4 yrs ago A was four yrs younger than B. 6 yrs hence ratio of ages of A to B will be 16:17. Find the sum of ages of both 4 yrs ago.

Answer

To solve the problem step by step, we will define the ages of A and B and use the information given in the question to find their ages.

### Step 1: Define the ages of A and B four years ago
Let the age of A four years ago be \( x \). Since A was four years younger than B at that time, we can express B's age four years ago as:
\[ \text{Age of B four years ago} = x + 4 \]

### Step 2: Express their current ages
Since four years have passed since then, we can express their current ages as:
\[ \text{Current age of A} = x + 4 \]
\[ \text{Current age of B} = (x + 4) + 4 = x + 8 \]

### Step 3: Express their ages six years hence
Six years from now, their ages will be:
\[ \text{Age of A six years hence} = (x + 4) + 6 = x + 10 \]
\[ \text{Age of B six years hence} = (x + 8) + 6 = x + 14 \]

### Step 4: Set up the ratio of their ages six years hence
According to the problem, the ratio of their ages six years hence will be \( 16:17 \). Therefore, we can write the equation:
\[ \frac{x + 10}{x + 14} = \frac{16}{17} \]

### Step 5: Cross-multiply to solve for x
Cross-multiplying gives us:
\[ 17(x + 10) = 16(x + 14) \]
Expanding both sides:
\[ 17x + 170 = 16x + 224 \]

### Step 6: Rearrange the equation to isolate x
Subtract \( 16x \) from both sides:
\[ 17x - 16x + 170 = 224 \]
This simplifies to:
\[ x + 170 = 224 \]
Now, subtract 170 from both sides:
\[ x = 224 - 170 \]
\[ x = 54 \]

### Step 7: Find the ages of A and B four years ago
Now that we have \( x = 54 \), we can find the ages of A and B four years ago:
\[ \text{Age of A four years ago} = x = 54 \]
\[ \text{Age of B four years ago} = x + 4 = 54 + 4 = 58 \]

### Step 8: Calculate the sum of their ages four years ago
Finally, we find the sum of their ages four years ago:
\[ \text{Sum of ages} = 54 + 58 = 112 \]

### Final Answer
The sum of the ages of A and B four years ago is \( \boxed{112} \).

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