Word problems with substitution/elimination

Question

The sum of the digits of a two-digit number is 11. If 45 is added to the number, the result is the number with its digits reversed. What’s the original number?

Answer 1

Let's say our number consists of a and b so the number is ab. We know the sum of the digits is 11 so that means a+b = 11. Let's call this equation 1.

The second part of the problem tells us that if you add 45 to our number (ab), then you get the number with the digits reversed (ba). So let's figure out how to write that in equation form. We know that a is the 10 digit of our number so that means we need to multiply the value of a by 10. Then we can sum the value of b to get our number. For instance, if we have the digits a=2 and b=1 then to get our number we'd say (2*10+1) so our number would be 21. So that means:

10a + b gives us our number

10b + a gives us our number with the digits reversed

Now we have all the information we need to write this second equation:

10a + b + 45 = 10b + a

Let's combine like terms to simplify this equation:

9a + 45 = 9b Let's call this equation 2.

Let's take our first equation of a + b = 11 and solve for a. We get a = 11 - b. Now we can substitute this value into equation 2:

9(11 - b) + 45 = 9b

99 - 9b + 45 = 9b

114=18b

b=8

Great, now that we have b, we can substitute it's value into the first equation to get a.

a = 11 - 8

a = 3

That means our original number is 38. Let's just make sure we didn't make any errors. According to the question, 38 + 45 should equal 83 and it does! So we did this correctly!

Please feel free to reach out to me if you have any additional questions about this.

Answer 2

To make this a little clearer, let's use the number 25 as an example. The number 25 can be described in 2 different ways:

1) as 2 and 5 (if we are talking about digits), and

2) as 20 and 5 (if we are talking about values)

the same applies here, the number we are looking for can be described as:

1) x and y; and

2) 10x and y

therefore:

1) x + y = 11

2) 10x + y + 45 = 10y + x

simplify equation 2,

10x - x = 10y - y - 45

9x = 9y - 45

divide all by 5, then x = y - 5

let's rewrite the 2 equations

1) x = 11 - y

2) x = y - 5

solve:

11 - y = y - 5

2y = 16

y = 8

and x = 3

the original number is xy or 38