LUANDATO wrote:The average of five natural numbers is 150. A particular number among the five exceeds another by 100. The rest three numbers lie between these two numbers and they are equal. How many different values can the largest number among the five take?
A. 59
B. 19
C. 21
D. 18
E. 42
The OA is B.
I'm really confused with this PS question. Experts, any suggestion about how can I solve it? Thanks in advance.
What's the source of the question?
Natural numbers term is not used in the GMAT. To refer natural numbers, it uses positive integers.
Say the smallest of the five numbers is x, thus, the largest is (x + 100). Say the three equal numbers between x and (x + 100) are y each.
Thus, the five numbers are x, y, y, y, (x + 100).
=> [x + y + y + y + (x + 100)] / 3= 150
x + y + y + y + (x + 100) = 450
2x + 3y = 350
y = (350 - 2x)/3
We must apply a hit and trial method here to find out the qualified positive integer values of x and y such that x < y < (x + 100).
1. At x = 1, we have y = 116, and x + 100 = 101; this is not a qualfied value as y is not less than x + 100.
2. Similarly, for x = 4, 7, and 10, though we get positive integer values of y, they are not less than x + 100.
3. The first qualified value of x is 13. At x = 13, we have y = 108, and x + 100 = 113; this is a qualfied value as y is less than x + 100.
4. Getting the last qualified value of x is time-consuming. Let's apply another approach.
We see that the values of x form a sequence: 1, 4, 7, 10, 13... We see that qualified values of x increase at an interval of 3 with 13 being the first qualified value.
Say x = y for the time being. Let's compute the value of x and see if it fits in the above sequence.
=> We have 2x + 3y = 350
Thus, at y = x, we have 2y + 3y = 350 => 5x = 350 => x = 70.
At x = 70, we have y = (350 - 2x)/3 => y = (350 - 2*70)/3 = 210/3 = 70.
This is not a qualified value of x as x = y = 70; we know that x < y, thus, the largest possible value of x would be 70 - common interval = 70 - 3 = 67.
Thus, the possible values of x are 13, 16, 19..., 67.
The number of values = 1 + (67 - 13)/3 = 1 + 54/3 = 1 + 18 = 19.
The correct answer:
B
Hope this helps!
-Jay
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