What is 5.618 repeating as a fraction?

Answer 1

I assume you mean the fraction for 5.618618618...

5.618618618... = 5206/333

= 1871/333

5.618618618... = 5 + 0.618618618...

So to solve the fractional bit 0.618618618....

let x be the fraction for the decimal; then:

x = 0.618618618....

multiply both sides by 1000:

1000x = 618.618618618....

Subtract the first from the second:

1000x - x = 618.618618618... - 0.618618618...

⇒ 999x = 618

⇒ 333x = 206

⇒ x = 206/333

which has solved the fractional bit, now the whole is:

5.618618618... = 5 + 0.618618618...

= 5 + 206/333

= 5206/333