Ratio of the age of A and B is 3 : 5 and ratio of the age of B and C is 4 : 7. Find the ratio of the age of A and C.

Answer

To find the ratio of the ages of A and C given the ratios of A to B and B to C, we can follow these steps:

### Step 1: Write down the given ratios
The ratio of the ages of A and B is given as:
\[ A : B = 3 : 5 \]

The ratio of the ages of B and C is given as:
\[ B : C = 4 : 7 \]

### Step 2: Express the ratios in terms of a common variable
Let’s express the ages of A, B, and C in terms of a common variable.

From the ratio \( A : B = 3 : 5 \), we can write:
- \( A = 3x \)
- \( B = 5x \)

From the ratio \( B : C = 4 : 7 \), we can write:
- \( B = 4y \)
- \( C = 7y \)

### Step 3: Make the age of B consistent in both ratios
Since B is represented in both ratios, we need to equate the two expressions for B:
\[ 5x = 4y \]

### Step 4: Solve for one variable in terms of the other
From the equation \( 5x = 4y \), we can express y in terms of x:
\[ y = \frac{5}{4}x \]

### Step 5: Substitute y back into the expression for C
Now, substitute \( y \) back into the expression for C:
\[ C = 7y = 7 \left(\frac{5}{4}x\right) = \frac{35}{4}x \]

### Step 6: Write the ages of A and C
Now we have:
- \( A = 3x \)
- \( C = \frac{35}{4}x \)

### Step 7: Find the ratio of A to C
Now we can find the ratio of A to C:
\[ A : C = 3x : \frac{35}{4}x \]

To simplify this, we can eliminate \( x \):
\[ A : C = 3 : \frac{35}{4} \]

### Step 8: Simplify the ratio
To simplify \( 3 : \frac{35}{4} \), we can multiply both sides by 4 to eliminate the fraction:
\[ A : C = 3 \times 4 : 35 = 12 : 35 \]

### Final Answer
Thus, the ratio of the ages of A and C is:
\[ A : C = 12 : 35 \]

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