3
To solve this problem, we start by understanding the information given:
Let's denote the present age of X as 4x and the present age of Y as 5x, based on the given ratio 4:5.
Step 1: Setting up the equation for k years ago:
The ages k years ago would be:
According to the problem, k years ago, the ratio was 1:2, hence:
\(\frac{4x - k}{5x - k} = \frac{1}{2}\)
Cross-multiplying gives us:
\(2(4x - k) = 1(5x - k)\)
This simplifies to:
\(8x - 2k = 5x - k\)
Rearranging terms, we get:
\(3x = k\)
Since k must be less than 10, feasible integer values for x such that 3x is an integer and less than 10 are:
Step 2: Calculating feasible values for k
These are all the possible values for k, as any value of x greater than 3 would result in k being 10 or more, which is not permissible by the problem statement.
Therefore, there are 3 possible values for k: 3, 6, and 9.
Conclusion: The correct answer is that there are 3 possible values of k.
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