AverageGuy123 wrote:

If the average(arithmetic mean) of four numbers is 10 ,how many of the numbers are greater than 10?

1) Precisely two numbers are equal to 10

2) The largest of the 4 numbers is 10 greater than the smallest of the 4 numbers.

Assume that \(a\), \(b\), \(c\) and \(d\) are four numbers, where \(a \leq b \leq c \leq d\).

Then we have 1 equation that \(a + b + c + d = 40\).

Let's translate the conditions to mathematical expressions.

1) \(b = c = 10\), \(a < 10\), \(d > 10\)

2) \(d = a + 10\)

For the first condition, only \(d\) is greater than 10 form the above mathematical expression.

Hence the condition 1) is sufficient.

For the second condition, we have following examples.

Let \(a = 5\) and \(d = 15\).

Then we have two kinds of cases as follows.

5, 10, 10, 15 : They have just 1 number of elements greater than 10, which is just 15.

5, 9, 11, 15 : They have 2 numbers of elements greater than 10, which are 11 and 15.

Hence, we don't have a unique solution about the question how many numbers are greater than 10.

Therefore, the second condition is not sufficient.

Please remind that we should forget everything about the condition 1), when we consider the second condition only.

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