The ratio A, B and C of last year income is 3:4:5. The ratio of A, B and C last year and curent year is 4:5, 2:3, 3 :4 respectively. If sum of income of A, B, and C is Rs. 78800, then find out curent income of A, B, and C separately.

Options

A. `18000,28800,32000`
B. `18000,28000,32000`
C. `18000,28000,32000`
D. `18000,28000,32000`

Answer

To solve the problem step by step, let's break it down:

### Step 1: Understand the Ratios
The income ratios of A, B, and C last year are given as:
- A : B : C = 3 : 4 : 5

### Step 2: Assign Variables
Let the incomes of A, B, and C last year be:
- A = 3x
- B = 4x
- C = 5x

### Step 3: Find the Total Income Last Year
The total income of A, B, and C last year can be calculated as:
\[ \text{Total Income} = A + B + C = 3x + 4x + 5x = 12x \]

### Step 4: Current Year Ratios
The ratios of their incomes from last year to the current year are:
- A last year : A current = 4 : 5
- B last year : B current = 2 : 3
- C last year : C current = 3 : 4

### Step 5: Express Current Incomes in Terms of Last Year Incomes
Using the ratios, we can express the current incomes:
- For A:
\[ \frac{3x}{A_{current}} = \frac{4}{5} \implies A_{current} = \frac{5}{4} \times 3x = \frac{15x}{4} \]
- For B:
\[ \frac{4x}{B_{current}} = \frac{2}{3} \implies B_{current} = \frac{3}{2} \times 4x = 6x \]
- For C:
\[ \frac{5x}{C_{current}} = \frac{3}{4} \implies C_{current} = \frac{4}{3} \times 5x = \frac{20x}{3} \]

### Step 6: Find the Total Current Income
Now, we can find the total current income:
\[ \text{Total Current Income} = A_{current} + B_{current} + C_{current} \]
Substituting the values:
\[ = \frac{15x}{4} + 6x + \frac{20x}{3} \]

### Step 7: Find a Common Denominator
To add these fractions, we need a common denominator. The least common multiple of 4, 1, and 3 is 12.
- Convert each term:
\[ \frac{15x}{4} = \frac{45x}{12}, \quad 6x = \frac{72x}{12}, \quad \frac{20x}{3} = \frac{80x}{12} \]
Now, adding them:
\[ \text{Total Current Income} = \frac{45x + 72x + 80x}{12} = \frac{197x}{12} \]

### Step 8: Set Up the Equation
We know the total current income is Rs. 78,800:
\[ \frac{197x}{12} = 78800 \]

### Step 9: Solve for x
Multiply both sides by 12:
\[ 197x = 78800 \times 12 \]
\[ 197x = 945600 \]
Now, divide by 197:
\[ x = \frac{945600}{197} = 4800 \]

### Step 10: Calculate Current Incomes
Now that we have x, we can find the current incomes:
- A's current income:
\[ A_{current} = \frac{15x}{4} = \frac{15 \times 4800}{4} = 18000 \]
- B's current income:
\[ B_{current} = 6x = 6 \times 4800 = 28800 \]
- C's current income:
\[ C_{current} = \frac{20x}{3} = \frac{20 \times 4800}{3} = 32000 \]

### Final Answer
- Current income of A = Rs. 18,000
- Current income of B = Rs. 28,800
- Current income of C = Rs. 32,000

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