Eighth Grade Rational and Irrational Numbers Quiz - Twinkl - Free Printable

It looks like you've uploaded a "Rational and Irrational Numbers Quiz" with two pages of problems. I'll solve each problem step by step, explain the reasoning, and provide the correct answers.

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🔹 Page 1: Rational and Irrational Numbers Quiz

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#### Problem 1: Write each decimal below as a fraction in simplest form.

a. -2.498

To convert a decimal to a fraction:
- Write it as a fraction over 1000 (since there are 3 decimal places):
$$
-2.498 = -\frac{2498}{1000}
$$
- Simplify the fraction:
- Divide numerator and denominator by 2:
$$
\frac{2498 ÷ 2}{1000 ÷ 2} = \frac{1249}{500}
$$
- So, $-\frac{1249}{500}$ is the simplified form.

✔ Answer: $-\frac{1249}{500}$

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b. 0.12

- Write as: $\frac{12}{100}$
- Simplify: divide numerator and denominator by 4 → $\frac{3}{25}$

✔ Answer: $\frac{3}{25}$

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#### Problem 2: Write each fraction as a decimal.

a. $\frac{17}{2}$

- Divide: $17 ÷ 2 = 8.5$

✔ Answer: $8.5$

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b. $\frac{18}{100}$

- This is already a decimal: $0.18$

✔ Answer: $0.18$

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#### Problem 3: Determine if each number is rational or irrational. Place a checkmark in the corresponding box.

| Number | Irrational | Rational |
|--------------|------------|----------|
| $\frac{2\pi}{3}$ | ✔ | |
| $-0.77$ | | ✔ |
| $\sqrt{61}$ | ✔ | |
| $2.8$ | | ✔ |
| $\frac{\sqrt{90}}{25}$ | ✔ | |

Let’s go through each:

1. $\frac{2\pi}{3}$:
- $\pi$ is irrational → any non-zero multiple of $\pi$ is irrational → this is irrational.

2. $-0.77$:
- Terminating decimal → can be written as $-\frac{77}{100}$ → rational.

3. $\sqrt{61}$:
- 61 is not a perfect square → $\sqrt{61}$ is irrational.

4. $2.8$:
- Terminating decimal → $\frac{28}{10} = \frac{14}{5}$ → rational.

5. $\frac{\sqrt{90}}{25}$:
- $\sqrt{90} = \sqrt{9 \cdot 10} = 3\sqrt{10}$ → $\sqrt{10}$ is irrational → so entire expression is irrational.

✔ Final Table:

| Number | Irrational | Rational |
|------------------|------------|----------|
| $\frac{2\pi}{3}$ | ✔ | |
| $-0.77$ | | ✔ |
| $\sqrt{61}$ | ✔ | |
| $2.8$ | | ✔ |
| $\frac{\sqrt{90}}{25}$ | ✔ | |

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#### Problem 4: Circle all of the rational numbers.

List:
a. $4\pi$
b. $\frac{1}{3}$
c. $\sqrt{25}$
d. $-2.32875321...$
e. $\sqrt{2}$
f. $\sqrt{121}$

Now evaluate each:

- a. $4\pi$:
- $\pi$ is irrational → $4\pi$ is irrational

- b. $\frac{1}{3}$:
- Fraction → rational

- c. $\sqrt{25}$:
- $\sqrt{25} = 5$ → integer → rational

- d. $-2.32875321...$:
- The dots suggest it's non-repeating and non-terminating → likely irrational (unless specified repeating). Assuming non-repeating → irrational

- e. $\sqrt{2}$:
- Classic irrational number → irrational

- f. $\sqrt{121}$:
- $\sqrt{121} = 11$ → integer → rational

✔ Rational numbers: b, c, f

So circle: b, c, f

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🔹 Page 2: Same quiz, but appears to be a duplicate or second version?

Looking at the second page, it's very similar — possibly just a reformatted version. Let's confirm the answers match.

But since both pages have the same content (slightly reorganized), we’ve already solved everything.

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✔ Final Answers Summary

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#### Problem 1: Decimals to Fractions
- a. $-2.498 = -\frac{1249}{500}$
- b. $0.12 = \frac{3}{25}$

#### Problem 2: Fractions to Decimals
- a. $\frac{17}{2} = 8.5$
- b. $\frac{18}{100} = 0.18$

#### Problem 3: Rational or Irrational?
| Number | Irrational | Rational |
|--------------------|------------|----------|
| $\frac{2\pi}{3}$ | ✔ | |
| $-0.77$ | | ✔ |
| $\sqrt{61}$ | ✔ | |
| $2.8$ | | ✔ |
| $\frac{\sqrt{90}}{25}$ | ✔ | |

#### Problem 4: Circle Rational Numbers
- Rational: b ($\frac{1}{3}$), c ($\sqrt{25}$), f ($\sqrt{121}$)

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📌 Key Concepts Recap:

- Rational numbers: Can be written as fractions $\frac{p}{q}$ where $p, q$ are integers and $q ≠ 0$. Include terminating and repeating decimals.
- Irrational numbers: Cannot be written as simple fractions. Include non-repeating, non-terminating decimals and roots of non-perfect squares (like $\sqrt{2}, \pi$, etc.).

Let me know if you'd like this formatted as a printable answer key!