Two numbers are in the ratio 2 : 3 . If 15 is added to both the numbers , then the ratio between numbers becomes `11/14` . Find the greater number

Answer

To solve the problem, we need to find two numbers that are in the ratio of 2:3. Let's denote these numbers as \( x \) and \( y \).

### Step 1: Set up the equations based on the ratio
Since the numbers are in the ratio of 2:3, we can express them as:
\[ x = 2k \quad \text{and} \quad y = 3k \]
where \( k \) is a common multiplier.

### Step 2: Set up the equation after adding 15
According to the problem, if we add 15 to both numbers, the new ratio becomes \( \frac{11}{14} \). Therefore, we can write:
\[ \frac{x + 15}{y + 15} = \frac{11}{14} \]

### Step 3: Cross-multiply to eliminate the fraction
Cross-multiplying gives us:
\[ 14(x + 15) = 11(y + 15) \]

### Step 4: Expand both sides
Expanding both sides of the equation, we get:
\[ 14x + 210 = 11y + 165 \]

### Step 5: Rearrange the equation
Rearranging the equation to isolate terms gives us:
\[ 14x - 11y = -45 \]

### Step 6: Substitute \( x \) and \( y \)
Now, substitute \( x = 2k \) and \( y = 3k \) into the equation:
\[ 14(2k) - 11(3k) = -45 \]
This simplifies to:
\[ 28k - 33k = -45 \]

### Step 7: Combine like terms
Combining the terms gives us:
\[ -5k = -45 \]

### Step 8: Solve for \( k \)
Dividing both sides by -5 gives:
\[ k = 9 \]

### Step 9: Find the values of \( x \) and \( y \)
Now, we can find \( x \) and \( y \):
\[ x = 2k = 2(9) = 18 \]
\[ y = 3k = 3(9) = 27 \]

### Step 10: Identify the greater number
The greater number is \( y \), which is:
\[ \text{Greater number} = 27 \]

### Final Answer:
The greater number is **27**.

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