Question

Some amount of billiard balls were arranged in an equilateral triangle. When the same set of billiard balls were arranged into a triangle in which each side has one more ball than in the first arrangement, there were 7 balls shortage. How many balls were in the set initially?

A) 3 balls
B) 5 balls
C) 13 balls
D) 16 balls

Answer 1

The initial set of billiard balls had 3 balls arranged in an equilateral triangle.

Step-by-step explanation:

Let's suppose that the initial arrangement of billiard balls formed an equilateral triangle with n balls on each side. The total number of balls in this arrangement would be the sum of the arithmetic sequence 1 + 2 + 3 + ... + n, which is equal to n(n + 1)/2.

According to the given information, when the same set of balls were arranged into a triangle with each side having 1 more ball, there was a shortage of 7 balls. This implies that the total number of balls in the second arrangement is n(n + 1)/2 - 7.

Setting up an equation, we have n(n + 1)/2 - 7 = n(n + 1)/2. Simplifying this equation, we get n(n + 1)/2 = 7. Solving for n, we find that n = 3.

Therefore, the initial set of balls had 3 balls. So, the correct answer is A) 3 balls.