Bunuel wrote:
If n is the product of the least and the greatest of 6 consecutive integers, what is the value of n ?
(1) The greatest of the 6 consecutive integers is 20.
(2) The average (arithmetic mean) of the 6 consecutive integers is 17.5.
Given: n = the product of the least and the greatest of 6 consecutive integers Target question: What is the value of n? Statement 1: The greatest of the 6 consecutive integers is 20. Since the 6 integers are
consecutive, we can work backwards to find the other 5 integers
They are {
15, 16, 17, 18, 19,
20}
So, the product of the least and the greatest integers = (
15)(
20) = 300
The answer to the target question is
n = 300Since we can answer the
target question with certainty, statement 1 is SUFFICIENT
Statement 2: The average (arithmetic mean) of the 6 consecutive integers is 17.5There are several different ways to approach the statement.
APPROACH #1The fastest approach is to recognize that for every possible set of 6 consecutive integers, we get a different average.
For example, the average of {2, 3, 4, 5, 6, 7} is 4.5, the average of {3, 4, 5, 6, 7, 8} is 5.5, the average of {4, 5, 6, 7, 8, 9} is 6.5, etc
So, if the average of of the 6 integers is 17.5, there must be EXACTLY ONE set of six consecutive integers that satisfies this information.
So, even if we were to just keep randomly checking various sets of 6 integers until we find the one set that has an average of 17.5, then we'd be able to determine the value of n.
Since we COULD use the information from statement 2 to answer the target question with certainty, statement 2 is SUFFICIENT
APPROACH #2Another approach to handle statement 2 is as follows:
let x = the first (smallest) number in the set
So, x+1 = the next number in the set
x+2 = the next number in the set
x+3 = the next number in the set
x+4 = the next number in the set
x+5 = the last (greatest) number in the set
If the average = 17.5, we can write: \(\frac{x + (x+1)+ (x+2)+ (x+3)+ (x+4)+ (x+5)}{6 = 17.5}\)
Multiply both sides of the equation by 6 to get: \(x + (x+1)+ (x+2)+ (x+3)+ (x+4)+ (x+5) = 105\)
Simplify the left side to get: \(6x + 15= 105\)
Subtract 15 from both sides: \(6x= 90\)
Divide both sides by 6 to get: \(x = 15\)
Since x represents the first (smallest) value in the set, the 6 integers are {
15, 16, 17, 18, 19,
20}
So, the product of the least and the greatest integers = (
15)(
20) = 300
The answer to the target question is
n = 300Since we can answer the
target question with certainty, statement 2 is SUFFICIENT
Answer: D
Cheers,
Brent