If `(7m+2n)/(7m-2n)=5/(3)` use properties of proportion to find
(i) `m:n`
(ii) `(m^(2)+n^(2))/(m^(2)-n^(2))`

Answer

To solve the problem, we start with the given equation:

\[ \frac{7m + 2n}{7m - 2n} = \frac{5}{3} \]

### Step 1: Cross-multiply the proportions
Cross-multiplying gives us:

\[ 3(7m + 2n) = 5(7m - 2n) \]

### Step 2: Expand both sides
Expanding both sides yields:

\[ 21m + 6n = 35m - 10n \]

### Step 3: Rearrange the equation
Rearranging the equation to bring all terms involving \(m\) on one side and all terms involving \(n\) on the other side gives:

\[ 21m - 35m = -10n - 6n \]

This simplifies to:

\[ -14m = -16n \]

### Step 4: Simplify the equation
Dividing both sides by -2 gives:

\[ 7m = 8n \]

### Step 5: Find the ratio \(m:n\)
From the equation \(7m = 8n\), we can express the ratio \(m:n\):

\[ \frac{m}{n} = \frac{8}{7} \]

Thus, the ratio \(m:n\) is:

\[ m:n = 8:7 \]

### Step 6: Calculate \(\frac{m^2 + n^2}{m^2 - n^2}\)
Using the ratio \(m:n = 8:7\), we can find \(m^2\) and \(n^2\):

\[ m^2 = (8k)^2 = 64k^2 \quad \text{and} \quad n^2 = (7k)^2 = 49k^2 \]

### Step 7: Substitute into the expression
Now substitute \(m^2\) and \(n^2\) into the expression:

\[ \frac{m^2 + n^2}{m^2 - n^2} = \frac{64k^2 + 49k^2}{64k^2 - 49k^2} \]

### Step 8: Simplify the expression
This simplifies to:

\[ \frac{113k^2}{15k^2} = \frac{113}{15} \]

### Final Answers
(i) The ratio \(m:n\) is \(8:7\).

(ii) The value of \(\frac{m^2 + n^2}{m^2 - n^2}\) is \(\frac{113}{15}\).

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