Ordering and Comparing Rational Numbers - Free Printable

Let's solve this step-by-step. The task is to compare rational numbers using the symbols >, <, or =.

We'll go through each problem, simplify where needed (especially with absolute values and fractions), and compare the two sides.

---

🔷 Key Concepts:

- Absolute value: $|a|$ means the distance from 0 on the number line, so it’s always non-negative.
- Example: $|-3| = 3$, $|2.5| = 2.5$
- Fractions to decimals:
- $ \frac{1}{4} = 0.25 $, $ \frac{1}{2} = 0.5 $, $ \frac{3}{4} = 0.75 $
- $ \frac{3}{2} = 1.5 $, $ \frac{6}{2} = 3 $, etc.
- Mixed numbers:
- $ 1\frac{1}{4} = 1.25 $, $ 3\frac{1}{2} = 3.5 $, etc.

---

## ✔ Part 1: Compare Rational Numbers

1) $ |-2| > 1\frac{1}{4} $

- $ |-2| = 2 $
- $ 1\frac{1}{4} = 1.25 $
- $ 2 > 1.25 $ → ✔️ Given as correct.

---

2) $ \frac{1}{4} \quad ? \quad 0.5 $

- $ \frac{1}{4} = 0.25 $
- $ 0.25 < 0.5 $ → <

✔ Answer: $ \frac{1}{4} < 0.5 $

---

3) $ 1 \quad ? \quad \frac{3}{3} $

- $ \frac{3}{3} = 1 $
- $ 1 = 1 $ → =

✔ Answer: $ 1 = \frac{3}{3} $

---

4) $ |-3.2| \quad ? \quad \frac{3}{2} $

- $ |-3.2| = 3.2 $
- $ \frac{3}{2} = 1.5 $
- $ 3.2 > 1.5 $ → >

✔ Answer: $ |-3.2| > \frac{3}{2} $

---

5) $ |-3| \quad ? \quad |-5| $

- $ |-3| = 3 $
- $ |-5| = 5 $
- $ 3 < 5 $ → <

✔ Answer: $ |-3| < |-5| $

---

6) $ 3\frac{1}{4} \quad ? \quad |-4| $

- $ 3\frac{1}{4} = 3.25 $
- $ |-4| = 4 $
- $ 3.25 < 4 $ → <

✔ Answer: $ 3\frac{1}{4} < |-4| $

---

7) $ |-7| \quad ? \quad 6\frac{1}{4} $

- $ |-7| = 7 $
- $ 6\frac{1}{4} = 6.25 $
- $ 7 > 6.25 $ → >

✔ Answer: $ |-7| > 6\frac{1}{4} $

---

8) $ -5 \quad ? \quad 3\frac{1}{2} $

- $ -5 $ is negative
- $ 3\frac{1}{2} = 3.5 $ is positive
- Negative < Positive → <

✔ Answer: $ -5 < 3\frac{1}{2} $

---

9) $ 1.5 \quad ? \quad 1\frac{1}{2} $

- $ 1\frac{1}{2} = 1.5 $
- So $ 1.5 = 1.5 $ → =

✔ Answer: $ 1.5 = 1\frac{1}{2} $

---

10) $ |-2.6| \quad ? \quad 3 $

- $ |-2.6| = 2.6 $
- $ 2.6 < 3 $ → <

✔ Answer: $ |-2.6| < 3 $

---

11) $ |-\frac{1}{4}| \quad ? \quad \frac{3}{4} $

- $ |-\frac{1}{4}| = \frac{1}{4} = 0.25 $
- $ \frac{3}{4} = 0.75 $
- $ 0.25 < 0.75 $ → <

