Lesson Sum of Digits Problems

Problem: We are told that a certain two-digit number has a value that is three times the sum of its digits. Also, the units digit is one more than tree times the tens digit. Find the number.
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Solution: Solving "sum of digits" problems hinges on remembering that any "number" is simply a visual representation of the number of ones, the number of tens, the number of hundreds, etc. A two-digit number, for example 23, really means 10*2 + 1*3 = 23.
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To solve this problem, let x = tens, and y = ones. We would write the number as xy = 23, Each x is valued at 10x, and y is valued at 1. So the representation 23 would mean 10*2 + 3*1.
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Note well that, for this problem, when we show 'xy', we only mean 'x' sitting next to 'y', not x times y.
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We also are told the two-digit number, xy, that has a value '3 times the sum of its digits'. That means the value of the expression is 3 * sum of digits.
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The sum of digits is simply x + y. So, 3 times the sum of digits is just 3*(x+y) or 3(x+y).
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Since we know each x is worth 10 and each y is worth 1, then we can say the numeric value is:
10x + y = 3 * (x + y)
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At this point we have only one equation, but it has two unknowns. That is unsolvable.
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But looking back at the problem we see another relationship is given. The units digit (y) is one more (+1) than 3 times the tens digit (3x). So, y = 3x+1
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Now we can substitute that fact into our equation.
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10x + 3x+1 = 3 * (x + 3x+1)
10x + 3x +1 = 3x + 9x + 3
13x + 1 = 12x + 3
x = 2
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We know from the setup that:
y = 3x + 1
y = 3*2 + 1
y = 7
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Now we have a solution: the 2-digit number is xy = 27.
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Again, remember this is not x times y, but just x beside y.
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Always check your work to determine the proposed solution is the correct answer.
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What is the sum of digits? 2+7 = 9
What is 3 times the sum of digits? 3*(2+7) = 27
Is that the same as xy? Yes.
So the solution is the answer we were seeking.
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Answer: The number is 27.