Question

How many two digit whole numbers are increased by 18 when their digits are reversed?
@phi

Answer 1

I look at this and think, what can I easily eliminate? So I ask you, what's your best guess? You have to start somewhere, so take some guesses and I'll help point you in the right direction.

Answer 2

The part that I find confusing is when it says about the number being increased by 18, but then asks about how many of them of are increased?

Answer 3

So all this is saying is suppose you have a number like 12 and reverse the digits, you get 21. That's an increase 9.

Answer 4

Can you explain to me more about how is it increased by 9?

Answer 5

12+9=21

Answer 6

Ok. And then what do i do?

Answer 7

I would write an equation.

if you have the "number" ab, it has the value 10a+b
it goes up by 18 if you reverse the digits
10a+b+18 = 10b + a

Answer 8

Then you use this nice little example I've laid out for you to think about it and figure it out.

Answer 9

@phi Why is it next to 10 (meaning ab)?

Answer 10

the number 23 has the value 2*10 + 3

Answer 11

Ok

Answer 12

simplify
10a+b+18 = 10b + a

to
a= b-2

now list all digits that b can be, and find the corresponding a

Answer 13

Would it be 9a+18=9b?

Answer 14

ab - ba= 10a + b - 10b - a = 18 --> 9(a-b) = 18 --> a-b=2.

In whole numbers, for example a, with 1 digit, we have totally 10 possible cases (0,1,2,3 ---9)

But a and b cannot be equal to 0, then only 9 possible cases left.

a - b = 2 which means a > 2 also, therefore a only takes 3 --> 9 (7 possible values)

For 7 possible values of a, we have 7 possible values of b. Therefore, totally the answer is we have 7 possible two-digit whole numbers (couples) which reverse for difference of 18

They are: 31 and 13 ; 42 and 24; 53 and 35; 64 and 46; 75 and 57; 86 and 68; 97 and 79

Answer 15

yes, now divide by 9 to get a+2 = b

Answer 16

I guess we have to leave out 02 (+18= 20) because 02 is not a two digit number.

Answer 17

but you can just use b= 0... 9 and toss out the invalid ones.
example
a+2 = 0 -> a= -2 not allowed...

Answer 18

or start with a=0 ... 9 and find b. Again, toss out invalid numbers
a=0--> b=2 but 02 is not allowed because it is not a two digit number.

Answer 19

I don't get about what I should do with a+2=b. Can you example to me this more?

Answer 20

let a=0 ... 9 and find b.
using a+2 = b

Answer 21

Like a=1 + 2 and then b=3.
1+2=3

Answer 22

***
Like a=1 + 2 and then b=3.
1+2=3
***

yes. that means your number is 13
if we add 18 to it we should get 31... and we do. so that is one of the solutions

Answer 23

Oh! Can you help me approach all the solutions?while I do the work though. I don't need answers yet :)

Answer 24

just keep going and writing down what you get... (the answers will be "ab" concatenated together)

Answer 25

Ok thanks

Answer 26

start with b= a+2

make a list of all digits a can be 0,1,2... up to 9
for each a digit, write the "b" digit next to it (b is a+2)
cross off any combination that does not make sense.

Answer 27

@phi you listed "0" as being a possible digit, but remember that's not possible if these are two digit numbers.

Answer 28

yes, we will cross off 0... but in the beginning I would list all possibilities. First do the mechanical part... then do the "thinking" part

Answer 29

Seems like once you've narrowed it down to a+2=b you've really got a nice clean shot by thinking rather than going through any kind of hardship.

a and b are both single digit numbers, so we can finish up by making these extra couple of equations:

0<a<10
0<b<10

And this combined with a+2=b will give you all the information you need.

Answer 30

good point... I was assuming it was self-evident that a and b are restricted to single digits
but it is always good to make that explicit.

Answer 31

Ok so I came up to like 32+18=50. That means that that won't work right?

Answer 32

@phi

Answer 33

How did you get 32 ?

Answer 34

3(10)+2=32

Answer 35

make this table

a b= a+2 ab
1 3 13
2 4 24
and so on...

Answer 36

Thank you soooooo much. I got my answer as 7. And that was correct!

Answer 37

14
A Mathematica solution is attached.