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

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20 Nov 2014 Number of Balls

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

Option 5)

Correct Option: 5 Best Solution (26)

Shreya Tiwari   10 years AGO

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

Like? Yes (25) | No (8)  

Kushagra Tiwari   10 years AGO

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

Like? Yes (19) | No (7)  

Dr.Dipin Singh   10 years AGO

If there are n rows in triangle of balls, then
n(n+1)/2 + 456 = (n-8)^2
solving it, we get n=49
No of initial balls = 49*50/2=1225

Like? Yes (6) | No (3)  

CHANDA VIJAYALAKSHMI   10 years AGO

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

Like? Yes (5) | No (3)  

CHANDAN KUMAR   10 years AGO

let side=n
no of balls=n/2(n+1)
n/2(n+1)+456=(n-8)^2
n=49
no of ballls=n/2(n+1)=49x25=1225

Like? Yes (1) | No   

purushothama   10 years AGO

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

Like? Yes (5) | No (4)  

Jayanta Dutta   10 years AGO

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

Like? Yes (7) | No (7)  

MAHARAJAN.G   10 years AGO

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

Like? Yes (4) | No (4)  

santosh kumar patel   10 years AGO

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

Like? Yes (5) | No (5)  

Ayushi Gupta   10 years AGO

as 1st row has 1 ball, second has 2 so, nth row will have n balls
total initial ball = (1+2+3+4..n) = n(n+1)/2
side of square = n-8 (n = side of a triangle)
area of square = (n-8)^2
so, n(n+1)/2 + 456 = (n-8)^2
n = 49
total initial balls = (49)(50)/2 = 1225

Like? Yes (3) | No (3)  

lakshmi   10 years AGO

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

Like? Yes (4) | No (4)  

jayalakshmi   10 years AGO

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

Like? Yes (4) | No (4)  

Anoop Tiwari   10 years AGO

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

Like? Yes (3) | No (4)  

P.Pandeeswari   10 years AGO

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

Like? Yes (3) | No (4)  

Anand   10 years AGO

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

Like? Yes (4) | No (5)  

ravi prakash   10 years AGO

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

Like? Yes (4) | No (5)  

bhagya   10 years AGO

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

Like? Yes (3) | No (4)  

RAKESH   10 years AGO

equate no. of balls
n(n+1)/2 + 456 = (n-8)^2

=> n^2 -33n -784 = 0
=> (n-49)(n+16) = 0
=> n = 49

initially n*(n+1)/2 = 49*25 = 1225 balls

Like? Yes (4) | No (6)  

BHASKAR MANDAL   10 years AGO

let there are n balls on each side of the triangle then we can write
n(n+1)/2 + 456 = (n-8)^2
n^2 + n + 912 = 2n^2 - 32 n + 128
=> n^2 -33 n - 784 = 0
=>n^2 - 49n + 16 n -784 = 0
=> n = 49.....we will take the positive value
to total number of balls = 49*50 / 2 = 49 * 25 = 1225
so the option is 5

Like? Yes (4) | No (6)  

Sundar Kumar   10 years AGO

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

Like? Yes (2) | No (4)  

Tark Patel   10 years AGO

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

Like? Yes (2) | No (4)  

Chirag Ranjan Lahiri   10 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 (2) | No (4)  

Abhishek Vishwakarma   10 years AGO

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

Like? Yes (2) | No (4)  

Sourabh Prasad   10 years AGO

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

Like? Yes (2) | No (4)  

Parag   10 years AGO

Number of balls in a triangle of length n = n(n+1)/2
and number of balls in square of length n = (n)*(n)
now from the given information we can say
number of balls in triangle of (n-1) length + 456 = number of balls in square of length [n-8]

that is n(n+1)/2 + 456 = (n-8)*(n-8)

u get a final quadratic eqn n^2 -33n - 784=0
finally n=49
this the length of one side of triangle.
total number of balls = 49*50/2=1225

Like? Yes (2) | No (4)  

Surit Datta   10 years AGO

let each side of the equilateral triangle be n

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

by solving this

n = 49

n(n+1)/2 = 49*50/2

=49*25

=1225

hence initial number of balls = 1225

Like? Yes (1) | No (5)