < 36 < 56

**Step 4:** So,

$\frac{1}{6}$ <

$\frac{3}{14}$ <

$\frac{2}{7}$ <

$\frac{4}{9}$

2) Write the order of the given fractions: $\frac{1}{5}$, $\frac{3}{25}$, $\frac{2}{9}$, $\frac{4}{3}$?

**Solution:**

**Step 1:** Least common multiple of the denominators (LCD) 5, 25, 9 and 3 = 225

**Step 2:** Make the fractions as equivalent fractions with the denominator as the LCD.

$\frac{1}{5}$ = $\frac{(1 \times 18)}{(5 \times 45)}$ = $\frac{18}{225}$

$\frac{3}{25}$ = $\frac{(3 \times 9)}{(25 \times 9)}$ = $\frac{27}{225}$

$\frac{2}{9}$ = $\frac{(2 \times 25)}{(9 \times 25)}$ = $\frac{50}{225}$

$\frac{4}{3}$ = $\frac{(4 \times 75)}{(3 \times 75)}$ = $\frac{300}{225}$

**Step 3:** Compare the numerators $\rightarrow$ 18 < 27 < 50 < 300

**Step 4:** So, $\frac{1}{5}$ < $\frac{3}{25}$ < $\frac{2}{9}$ < $\frac{4}{3}$

**Practice Problems on Ordering Proper Fractions:**

**Problem 1:** Write the order of the given fractions: $\frac{2}{7}$, $\frac{1}{5}$, $\frac{3}{14}$, $\frac{4}{21}$?

**Problem 2:** Write the order of the given fractions: $\frac{1}{2}$, $\frac{3}{4}$, $\frac{5}{6}$, $\frac{7}{8}$?