HarveyKlaus wrote:

If x is a positive three-digit integer, what is the tens digit of x?

1) x is a multiple of 9

2) The hundreds digit of x is 9 times the units digit of x

Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

\(x = 100a + 10b + c\) where \(a\), \(b\), \(c\) are integers such that \(1 \le a \le 9\), \(0\le b,c \le 9\).

The question asks what the value of \(b\).

There are 4 variables and 1 equation. Thus E is the answer most likely.

Condition 1) : \(a + b + c\) is a multiple of 9.

This is not sufficient, since there are many combination of \(a\), \(b\) and \(c\).

For example, \(a = b = c = 3\) and \(a = b = c = 9\).

Condition 2) \(a = 9c\)

We have \(a = 9\) and \(c= 1\) from this condition.

But we can not get anything else about \(b\).

Condition 1) & 2)

There is a unique solution for these conditions, which is \(a = 9\) and \(c= 1\) and \(b = 8\), since \(a + b + c\) is a multiple of 9.

This is sufficient.

Therefore, the answer is C).

For cases where we need 3 more equations, such as original conditions with “3 variables”, or “4 variables and 1 equation”, or “5 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 80 % chance that E is the answer, while C has 15% chance and A, B or D has 5% chance. Since E is most likely to be the answer using 1) and 2) together according to DS definition. Obviously there may be cases where the answer is A, B, C or D.

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