The income of A B and C are in the ratio `7 : 9 : 12` and their spendings are in the ratio `8:9:15`. If A saves `1/4` th of his income, then the savings of A, B and C are in the ratio of:

Options

A. `56: 99: 69`
B. `69: 56: 99`
C. `99: 56: 69`
D. `99: 69: 56`

Answer

To solve the problem step by step, we will follow the given ratios of income and spending for A, B, and C, and then calculate their savings based on the information provided.

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

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

### Step 3: Calculate A's Savings
It is given that A saves 1/4 of his income. Therefore, we can calculate A's savings as:
- Savings of A = Income of A - Spending of A
- Savings of A = 7x - 8y

Since A saves 1/4 of his income:
- Savings of A = (1/4) * (7x) = (7x)/4

### Step 4: Set Up the Equation
Now we can set up the equation based on the savings of A:
\[ 7x - 8y = \frac{7x}{4} \]

### Step 5: Solve for y in terms of x
To solve for y, we will first eliminate the fraction by multiplying the entire equation by 4:
\[ 4(7x - 8y) = 7x \]
This simplifies to:
\[ 28x - 32y = 7x \]
Now, rearranging gives:
\[ 28x - 7x = 32y \]
\[ 21x = 32y \]
Thus, we can express y in terms of x:
\[ y = \frac{21x}{32} \]

### Step 6: Substitute y back into the Spending Expressions
Now we can substitute y back into the spending expressions:
- Spending of A = 8y = 8 * (21x/32) = \frac{168x}{32} = \frac{21x}{4}
- Spending of B = 9y = 9 * (21x/32) = \frac{189x}{32}
- Spending of C = 15y = 15 * (21x/32) = \frac{315x}{32}

### Step 7: Calculate the Savings for B and C
Now we can calculate the savings for B and C:
- Savings of B = Income of B - Spending of B
\[ Savings of B = 9x - \frac{189x}{32} = \frac{288x}{32} - \frac{189x}{32} = \frac{99x}{32} \]

- Savings of C = Income of C - Spending of C
\[ Savings of C = 12x - \frac{315x}{32} = \frac{384x}{32} - \frac{315x}{32} = \frac{69x}{32} \]

### Step 8: Find the Ratio of Savings
Now we have the savings:
- Savings of A = \(\frac{7x}{4}\)
- Savings of B = \(\frac{99x}{32}\)
- Savings of C = \(\frac{69x}{32}\)

To find the ratio of A, B, and C's savings, we can express them with a common denominator:
- Savings of A = \(\frac{56x}{32}\) (since \(\frac{7x}{4} = \frac{56x}{32}\))
- Savings of B = \(\frac{99x}{32}\)
- Savings of C = \(\frac{69x}{32}\)

Thus, the ratio of savings is:
\[ 56 : 99 : 69 \]

### Final Answer
The savings of A, B, and C are in the ratio of **56:99:69**.