Question

The sum of the digits of a two digit number is 12 the number formed by interchaning the digits is 54 more than the original number what is the original number?
A 39
B48
C57

Answer 1

let's call the number ab where a is the tens digit and b is the ones digit

Answer 2

a+b=12 (the sum of the digits = 12)

Answer 3

and the number itself can be represented as a*10+b (a gets multiplied by 10 because it is the tens digit)

Answer 4

the interchanged number is b*10+a and we're told that the interchanged number is 54 more than the original

so we have:
a+b=12
10b+a=10a+b+54

that's a system of equations

Answer 5

Wait where did the 54 come from? ._.

Answer 6

the question. do you know what interchanging is?

Answer 7

I'm confused on exactly what it is. /).(\

Answer 8

no shame.

basically if we have a number ab, the interchanged number is ba

so if we have 36, the interchanged version would be 63

Answer 9

and so that's what I mean when I did the thing with times 10.

If our number is 36, it can be written as \((3*10)+6\), aka \(36\) right?

the interchanged number, 63, can be written as \((6*10)+3\) aka 63

Answer 10

I really suck at explaining this but this is the site I used to look this up http://www.onlinemathlearning.com/digit-problems.html

Answer 11

I get a bit of what you're saying! :3 Thank you for helping me. :)

Answer 12

so can you solve the system?

Answer 13

It would be 57 right?

Answer 14

well to check it, compare 57 and the interchanged value
to start, what is 57 interchanged?

Answer 15

5x10+7?

Answer 16

that's just 57, to interchange it, we flip the digits around

Answer 17

OHHHH! So it would be 75?

Answer 18

yeahhh, and so now compare 57 and 75

is 75-57 = 54?

Answer 19

noo.

Answer 20

which means 57 is not our answer
if you're really lazy you could bruteforce your way through and just check each choice like that

Answer 21

It'd be 39! :D

Answer 22

yeah, let me show you how it works if you solve the system tho

Answer 23

okay :)

Answer 24

a+b=12
10b+a=10a+b+54

let's use substitution. we'll start by solving for a: \(a=12-b\)
is that much clear?

Answer 25

Mhmm.

Answer 26

we can then plug it into the second equation.
\(10b+(12-b)=10(12-b)+b+54\) simplify that and we get
\(9b+12=120-10b+b+54\\9b+12=174-9b\\18b=162\\b=\dfrac{162}{18}=9\)

Answer 27

and so that means that our "ones" digit has to be a 9. you can get the tens digit by solving for b and substituting

Answer 28

Ohhh okay.

Answer 29

I can't seem to get a=3, but this calculator agrees with our answer http://www.wolframalpha.com/input/?i=a%2Bb%3D12+%2C10b%2Ba%3D10a%2Bb%2B54

Answer 30

Okay! :D Thank you so much! <3

Answer 31

\(\huge \color{pink}{\heartsuit}\) <3