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

Options

A. `4:3`
B. `8:6`
C. `3:4`
D. `7:8`

Answer

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

### Step-by-Step Solution:

1. **Set Up the Ratios**:
We are given the ratios:
\[ m+n : m-n : mn = 7 : 1 : 60 \]
We can express these ratios in terms of a common variable \( k \):
\[ m+n = 7k, \quad m-n = k, \quad mn = 60k \]

2. **Express \( m \) and \( n \)**:
From the equations \( m+n = 7k \) and \( m-n = k \), we can solve for \( m \) and \( n \).
Adding the two equations:
\[ (m+n) + (m-n) = 7k + k \implies 2m = 8k \implies m = 4k \]
Subtracting the second equation from the first:
\[ (m+n) - (m-n) = 7k - k \implies 2n = 6k \implies n = 3k \]

3. **Calculate \( mn \)**:
Now, we can calculate \( mn \):
\[ mn = m \cdot n = (4k)(3k) = 12k^2 \]
We also know from the ratio that \( mn = 60k \). Setting these equal gives:
\[ 12k^2 = 60k \]
Dividing both sides by \( k \) (assuming \( k \neq 0 \)):
\[ 12k = 60 \implies k = 5 \]

4. **Find \( m \) and \( n \)**:
Now substituting \( k = 5 \) back into the expressions for \( m \) and \( n \):
\[ m = 4k = 4 \times 5 = 20 \]
\[ n = 3k = 3 \times 5 = 15 \]

5. **Calculate \( \frac{1}{m} : \frac{1}{n} \)**:
Now we need to find \( \frac{1}{m} : \frac{1}{n} \):
\[ \frac{1}{m} = \frac{1}{20}, \quad \frac{1}{n} = \frac{1}{15} \]
To find the ratio:
\[ \frac{1}{m} : \frac{1}{n} = \frac{1/20}{1/15} = \frac{15}{20} = \frac{3}{4} \]

### Final Answer:
\[ \frac{1}{m} : \frac{1}{n} = 3 : 4 \]