Math worksheet for comparing rational numbers, presented by Codeyoung, with exercises to determine which number in each pair is greater or smaller.

Let's solve the problem step by step.

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Part 1: Which of the two rational numbers in each pair is greater?

We will simplify and compare each pair.

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#### I. $ \frac{-7}{12} $ or $ \frac{5}{-8} $

Simplify:
- $ \frac{-7}{12} = -\frac{7}{12} $
- $ \frac{5}{-8} = -\frac{5}{8} $

Now compare $ -\frac{7}{12} $ and $ -\frac{5}{8} $

To compare negative numbers, the one with the smaller absolute value is greater.

Compute absolute values:
- $ \left| -\frac{7}{12} \right| = \frac{7}{12} \approx 0.583 $
- $ \left| -\frac{5}{8} \right| = \frac{5}{8} = 0.625 $

Since $ 0.583 < 0.625 $, then $ -\frac{7}{12} > -\frac{5}{8} $

✔ Answer: $ \frac{-7}{12} $ is greater

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#### II. $ \frac{2}{3} $ or $ \frac{3}{4} $

Compare positive fractions.

Find common denominator: LCM of 3 and 4 is 12

- $ \frac{2}{3} = \frac{8}{12} $
- $ \frac{3}{4} = \frac{9}{12} $

Since $ \frac{8}{12} < \frac{9}{12} $, so $ \frac{2}{3} < \frac{3}{4} $

✔ Answer: $ \frac{3}{4} $ is greater

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#### III. $ \frac{4}{-9} $ or $ \frac{-3}{-7} $

Simplify:
- $ \frac{4}{-9} = -\frac{4}{9} $
- $ \frac{-3}{-7} = \frac{3}{7} $ (negative divided by negative = positive)

So we are comparing $ -\frac{4}{9} $ and $ \frac{3}{7} $

One is negative, one is positive → positive is greater

✔ Answer: $ \frac{-3}{-7} $ is greater

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#### IV. $ \frac{-1}{2} $ or $ -1 $

Convert both to decimals:
- $ \frac{-1}{2} = -0.5 $
- $ -1 = -1.0 $

On number line: $ -0.5 > -1 $

✔ Answer: $ \frac{-1}{2} $ is greater

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#### V. $ \frac{-5}{8} $ or $ \frac{3}{-4} $

Simplify:
- $ \frac{-5}{8} = -\frac{5}{8} $
- $ \frac{3}{-4} = -\frac{3}{4} $

Compare $ -\frac{5}{8} $ and $ -\frac{3}{4} $

Absolute values:
- $ \frac{5}{8} = 0.625 $
- $ \frac{3}{4} = 0.75 $

Since $ 0.625 < 0.75 $, then $ -\frac{5}{8} > -\frac{3}{4} $

✔ Answer: $ \frac{-5}{8} $ is greater

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#### VI. $ \frac{5}{9} $ or $ \frac{-3}{-8} $

Simplify:
- $ \frac{5}{9} $ is positive
- $ \frac{-3}{-8} = \frac{3}{8} $ (positive)

Compare $ \frac{5}{9} $ and $ \frac{3}{8} $

Common denominator: LCM of 9 and 8 is 72

- $ \frac{5}{9} = \frac{40}{72} $
- $ \frac{3}{8} = \frac{27}{72} $

$ \frac{40}{72} > \frac{27}{72} $, so $ \frac{5}{9} > \frac{3}{8} $

✔ Answer: $ \frac{5}{9} $ is greater

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✔ Part 1 Summary:

| Pair | Greater Number |
|------|----------------|
| I | $ \frac{-7}{12} $ |
| II | $ \frac{3}{4} $ |
| III | $ \frac{-3}{-7} $ |
| IV | $ \frac{-1}{2} $ |
| V | $ \frac{-5}{8} $ |
| VI | $ \frac{5}{9} $ |

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Part 2: Which of the two rational numbers in each pair is smaller?

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#### I. $ \frac{-4}{3} $ or $ \frac{8}{-7} $

Simplify:
- $ \frac{-4}{3} = -\frac{4}{3} \approx -1.333 $
- $ \frac{8}{-7} = -\frac{8}{7} \approx -1.142 $

Compare: $ -1.333 $ vs $ -1.142 $

The more negative number is smaller.

Since $ -1.333 < -1.142 $, so $ -\frac{4}{3} $ is smaller

✔ Answer: $ \frac{-4}{3} $ is smaller

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#### II. $ \frac{9}{-13} $ or $ \frac{7}{-12} $

Simplify:
- $ \frac{9}{-13} = -\frac{9}{13} \approx -0.692 $
- $ \frac{7}{-12} = -\frac{7}{12} \approx -0.583 $

Compare: $ -0.692 $ vs $ -0.583 $

$ -0.692 < -0.583 $ → so $ -\frac{9}{13} $ is smaller

✔ Answer: $ \frac{9}{-13} $ is smaller

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#### III. $ \frac{-12}{5} $ or $ -3 $

Simplify:
- $ \frac{-12}{5} = -2.4 $
- $ -3 = -3.0 $

Compare: $ -2.4 $ vs $ -3.0 $

$ -3.0 < -2.4 $ → so $ -3 $ is smaller

✔ Answer: $ -3 $ is smaller

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#### IV. $ \frac{4}{-5} $ or $ \frac{-7}{10} $

Simplify:
- $ \frac{4}{-5} = -\frac{4}{5} = -0.8 $
- $ \frac{-7}{10} = -0.7 $

Compare: $ -0.8 $ vs $ -0.7 $

$ -0.8 < -0.7 $ → so $ -\frac{4}{5} $ is smaller

✔ Answer: $ \frac{4}{-5} $ is smaller

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#### V. $ \frac{-12}{5} $ or $ -3 $

This is same as part III.

- $ \frac{-12}{5} = -2.4 $
- $ -3 = -3.0 $

$ -3.0 < -2.4 $ → so $ -3 $ is smaller

✔ Answer: $ -3 $ is smaller

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✔ Part 2 Summary:

| Pair | Smaller Number |
|------|----------------|
| I | $ \frac{-4}{3} $ |
| II | $ \frac{9}{-13} $ |
| III | $ -3 $ |
| IV | $ \frac{4}{-5} $ |
| V | $ -3 $ |

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✔ Final Answers:

#### Which is greater?
1. $ \frac{-7}{12} $
2. $ \frac{3}{4} $
3. $ \frac{-3}{-7} $
4. $ \frac{-1}{2} $
5. $ \frac{-5}{8} $
6. $ \frac{5}{9} $

#### Which is smaller?
1. $ \frac{-4}{3} $
2. $ \frac{9}{-13} $
3. $ -3 $
4. $ \frac{4}{-5} $
5. $ -3 $

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