# Math

sum of the digits of a three-digit number is 20. If the hundreds and unit digits were interchanged, the resulting number is 297 less than the original. Find the number if the hundreds digit exceeds the tens digit by one.

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1. let the unit digit be x
let the tens digit be y
let the hundred digit by (y+1)

y+1 + y + x = 20
2y+ x = 19
x = 19-2y, where both x and y must be whole numbers from 0 to 9.

the number is 100(y+1) + 10y + x
= x + 110y + 100

reverse the hundreds and unit digit:
100x + 10y + y+1
= 100x + 11y + 1

x + 110y + 100 - (100x + 11y + 1) = 297
110y+x+100-100x-11y-1 = 297
99y - 99x = 198
y - x = 2
y - 2(19-2y) = 2
3y=21
y=7

So our number is 875

check:
the digits of 875 add up to 20 ? YES
let's interchange the 8 and 5 ---> 578
what is the difference?
875-578 = 297 , all is good

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