✔ Answer: $ |-\frac{1}{4}| < \frac{3}{4} $

---

12) $ |-12| \quad ? \quad |-9| $

- $ |-12| = 12 $
- $ |-9| = 9 $
- $ 12 > 9 $ → >

✔ Answer: $ |-12| > |-9| $

---

13) $ 0.37 \quad ? \quad 0.4 $

- $ 0.37 < 0.4 $ → <

✔ Answer: $ 0.37 < 0.4 $

---

14) $ |-0.6| \quad ? \quad \frac{1}{2} $

- $ |-0.6| = 0.6 $
- $ \frac{1}{2} = 0.5 $
- $ 0.6 > 0.5 $ → >

✔ Answer: $ |-0.6| > \frac{1}{2} $

---

15) $ |-\frac{3}{4}| \quad ? \quad 0.75 $

- $ |-\frac{3}{4}| = \frac{3}{4} = 0.75 $
- $ 0.75 = 0.75 $ → =

✔ Answer: $ |-\frac{3}{4}| = 0.75 $

---

16) $ -\frac{3}{2} \quad ? \quad -2.4 $

- $ -\frac{3}{2} = -1.5 $
- $ -1.5 > -2.4 $ because -1.5 is closer to zero → >

✔ Answer: $ -\frac{3}{2} > -2.4 $

---

17) $ -0.75 \quad ? \quad -0.6 $

- Both negative
- $ -0.75 < -0.6 $ (because -0.75 is farther left)
→ <

✔ Answer: $ -0.75 < -0.6 $

---

18) $ |-0.38| \quad ? \quad |-0.5| $

- $ |-0.38| = 0.38 $
- $ |-0.5| = 0.5 $
- $ 0.38 < 0.5 $ → <

✔ Answer: $ |-0.38| < |-0.5| $

---

19) $ |-\frac{1}{2}| \quad ? \quad 0.27 $

- $ |-\frac{1}{2}| = 0.5 $
- $ 0.5 > 0.27 $ → >

✔ Answer: $ |-\frac{1}{2}| > 0.27 $

---

20) $ -0.8 \quad ? \quad -\frac{8}{10} $

- $ -\frac{8}{10} = -0.8 $
- So $ -0.8 = -0.8 $ → =

✔ Answer: $ -0.8 = -\frac{8}{10} $

---

21) $ |2.6| \quad ? \quad |-3.2| $

- $ |2.6| = 2.6 $
- $ |-3.2| = 3.2 $
- $ 2.6 < 3.2 $ → <

✔ Answer: $ |2.6| < |-3.2| $

---

22) $ |-0.6| \quad ? \quad 1\frac{1}{4} $

- $ |-0.6| = 0.6 $
- $ 1\frac{1}{4} = 1.25 $
- $ 0.6 < 1.25 $ → <

✔ Answer: $ |-0.6| < 1\frac{1}{4} $

---

23) $ \frac{3}{2} \quad ? \quad |-1\frac{1}{2}| $

- $ \frac{3}{2} = 1.5 $
- $ |-1\frac{1}{2}| = | -1.5 | = 1.5 $
- $ 1.5 = 1.5 $ → =

✔ Answer: $ \frac{3}{2} = |-1\frac{1}{2}| $

---

24) $ -2.5 \quad ? \quad -3 $

- Both negative
- $ -2.5 > -3 $ (closer to zero) → >

✔ Answer: $ -2.5 > -3 $

---

25) $ |-0.65| \quad ? \quad |-0.8| $

- $ |-0.65| = 0.65 $
- $ |-0.8| = 0.8 $
- $ 0.65 < 0.8 $ → <

✔ Answer: $ |-0.65| < |-0.8| $

---

26) $ -\frac{1}{4} \quad ? \quad -\frac{1}{2} $

- $ -\frac{1}{4} = -0.25 $
- $ -\frac{1}{2} = -0.5 $
- $ -0.25 > -0.5 $ → >

✔ Answer: $ -\frac{1}{4} > -\frac{1}{2} $

---

27) $ |-\frac{1}{3}| \quad ? \quad |-\frac{1}{5}| $

- $ |-\frac{1}{3}| = \frac{1}{3} \approx 0.333 $
- $ |-\frac{1}{5}| = \frac{1}{5} = 0.2 $
- $ 0.333 > 0.2 $ → >

✔ Answer: $ |-\frac{1}{3}| > |-\frac{1}{5}| $

---

28) $ \frac{6}{2} \quad ? \quad |-3| $

- $ \frac{6}{2} = 3 $
- $ |-3| = 3 $
- $ 3 = 3 $ → =

✔ Answer: $ \frac{6}{2} = |-3| $

---

29) $ |-0.62| \quad ? \quad -0.8 $

- $ |-0.62| = 0.62 $
- $ -0.8 $ is negative
- Positive > Negative → >

✔ Answer: $ |-0.62| > -0.8 $

---

30) $ |-0.06| \quad ? \quad |-0.1| $

- $ |-0.06| = 0.06 $
- $ |-0.1| = 0.1 $
- $ 0.06 < 0.1 $ → <

✔ Answer: $ |-0.06| < |-0.1| $

---

## ✔ Part 2: Compare these amounts

Now we evaluate expressions on both sides.

