Question 243221: A certain two-digit number has a value that is three times the sum of its digits. The units digit is one more than tree times the tens digit. Find the number.
Answer by oberobic(2304)   (Show Source):
You can put this solution on YOUR website! Solving "sum of digits" problems hinges on remembering that any number is simply a representation of the number of ones, the number of tens, the numbers of hundreds, etc. A 2-digit number like 23 really means 2*10 + 3*1 = 23.
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Let x = tens, and y = ones. We could say the number xy = 23, Each x is valued at 10y, and y is valued at 1. So the representation 23 would mean x = 2 = 2*10y = 2*10*1 = 20. True.
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In this case we have a 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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N.B. When we show 'xy' in this context, we only mean 'x' sitting next to 'y', not x times y.
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The sum of digits is simply x + y.
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Since we know each x is worth 10 and each y is worth 1, then we can say
10x + y = 3 * (x + y)
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At this point we have only one equation, and it has two unknowns. That is unsolvable. But looking back at the problem we see another relationship is given. y = 3x+1. (The units digit is one more (+1) than 3 times the tens digit (3x).)
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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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So the 2-digit number is xy = 27.
Remember! This is NOT x times y, but just x beside y.
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Now we have to check our work.
What is the sum of digits? 2+7 = 9
What is 3 times the sum of digits? 3*9 = 27
Is that the same as xy? YES!
Done.
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