This is easiest to see if we just pretend that there are 100 kids at the school

So......8%  of 100  = 8   kids bring both

Let the number bringing raincoats   = x

Let the number bringing umbrellas  = 2x

And the number bringing  raincoats  only  = (x - 8)

And the number bringing  umbrellas only    =  (2x - 8)

So we have that

Number bringing raincoats only  +  number bringing umbrellas only +  number  bringing  both  = 100

(x  - 8)   +    ( 2x  - 8)   +   8    =  100   simplify

3x  - 8   = 100     add 8 to both sides

3x  = 108      divide both sides by   3

x  = 36  = number bringing raincoats     ...so....    2 (36)  = 72  = number bringing umbrellas

Number bringing umbrellas only  =  (  2x - 8)  = (2 (72)  - 8 )   = 64  = 64 / 100   =  64%

Actually, it all depends on how you interpret the question. Either we assume that when double the kids bringing umbrellas versus those bringing raincoats is from the remainder. Or as CPhill did, we assume that when double the kids versus those bringing raincoats goes for everyone instead of just those left from the 92%

I think that CPhill is actually correct but here is my alternate way of working if you interpret thing different

Simple Working:

Assumptions:

Make x = the number of kids bringing raincoats only

Make number of kids bringing raincoats only = x

Since twice as many kids only bring umbrellas opposed to raincoat: Number of kids with only umbrellas = 2x

If there were 100 kids then the 8% bringing both would be 8 kids

Working:

So with x kids with raincoats and 2x kids with umbrellas and 8 kids with both

x + 2x + 8 =100

3x + 8 = 100

3x = 100 - 8

3x = 92

x = 92/3

x = 30.66666666...

Since twice as many kids have only umbrellas:

Number of kids with only umbrellas (out of 100) = twice as many kids than raincoats only

2 times (92/3 or 30.66666666...) = 184/3 or 61.33333333...

We can also do it using probability:

Assumptions:

Make number of kids bringing raincoats only = x

Since twice as many kids only bring umbrellas opposed to raincoat: Number of kids with only umbrellas = 2x

8% bring both

Working:

(8% of kids have both)

100% - 8% = 92%

So 92% of kids have either raincoat or umbrella

x + 2x =92%

3x = 92%

x = 92% / 3

x = 30.66666666%

Since twice as many kids have only umbrellas:

Number of kids with only umbrellas (out of 100) = twice as many kids than raincoats only

2 times (30.66666666%) = 61.33333333%

Conclusion:

61.33333333...% of kids bring umbrellas only