Balls of equal size are arranged in rows to form an equilateral triangle. the top most row consists of one ball, the 2nd row of two balls and so on. If 424 balls are added, then all the balls can be arranged in the shape of square and each of the sides of the square contain 8 balls less than the each side of the triangle did. How many balls made up the triangle?  Choose the correct choice from the following answers:  A) 1176  B) 2209  C) 2878  D) 1210  E) 1560

Rationale: as expected , this question boils down to 2 equation , consider total number of balls in triangle = t and number of balls in last row = x . 1 + 2 + 3 + . . . + x = t x ( x + 1 ) / 2 = t - - - - ( a ) as mentioned in the question , side of a square will be ( x - 8 ) and total number of balls in square will be ( t + 424 ) ( x - 8 ) ^ 2 = t + 424 - - - - - ( b ) now the hardest part of the question will be to solve these 2 equations and this looks like time consuming but the easy way will be plug and play . also , we ' ve to find a value of t ( from 5 optiosn given below ) which can make a square of a a number . one we know this , it will be a cake walk . we can see that option a fits this criteria in eq ( b ) . add - 1176 + 424 = 1600 = 40 ^ 2 = ( x - 8 ) ^ 2 hence , x = 48 . cross check by putting in eq ( a ) = x ( x + 1 ) / 2 = t = > 48 * 49 / 2 = 1176 hence , answer is a .
Answer: A