31) $ |3 - 5| \quad ? \quad 2^2 $

- $ |3 - 5| = |-2| = 2 $
- $ 2^2 = 4 $
- $ 2 < 4 $ → <

✔ Answer: $ |3 - 5| < 2^2 $

---

32) $ |-\frac{1}{4}| \quad ? \quad 1 - \frac{1}{4} $

- $ |-\frac{1}{4}| = \frac{1}{4} = 0.25 $
- $ 1 - \frac{1}{4} = \frac{3}{4} = 0.75 $
- $ 0.25 < 0.75 $ → <

✔ Answer: $ |-\frac{1}{4}| < 1 - \frac{1}{4} $

---

33) $ \frac{3}{8} + \frac{5}{8} \quad ? \quad |6 - 7| $

- $ \frac{3}{8} + \frac{5}{8} = \frac{8}{8} = 1 $
- $ |6 - 7| = |-1| = 1 $
- $ 1 = 1 $ → =

✔ Answer: $ \frac{3}{8} + \frac{5}{8} = |6 - 7| $

---

34) $ 1.5 \quad ? \quad |2 - 4| $

- $ |2 - 4| = |-2| = 2 $
- $ 1.5 < 2 $ → <

✔ Answer: $ 1.5 < |2 - 4| $

---

35) $ 2 \times 0.8 \quad ? \quad \frac{1}{2} \times 3 $

- $ 2 \times 0.8 = 1.6 $
- $ \frac{1}{2} \times 3 = 1.5 $
- $ 1.6 > 1.5 $ → >

✔ Answer: $ 2 \times 0.8 > \frac{1}{2} \times 3 $

---

36) $ |2 - 7| \quad ? \quad |-2| \times 3 $

- $ |2 - 7| = |-5| = 5 $
- $ |-2| = 2 $, so $ 2 \times 3 = 6 $
- $ 5 < 6 $ → <

✔ Answer: $ |2 - 7| < |-2| \times 3 $

---

37) $ |4 - 11| \quad ? \quad |-5| \times |-1| $

- $ |4 - 11| = |-7| = 7 $
- $ |-5| = 5 $, $ |-1| = 1 $, so $ 5 \times 1 = 5 $
- $ 7 > 5 $ → >

✔ Answer: $ |4 - 11| > |-5| \times |-1| $

---

38) $ \frac{1}{4} \times 2 \quad ? \quad 1 \div 2 $

- $ \frac{1}{4} \times 2 = \frac{2}{4} = 0.5 $
- $ 1 \div 2 = 0.5 $
- $ 0.5 = 0.5 $ → =

✔ Answer: $ \frac{1}{4} \times 2 = 1 \div 2 $

---

39) $ \frac{1}{8} + \frac{3}{8} \quad ? \quad 1 - \frac{3}{8} $

- $ \frac{1}{8} + \frac{3}{8} = \frac{4}{8} = 0.5 $
- $ 1 - \frac{3}{8} = \frac{5}{8} = 0.625 $
- $ 0.5 < 0.625 $ → <

✔ Answer: $ \frac{1}{8} + \frac{3}{8} < 1 - \frac{3}{8} $

---

40) $ |5 - 16| \quad ? \quad 3^2 $

- $ |5 - 16| = |-11| = 11 $
- $ 3^2 = 9 $
- $ 11 > 9 $ → >

✔ Answer: $ |5 - 16| > 3^2 $

---

## ✔ Final Answers:

Part 1:

1) >
2) <
3) =
4) >
5) <
6) <
7) >
8) <
9) =
10) <
11) <
12) >
13) <
14) >
15) =
16) >
17) <
18) <
19) >
20) =
21) <
22) <
23) =
24) >
25) <
26) >
27) >
28) =
29) >
30) <

Part 2:

31) <
32) <
33) =
34) <
35) >
36) <
37) >
38) =
39) <
40) >

---

Let me know if you'd like this formatted into a printable answer sheet!