Solve the following:
(i) Three consecutive natural numbers are such that the square of the middle number exceeds the difference of the squares of the other two by 60. Find the numbers.
(ii) A train travels 360 km at a uniform speed. If the speed had been 5 km/h more, it would have taken 1 hour less for the same journey. Find the speed of the train.

Options

A. `{:(i,ii),("10, 11, 12",45km//h):}`
B. `{:(i,ii),("9, 10, 11",40 km//h):}`
C. `{:(i,ii),("8, 10, 12",30 km//h):}`
D. `{:(i,ii),("9, 10, 11",35 km//h):}`

Answer

### Step-by-Step Solution

#### Part (i)

1. **Define the Consecutive Natural Numbers**:
Let the three consecutive natural numbers be \( x - 1 \), \( x \), and \( x + 1 \), where \( x \) is the middle number.

2. **Set Up the Equation**:
According to the problem, the square of the middle number exceeds the difference of the squares of the other two by 60. This can be expressed mathematically as:
\[ x^2 = (x + 1)^2 - (x - 1)^2 + 60 \]

3. **Expand the Squares**:
Expanding the squares gives:
\[ (x + 1)^2 = x^2 + 2x + 1 \]
\[ (x - 1)^2 = x^2 - 2x + 1 \]
Therefore, the equation becomes:
\[ x^2 = (x^2 + 2x + 1) - (x^2 - 2x + 1) + 60 \]

4. **Simplify the Equation**:
Simplifying the right side:
\[ x^2 = (x^2 + 2x + 1 - x^2 + 2x - 1) + 60 \]
This simplifies to:
\[ x^2 = 4x + 60 \]

5. **Rearrange the Equation**:
Rearranging gives:
\[ x^2 - 4x - 60 = 0 \]

6. **Factor the Quadratic Equation**:
We need to factor the quadratic:
\[ (x - 10)(x + 6) = 0 \]
This gives us:
\[ x = 10 \quad \text{or} \quad x = -6 \]

7. **Determine Valid Natural Numbers**:
Since we are looking for natural numbers, we take \( x = 10 \). Thus, the three consecutive natural numbers are:
\[ 9, 10, 11 \]

#### Part (ii)

1. **Define Variables**:
Let the speed of the train be \( x \) km/h. The distance is given as 360 km.

2. **Set Up the Time Equation**:
The time taken at speed \( x \) is:
\[ \text{Time} = \frac{360}{x} \]
If the speed increases by 5 km/h, the new speed is \( x + 5 \) km/h, and the time taken becomes:
\[ \text{New Time} = \frac{360}{x + 5} \]

3. **Set Up the Equation Based on Time**:
According to the problem, the new time is 1 hour less than the original time:
\[ \frac{360}{x} - 1 = \frac{360}{x + 5} \]

4. **Clear the Fractions**:
Multiply through by \( x(x + 5) \) to eliminate the denominators:
\[ 360(x + 5) - x(x + 5) = 360x \]

5. **Expand and Rearrange**:
Expanding gives:
\[ 360x + 1800 - x^2 - 5x = 360x \]
Simplifying leads to:
\[ -x^2 - 5x + 1800 = 0 \]
Rearranging gives:
\[ x^2 + 5x - 1800 = 0 \]

6. **Factor the Quadratic Equation**:
We need to factor the quadratic:
\[ (x + 45)(x - 40) = 0 \]
This gives us:
\[ x = 40 \quad \text{or} \quad x = -45 \]

7. **Determine Valid Speed**:
Since speed cannot be negative, we take \( x = 40 \). Thus, the speed of the train is:
\[ 40 \text{ km/h} \]

### Final Answers
- The three consecutive natural numbers are **9, 10, 11**.
- The speed of the train is **40 km/h**.