Question 122157: A two-digit number is 11 times its units digit. The sum of the digits is 12. Find the number.
Answer by bucky(2189)   (Show Source):
You can put this solution on YOUR website! To work this problem, you have to think a little bit about what numbers mean.
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You are told that you have a two-digit number. That means the number contains some tens (call
the number of tens T) and some units (call the number of units U).
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This means that the unknown number is written as TU.
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The last part of the problem tells you that the sum of the digits in the number is 12. This
means that T + U = 12.
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Since T is the number of tens in the number and U is the number of units in the number, the
value of the number is T times ten plus U times 1 .... or just 10T + U. [For example,
the number 23 contains 2 tens plus 3 units.]
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So again, the number consists of 10T + U and the problem tells you that this is equal to
11 times U. In equation form this is written as:
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10T + U = 11U
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Now you have two equations with two unknowns:
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T + U = 12
10T + U = 11U
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Let's get the bottom equation in standard form by getting the two terms with the variables
all on one side of the equation. Do that by subtraction 11U from both sides of the equation.
When you do that subtraction the bottom equation becomes:
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10T - 10U = 0
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and the two equations that we have are now:
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T + U = 12
10T - 10U = 0
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Let's plan to solve this by variable elimination. One way we can do that is to multiply
the entire top equation (both sides and all terms) by 10. This will make the top equation
become 10T + 10U = 120 and the pair of equations is then:
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10T + 10U = 120
10T - 10U = 0
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If we then add the two equations vertically in columns, notice that the +10U and the -10U
cancel each other out ... so the terms containing U are gone. The vertical addition results
in the equation:
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20T = 120
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You can solve this for T by dividing both sides of this equation by 20 to get:
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T = 120/20 = 6
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This tells us that the number we are looking for contains 6 tens ... so it is in the sixties.
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We can return to the equation T + U = 12 and since we now know that T is 6, this equation
becomes 6 + U = 12. You can solve this equation for U by subtracting 6 from both sides to
get rid of the 6 on the left side. When you do that subtraction, the result is U = 6.
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So now we know that the number TU is 66.
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Let's check it to see that it satisfies the original problem:
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Is the number equal to 11 times its units number? That means, does the number 66 equal 11 times
the units number of 6. Yes, 11 times 6 is equal to 66.
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Does the sum of its digits equal 12? Yes, 6 + 6 does equal 12.
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So we have checked the problem and we know for certain that the two-digit number we were asked to
find is 66.
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Hope this helps you to understand the problem and a method that can be used for solving it.
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