SOLUTION: The product of the two-digit number and its tens digit is 54. Find the number if the sum of the digits when added to the number gives a result of 36?

Question 164608: The product of the two-digit number and its tens digit is 54. Find the number if the sum of the digits when added to the number gives a result of 36?
Found 2 solutions by stanbon, gonzo:
Answer by stanbon(75887) (Show Source): You can put this solution on YOUR website!
The product of the two-digit number and its tens digit is 54. Find the number if the sum of the digits when added to the number gives a result of 36?
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Let the number be 10t+u where t is the tens digit and u is the ones digit.
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EQUATION:
t(10t+u) = 54
(t+u)+ 10t+u = 36
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Rearrange the equations:
10t^2 + tu = 54
11t + 2u = 36
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u = (36-11t)/2
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Substitute into the quadratic to get:
10t^2 + t[(36-11t)/2] = 54
20t^2 + 36t - 11t^2 = 108
9t^2 + 36t - 108 = 0
t^2 + 4t - 12 = 0
(t+6)(t-2) = 0
t = 2 (the tens digit is 2)
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u = (36-11*2)/2 = 7 (the units digit is 7)
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The number is 27
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Cheers,
Stan H.

Answer by gonzo(654) (Show Source): You can put this solution on YOUR website!
let x = the tens digit.
let y = the ones digit.
number is 10*x + y
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product of the two digit number and its tens digit is 54.
this implies that
x * (10*x + y) = 54 (first equation)
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sum of the digits when added to the number gives a result of 36.
this implies that
(10*x + y) + x + y = 36 (second equation)
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it looks like we can eliminate one of the unknowns by solving for y in the second equation.
second equation again is:
(10*x + y) + x + y = 36
remove the parentheses.
10*x + y + x + y = 36
combine like terms.
11*x + 2*y = 36
subtract 11*x from both sides.
2*y = 36 - 11*x
divide both sides by two.
y = (36 - 11*x)/2
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first equation again is:
x * (10*x + y) = 54
substitute for y.
x * (10*x + (36-11*x)/2) = 54
remove the parentheses by multiplying out.
10*x^2 + (36*x - 11*x^2)/2 = 54
multiply both sides of equation by 2.
20*x^2 + 36*x - 11*x^2 = 108
combine like terms.
9*x^2 + 36*x = 108
subtract 108 from both sides.
9*x^2 + 36*x - 108 = 0
divide both sides by 9.
x^2 + 4*x - 12 = 0
factor the quadratic equation on the left.
(x-2)*(x+6) = 0
roots are either:
x = 2
or
x = -6
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x = -6 doesn't look good since the tens digit needs to be positive.
solving for y anyway to see what happens.
second equation reproduced here is:
(10*x + y) + x + y = 36
substitute -6 for x.
10*(-6) + y + (-6) + y = 36
-60 + y - 6 + y = 36
combine like terms.
-66 + 2*y = 36
add 66 to both sides
2*y = 102
y = 51
impossible since digits must be 0 to 9 only.
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x = 2 might be the answer.
second equation reproduced here again is:
(10*x + y) + x + y = 36
substituting 2 for x.
10*2 + y + 2 + y = 36
multiplying and combining like terms gets
22 + 2*y = 36
subtract 22 from both sides.
2*y = 14
divided both sides by 2.
y = 7
this number is possible.
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tens digit = 2
ones digit = 7
number is 27
27 * tens digit is 27*2 = 54 which satisfies the first equation requirements.
27 + 2 + 7 = 36 which satisfies the second equation requirements.
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number is 27.

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