Rational and Irrational Numbers Quiz worksheet with multiple-choice and fill-in-the-blank questions.

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.

---

🔹 Page 1: Rational and Irrational Numbers Quiz

---

#### 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: $-\frac{1249}{500}$

---

b. 0.12

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

✔ Answer: $\frac{3}{25}$

---

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

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

- Divide: $17 ÷ 2 = 8.5$

✔ Answer: $8.5$

---

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

- This is already a decimal: $0.18$

✔ Answer: $0.18$

---

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

---

#### 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 ellipsis (...) suggests a non-repeating, non-terminating decimal → likely irrational
- If it's truly non-repeating, it's irrational

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

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

✔ Rational numbers: b. $\frac{1}{3}$, c. $\sqrt{25}$, f. $\sqrt{121}$

So, circle: b, c, f

---

🔹 Page 2: Additional Problems

This appears to be a duplicate or continuation of the same quiz.

Let me confirm that all problems on both pages are covered.

But wait — Page 2 has:

> Problem 3: Same as above (already solved)

> Problem 4: Same as above (already solved)

So both pages seem to have overlapping content. Likely, one is a copy or version.

---

✔ Final Answers Summary

---

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

---

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

---

#### Problem 3: Rational vs 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. $\sqrt{121}$

---

📌 Key Concepts Explained:

- Rational Numbers: Can be written as a fraction $\frac{p}{q}$ where $p$ and $q$ are integers and $q \ne 0$. Includes terminating decimals, repeating decimals, and integers.
- Irrational Numbers: Cannot be expressed as a fraction. Non-repeating, non-terminating decimals. Examples: $\pi$, $\sqrt{2}$, $\sqrt{61}$, etc.

---

If you'd like, I can generate a clean printable version of the completed quiz! Let me know.