Three numbers are in the ratio `1 : 2 : 3`. By adding 5 to each of them, the new numbers are in the ratio `2:3:4`. The numbers are:

Options

A. 10, 20, 30
B. 15, 30, 45
C. 1, 2, 3
D. 5,10,15

Answer

To solve the problem step by step, we will follow the given information about the ratios of the numbers and the changes made to them.

### Step 1: Define the Numbers
Let the three numbers be represented as:
- First number = \( x \)
- Second number = \( 2x \)
- Third number = \( 3x \)

This is based on the ratio \( 1:2:3 \).

### Step 2: Add 5 to Each Number
According to the problem, we add 5 to each of these numbers:
- New first number = \( x + 5 \)
- New second number = \( 2x + 5 \)
- New third number = \( 3x + 5 \)

### Step 3: Set Up the New Ratio
We know that after adding 5, the new numbers are in the ratio \( 2:3:4 \). Therefore, we can write the following equation based on the new ratios:
\[ \frac{x + 5}{2} = \frac{2x + 5}{3} = \frac{3x + 5}{4} \]

### Step 4: Solve the First Two Parts of the Ratio
We will first equate the first two parts of the ratio:
\[ \frac{x + 5}{2} = \frac{2x + 5}{3} \]

Cross-multiplying gives:
\[ 3(x + 5) = 2(2x + 5) \]
Expanding both sides:
\[ 3x + 15 = 4x + 10 \]
Rearranging gives:
\[ 15 - 10 = 4x - 3x \]
\[ 5 = x \]

### Step 5: Find the Values of the Original Numbers
Now that we have \( x = 5 \), we can find the original numbers:
- First number = \( x = 5 \)
- Second number = \( 2x = 2 \times 5 = 10 \)
- Third number = \( 3x = 3 \times 5 = 15 \)

### Step 6: Conclusion
Thus, the three numbers are:
- First number = 5
- Second number = 10
- Third number = 15

### Final Answer
The numbers are \( 5, 10, \) and \( 15 \).

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