The ratio of age between A and B is 6 : 5 and the age of each C and D is 9/10 times that of B. Age of F is less than A but greater than B. The ratio of ages between B and E is 2 : 3 also age of A is 3 years less than E. What is the ratio of ages of A and F if all the ages are in integers?

Options

A. 0.50763888888889
B. 0.37986111111111
C. 24:19
D. 0.50902777777778

Answer

To solve the problem step by step, we will define the ages of A, B, C, D, E, and F based on the given ratios and relationships.

### Step 1: Define Ages Based on Ratios
The ratio of ages between A and B is given as 6:5. We can represent their ages as:
- Age of A = 6x
- Age of B = 5x

### Step 2: Calculate Ages of C and D
The ages of C and D are 9/10 times that of B. Therefore:
- Age of C = (9/10) * Age of B = (9/10) * 5x = 4.5x
- Age of D = (9/10) * Age of B = (9/10) * 5x = 4.5x

### Step 3: Define Age of E
The ratio of ages between B and E is 2:3. We can express the age of E as:
- Age of E = (3/2) * Age of B = (3/2) * 5x = 7.5x

### Step 4: Relate Age of A and E
It is given that the age of A is 3 years less than the age of E. Therefore, we can write:
- Age of A = Age of E - 3
Substituting the values we have:
- 6x = 7.5x - 3
Now, we can solve for x:
- 6x + 3 = 7.5x
- 3 = 7.5x - 6x
- 3 = 1.5x
- x = 2

### Step 5: Calculate Actual Ages
Now that we have the value of x, we can calculate the actual ages:
- Age of A = 6x = 6 * 2 = 12 years
- Age of B = 5x = 5 * 2 = 10 years
- Age of C = 4.5x = 4.5 * 2 = 9 years
- Age of D = 4.5x = 4.5 * 2 = 9 years
- Age of E = 7.5x = 7.5 * 2 = 15 years

### Step 6: Define Age of F
It is stated that the age of F is less than A but greater than B. Therefore, the possible integer age for F can be:
- Age of F can be 11 years (since it must be greater than B's age of 10 and less than A's age of 12).

### Step 7: Calculate Ratio of Ages of A and F
Now we can find the ratio of ages of A and F:
- Ratio of A to F = Age of A : Age of F = 12 : 11

### Final Answer
The ratio of ages of A and F is 12:11.

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