peter has written down four natural numbers . if he choose three of his numbers at a time and adds up each triple he obtains totals of 186, 206, 215, and 194. The largest number peter has written is 

(A) 93      (B) 103   (C) 81  (D) 73

Dear Student,

Let a, b, c, d be the four natural numbers.

$\Rightarrow$a + b + c = 186 ................ (1)
$\Rightarrow$a + b + d = 206 ................ (2)
$\Rightarrow$b + c + d = 215  ............... (3)
$\Rightarrow$a + c + d = 194  ............... (4)

Now, (2) - (1) $\Rightarrow$
$\Rightarrow$(a+b+d) - (a+b+c) = 206 - 186 
$\Rightarrow$d - c = 20       ................... (5) 

From (1) $\Rightarrow$
$\Rightarrow$b = 186 - a - c 
Substituting this in (3)
$\Rightarrow$186 - a - c + c + d = 215
$\Rightarrow$d - a = 29 ............. (6) 
Now (4) $\Rightarrow$
$\Rightarrow$a + c + d = 194 
$\Rightarrow$a + d - 20 + d = 194         (From (5) )
$\Rightarrow$a + d = 194 + 20
$\Rightarrow$a + 2d = 214  ...................... (7)
Adding (6) and (7) $\Rightarrow$
$\Rightarrow$d - a + a + 2d = 29 + 214
$\Rightarrow$3d = 243
$\Rightarrow$d = 81
(5) $\Rightarrow$
$\Rightarrow$d - c = 20
$\Rightarrow$c = 81 - 20
$\Rightarrow$c = 61

(6) $\Rightarrow$
$\Rightarrow$d - a = 29
$\Rightarrow$a = 81 - 29
$\Rightarrow$a = 52
(1) $\Rightarrow$
$\Rightarrow$a + b + c = 186
$\Rightarrow$52 + b + 61 = 186 
$\Rightarrow$b = 186 - 52 - 61
$\Rightarrow$b = 73
Therefore, a = 52 , b = 73 , c = 61 and d = 81

Hence, largest number is 81

Regards