6. How many kg. of salt costing ₹ 6.00 a kg must be mixed with 16 kg of salt costing $$ ₹ 8.25 $$ per kg so that 25 percent may be gained by selling the mixture at $$ ₹ 9.00 \mathrm{a} \mathrm{kg} $$ ? (1) 14 kg (2) 12 kg (3) 10 kg (4) 15 kg 7. A room is 20 metres long and 15 metres broad. Find the length of the carpet 75 cm . wide which is required to cover the floor of the room. (1) 400 m (2) 300 m (3) 380 m (4) 440 m 8. How many spherical bullets can be made out of a lead cylinder 28 cm high and 6 cm radius, each bullet being 1.5 cm in diameter? (1) 1972 (2) 1279 (3) 1792 (4) Can't be determined 9. A cylinder, a hemisphere and a cone is described on the same base and within the same height. Find the ratio of their volumes in the same order. (1) $$ 2: 1: 3 $$ (2) $$ 1: 2: 3 $$ (3) $$ 3: 2: 1 $$ (4) $$ 3: 1: 2 $$ The number of boys and girls in a school are 576 and 480 respectively. Express the ratio of the number of boys to that of girls in the simplest form. (1) $$ \frac{7}{9} $$ (2) $$ \frac{6}{5} $$ (3) $$ \frac{1}{8} $$ (4) $$ \frac{4}{3} $$ 51. If the markup percentage of an article is $$ 50 \% $$ and discount percentage is also $$ 20 \% $$, then the profit percentage will be $$ \qquad $$ (1) $$ 10 \% $$ (2) $$ 25 \% $$ (3) $$ 30 \% $$ (4) $$ 20 \% $$ 62. Find the probability of getting a head in a throw of a coin. (1) $$ \frac{1}{2} $$ (2) 1 (3) 2 (4) None of these 63. One card is drawn from a pack of 52 cards, each of the 52 cards being equally likely to be drawn. Find the probability that the card drawn is black. (1) $$ \frac{1}{2} $$ (2) $$ \frac{1}{4} $$ (3) $$ \frac{8}{13} $$ (4) Can't be determined

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Let's tackle each of your questions one by one:

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**6. How many kg. of salt costing ₹6.00 a kg must be mixed with 16 kg of salt costing ₹8.25 per kg so that 25 percent may be gained by selling the mixture at ₹9.00 per kg?**

**Options:**
1. 14 kg
2. 12 kg
3. 10 kg
4. 15 kg

**Solution:**

- **Let** $$ x $$ **be the amount of ₹6 salt needed.**
- **Total Cost Price (CP):** $$ 6x + 8.25 \times 16 = 6x + 132 $$ ₹
- **Total Weight:** $$ x + 16 $$ kg
- **Desired Selling Price with 25% profit:** $$ 9(x + 16) $$ ₹

Set up the equation for 25% profit:
$$
9(x + 16) = 1.25(6x + 132)
$$
$$
9x + 144 = 7.5x + 165
$$
$$
1.5x = 21
$$
$$
x = 14 \text{ kg}
$$

**Answer:** **(1) 14 kg**

---

**7. A room is 20 metres long and 15 metres broad. Find the length of the carpet 75 cm wide which is required to cover the floor of the room.**

**Options:**
1. 400 m
2. 300 m
3. 380 m
4. 440 m

**Solution:**

- **Area of the room:** $$ 20 \times 15 = 300 $$ m²
- **Width of carpet:** 75 cm = 0.75 m
- **Length of carpet needed:** $$ \frac{300}{0.75} = 400 $$ m

**Answer:** **(1) 400 m**

---

**8. How many spherical bullets can be made out of a lead cylinder 28 cm high and 6 cm radius, each bullet being 1.5 cm in diameter?**

