Let's go through these questions one by one.

1. Volume of a Cube with Surface Area 294

The surface area of a cube is given by the formula:

$\text{Surface Area} = 6s^2$

where $s$ is the side length of the cube. To find the side length, we solve:

$6s^2 = 294 \quad \Rightarrow \quad s^2 = \frac{294}{6} = 49 \quad \Rightarrow \quad s = \sqrt{49} = 7$

Now, the volume $V$ of the cube is:

$V = s^3 = 7^3 = 343$

So, the correct answer is (D) 343.

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2. Ratio of Boys to Girls in the Classroom

Let the total number of people in the class be $x$. The ratio of boys to girls is $2:3$, meaning the boys represent $\frac{2}{5}$ of the total class, and girls represent $\frac{3}{5}$ of the total class. According to the problem, boys are 5 more than one-third of the class. We set up the equation:

$\frac{2}{5}x = \frac{1}{3}x + 5$

Multiply through by 15 to eliminate the fractions:

$6x = 5x + 75$

Solving for $x$:

$x = 75$

So, the total number of people in the class is (C) 75.

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3. Average of 530, 560, 517, 513, and 590

The average is the sum of the numbers divided by the count of numbers. First, sum the values:

$530 + 560 + 517 + 513 + 590 = 2710$

Now, divide by the number of terms (5):

$\text{Average} = \frac{2710}{5} = 542$

So, the average is 542, but this option doesn't seem listed. Let me know if there's a different set of choices to check!

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4. Solve the inequality $p + 1 < 3p + 5$

First, subtract $p$ from both sides:

$1 < 2p + 5$

Now subtract 5 from both sides:

$-4 < 2p$

Divide by 2:

$p > -2$

So the correct answer is (B) p > -2.

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5. Ratio of Side Lengths of an Equilateral Triangle and a Square

Let the side length of the equilateral triangle be $s_{\text{triangle}}$ and the side length of the square be $s_{\text{square}}$. The perimeter of an equilateral triangle is $3s_{\text{triangle}}$ and the perimeter of a square is $4s_{\text{square}}$. Since the perimeters are equal:

$3s_{\text{triangle}} = 4s_{\text{square}}$

Solving for the ratio:

$\frac{s_{\text{triangle}}}{s_{\text{square}}} = \frac{4}{3}$

So, the correct ratio is (B) 4:3.

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Recap:

1. (D) 343
2. (C) 75
3. Average: 542 (no matching option)
4. (B) p > -2
5. (B) 4:3

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Do you want any further details or have questions on any part?

Relative Questions:

1. How do you find the surface area of a rectangular prism?
2. How does the ratio of girls to boys change if 5 more boys join the class?
3. What is the median of a dataset, and how does it differ from the average?
4. Can inequalities have more than one solution?
5. How does the perimeter of a polygon affect its area?

Tip: When dealing with ratios, always simplify or cross-multiply to make comparisons easier.