The number of all values of n, (whre `n` is a whole number ) for which the equation ` (x-6)/(n -10) =n /x` has no solution.

Answer

To solve the problem, we need to determine the number of whole number values of \( n \) for which the equation

\[ \frac{x-6}{n-10} = \frac{n}{x} \]

has no solution.

### Step-by-step Solution:

1. **Cross Multiply the Equation**:
Start by cross-multiplying the given equation:
\[ (x - 6) \cdot x = n \cdot (n - 10) \]
This simplifies to:
\[ x^2 - 6x = n^2 - 10n \]

2. **Rearranging the Equation**:
Rearranging gives us a standard quadratic equation in \( x \):
\[ x^2 - 6x - (n^2 - 10n) = 0 \]
or
\[ x^2 - 6x + (10n - n^2) = 0 \]

3. **Identifying Coefficients**:
Here, the coefficients are:
- \( a = 1 \)
- \( b = -6 \)
- \( c = 10n - n^2 \)

4. **Finding the Discriminant**:
For the quadratic equation to have no solution, the discriminant \( D \) must be less than zero:
\[ D = b^2 - 4ac < 0 \]
Substituting the values:
\[ D = (-6)^2 - 4(1)(10n - n^2) < 0 \]
This simplifies to:
\[ 36 - 4(10n - n^2) < 0 \]

5. **Simplifying the Inequality**:
Expanding the inequality:
\[ 36 - 40n + 4n^2 < 0 \]
Rearranging gives:
\[ 4n^2 - 40n + 36 < 0 \]
Dividing through by 4 simplifies to:
\[ n^2 - 10n + 9 < 0 \]

6. **Factoring the Quadratic**:
Factoring the quadratic:
\[ (n - 1)(n - 9) < 0 \]

7. **Finding the Roots**:
The roots of the equation are \( n = 1 \) and \( n = 9 \).

8. **Analyzing the Sign**:
To find where the product is negative, we analyze the intervals:
- For \( n < 1 \): both factors are negative, product is positive.
- For \( 1 < n < 9 \): one factor is positive and one is negative, product is negative.
- For \( n > 9 \): both factors are positive, product is positive.

Thus, \( (n - 1)(n - 9) < 0 \) holds true for:
\[ 1 < n < 9 \]

9. **Identifying Whole Number Solutions**:
The whole numbers in the interval \( (1, 9) \) are:
\[ 2, 3, 4, 5, 6, 7, 8 \]
This gives us a total of 7 values.

### Final Answer:
The total number of whole number values of \( n \) for which the equation has no solution is **7**.