Number Theory Questions...?

Question

The Sum of the digits of a two-digit number is 14. If the digits are reversed, the new number is 18 less thatn the original number. Find the original number.

I know the answer to this one is 86 by a wild guess but how do you work out this question? I need an explanation
and..

The ratio of the tens digit to the units digit of a two digit number is 1:4. If the digits are reversed, the sum of the new number and the original number is 110. Find the original number.

I also need an explanation on this thanks

Answer 1

Let's assume the first digit of the number is x and the second number of the digit is y

sum of the digits is 14....x+y=14
solve for x; x = 14 - y

To get the actual value of the number you must multiply the tens digit by ten and then add it to the units digit:
original number....10(x) + y
reversed digits.... 10(y) + x

18 less than original....[10(x) + y] - 18

whole equation:
10(y) + x = [10(x) + y] - 18

plug 14 - y in for all x's and solve

10(y) + (14-y) = 10(14-y) + y - 18
10y + 14 - y = 140 - 10y + y - 18
9y + 14 = 122 - 9y.....subtract 14 from both sides
9y = 108 -9y........add 9y to both sides
18y = 108...........divide by 18
y = 6; plug y back into original equation of x + y = 14
x + 6 = 14
x = 14 - 6 = 8
originial number is 86.
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Assuming that the first digit is x and the second is y:
ratio of the tens digit to the units digit is 1:4
expressed as an equation that would be; x = 4y

original number equation would be; 10(x) + y
reversed digits equation; 10(y) + x
sum of original number and reversed number is 110
[10(x) + y] + [10(y) + x] = 110; now plug 4y in for the x's
10(4y) + y + 10y + 4y = 110; simplify
40y + y + 10y + 4y = 110
55y = 110; divide by 55
y = 2; plug y into original equation x = 4y
x = 4(2) = 8
originial number 82

Answer 2

Simply solve the simultaneous equations:

x+y=14
10y+x=(10x+y)-18

The second problem is solved the same way.