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.

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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) About Me  (Show Source):
You can put this solution on YOUR website!
If I am reading this problem right,
------------------------------
Let the tens digit = +t+
Let the units digit = +u+
-------------------------
(1) +abs%28+t+-+u+%29+=+2+
I have to use absolute value because I
don't know which digit is bigger
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The actual value of the number is +10t+%2B+u+
(2) +10t+%2B+u+-+%283%2F2%29%2A%28+t+%2B+u+%29+=+10u+%2B+t+
(2) +2%2A%28+10t+%2B+u+%29+-+3%2A%28+t+%2B+u+%29+=+10u+%2B+t+
(2) +20t+%2B+2u+-+3t+-+3u+=+20u+%2B+2t+
(2) +17t+-+u+=+20u+%2B+2t+
(2) +15t+=+21u+
(2) +5t+=+7u+
The only way for this to be true with whole numbers
is if t+=+7+ and +u+=+5+
( and the difference between them is +2+ )
The number is 75
-------------------------
check:
(2) +10%2A7+%2B+5+-+%283%2F2%29%2A%28+7+%2B+5+%29+=+10%2A5+%2B+7+
(2) +75+-+%283%2F2%29%2A12+=+57+
(2) +150+-+3%2A12+=+114+
(2) +150+-+36+=+114+
(2) +114+=+114+
OK

Answer by KMST(5328) About Me  (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
t= the tens digit,
u= the units digit,
and we would say that t%3Eu ).
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 42 as a first try
The sum of its digits is
4%2B2=6 .
3%2F2 of that is 3%2F2%29%2A6=9 .
If 9 is diminished from 42 , we get
42-9=33 .
We were expecting to get 24 , which is 42 with the digits reversed, and 33%7D%7D+%7D+is+much+larger+that+%7B%7B%7B24 .
Let's say we try 64 as a second try
The sum of its digits is
6%2B4=10 .
3%2F2 of that is 3%2F2%29%2A10=15 .
If 15 is diminished from 64 , we get
64-15=49 .
We were expecting to get 46 , which is 64 with the digits reversed, and 49%7D%7D+%7D+is+larger+that+%7B%7B%7B46 , but we are getting closer.
Let's say we try 75 as a third try
The sum of its digits is
7%2B5=12 .
3%2F2 of that is 3%2F2%29%2A12=18 .
If 18 is diminished from 75 , we get
75-15=57 .
We were expecting to get 57 , which is 75 with the digits reversed, and that is exactly what we got, so the original number is highlight%2875%29.

THE ALGEBRA WAY:
We can set define variables, and set up equations based on the information given.
We define
t= the tens digit, and
u= the units digit.
From that we can write that
10t%2Bu= the value of the number
10u%2Bt= the value of the number that results from reversing the digits
t%2Bu= the sum of the digits
%283%2F2%29%28t%2Bu%29= 3/2 times the sum of the digits
, so
abs%28t-u%29=t-u= the difference between the digits.
The problem tells us that t-u=2 .
It also says that
.
We have the equations
t-u=2 and that 10t%2Bu-%283%2F2%29%28t%2Bu%29=10u%2Bt .
That is a system of two linear equations with two variables.
We can solve that.
t-u=2---> t=u%2B2
We can substitute the expression u%2B2 for t
in equation 10t%2Bu-%283%2F2%29%28t%2Bu%29=10u%2Bt .
We get
10%28u%2B2%29%2Bu-%283%2F2%29%28u%2B2%2Bu%29=10u%2Bu%2B2
10u%2B20%2Bu-%283%2F2%29%282u%2B2%29=11u%2B2
11u%2B20-%283%2F2%29%2A2%28u%2B1%29=11u%2B2
20-%283%2F2%29%2A2%28u%2B1%29=2
20-3%28u%2B1%29=2
20-3u-3=2
20-3-2-3u=0
15-3u=0
3u=15
u=15%2F3
highlight%28u=5%29
substituting the value 5 fur u in t=u%2B2 we get
t=5%2B2---> highlight%28t=7%29
The number is highlight%2875%29 .