Question

each of the 33 cards is printed with. different number .the largest number is 48.when the number are added two at a time the sum are 64 ,76and 84.what is the smallest number.

Answer 1

As each of the 33 cards are printed with a different number, the largest number is 48.
and the numbers when added two at a time, the sums are 64, 76,and 84.

∴ let the numbers be a,b and c.

so, a + b = 64 -----------eq (i)
     a + c = 76 ------------eq (ii)
     b + c = 84 ------------eq (iii)

Now we can add equation i, ii, iii 
and get

a + b = 64
a + c = 76
b + c = 84
+    +    +
2a + 2b +2c = 224

2 ( a + b + c ) = 224

a + b + c = 224/2

∴ a + b + c = 112 ---------------------------eq (iv)

∴ Now we can subtract equation (i) from equation (iv)
 
and get

a + b + c = 112
a + b      =    64
-    -            -
c = 48

and when we subtract equation (ii) from equation (iv) we will get

a + b + c = 112
a    +    c =  76
-          -      -
b = 36

∴ in the same manner when we subtract equation (iii) from equation (iv)

we will get

a + b + c = 112
     b + c =    84
-        -      -
a = 28

∴ the smallest number is 28.