SOLUTION: The sum of the digits in a certain three digit number is 11. The difference of the first and the third digits is one less than the middle digit. If the hundreds and tens digits are

Algebra ->  Coordinate Systems and Linear Equations  -> Linear Equations and Systems Word Problems -> SOLUTION: The sum of the digits in a certain three digit number is 11. The difference of the first and the third digits is one less than the middle digit. If the hundreds and tens digits are      Log On


   

Question 1166205: The sum of the digits in a certain three digit number is 11. The difference of the first and the third digits is one less than the middle digit. If the hundreds and tens digits are interchanged, the number is reduced by 90. Find the number.
Found 2 solutions by josgarithmetic, greenestamps:
Answer by josgarithmetic(39621) About Me  (Show Source):
You can put this solution on YOUR website!
100h%2B10t%2Bu, the number

h%2Bt%2Bu=11%2Ch-u=t-1-----not yet finished analyzing the whole description.
-
h=u%2Bt-1
u%2Bt-1=h
t%2Bu-h=1

Expressing the system of those two equations again as:
system%28h%2Bt%2Bu=11%2Ch-t-u=-1%29
ADD corresponding members.
2h=10
highlight_green%28h=5%29-----------the hundreds digit appears to be 5.

Looking at the original two-equation system again sub for h
5%2Bt%2Bu=11%2C5-u=t-1
-
highlight_green%28t%2Bu=6%29


-----------------------------------------------------------------
If the hundreds and tens digits are interchanged, the number is reduced by 90.
--------------------------------------------------------------

100t%2B10h%2Bu%2B90=100h%2B10t%2Bu

100t%2B50%2Bu%2B90=500%2B10t%2Bu
100t%2B140=500%2B10t
90t=500-140
9t=50-14
9t=36
highlight_green%28t=4%29

.
.
highlight_green%28u=2%29

Original Number, highlight%28highlight%28542%29%29

Answer by greenestamps(13203) About Me  (Show Source):
You can put this solution on YOUR website!


I didn't look closely at the solution by the other tutor. There are many ways to solve the problem using formal algebra; that solution shows one of them.

If a formal algebraic solution is not required, a bit of logical reasoning and some simple mental arithmetic will find the answer quickly.

Logical analysis should easily show that, since switching the hundreds and tens digit reduces the number by 90, the hundreds digit is 1 more than the tens digit.

Then the requirement that the sum of the digits be 11 leaves only a very few possibilities for the number:
218
326
434
542
650

In only one of those numbers is the other requirement satisfied -- that the difference between the first and third digits is one less than the middle digit: 542.

ANSWER: 542