The sum of income of A, B and C is Rs. 72000. A, B and C spends `80% , 85% and 75%` of his income. The ratio of their saving is 8 : 9 : 20. Find the income of A.

Answer

To find the income of A, we will follow these steps:

### Step 1: Understand the problem
We know the total income of A, B, and C is Rs. 72,000. Each person spends a certain percentage of their income, and we need to find the income of A based on the given savings ratio.

### Step 2: Define the savings
- A spends 80% of his income, so he saves 20% of his income.
- B spends 85% of his income, so he saves 15% of his income.
- C spends 75% of his income, so he saves 25% of his income.

### Step 3: Set up the savings in terms of a variable
Let the savings of A, B, and C be represented as:
- A's savings = 20% of A's income = 0.2A
- B's savings = 15% of B's income = 0.15B
- C's savings = 25% of C's income = 0.25C

According to the problem, the ratio of their savings is given as 8:9:20. We can express their savings in terms of a common variable \( k \):
- A's savings = 8k
- B's savings = 9k
- C's savings = 20k

### Step 4: Set up equations based on savings
From the savings equations, we can write:
1. \( 0.2A = 8k \) (1)
2. \( 0.15B = 9k \) (2)
3. \( 0.25C = 20k \) (3)

### Step 5: Express incomes in terms of k
From these equations, we can express A, B, and C's incomes in terms of \( k \):
1. From (1): \( A = \frac{8k}{0.2} = 40k \)
2. From (2): \( B = \frac{9k}{0.15} = 60k \)
3. From (3): \( C = \frac{20k}{0.25} = 80k \)

### Step 6: Set up the total income equation
Now, we know that the total income of A, B, and C is Rs. 72,000:
\[ A + B + C = 72000 \]
Substituting the values we found:
\[ 40k + 60k + 80k = 72000 \]
This simplifies to:
\[ 180k = 72000 \]

### Step 7: Solve for k
To find \( k \), divide both sides by 180:
\[ k = \frac{72000}{180} = 400 \]

### Step 8: Find A's income
Now that we have \( k \), we can find A's income:
\[ A = 40k = 40 \times 400 = 16000 \]

### Final Answer
The income of A is Rs. 16,000.

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