# The sum of the digits of a number of three digits is 14, the ten's digit being 4 more than the unit's digit. if 99 is subtracted from the number, the digits will be reversed. find the number.

1
by charrieacedo

2014-12-29T18:06:19+08:00

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Well, it looks like this will really take a very long time to solve. So, I think, I am going to make the linear equation of 2 variables write away. Let x be the units digit of the 3 digit number:

Equation:

1.)
a.) y + 10 ( x + 4 ) + x = 14
b.) { [ 100y + 10 ( x + 4 ) + x ] - 99 } = y + 10 ( x + 4 ) + 100x

a.) y + ( x + 4 ) + x = 14
y + x + 4 + x = 14
y + 2x + 4 = 14
y + 2x = 14 - 4
y + 2x = 10
y + 2x - 10 = 0
2x - 10 = -y

-2x + 10 = y

b.) { [ 100 y + 10 ( x + 4 ) + x ] - 99 } = y + 10 ( x + 4 ) + 100x
{ [ 100y + 10x + 40 + x ] - 99 } = y + 10x + 40 + 100x
[ ( 100 y + 11x + 40 ) - 99 ] = y + 110x + 40
100y + 11x + 40 = y + 110x + 40 + 99
100y + 11x + 40 = y + 110 x + 139
100 ( - 2x + 10 ) + 11x + 40 = -2x + 10 + 110x + 139
-200x + 1, 000 + 11x + 40 = 108x + 10 + 139
-189x + 1, 040 = 108x + 149
-189x + 1, 040 - 149 = 108x
-189x + 891 = 108x
891 = 108x + 189x
891 = 297x

3  = x
x = 3 = unit's digit
x + 4 = 3 + 4 = 7 = ten's digit
-2x + 10 = -2 ( 3 ) + 10 = -6 + 10 = 4 = y = hundred's digit

Hundred's digit = 4
Ten's digit = 7
Unit's digit = 3

Number formed = 473

Check:

1.) Ten's digit is 4 more than the unit's digit:
3 + 4 = 7
7 = 7, CORRECT!!!

2.) The number formed - 99 = digits will be reversed:
473 - 99 = 374
473 and 374, REVERSED!!!, CORRECT!!!