**Options:**
1. 1972
2. 1279
3. 1792
4. Can't be determined

**Solution:**

- **Volume of the cylinder:**
$$
V_{\text{cyl}} = \pi r^2 h = \pi \times 6^2 \times 28 = 1008\pi \text{ cm}^3
$$
- **Volume of one bullet (sphere):**
$$
V_{\text{sphere}} = \frac{4}{3} \pi \left(\frac{1.5}{2}\right)^3 = \frac{4}{3} \pi \times 0.75^3 = 1.6875\pi \text{ cm}^3
$$
- **Number of bullets:**
$$
\frac{1008\pi}{1.6875\pi} \approx 597 \text{ bullets}
$$

**However, none of the provided options match this calculation.**

**Answer:** **(4) Can't be determined**

---

**9. A cylinder, a hemisphere, and a cone are described on the same base and within the same height. Find the ratio of their volumes in the same order.**

**Options:**
1. $$ 2 : 1 : 3 $$
2. $$ 1 : 2 : 3 $$
3. $$ 3 : 2 : 1 $$
4. $$ 3 : 1 : 2 $$

**Solution:**

Assuming all have the same base radius $$ r $$ and height $$ h $$:

- **Volume of Cylinder:**
$$
V_{\text{cyl}} = \pi r^2 h
$$
- **Volume of Hemisphere:**
$$
V_{\text{hemi}} = \frac{2}{3} \pi r^3
$$
- **Volume of Cone:**
$$
V_{\text{cone}} = \frac{1}{3} \pi r^2 h
$$

Given that all shapes share the same base and height, and assuming $$ r = h $$ (for the hemisphere to have height $$ h $$):
$$
V_{\text{cyl}} : V_{\text{hemi}} : V_{\text{cone}} = 3 : 2 : 1
$$

**Answer:** **(3) $$ 3 : 2 : 1 $$**

---

**The number of boys and girls in a school are 576 and 480 respectively. Express the ratio of the number of boys to that of girls in the simplest form.**

**Options:**
1. $$ \frac{7}{9} $$
2. $$ \frac{6}{5} $$
3. $$ \frac{1}{8} $$
4. $$ \frac{4}{3} $$

**Solution:**

- **Simplify the ratio 576:480**
$$
\frac{576}{480} = \frac{6}{5}
$$

**Answer:** **(2) $$ \frac{6}{5} $$**

---

**51. If the markup percentage of an article is 50% and discount percentage is also 20%, then the profit percentage will be**

**Options:**
1. 10%
2. 25%
3. 30%
4. 20%

**Solution:**

- **Assume Cost Price (CP) = ₹100**
- **Marked Price (MP) = CP + 50% of CP = 150**
- **Selling Price (SP) after 20% discount = 150 - 20% of 150 = 120**
- **Profit = SP - CP = 120 - 100 = 20%**

**Answer:** **(4) 20%**

---

**62. Find the probability of getting a head in a throw of a coin.**

**Options:**
1. $$ \frac{1}{2} $$
2. 1
3. 2
4. None of these

**Solution:**

A fair coin has two outcomes: Head or Tail.

$$
P(\text{Head}) = \frac{1}{2}
$$

**Answer:** **(1) $$ \frac{1}{2} $$**

---

**63. One card is drawn from a pack of 52 cards, each of the 52 cards being equally likely to be drawn. Find the probability that the card drawn is black.**

**Options:**
1. $$ \frac{1}{2} $$
2. $$ \frac{1}{4} $$
3. $$ \frac{8}{13} $$
4. Can't be determined

**Solution:**

- **In a standard deck, there are 26 black cards and 26 red cards.**
$$
P(\text{Black}) = \frac{26}{52} = \frac{1}{2}
$$

**Answer:** **(1) $$ \frac{1}{2} $$**

---

If you have any further questions or need more detailed explanations, feel free to ask!
**6.** 14 kg **7.** 400 m **8.** Can't be determined **9.** 3 : 2 : 1 **10.** 6 : 5 **51.** 20% **62.** 1/2 **63.** 1/2

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