SOLUTION: the sum of the digits is 7. when the digits are interchanged, and the resulting number is subtracted from the original one, the difference is 45. what are the digits?

Question 693762: the sum of the digits is 7. when the digits are interchanged, and the resulting number is subtracted from the original one, the difference is 45. what are the digits?
Found 2 solutions by RedemptiveMath, MathTherapy:
Answer by RedemptiveMath(80) (Show Source): You can put this solution on YOUR website!
These problems can be difficult because they are not like basic problems that want you to find unknown numbers by a given sum or property (e.g. "these numbers are even" or "these are consecutive numbers"). These problems just give you one number that you need to find by dealing with its digits. The first characteristic about this number is that its digits add up to 7. Let's use x and y for our digits. The digit x will be the tens' place, and the digit y will be the ones' place. So,

x + y = 7.

Since the problem didn't tell us that this number has two or three digits, we'll start by trying to find a two-digit number that meets the requirements. The next part may be difficult to understand at first. We need to find another equation that describes the number's other characteristic given in this problem. How can we write in algebraic language the difference of the original number and its interchanged state equaling 45? Here's my equation:

(10x + y) - (10y + x) = 45.

The first parenthetical, (10x + y), describes the original number. There is a certain property of numbers that holds true for any natural case. When we are dealing with digits of natural numbers, a two-digit number can be written as (10x + y). How? Any number that is in the tens' place (1-9) gets there by multiplying by 10. For example, a 9 in the tens' place is any number between 90 and 99. A 4 in the tens' place is any number between 40 and 49. These two numbers get there by multiplying by 10. (Even though the only numbers that can actually be achieved by multiplying these numbers by 10 are 40 and 90, bear with me as I explain the next part). That is why we write 10x. However, since we need to know what the ones' place is (not every number has a 0 in the ones' place), we need to add a number to 10x to get the number. Since x does not have to be the same in the ones place, we use another variable y. That is how a two-digit number can be written as a quantity (10x + y). For examples, 25 can be written as (20 + 5) with x = 2 and y = 5, and 56 can be written as (50 + 6) with x = 5 and y = 6. As you can see, x and y stand for the digits of these numbers. This is what we want since our first equation, x + y = 7, has x and y as the numbers digits. The second quantity, (10y + x), is the interchanged digits. For example, 46 is the interchanged digits of 64. We can notice that when we interchanged digits, all we do is switch the tens' and ones' place (for two-digit numbers). Therefore, we just switch x and y in our quantity.

Now we can finally solve this problem by using substitution or elimination. I will show substitution. Before we can do this, let us simplify the second equation:

10x + y - 10y - x = 45
9x - 9y = 45
9x = 9y + 45
x = y + 5

Now we can just plug what we have found for x into the first equation to solve for y:

y + 5 + y = 7
2y = 2
y = 1

Plugging y = 1 into the first or second equation, we will find what x is:

x + 1 = 7
x = 6

Remember that we still need to place these two numbers side by side to form a two-digit number. We have limited our possibilities to either 16 or 61. Both have its digits' sum of 7. The problem states that the original number minus the interchanged number will give us a value of 45, so the original number must be bigger than the interchanged number. Therefore, our number is 61.
Answer by MathTherapy(10858) (Show Source): You can put this solution on YOUR website!
the sum of the digits is 7. when the digits are interchanged, and the resulting number is subtracted from the original one, the difference is 45. what are the digits?

Let the tens and units digits be T and U, respectively

Then: T + U = 7 ----- eq (i)

Original number = 10T + U, and interchanged number = 10U + T

Therefore, 10T + U - (10U + T) = 45 ---- 10T + U - 10U - T = 45 --- 9T - 9U = 45 ---- 9(T - U) = 9(5) ---- T - U = 5 ---- eq (ii)

T + U = 7 ----- eq (i)
T - U = 5 ----- eq (ii)
2T = 12 ----- Adding eqs (ii) & (i)
T, or tens digit = , or

6 + U = 7 ------ Substituting 6 for T in eq (i)
U, or units digit = 7 - 6, or

You can do the check!!

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