Question

The product of two numbers is 48 and their sum is 19.
what are the numbers????

Answer 1

we should algebra to find the numbers logically.  not use trial and error or guessing game.

numbers  x, y.
x + y = 19       ---- (1)
   x y = 48

(x - y)² = (x + y)² - 4 xy  = 19² - 4*48 =  361 - 192 = 169
so     x - y = 13   ---- (2)

add   (1) and (2)...  2x = 32     so  x = 16

from (1) , we get    y = 3.

So the numbers are  16 and 3.

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another way:

48 = 1 * 2 * 3 * 4 * 6 * 8 * 12 * 16 * 24 * 48

In the factors on the right side,  find two factors  which will sum to 19.  So we can find  3 and 16.

Answer 2

Let the two numbers be x and y.
xy=48
x+y=19
⇒ y=19-x

⇒x(19-x)=48
⇒19x-x²=48
x²-19x+48=0
⇒(x-3)(x-16)=0
Therefore, x=16 (or) x=3.
which implies,
one number is 3,
and the other number is 16.