For any two numbers (m+n) : (m-n) : mn = 7 : 1: 60, then find the value of 1/m:1/n

Options

A. 0.16875
B. 0.3375
C. 0.12777777777778
D. 0.29722222222222

Answer

To solve the problem, we need to find the value of \( \frac{1}{m} : \frac{1}{n} \) given the ratios \( (m+n) : (m-n) : mn = 7 : 1 : 60 \).

### Step-by-Step Solution:

1. **Assign Variables Based on Ratios**:
- Let \( m+n = 7k \)
- Let \( m-n = 1k \)
- Let \( mn = 60k \)

2. **Use the Identity**:
We use the identity:
\[ (m+n)^2 = (m-n)^2 + 4mn \]
Substituting the values we assigned:
\[ (7k)^2 = (1k)^2 + 4(60k) \]

3. **Expand and Simplify**:
- Expanding both sides:
\[ 49k^2 = k^2 + 240k \]
- Rearranging gives:
\[ 49k^2 - k^2 - 240k = 0 \]
\[ 48k^2 - 240k = 0 \]

4. **Factor Out Common Terms**:
- Factoring out \( k \):
\[ k(48k - 240) = 0 \]
- This gives us two solutions: \( k = 0 \) or \( 48k - 240 = 0 \). We discard \( k = 0 \) since it does not make sense in this context.

5. **Solve for \( k \)**:
\[ 48k = 240 \implies k = \frac{240}{48} = 5 \]

6. **Find Values of \( m \) and \( n \)**:
- Substitute \( k \) back into the equations for \( m+n \) and \( m-n \):
\[ m+n = 7k = 7 \times 5 = 35 \]
\[ m-n = 1k = 1 \times 5 = 5 \]

7. **Set Up the System of Equations**:
- We now have the system:
\[ m+n = 35 \quad (1) \]
\[ m-n = 5 \quad (2) \]

8. **Add and Subtract the Equations**:
- Adding (1) and (2):
\[ 2m = 40 \implies m = 20 \]
- Subtracting (2) from (1):
\[ 2n = 30 \implies n = 15 \]

9. **Calculate \( \frac{1}{m} : \frac{1}{n} \)**:
- Now, we find \( \frac{1}{m} \) and \( \frac{1}{n} \):
\[ \frac{1}{m} = \frac{1}{20}, \quad \frac{1}{n} = \frac{1}{15} \]
- The ratio \( \frac{1}{m} : \frac{1}{n} = \frac{1/20}{1/15} = \frac{15}{20} = \frac{3}{4} \).

10. **Final Answer**:
\[ \frac{1}{m} : \frac{1}{n} = 3 : 4 \]