What is the volume of a cube whose surface area is 294? (A) 125 (B) 216
C) 294 (D) 343
4. The ratio of boys to girls in a certain classroom was 2 : 3. If boys represented five more than one-third of the class, how many people were in the classroom?
 (A) 30 (C) 75
5. What is the average of 530, 560, 517, 513 and 590? (A) 5210
(C) 529 +559 +516 +512 +589 6. Ifp+1<3p+5,then:
(A) p<-2
(C) p=0
(B) 45 (D) 60
(B) 5209 (D) 5205
(B) p>-2
(D) p>2
  7. If an equilateral triangle and a square have the same perimeter, what is the ratio of the lengths of the sides of the equilateral triangle to the lengths of the sides of the square?
(A) 3 : 4 (C) 1 : 2
(B) 4 : 3 (D) 1 : 3

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**.

---

### 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**.

---

### 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!

---

### 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**.

---

### 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**.

---

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

---

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.

What is the volume of a cube whose surface area is 294? (A) 125 (B) 216 (C) 294 (D) 343
4. The ratio of boys to girls in a certain classroom was 2 : 3. If boys represented five more than one-third of the class, how many people were in the classroom? (A) 30 (B) 45 (C) 75 (D) 60
5. What is the average of 530, 560, 517, 513 and 590? (A) 5210 (B) 5209 (C) 529 + 559 + 516 + 512 + 589 (D) 5205
6. If p + 1 < 3p + 5, then: (A) p < -2 (B) p > -2 (C) p = 0 (D) p > 2
7. If an equilateral triangle and a square have the same perimeter, what is the ratio of the lengths of the sides of the equilateral triangle to the lengths of the sides of the square? (A) 3 : 4 (B) 4 : 3 (C) 1 : 2 (D) 1 : 3

Solve a variety of math problems including finding the volume of a cube with surface area 294, determining the ratio of boys to girls in a classroom, calculating the average of a series of numbers, solving an inequality, and comparing the perimeters of an equilateral triangle and a square. Suitable for students in Grades 6-8.

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