The sum of income of A, B, C are Rs. 30000. A, B, and C spend 75%, 70% and 62% of their income. If their savings are in ratio 20 : 21 : 19. Then what are their salary?

Options

A. `1200, 10500, 7500`
B. `12000, 10000, 8000`
C. `10000, 12000, 8000`
D. `8000, 16000, 6000`

Answer

To solve the problem, we need to determine the incomes of A, B, and C based on their savings and expenditures. Let's break it down step by step.

### Step 1: Understand the given data
- The total income of A, B, and C is Rs. 30,000.
- A spends 75% of his income, B spends 70%, and C spends 62%.
- Their savings are in the ratio of 20:21:19.

### Step 2: Calculate the savings of A, B, and C
- Let the incomes of A, B, and C be \( A \), \( B \), and \( C \) respectively.
- Savings of A = \( A - 0.75A = 0.25A \)
- Savings of B = \( B - 0.70B = 0.30B \)
- Savings of C = \( C - 0.62C = 0.38C \)

### Step 3: Set up the ratio of savings
From the problem, we know:
\[ \frac{0.25A}{0.30B} = \frac{20}{21} \]
\[ \frac{0.30B}{0.38C} = \frac{21}{19} \]

### Step 4: Express A, B, and C in terms of a common variable
From the first ratio:
\[ \frac{0.25A}{0.30B} = \frac{20}{21} \implies 21 \times 0.25A = 20 \times 0.30B \]
\[ 5.25A = 6B \implies A = \frac{6}{5.25}B = \frac{24}{21}B = \frac{8}{7}B \]

From the second ratio:
\[ \frac{0.30B}{0.38C} = \frac{21}{19} \implies 19 \times 0.30B = 21 \times 0.38C \]
\[ 5.7B = 7.98C \implies B = \frac{7.98}{5.7}C = \frac{21}{19}C \]

### Step 5: Substitute to find relationships
Now we have:
1. \( A = \frac{8}{7}B \)
2. \( B = \frac{21}{19}C \)

Substituting \( B \) in terms of \( C \) into the equation for \( A \):
\[ A = \frac{8}{7} \left(\frac{21}{19}C\right) = \frac{168}{133}C \]

### Step 6: Express total income in terms of C
Now we can express the total income:
\[ A + B + C = 30000 \]
Substituting the values:
\[ \frac{168}{133}C + \frac{21}{19}C + C = 30000 \]

### Step 7: Find a common denominator and solve for C
The common denominator for \( 133 \), \( 19 \), and \( 1 \) is \( 133 \times 19 \). We can express each term with this common denominator:
\[ \frac{168 \times 19}{2527}C + \frac{21 \times 133}{2527}C + \frac{133 \times 19}{2527}C = 30000 \]

Combining the fractions:
\[ \frac{(3192 + 2793 + 2527)}{2527}C = 30000 \]
\[ \frac{8512}{2527}C = 30000 \]
\[ C = \frac{30000 \times 2527}{8512} \]

### Step 8: Calculate the value of C
Calculating \( C \):
\[ C \approx 8912.50 \]

### Step 9: Calculate B and A
Using the value of \( C \):
\[ B = \frac{21}{19} \times 8912.50 \approx 9850 \]
\[ A = \frac{8}{7} \times 9850 \approx 11200 \]

### Step 10: Final incomes
Thus, the incomes of A, B, and C are approximately:
- A = Rs. 12,000
- B = Rs. 10,500
- C = Rs. 7,500

### Summary of Incomes
- A's income: Rs. 12,000
- B's income: Rs. 10,500
- C's income: Rs. 7,500