The ratio of salary of 4 men is 3 : 2:1:4. If they increase their salary by 10%, 15%, 12%, 20% respectively. Find out ratio of these income.

Options

A. `165:115:56:240`
B. `155:110:54:24`
C. `165:156: 115:32`
D. `115:165:56:24`

Answer

To solve the problem step by step, we will first express the salaries of the four men in a more manageable form, calculate their new salaries after the percentage increases, and then find the ratio of their new salaries.

### Step 1: Express the salaries in terms of a common factor
The ratio of the salaries of the four men is given as 3:2:1:4. To make calculations easier, we can assign a common factor \( k \) to these ratios. Thus, we can express their salaries as follows:
- Salary of A = \( 3k \)
- Salary of B = \( 2k \)
- Salary of C = \( 1k \)
- Salary of D = \( 4k \)

### Step 2: Calculate the new salaries after the percentage increases
Next, we will calculate the new salaries after applying the respective percentage increases.

1. **For A:**
- Old Salary = \( 3k \)
- Increase = 10% of \( 3k \) = \( \frac{10}{100} \times 3k = 0.3k \)
- New Salary of A = \( 3k + 0.3k = 3.3k \)

2. **For B:**
- Old Salary = \( 2k \)
- Increase = 15% of \( 2k \) = \( \frac{15}{100} \times 2k = 0.3k \)
- New Salary of B = \( 2k + 0.3k = 2.3k \)

3. **For C:**
- Old Salary = \( 1k \)
- Increase = 12% of \( 1k \) = \( \frac{12}{100} \times 1k = 0.12k \)
- New Salary of C = \( 1k + 0.12k = 1.12k \)

4. **For D:**
- Old Salary = \( 4k \)
- Increase = 20% of \( 4k \) = \( \frac{20}{100} \times 4k = 0.8k \)
- New Salary of D = \( 4k + 0.8k = 4.8k \)

### Step 3: Write the new salaries in ratio form
Now we have the new salaries:
- New Salary of A = \( 3.3k \)
- New Salary of B = \( 2.3k \)
- New Salary of C = \( 1.12k \)
- New Salary of D = \( 4.8k \)

The ratio of their new salaries is:
\[ 3.3k : 2.3k : 1.12k : 4.8k \]
This simplifies to:
\[ 3.3 : 2.3 : 1.12 : 4.8 \]

### Step 4: Eliminate the common factor \( k \)
Since \( k \) is a common factor in all terms, we can ignore it when calculating the ratio:
\[ 3.3 : 2.3 : 1.12 : 4.8 \]

### Step 5: Convert to whole numbers
To eliminate decimals, we can multiply each term by 100:
\[ 330 : 230 : 112 : 480 \]

### Step 6: Simplify the ratio
Now we will simplify the ratio \( 330 : 230 : 112 : 480 \). To do this, we can find the greatest common divisor (GCD) of these numbers.

Calculating the GCD:
- The GCD of \( 330, 230, 112, \) and \( 480 \) is \( 2 \).

Now we divide each term by \( 2 \):
\[ \frac{330}{2} : \frac{230}{2} : \frac{112}{2} : \frac{480}{2} = 165 : 115 : 56 : 240 \]

### Final Answer:
The final ratio of their new salaries is:
\[ \boxed{165 : 115 : 56 : 240} \]