Two numbers are in the ratio `1(1/2) : 2(2/3)`. When each of these is increased by 15, they become in the ratio `1(2/3): 2(1/2)`. The greater of the numbers is :

Options

A. 27
B. 36
C. 48
D. 64

Answer

To solve the problem, we will follow these steps:

### Step 1: Convert the ratios to improper fractions
The given ratio is \(1 \frac{1}{2} : 2 \frac{2}{3}\).

1. Convert \(1 \frac{1}{2}\) to an improper fraction:
\[ 1 \frac{1}{2} = \frac{3}{2} \]

2. Convert \(2 \frac{2}{3}\) to an improper fraction:
\[ 2 \frac{2}{3} = \frac{8}{3} \]

Thus, the ratio can be expressed as:
\[ \frac{3}{2} : \frac{8}{3} \]

### Step 2: Set up the equations
Let the two numbers be \(A\) and \(B\). From the ratio, we can express:
\[ \frac{A}{B} = \frac{3/2}{8/3} \]

Cross-multiplying gives:
\[ 3B = 16A \quad \text{(1)} \]

### Step 3: Increase both numbers by 15
According to the problem, when both numbers are increased by 15, they become in the ratio \(1 \frac{2}{3} : 2 \frac{1}{2}\).

1. Convert \(1 \frac{2}{3}\) to an improper fraction:
\[ 1 \frac{2}{3} = \frac{5}{3} \]

2. Convert \(2 \frac{1}{2}\) to an improper fraction:
\[ 2 \frac{1}{2} = \frac{5}{2} \]

Thus, the new ratio can be expressed as:
\[ \frac{A + 15}{B + 15} = \frac{5/3}{5/2} \]

### Step 4: Set up the second equation
Cross-multiplying gives:
\[ 2(A + 15) = 3(B + 15) \]
Expanding this results in:
\[ 2A + 30 = 3B + 45 \quad \text{(2)} \]

### Step 5: Solve the equations
Now we have two equations:
1. \(3B = 16A\) (Equation 1)
2. \(2A + 30 = 3B + 45\) (Equation 2)

Substituting \(B\) from Equation 1 into Equation 2:
\[ 2A + 30 = 3\left(\frac{16A}{3}\right) + 45 \]
This simplifies to:
\[ 2A + 30 = 16A + 45 \]
Rearranging gives:
\[ 30 - 45 = 16A - 2A \]
\[ -15 = 14A \]
\[ A = \frac{-15}{14} \]

### Step 6: Find \(B\)
Substituting \(A\) back into Equation 1 to find \(B\):
\[ 3B = 16\left(\frac{-15}{14}\right) \]
\[ 3B = \frac{-240}{14} \]
\[ B = \frac{-80}{14} = \frac{-40}{7} \]

### Step 7: Find the greater of the two numbers
The greater number is \(B\) since \(B\) is larger in magnitude than \(A\).

### Final Answer
Thus, the greater of the two numbers is:
\[ \text{Greater number} = B = \frac{-40}{7} \]