Question

Some identical balls are arranged in rows to form an equilateral triangle. The first row consists of one ball, the second row consists of two balls and so on. If 99 more identical balls are added to the total number of balls used in forming the equilateral triangle, then all these balls can be arranged in a square whose each side contains exactly 2 balls less than the number of balls each side of the triangle contains. Then the number of balls used to form the side of the equilateral triangle is:\[\]
A.19\[\]
B.26\[\]
C.22\[\]
D.25\[\]

Answer 1

Hint: We observe that the number of rows is the same as the number of balls at each side of the equilateral triangle. We assume that $n$ numbers of balls required to form the side of the equilateral triangle and get the number of balls at the side of square $n-2$. We add 99 to the total number of balls required to form the triangle and equate to the number of balls in the square in accordance with the question. We solve the equation to find $n$.\[\]

Complete step by step answer:

 We are given in the question that the identical balls arranged in rows to form an equilateral triangle. The first row consists of one ball, the second row consists of two balls, and so on. We cannot form an equilateral triangle with 1 ball but we can form an equilateral triangle with 2 balls whose each side will have 2 balls each. Similarly if we put 3 balls in the third row we can make an equilateral triangle with 3 balls at each side of the triangle. So the number of rows is equal to the number of balls at the side. \[\]

We see that the number of balls required to form an equilateral triangle with 2 balls at each side is $1+2=3$. The number of balls required to form an equilateral triangle with 3 balls at each side is $1+2+3=6$. $1+2=3$. The number of balls required to form an equilateral triangle with $n$ balls at each side is $1+2+3+...=n$ which is the sum of first $n$ numbers, given by $\dfrac{n\left( n+1 \right)}{2}$.\[\]
We are further given that if we add 99 balls we from the square whose side contains exactly 2 balls less than in the side of the equilateral triangle. So if the equilateral triangle has $n$ balls at the side then the number of balls at the side of the square is $n-2$.\[\]
 We also observe here the number of balls at the side is analogous to length of the side and the total number balls in the triangle or the square is analogous to area. We use the are formula of square to find the number of balls in the square which is $\left( n-2 \right)\left( n-2 \right)={{\left( n-2 \right)}^{2}}$. We are given that
\[\begin{align}
  & \dfrac{n\left( n+1 \right)}{2}+99={{\left( n-2 \right)}^{2}} \\
 & \Rightarrow {{n}^{2}}+n+198=2\left( {{n}^{2}}-4n+4 \right) \\
 & \Rightarrow {{n}^{2}}-9n-190=0 \\
\end{align}\]
We split the middle term of the above quadratic equation and get,
\[\begin{align}
  & \Rightarrow {{n}^{2}}-19n+10n-190=0 \\
 & \Rightarrow \left( n-19 \right)\left( n+10 \right)=0 \\
\end{align}\]
We reject the negative value and conclude that $n=19$. So the number of balls required to form the side of the triangle is 19 which is in the option A. \[\]
Note:
 We can also find the number of balls of required to form the whole triangle with $n=19$balls at each side which is $\dfrac{19\left( 19+1 \right)}{2}=19\times 10=190$. We also should not use the formula for the area of an equilateral triangle which is $\dfrac{\sqrt{3}}{4}{{n}^{2}}$ because the square root $\sqrt{3}$ is involved.