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Question 525708: The sum of a 3-digit number and a 1-digit number is 217. The product of the numbers is 642. If one number is between 200 and 225, what are the numbers?
Answer by lmeeks54(111)   (Show Source):
You can put this solution on YOUR website! There are two different ways to solve this (three if you count trial and error): one using algebra and one using a quickly built MS Excel spreadsheet (almost all of life can be modeled in Excel!) 8-)...
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We can derive two equations with two unknowns, which are easy to solve:
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a + b = 217
a * b = 642
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There are also two other constraints:
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Either a or b is a 3-digit number, and the other has 1 digit
the 3-digit number is 200 < a < 225
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The classic method for two equations with two unknowns, in one equation, set one of the unknowns in terms of the other and insert that equality into the other equation so you are solving 1 equation with 1 unknown:
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b = 217 - a
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a * (217 - a) = 642
217a - a^2 = 642
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subtract 642 from both sides to restate as a quadratic equation = 0:
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-a^2 + 217a - 642 = 0
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factor this equation:
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(-a - 214)*(a - 3) = 0
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a = -214 or 3
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-214 is a nonsensical answer given our original problem, so we throw out that solution. We are left with a = 3
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Plug a = 3 into the original a + b = 217 equation:
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3 + b = 217
b = 217 - 3
b = 214
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check our work with the other equation:
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a * b = 642
3 * 214 = 642
642 = 642 checks...
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cheers,
Lee
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PS you could have done this faster with Excel. Set up a spreadsheet with 4 columns:
a, b, sum of a + b, product of a * b
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fill the first column with all possible a's from 201 to 224
put a formula for b that equals 217 - a to fill that column
the 3rd column will always be 217
the 4th column will be the product of all possible a's and b's
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look down the product column for 642. It appears where a = 214 and b = 3 (yes, I know, the reverse of how I labeled the variables above)...
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The 3rd method was trial and error. Using some logic, you can eliminate most of the possible solutions so you don't have to check too many.
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Of the 224 possible solutions given the constraint 200 < a < 225, the range of:
a = {201, 202, 203, 204, 205, 206, 207} yields a b that is not a 1-digit number. Those don't work.
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Of the possible solutions with a > 217, b will be 0 (product doesn't work) or a negative number (product will go in the wrong direction)
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Of the remaining potential solutions, any with a high b (b > 4 or 5) are going to produce a really large product of a and b.
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Once you start thinking about this, you get to the solution very quickly.
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8-)
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