Question 456236: The sum of the digits of a 3-digit number 15. If the tens and the units are interchanged, the original number will be 9 more than the new number. If the units and the hundreds digits are interchanged, the new number will be 99 more than the original number. What is the original number?
Answer by ptaylor(2198)   (Show Source):
You can put this solution on YOUR website! Let x= the hundreds digit
and let y=the tens digit
and z=the units digit
Now we are told that x+y+z=15----eq1
We know that the original number is:
100x+10y+z
If the tens and units digits are interchanged, the new number will be:
100x+10z+y
Now we are also told that:
100x+10y+z=100x+10z+y+9 simplify by subtracting 100x+10z+y from each side
100x-100x+10y-y+z-10z=9 collect like terms
9y-9z=9 or
y-z=1-------------------------------eq2
If the units and hundreds digits are interchanged, we have:
100z+10y+x and this number is 99 more than the original number, so:
100z+10y+x=100x+10y+z+99 simplify by subtracting 100x+10y+z from each side
100z-z+10y-10y+x-100x=99
99z-99x=99 or
z-x=1------------------------------eq3
From eq2, y=1+z and from eq3, x=z-1
substitute these values into eq1 for for x and y and we get:
z-1+z+1+z=15
3z=15
z=5
Now from eq2:
y-5=1
y=6
and from eq3
x=5-1=4
So the original number is:
465------------------------------ans
CK
465 is 9 more than 456, and
564 is 99 more than 465
Hop this helps---ptaylor
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