MHB What numbers is Ron thinking of for his Mystery Number?

[Marcelo Arevalo]

Ron is thinking of a 3-digit number less than 500. If he exchanges the ones and hundreds digits, the new number is 396 more. If he exchanges the ones and tens, the new number is 18 more. Find as many numbers as you can of which Ron is thinking.

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I am using a Trial & Error method for this type of problem. its taking too much time.
here it is:
abc - cba = abc + 396
125 - 521 = -396
am i doing this correctly?

in the second situation ;
abc -acb = abc + 18
125 - 152 = -27 (Did not satisfy the second situation) definitely my trial is wrong.. or did I analyzed it correctly?

 

[MarkFL]

I would let Ron's number be:

$$N=100a+10b+c$$

So, first we are told

$$(100c+10b+a)-(100a+10b+c)=396$$

or:

$$99c-99a=396$$

$$c-a=4\tag{1}$$

Next, we are told:

$$(100a+10c+b)-(100a+10b+c)=18$$

or:

$$9c-9b=18$$

$$c-b=2\tag{2}$$

Thus, we know we must have:

$$N=100a+10(a+2)+(a+4)=111a+24$$

Since we know $0<a<5$, can you now find the numbers? :)

 

[Greg]

Alternatively (but not as "tight" as Mark's solution),

(1) 100c + 10b + a = 100a + 10b + c + 396
(2) 100a + 10c + b = 100a + 10b + c + 18

(1) - (2):
90c + 9b - 99a = 378

Divide both sides by 9:

10c + b - 11a = 42
10c + b = 42 + 11a [for a = (1, 2, 3, 4)]

 

[Wilmer]

Hmmm...a can also equal 5, right?
Try 579.

 

[Greg]

abc < 500

 

[Wilmer]

abc < 500

Smarty pants :)

 

[Marcelo Arevalo]

Oh, Got it..
Thanks a Lot.
my answers are:
135
246
357
468
(Handshake)