Eighth Grade Compare and Order Rational and Irrational Numbers ... - Free Printable

Let's solve the problems on both pages of the worksheet titled "Compare and Order Rational and Irrational Numbers." We'll go step by step, comparing each pair and then ordering the sets in the final questions.

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✔ Page 1: Compare Rational and Irrational Numbers

We are to use `<`, `>`, or `=` to compare each pair.

#### 1. $\frac{3}{5} \quad \boxed{} \quad 0.65$

- $\frac{3}{5} = 0.6$
- $0.6 < 0.65$
- So: $\frac{3}{5} < 0.65$

✔ Answer: $<$

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#### 2. $\sqrt{13} \quad \boxed{} \quad -3.91$

- $\sqrt{13} \approx 3.605$ (positive)
- $-3.91$ is negative
- Positive > Negative
- So: $\sqrt{13} > -3.91$

✔ Answer: $>$

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#### 3. $\frac{8}{9} \quad \boxed{} \quad \sqrt{721}$

- $\frac{8}{9} \approx 0.888...$
- $\sqrt{721} \approx ?$
- $\sqrt{729} = 27$, so $\sqrt{721} < 27$
- Try $26^2 = 676$, $27^2 = 729$, so $\sqrt{721} \approx 26.85$
- Clearly, $0.888 < 26.85$
- So: $\frac{8}{9} < \sqrt{721}$

✔ Answer: $<$

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#### 4. $-\frac{1}{4} \quad \boxed{} \quad -0.25$

- $-\frac{1}{4} = -0.25$
- So: $-\frac{1}{4} = -0.25$

✔ Answer: $=$

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#### 5. $\sqrt{36} \quad \boxed{} \quad \frac{61}{10}$

- $\sqrt{36} = 6$
- $\frac{61}{10} = 6.1$
- $6 < 6.1$
- So: $\sqrt{36} < \frac{61}{10}$

✔ Answer: $<$

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#### 6. $0.07 \quad \boxed{} \quad 0.007$

- $0.07 = 0.070$, $0.007 = 0.007$
- $0.070 > 0.007$
- So: $0.07 > 0.007$

✔ Answer: $>$

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#### 7. $\frac{3}{100} \quad \boxed{} \quad \sqrt{2}$

- $\frac{3}{100} = 0.03$
- $\sqrt{2} \approx 1.414$
- $0.03 < 1.414$
- So: $\frac{3}{100} < \sqrt{2}$

✔ Answer: $<$

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#### 8. $7.56 \quad \boxed{} \quad \sqrt{61}$

- $\sqrt{61} \approx ?$
- $7^2 = 49$, $8^2 = 64$, so $\sqrt{61} \approx 7.81$
- $7.56 < 7.81$
- So: $7.56 < \sqrt{61}$

✔ Answer: $<$

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#### 9. $\sqrt{86} \quad \boxed{} \quad -9.3$

- $\sqrt{86} \approx 9.27$ (since $9^2 = 81$, $10^2 = 100$)
- $-9.3$ is negative
- Positive > Negative
- So: $\sqrt{86} > -9.3$

✔ Answer: $>$

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✔ Page 2: Ordering Problems

Now, order the numbers from least to greatest.

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#### 11. $6.6,\ \pi,\ \frac{13}{2},\ 6.002,\ \sqrt{37}$

First, convert all to decimal approximations:

- $6.6$ → $6.6$
- $\pi \approx 3.1416$
- $\frac{13}{2} = 6.5$
- $6.002$ → $6.002$
- $\sqrt{37} \approx ?$
- $6^2 = 36$, $7^2 = 49$, so $\sqrt{37} \approx 6.082$

Now list them with approx values:

| Number | Approx |
|--------|--------|
| $\pi$ | 3.1416 |
| $6.002$ | 6.002 |
| $\sqrt{37}$ | 6.082 |
| $\frac{13}{2}$ | 6.5 |
| $6.6$ | 6.6 |

So, order from least to greatest:

$$
\pi,\ 6.002,\ \sqrt{37},\ \frac{13}{2},\ 6.6
$$

✔ Answer: $\pi,\ 6.002,\ \sqrt{37},\ \frac{13}{2},\ 6.6$

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#### 12. $\frac{27}{4},\ -\pi,\ -2.8,\ \sqrt{8}$

Convert to decimals:

- $\frac{27}{4} = 6.75$
- $-\pi \approx -3.1416$
- $-2.8$ → $-2.8$
- $\sqrt{8} = \sqrt{4 \cdot 2} = 2\sqrt{2} \approx 2 \times 1.414 = 2.828$

List with approx:

| Number | Approx |
|--------|--------|
| $-\pi$ | -3.1416 |
| $-2.8$ | -2.8 |
| $\sqrt{8}$ | 2.828 |
| $\frac{27}{4}$ | 6.75 |

Order from least to greatest:

$$
-\pi,\ -2.8,\ \sqrt{8},\ \frac{27}{4}
$$

✔ Answer: $-\pi,\ -2.8,\ \sqrt{8},\ \frac{27}{4}$

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✔ Final Answers Summary:

#### Comparison Section (Page 1):

1. $\frac{3}{5} < 0.65$
2. $\sqrt{13} > -3.91$
3. $\frac{8}{9} < \sqrt{721}$
4. $-\frac{1}{4} = -0.25$
5. $\sqrt{36} < \frac{61}{10}$
6. $0.07 > 0.007$
7. $\frac{3}{100} < \sqrt{2}$
8. $7.56 < \sqrt{61}$
9. $\sqrt{86} > -9.3$

#### Ordering Section (Page 2):

11. $\pi,\ 6.002,\ \sqrt{37},\ \frac{13}{2},\ 6.6$
12. $-\pi,\ -2.8,\ \sqrt{8},\ \frac{27}{4}$

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