Question

Express 0.55555 in the form of p and q
where P and Q are integers andQ≠0.

Answer 1

I am assuming the question is worn and I am giving the answer for the right ones (1/x +1/y =1/z) (and not 1/3)

First prove similarity for the Triangles PAC and QBC

Angle C is common then angle B = Angle A as mentioned in the question (perpendicular)

Hence triangle PAC and triangle QBC are similar so by using proportionality here we get BC/AC =QB/AP which is nting but b/a+b =z/x hence x=z * a+b/b

Similarly prove triangle RAC is similar to triangle QAB

Hence by proportionality we get a/a+b=z/y

Hence y=z * a+b/a

Hence 1/x = b/z(a+b)......(1)

1/y = a/z(a+b)......(2)

Adding both (1) and (2)

1/x + 1/y = b/ z(a+b) + a/z(a+b)

= b+a/z(a+b) (Since common denominator

= a+b/z(a+b)

= 1/z

And that is what we have to prove .

Hope my explanation is helpful to you

Answer 2

Answer:

Let $x=0.55..$... which is approximately equal to 0.3.

i.e, $x=0.55...=0.5$..................................( 1 )

Multiply 10 on both sides we obtain,

$10 x = 5.555....= 5.5$............................( 2 )

Subtracting equation ( 2 ) with ( 1 ), we get

$9x = 5.000.....= 5$

solving for x, we get

$x=\frac{5}{9}$

Hence we can express 0.55555... in the form of  $\frac{P}{Q}$  where P=5 and Q=9.