MATHS(PART THREE)

5. "the sum of its digits is 8." => x+y = 8 ...........(1)
"the digits are reversed, the number is increased by 18"
=> yx means 10y + x
=> xy means 10x + y
=> (10y+x) + 18 = (10x+y) .................(2)

Combine (1) and (2)
=> [10(8-x) + x] +18 = [10x + (8-x)]
=> 80-10x+x + 18 = 10x+8-x
=> -18x = -90
ANSWER => x=5,y =3
Check 5+3 = 8, deifference between 35 and 53 = 18

6. Let the orginal number called ab i.e. 10a+b
New number = 10b+a

"a two-digit number is equal to 8 times the sum of its digits."
=> 10a+b = 8(a+b)
=> 2a = 7b ...(1)

"the number formed by reversing its digits is greater than one-third of the original number by 3."
=> 10b+a = 1/3 (10a+b) + 3 ...(2)
=> 9a = 63
=> a=7, b=2
ANS = 72
Check: 72 = 8(2+7) .... (1)
27 = 1/3(72) + 3 => 27 = 24+3 ...(2)

a.
The difference between xy and yx is always divisible by 9. The quotient represent the difference between x and y.
So the answer could be 24/42, 35/53, 46/64, 57/75...
Since the sum of the digits is 8, there is only one answer: 35.

By Algebra:
10y+x=(10x+y)+18
x+y=8
Solve for x and y to give x=3, y=5

b.
Let the number be xy.
Then
10x+y=8(x+y)
10y+x=(10x+y)/3+3
Solve the simultaneous equations to give x=7 and y=2