If the average (arithmetic mean) cost per sweater for 3 pullover sweat

Given: The average (arithmetic mean) cost per sweater for 3 pullover sweaters and 1 cardigan sweater was $65
Let P = the combined price of ALL 3 pullover sweaters
Let C = the price of 1 cardigan sweater
Aside: It would be nice to let P = the price of ONE pullover, which would mean 3P = the price of THREE pullovers. However, we don't know whether all pullovers have the same price

The average price of all 4 sweaters is $65
So, we can write: (P + C)/4 = 65
Multiply both sides of the equation by 4 to get: P + C = 260

Target question: What is the value of C (the cost of the cardigan sweater)?

Statement 1: The average cost per sweater for the 3 pullover sweaters was $55.
Since P = the combined price of ALL 3 pullover sweaters, we can write: P/3 = 55
Multiply both sides of the equation by 3 to get: P = 165
He already know that: P + C = 260
Replace P to get: 165 + C = 260
Solve, to get: C = 95
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: The most expensive of the 3 pullover sweaters cost $30 more than the least expensive.
There are several scenarios that satisfy statement 2 (and the given information). Here are two:
Case a: The prices of the three pullovers are $10, $20 and $40, which means P = $10 + $20 + $40 = $70. Since we also know that P + C = 260, we get: 70 + C = 260, in which case the answer to the target question is C = 190
Case b: The prices of the three pullovers are $30, $40 and $60, which means P = $30 + $40 + $60 = $130. Since we also know that P + C = 260, we get: 130 + C = 260, in which case the answer to the target question is C = 130
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Answer: A

Cheers,
Brent

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The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question. We should simplify conditions if necessary.

Assume \(p_1, p_2\) and \(p_3\) are costs with \(p_1 ≤ p_2 ≤ p_3\) for thee different pullover sweaters and c is that of the cardigan sweater.
Then we have \(\frac{{p_1 + p_2 + p_3 + c} }{ 4} = 65\) or \(p_1 + p_2 + p_3 + c = 260\).

The question asks the value of \(c\).

Since condition 1) tells \(\frac{{p_1 + p_2 + p_3} }{ 3} = 55\) or \(p_1 + p_2 + p_3 = 165\), we have \(165 + c = 260\) or \(c = 96\).
Thus condition 1) is sufficient.

Condition 2)
Condition 2) tells \(p_3 = p_1 + 30\).
Since we don’t know \(p_2\), condition 2) does not yield a unique solution and it is not sufficient.

Therefore, A is the answer.

If the original condition includes “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations” etc., one more equation is required to answer the question. If each of conditions 1) and 2) provide an additional equation, there is a 59% chance that D is the answer, a 38% chance that A or B is the answer, and a 3% chance that the answer is C or E. Thus, answer D (conditions 1) and 2), when applied separately, are sufficient to answer the question) is most likely, but there may be cases where the answer is A,B,C or E.

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