Question 622669: The sum of the two digits of a number is 14.When 31 is subtracted from the number, the digits become equal. Find the number.
Answer by math-vortex(648)   (Show Source):
You can put this solution on YOUR website! Hi, there--
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We can use algebra to solve this problem.
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Let x be the units digit.
Let y be the tens digit.
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Now we need to write two equations using the information in the problem to model the situation.
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In algebra, the phrase "the sum of the two digits...is 14" can be written as
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Let's think for a minute about what the digits could be. Of course, the digits in a number are always the whole numbers from 0 to 9. However in this case, the digits can only be from 5 to 9 because their sum must be 14. (This will be important in the next part of the problem.)
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We know that "when 31 is subtracted from the number, the digits become equal." When we subtract two numbers, we subtract the units digits and the tens digits separately. Sometimes we need to regroup in subtraction. We know that we won't need to regroup in this case because the units digit in our number must be at least a 5.
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We want to show that the units digit equals the tens digit after subtracting 31. In other words,
[the difference in the units digits] = [the difference in the tens digits]
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In algebra, we can write this as,
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Now we have two equations. We will use the substitution method to find x and y.
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Rewrite the first equation in a "y=" format.
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Substitute 14-x for y in the second equation.
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Simplify by combining like terms and solve for x.
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Subtract 11 from both sides of the equation.
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Subtract x from both sides of he equation.
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Divide both sides of the equation by -2.
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In the context of this problem, x=6 means that the units digit is 6. Since the sum of the digits is 14, the tens digit must be 8. Our number is 86.
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We need to check that 86 satisfies all the requirements of the problem. When we subtract 31 from 86 we get 55. The digits in 55 are equal so everything checks out.
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That's it. Feel free to email me if you have questions about any part of the explanation.
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Ms.Figgy
math.in.the.vortex@gmail.com
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