The incomes of A,B,C are in the ratio of `12:9:7` and their spendings are in the ratioi `15:9:8`. If A saves 25% of his income. What is the ratio of the savings of A,B and C?

Answer

To solve the problem step by step, we will follow the given ratios of incomes and expenditures, and then calculate the savings for A, B, and C.

### Step 1: Define the incomes based on the given ratio
The incomes of A, B, and C are in the ratio of 12:9:7. We can express their incomes as:
- Income of A = 12x
- Income of B = 9x
- Income of C = 7x

### Step 2: Define the expenditures based on the given ratio
The expenditures of A, B, and C are in the ratio of 15:9:8. We can express their expenditures as:
- Expenditure of A = 15y
- Expenditure of B = 9y
- Expenditure of C = 8y

### Step 3: Calculate A's savings
We know that A saves 25% of his income. Therefore, A's savings can be calculated as:
- Savings of A = Income of A - Expenditure of A
- Savings of A = 12x - 15y

Since A saves 25% of his income:
- Savings of A = 0.25 × Income of A = 0.25 × 12x = 3x

Setting the two expressions for A's savings equal to each other:
\[ 12x - 15y = 3x \]
Rearranging gives:
\[ 12x - 3x = 15y \]
\[ 9x = 15y \]
Dividing both sides by 3:
\[ 3x = 5y \]
From this, we can express y in terms of x:
\[ y = \frac{3x}{5} \]

### Step 4: Calculate B's savings
Now we can calculate B's savings:
- Savings of B = Income of B - Expenditure of B
- Savings of B = 9x - 9y

Substituting the value of y:
\[ Savings of B = 9x - 9\left(\frac{3x}{5}\right) \]
\[ = 9x - \frac{27x}{5} \]
Finding a common denominator:
\[ = \frac{45x}{5} - \frac{27x}{5} \]
\[ = \frac{18x}{5} \]

### Step 5: Calculate C's savings
Now we can calculate C's savings:
- Savings of C = Income of C - Expenditure of C
- Savings of C = 7x - 8y

Substituting the value of y:
\[ Savings of C = 7x - 8\left(\frac{3x}{5}\right) \]
\[ = 7x - \frac{24x}{5} \]
Finding a common denominator:
\[ = \frac{35x}{5} - \frac{24x}{5} \]
\[ = \frac{11x}{5} \]

### Step 6: Calculate the ratio of savings
Now we have the savings for A, B, and C:
- Savings of A = 3x
- Savings of B = \(\frac{18x}{5}\)
- Savings of C = \(\frac{11x}{5}\)

To find the ratio of their savings:
\[ \text{Ratio of savings} = 3x : \frac{18x}{5} : \frac{11x}{5} \]

To eliminate the fractions, we can multiply the entire ratio by 5:
\[ = 15x : 18x : 11x \]

Now, we can simplify the ratio:
\[ = 15 : 18 : 11 \]

### Final Answer
The ratio of the savings of A, B, and C is \(15 : 18 : 11\).

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