Balls are arranged in rows to form an equilateral triangle. The first row consists of one ball,

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16 Mar 2010 Balls in equilateral triangle?

Balls are arranged in rows to form an equilateral triangle. The first row consists of one ball,
the second of two balls, the third of 3 balls and so on. If 456 more balls are added then all
the balls can be arranged in the shape of a square and each of its sides then contains 8 balls
less than the balls in each side of the triangle. Determine the initial number of balls.

OPtion

1) 913
2) 1144
3) 1065
4) 988
5) 1225
6) 840
7) 769
8) 700
9) 633
10) 568

Solution

Let number of balls in a side of the triangle are n
Thus total number of balls are
x = 1 + 2 + 3 +........+(n-2) +(n-1) + n (1)

or x = n + (n-1) + (n-2) +.......+ 3 + 2 + 1 (2)

adding (1) and (2)

2x =(n+1) +(n+1) + (n+1)+.......+(n+1) +(n+1)+(n+1)

or 2x = n.(n+1)

or x =n(n+1)/2

According to given condition n(n+1)/2 + 456 = (n-8)2

On solving or checking the options we get n = 49

Thus x = 49*50/2 = 1225

Correct Option: 5 Best Solution (6)

S. Singh   14 years AGO

balls in triangle of 49 balls in a side=49x25=1225
balls in aside of the square =49-8=41

total balls in the square=41x41=1681
this equals the no.=1225+456=1681

Like? Yes (2) | No (1)  

Dr.Dipin Singh   14 years AGO

If n is number of rows in equilateral triangle, then (n-8) is the number of balls in square arrangement, then

(n/2) ( 2+n-1) + 456 = (n-8)^2

solving this , we get .. n= 49.

hence total no of balls = 49*50/2= 49*25 = 1225

Like? Yes (2) | No (1)  

Amar   14 years AGO

Let x is number of rows in triangular arrangement , then (x-8) is the number of balls in square arrangement after adding 456 balls , then

(x/2)(2+x-1) + 456 = (x-8)^2

solving this , we get .. x= 49.
and
number of initial balls = 1225

Like? Yes (2) | No (1)  

Jagat   14 years AGO

Let x is number of rows in equilateral triangle, then (x-8) is the number of balls in square arrangement after adding balls , then

(x/2)(2+x-1) + 456 = (x-8)^2

solving this , we get .. x= 49.

So number of initial balls = (49-8)^2-456 = 41^2-456 = 1225

Like? Yes (2) | No (2)  

sivakumar   14 years AGO

1+2+3+...+49=1225
1225+456=1681
sqrt of 1681 is 41
41 is 8 less than 49

so initial number of balls are 1225.

Like? Yes (1) | No (2)  

anand balaji   14 years AGO

(N+456)= (b-8)^2

N = Total number of ball
b = side of a triangle

Like? Yes (1) | No (3)