What will be sum of two numbers?
A. Among the two numbers, the bigger number is greater than the smaller number by 6.
B. `40%` of the smaller number is equal to `30%` of the bigger number.
C. The ratio between half of the bigger number and `1/3rd` of the smaller number is `2:1.`

Options

A. Only B and C together are necessary
B. Only A and B together are necessary
C. Out of A, B and C any two together are ncecessary
D. All three A, B and C together are necessary

Explanation

To solve the problem of finding the sum of two numbers based on the given statements, we can follow these steps:

### Step 1: Define the Variables
Let the bigger number be \( x \) and the smaller number be \( y \).

### Step 2: Analyze Statement A
From Statement A, we know:
- The bigger number is greater than the smaller number by 6.
- This can be expressed mathematically as:
\[ x = y + 6 \]

### Step 3: Analyze Statement B
From Statement B, we have:
- \( 40\% \) of the smaller number is equal to \( 30\% \) of the bigger number.
- This can be expressed as:
\[ 0.4y = 0.3x \]
- To eliminate the decimals, we can multiply both sides by 100:
\[ 40y = 30x \]
- Rearranging gives:
\[ 4y = 3x \]

### Step 4: Substitute from Statement A into Statement B
Now we can substitute the expression for \( x \) from Statement A into the equation from Statement B:
- Substitute \( x = y + 6 \) into \( 4y = 3x \):
\[ 4y = 3(y + 6) \]
- Distributing the 3:
\[ 4y = 3y + 18 \]
- Rearranging gives:
\[ 4y - 3y = 18 \implies y = 18 \]

### Step 5: Find \( x \)
Now that we have \( y \), we can find \( x \):
- Using \( x = y + 6 \):
\[ x = 18 + 6 = 24 \]

### Step 6: Calculate the Sum
Now we can find the sum of the two numbers:
\[ x + y = 24 + 18 = 42 \]

### Conclusion
The sum of the two numbers is \( 42 \).

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