Comparing and Ordering Rational Numbers Worksheet by MathGuyMan | TPT - Free Printable

Problem Overview:

The task involves comparing and ordering rational numbers, which include fractions and decimals. The goal is to convert fractions to decimals (or vice versa), match equivalent forms, and order the numbers from least to greatest.

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Step-by-Step Solution:

#### Part 1: Write the Equivalent Decimal or Fraction for Each Number

1. $ x = \_\_\_\_\_ $
- No information is provided about $ x $. This part seems incomplete or missing context.

2. $ \frac{7}{8} = \_\_\_\_\_ $
- Convert the fraction to a decimal:
$$
\frac{7}{8} = 0.875
$$

3. $ 3.3 = \_\_\_\_\_ $
- Convert the decimal to a fraction:
$$
3.3 = 3 + 0.3 = 3 + \frac{3}{10} = \frac{30}{10} + \frac{3}{10} = \frac{33}{10}
$$

4. $ \frac{4}{9} = \_\_\_\_\_ $
- Convert the fraction to a decimal:
$$
\frac{4}{9} = 0.\overline{4} \quad (\text{repeating decimal})
$$

5. $ 4 = \_\_\_\_\_ $
- Convert the whole number to a fraction:
$$
4 = \frac{4}{1}
$$

6. $ \frac{6}{7} = \_\_\_\_\_ $
- Convert the fraction to a decimal:
$$
\frac{6}{7} \approx 0.8571 \quad (\text{rounded to four decimal places})
$$

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#### Part 2: Match the Fraction to a Decimal from the Box

The box contains the following decimals: $ 0.25 $, $ 0.3 $, $ 0.6 $, $ 1.25 $, $ 1.3 $, $ 1.6 $.

7. $ \frac{1}{4} = \_\_\_\_\_ $
- Convert $ \frac{1}{4} $ to a decimal:
$$
\frac{1}{4} = 0.25
$$

8. $ \frac{3}{10} = \_\_\_\_\_ $
- Convert $ \frac{3}{10} $ to a decimal:
$$
\frac{3}{10} = 0.3
$$

9. $ \frac{3}{2} = \_\_\_\_\_ $
- Convert $ \frac{3}{2} $ to a decimal:
$$
\frac{3}{2} = 1.5
$$
- Note: $ 1.5 $ is not in the box, so this might be an error or typo.

10. $ \frac{5}{4} = \_\_\_\_\_ $
- Convert $ \frac{5}{4} $ to a decimal:
$$
\frac{5}{4} = 1.25
$$

11. $ \frac{13}{10} = \_\_\_\_\_ $
- Convert $ \frac{13}{10} $ to a decimal:
$$
\frac{13}{10} = 1.3
$$

12. $ \frac{8}{5} = \_\_\_\_\_ $
- Convert $ \frac{8}{5} $ to a decimal:
$$
\frac{8}{5} = 1.6
$$

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#### Part 3: Order the Following from Least to Greatest

13. $ 0.555, 0.52, -0.53, 0 $
- Compare the numbers:
- Negative numbers are smaller than positive numbers.
- Among positive numbers, compare digits from left to right.
- Ordered list:
$$
-0.53, 0, 0.52, 0.555
$$

14. $ \frac{3}{8}, 2.4, 2.2, \frac{3}{2} $
- Convert all numbers to decimals for easy comparison:
$$
\frac{3}{8} = 0.375, \quad \frac{3}{2} = 1.5
$$
- Ordered list:
$$
\frac{3}{8} (0.375), \frac{3}{2} (1.5), 2.2, 2.4
$$

15. $ \frac{1}{4}, \frac{1}{2}, 100, -\infty $
- Compare the numbers:
- $ -\infty $ is the smallest number.
- Convert fractions to decimals:
$$
\frac{1}{4} = 0.25, \quad \frac{1}{2} = 0.5
$$
- Ordered list:
$$
-\infty, \frac{1}{4} (0.25), \frac{1}{2} (0.5), 100
$$

16. $ 1.55, \frac{7}{5}, 1.75 $
- Convert all numbers to decimals:
$$
\frac{7}{5} = 1.4
$$
- Ordered list:
$$
\frac{7}{5} (1.4), 1.55, 1.75
$$

17. $ -0.6, -0.34, \frac{7}{8} $
- Convert all numbers to decimals:
$$
\frac{7}{8} = 0.875
$$
- Ordered list:
$$
-0.6, -0.34, \frac{7}{8} (0.875)
$$

---

Final Answers:

1. $ x = \_\_\_\_\_ $ (Incomplete)
2. $ \frac{7}{8} = 0.875 $
3. $ 3.3 = \frac{33}{10} $
4. $ \frac{4}{9} = 0.\overline{4} $
5. $ 4 = \frac{4}{1} $
6. $ \frac{6}{7} \approx 0.8571 $

Matched Fractions:
7. $ \frac{1}{4} = 0.25 $
8. $ \frac{3}{10} = 0.3 $
9. $ \frac{3}{2} = 1.5 $ (Not in the box)
10. $ \frac{5}{4} = 1.25 $
11. $ \frac{13}{10} = 1.3 $
12. $ \frac{8}{5} = 1.6 $

Ordered Lists:
13. $ -0.53, 0, 0.52, 0.555 $
14. $ \frac{3}{8}, \frac{3}{2}, 2.2, 2.4 $
15. $ -\infty, \frac{1}{4}, \frac{1}{2}, 100 $
16. $ \frac{7}{5}, 1.55, 1.75 $
17. $ -0.6, -0.34, \frac{7}{8} $

$$
\boxed{
\begin{aligned}
&2. \, 0.875 \\
&3. \, \frac{33}{10} \\
&4. \, 0.\overline{4} \\
&5. \, \frac{4}{1} \\
&6. \, 0.8571 \\
&7. \, 0.25 \\
&8. \, 0.3 \\
&9. \, 1.5 \\
&10. \, 1.25 \\
&11. \, 1.3 \\
&12. \, 1.6 \\
&13. \, -0.53, 0, 0.52, 0.555 \\
&14. \, \frac{3}{8}, \frac{3}{2}, 2.2, 2.4 \\
&15. \, -\infty, \frac{1}{4}, \frac{1}{2}, 100 \\
&16. \, \frac{7}{5}, 1.55, 1.75 \\
&17. \, -0.6, -0.34, \frac{7}{8}
\end{aligned}
}
$$