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Question:

# Order the following fractions from LEAST to GREATEST: $\frac{-5}{6}$; $\frac{1}{6}$; $\frac{1}{2}$; $\frac{-1}{9}$

A $\frac{-5}{6}$ ; $\frac{-1}{9}$ ; $\frac{1}{6}$ ; $\frac{1}{2}$
explanation

Remember that when we are dealing with negative numbers, the greater its absolute value (a number’s distance from zero), the smaller the number itself.
Step 1: Organize and sort our numbers into groups Since none of these numbers has any whole number in front, we will sort these numbers into positive and negative numbers.
Positive Numbers: $\frac{1}{6}$ and $\frac{1}{2}$
Both $\frac{1}{6}$ and $\frac{1}{2}$ are greater than any of the negative fractions. We will compare these fractions using common denominators. (Step 2)
Negative Numbers: $\frac{-5}{6}$ and $\frac{-1}{9}$
These two we will also compare using common denominators. (Step 3)
Step 2: Compare positive fractions using Common Denominators & Equivalent Fractions
We are comparing the fractions $\frac{1}{6}$ and $\frac{1}{2}$
Let’s first find the Least Common Multiple of 6 and 2:
Multiples of 6: 6, 12, 18, 24
Multiples of 2: 2, 4, 6, 8, 10
LCM: 6
Now we create equivalent fractions with 6 as the new denominator.
Start with $\frac{1}{2}$
Since 2 × 3 = 6, we also multiply the numerator by 3
(1×3)/(2×3) = 3/6
$\frac{1}{6}$ has already the denominator of 6
Now we can compare:
$\frac{3}{6}$ is larger than $\frac{1}{6}$ therefore $\frac{1}{2}$ is larger than $\frac{1}{6}$
Step 3: Compare negative fractions using Common Denominators & Equivalent Fractions
We are comparing the fractions $\frac{5}{6}$ and $\frac{1}{9}$
Let’s first find the Least Common Multiple of 6 and 9:
Multiples of 6: 6, 12, 18, 24
Multiples of 9: 9, 18, 27, 36, 45
LCM: 18
Now we create equivalent fractions with 18 as the new denominator.
First Up: $\frac{5}{6}$
Since 6 × 3 = 18, we also multiply the numerator by 3
(5×3)/(6×3)= $\frac{15}{18}$
Second Up: $\frac{1}{9}$
Since 9 × 2 = 18, we also multiply the numerator by 2
(1×2)/(9×2) = $\frac{2}{18}$
Now we can compare:
$\frac{15}{18}$ is larger than $\frac{2}{18}$ therefore $\frac{-5}{6}$ is further from zero than $\frac{-1}{9}$ which means $\frac{-5}{6}$ / is less than $\frac{-1}{9}$
Step 4: Organize our fractions from least to greatest.
$\frac{-5}{6}$ is smaller than $\frac{-1}{9}$
$\frac{1}{6}$ is smaller than $\frac{1}{2}$
The order of the fractions from least to greatest is $\frac{-5}{6}$ ; $\frac{-1}{9}$ ; $\frac{1}{6}$ ; $\frac{1}{2}$

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