If `(x : y) = (3 : 4)` then `(7x + 3y) : (7x - 3y)` = ?

Answer

To solve the problem where \((x : y) = (3 : 4)\) and we need to find \((7x + 3y) : (7x - 3y)\), we can follow these steps:

### Step 1: Express \(x\) and \(y\) in terms of a common variable
Since the ratio \(x : y = 3 : 4\), we can express \(x\) and \(y\) in terms of a variable \(k\):
- Let \(x = 3k\)
- Let \(y = 4k\)

### Step 2: Substitute \(x\) and \(y\) into the expressions for \(7x + 3y\) and \(7x - 3y\)
Now we substitute \(x\) and \(y\) into the expressions:
- \(7x + 3y = 7(3k) + 3(4k)\)
- \(7x - 3y = 7(3k) - 3(4k)\)

### Step 3: Simplify the expressions
Now, let's simplify both expressions:
- For \(7x + 3y\):
\[ 7(3k) + 3(4k) = 21k + 12k = 33k \]
- For \(7x - 3y\):
\[ 7(3k) - 3(4k) = 21k - 12k = 9k \]

### Step 4: Form the ratio
Now we can form the ratio:
\[ (7x + 3y) : (7x - 3y) = 33k : 9k \]

### Step 5: Simplify the ratio
Since \(k\) is common in both terms, we can cancel it out:
\[ 33k : 9k = 33 : 9 \]

### Step 6: Further simplify the ratio
Now, we can simplify \(33 : 9\) by dividing both sides by 3:
\[ \frac{33}{3} : \frac{9}{3} = 11 : 3 \]

### Final Answer
Thus, the final answer is:
\[ (7x + 3y) : (7x - 3y) = 11 : 3 \]

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