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? Choose the correct choice from the following answers.
A) 59  B) 19  C) 21  D) 18  E) 42
We basically have 5 positive numbers - x, y, y, y, x+100 with sum = 750   so 2x + 3y = 650 and x&lt;y&lt;x+100.   Take extreme cases of inequality and we get for x=y case, 5x=650 so x = 130.   Another case, y=x+100, 5x+300 = 650. x=70.   BOTH THESE CASES are not allowed since y lies between x and x+100. But these are the bounds.   Once we have that, it is only a matter of finding the allowed values of x. (650-2x) should be divisible by 3, so changes allowed in x should be only in factors of 3. Start writing numbers from 73, 76....upto 127 and we get 19!   Answer is 19,
<result>B</result>
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?