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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
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