Quiz worksheet on rational and irrational numbers featuring multiple-choice questions, number line comparisons, and expressions involving square roots and fractions.

Let's solve each question one by one from the quiz titled "Rational & Irrational Numbers".

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Question 1:

The √42 falls between which two integers?

We need to find two consecutive integers such that:
$$
a < \sqrt{42} < b
$$

Find perfect squares near 42:
- $6^2 = 36$
- $7^2 = 49$

So,
$$
\sqrt{36} = 6,\quad \sqrt{49} = 7
\Rightarrow \sqrt{42} \text{ is between } 6 \text{ and } 7
$$

✔ Answer: A) 6 and 7

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Question 2:

This appears to be a table with ingredients and their amounts. The question asks:

> "Rewrite the amount of marshmallow that is magic (in grams) as a decimal."

From the table:
- Magic Marshmallows: 0.464646... (repeating)

This is a repeating decimal: $0.\overline{46}$

But the options are:
A) 1.57
B) 0.464646
C) 7.15
D) 0.46666

Since the value is 0.464646..., which matches Option B, even though it’s not written with a bar, it's likely meant to represent the repeating decimal.

Note: Option D is 0.4666..., which is different (that would be $0.4\overline{6}$).

✔ Answer: B) 0.464646

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Question 3:

Fill in the blank with <, >, or =:

$$
\frac{11}{4} \quad \boxed{?} \quad \frac{2}{3} \left( \sqrt[3]{27} \right)
$$

First, simplify both sides.

Left side:
$$
\frac{11}{4} = 2.75
$$

Right side:
- $\sqrt[3]{27} = 3$ (since $3^3 = 27$)
- Then: $\frac{2}{3} \times 3 = 2$

So:
$$
2.75 \quad ? \quad 2
\Rightarrow 2.75 > 2
$$

✔ So, fill in with >

Answer: >

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Question 4:

There is a number line with points labeled M, K, J, L at various positions.

We need to identify which point corresponds to a certain value — but the actual number is missing.

Wait, let's look carefully.

It says:
> “Label the point on the number line that represents -1.5.”

But the image shows a number line from -3 to 3, with tick marks every 0.5 units.

Let’s assume the task is to identify which letter corresponds to -1.5.

On the number line:
- -1.5 is halfway between -1 and -2.

Now check the labels:
- Let's suppose the points are placed as follows (from left to right):
- M → near -2.5
- K → near -1.5
- J → near 0
- L → near 1.5

So if K is at -1.5, then answer is K

✔ Answer: B) K

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Question 5:

Which of the following is true?

Let’s analyze each option:

#### Option A:
> $-\sqrt{64}$ is an irrational number because it can be expressed as a ratio of two integers.

But:
- $\sqrt{64} = 8$, so $-\sqrt{64} = -8$
- $-8$ is rational (it's $-8/1$), and can be expressed as a ratio of two integers.
- But the statement says it's irrational — this is false.

✘ Incorrect

#### Option B:
> $-\sqrt{64}$ is a rational number because it cannot be expressed as a ratio of two integers.

- Again, $-\sqrt{64} = -8$, which can be expressed as a ratio of two integers.
- But this option says it cannot — that’s false.

✘ Incorrect

#### Option C:
> $-\sqrt{169}$ is an irrational number because it cannot be expressed as a ratio of two integers.

- $\sqrt{169} = 13$, so $-\sqrt{169} = -13$
- $-13$ is rational ($-13/1$)
- So it is expressible as a ratio → so calling it irrational is wrong.

✘ Incorrect

#### Option D:
> $-\sqrt{169}$ is a rational number because it can be expressed as a ratio of two integers.

- Yes! $-\sqrt{169} = -13$, and $-13 = -13/1$ → rational
- And the reason is correct.

✔ Answer: D

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✔ Final Answers:

1. A – 6 and 7
2. B – 0.464646
3. >
4. B – K
5. D – $-\sqrt{169}$ is rational because it can be expressed as a ratio of two integers

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Let me know if you'd like explanations for any other questions!