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

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Question:

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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