The problem requires identifying a consistent mathematical operation between number pairs in the format A : B : : C : D.

Identifying the Pattern Logic

The structure is given as \(X : 98 : : 196 : Y\). This implies two pairs:

• The first pair involves \(X\) and \(98\).
• The second pair involves \(196\) and \(Y\).

The core rule is that the mathematical operation applied to get the second number from the first number in the left pair (\(X : 98\)) must be the same as the operation applied in the right pair (\(196 : Y\)).

Let the operation be denoted by 'op'. According to the problem, the second number is obtained by applying 'op' to the first number. Therefore:

1. \(98 = op(X)\)
2. \(Y = op(196)\)

Testing Options to Find the Pattern

We need to find values for \(X\) and \(Y\) that satisfy a consistent pattern. Let's test the first option (Option A): \(X = 116, Y = 178\).

Analyzing Pair 1 (X : 98)

Substituting \(X = 116\), the pair becomes \(116 : 98\).

Let's determine the operation:

\(op(116) = 98\)

The difference is \(116 - 98 = 18\). So, the operation appears to be subtracting 18.

This can be written as: \(98 = 116 - 18\).

Analyzing Pair 2 (196 : Y)

Substituting \(Y = 178\), the pair becomes \(196 : 178\).

Let's apply the same operation (subtracting 18) to the first number, \(196\):

\(op(196) = 196 - 18\)

Calculation: \(196 - 18 = 178\).

This matches the value of \(Y\) given in the option (\(Y=178\)).

Confirming the Pattern and Deriving X and Y

Since the operation "subtract 18" is consistent for both pairs using the values from Option A, this is the correct pattern.

The pattern is: Second Number = First Number - 18.

Calculating X

Using the pattern for the first pair \(X : 98\):

\(98 = X - 18\)

To find \(X\), we rearrange the equation:

\(X = 98 + 18\)

\(X = 116\)

Calculating Y

Using the pattern for the second pair \(196 : Y\):

\(Y = 196 - 18\)

\(Y = 178\)

Therefore, the correct values are \(X = 116\) and \(Y = 178\).