SOLUTION: The difference between the two digits of a number less than 100 is 2. If 3/2 times the sum of the digits be diminished from it, the digits will be reversed. Find the number.

Question 1043321: The difference between the two digits of a number less than 100 is 2. If 3/2 times the sum of the digits be diminished from it, the digits will be reversed. Find the number.

Found 2 solutions by josmiceli, KMST:
Answer by josmiceli(19441) (Show Source): You can put this solution on YOUR website!
If I am reading this problem right,
------------------------------
Let the tens digit =
Let the units digit =
-------------------------
(1)
I have to use absolute value because I
don't know which digit is bigger
-------------------------------
The actual value of the number is
(2)
(2)
(2)
(2)
(2)
(2)
The only way for this to be true with whole numbers
is if and
( and the difference between them is )
The number is 75
-------------------------
check:
(2)
(2)
(2)
(2)
(2)
OK
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
THE GUESS AND CHECK OPTION:
"If 3/2 times the sum of the digits be diminished from (the number), the digits will be reversed."
Obviously, the number with the digits reverses is smaller that the original number, so the units digits must be smaller than the tens digit.
(For an algebra solution, we would define variables as
= the tens digit,
= the units digit,
and we would say that ).
Also, the difference between the digits is 2,
so the units digit must be at least 0,
and the tens digit is 2 more than the units digit,
but no more than 9.
That clue narrows the choices.
The number could only be
20, or 31, or 42, or 53, or 64, or 75, or 86, or 97.
That is only eight numbers to try, so guess and check is a reasonable option.
Let's say you try as a first try
The sum of its digits is
.
of that is .
If is diminished from , we get
.
We were expecting to get , which is with the digits reversed, and .
Let's say we try as a second try
The sum of its digits is
.
of that is .
If is diminished from , we get
.
We were expecting to get , which is with the digits reversed, and , but we are getting closer.
Let's say we try as a third try
The sum of its digits is
.
of that is .
If is diminished from , we get
.
We were expecting to get , which is with the digits reversed, and that is exactly what we got, so the original number is .

THE ALGEBRA WAY:
We can set define variables, and set up equations based on the information given.
We define
= the tens digit, and
= the units digit.
From that we can write that
= the value of the number
= the value of the number that results from reversing the digits
= the sum of the digits
= 3/2 times the sum of the digits
, so
= the difference between the digits.
The problem tells us that .
It also says that
.
We have the equations
and that .
That is a system of two linear equations with two variables.
We can solve that.
--->
We can substitute the expression for
in equation .
We get

substituting the value fur in we get
--->
The number is .
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