Question 184700: the sum of 4 numbers got by aDDING,SUBTRACTING,DIVIDING AND MULTIPLYING 2 NUMBERS IS 441. FIND THE 2 NUMBERS
Answer by J2R2R(94)   (Show Source):
You can put this solution on YOUR website! Let the first number be X and the second number be Y.
We have
X + Y ..........................(1)
X – Y ..........................(2)
X times Y ......................(3)
X / Y ..........................(4)
Since the sum is a whole number, then X has to be a multiple of Y (for X/Y to be an integer).
One of the most obvious choices is if X and Y are equal, so let us try it:
the first would be 2X
the second would be 0
the third would be X^2
the fourth would be 1
Solve X^2 + 2X + 1 = 441
(X + 1)^2 = 441
X + 1 = +/- 21
Therefore X = 20 (or -22), and Y is the same as X, which gives:
X = 20, Y = 20
or
X = -22, Y= -22
20+20; 20-20; 20x20; 20/20 : 40 + 0 + 400 + 1 = 441
-22-22; -22--22; -22x-22; -22/-22 : -44 + 0 + 484 + 1 = 441
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Here is a more in depth solution where X and Y are not assumed equal:
You can do it without supposing they are equal but remember since X/Y is whole, X is a whole multiple (k) of Y so we could say Let X = kY
(1) would be kY + Y
(2) would be kY – Y
(3) would be kYY = kY^2
(4) would be kY/Y = k
Adding all together would give:
kY^2 + 2kY + k = 441
k(Y^2 + 2Y + 1) = 441
k(Y + 1)^2 = 441
With 441 and (Y + 1)^2 being squared we need k to also be squared and whole.
The factors of 441 which are square are 1, 9, 49 and 441
So by writing the products of 441 as k(Y + 1)^2 using these values for k we have
1 x 21^2 giving k = 1, Y+1 = +/-21; Y = 20 and X = 20 or Y = -22 and X = -22.
9 x 7^2 giving k = 9, Y+1 = +/-7; Y = 6 and X = 54 or Y = -8 and X = -72.
49 x 3^2 giving k = 49, Y+1 = +/-3; Y = 2 and X = 98 or Y = -4 and X = -196.
441 x 1^2 giving k = 441, Y+1 = +/-1, Y = 0 and X = 0 or Y = -2 and X = -882, but we cannot accept 0 as a value for Y since anything divided by 0 is undefined, so only Y = -2, X = -882 is accepted here.
Therefore the solutions are:
X = 20 ; Y = 20 ; X + Y = 40 ; X - Y = 0 ; XY = 400 ; X/Y = 1 ; Total = 441
X = -22 ; Y = -22 ; X + Y = -44 ; X - Y = 0 ; XY = 484 ; X/Y = 1 ; Total = 441
X = 54 ; Y = 6 ; X + Y = 60 ; X - Y = 48 ; XY = 324 ; X/Y = 9 ; Total = 441
X = -72 ; Y = -8 ; X + Y = -80 ; X - Y = -64 ; XY = 576 ; X/Y = 9 ; Total = 441
X = 98 ; Y = 2 ; X + Y = 100 ; X - Y = 96 ; XY = 196 ; X/Y = 49 ; Total = 441
X = -196 ; Y = -4 ; X + Y = -200 ; X - Y = -192 ; XY = 784 ; X/Y = 49 ; Total = 441
X = -882 ; Y = -2 ; X + Y = -884 ; X - Y = -880 ; XY = 1764 ; X/Y = 441 ; Total = 441
So, although k=1 was an obvious guess where X and Y are equal, it simplifies things but guessing 9, 49 and 441 isn’t so obvious.
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