Well, I’ll try to explain. As for rationals with a finite decimal expansion, the job is simple. Select p and q from the decimal expansion, here “37818181”, find the next larger potency of 10, and that’s it:

p=37818181,q=100000000

(which cannot be further simplified, because q is only divisible by the primes 2 and 5, and neither 2 nor 5 divide p.)

Or did you mean “what is the p/q form of 0.378181(81)…, i.e. a number with an endlessly repeating sequence of the digits 81? This case is a bit more difficult, but not really complicated.

Let x=0.378181(81)…

Then, 100·x=37.8181(81), and 10000·x=3781.8181(81)…

Subtract 100·x from 10000·x:

9900·x=3781.8181(81)…-37.8181(81)…=3781–37=3744

Now divide by 9900:

x=3744/9900

and you have a p/q form for x. In this case, however, it can be simplified; apparently p and q can both be divided by 4, which gives another possible form:

x=936/2475

which can still be simplified further (because 9 divides 936 as well as 2475):

## Answers ( )

2x/5Step-by-step explanation:-Let the rational number be , x

2/5 of x = 2x/5

P/Q form of 2/5 of a rational number is 2x/5

Well, I’ll try to explain. As for rationals with a finite decimal expansion, the job is simple. Select p and q from the decimal expansion, here “37818181”, find the next larger potency of 10, and that’s it:

p=37818181,q=100000000

(which cannot be further simplified, because q is only divisible by the primes 2 and 5, and neither 2 nor 5 divide p.)

Or did you mean “what is the p/q form of 0.378181(81)…, i.e. a number with an endlessly repeating sequence of the digits 81? This case is a bit more difficult, but not really complicated.

Let x=0.378181(81)…

Then, 100·x=37.8181(81), and 10000·x=3781.8181(81)…

Subtract 100·x from 10000·x:

9900·x=3781.8181(81)…-37.8181(81)…=3781–37=3744

Now divide by 9900:

x=3744/9900

and you have a p/q form for x. In this case, however, it can be simplified; apparently p and q can both be divided by 4, which gives another possible form:

x=936/2475

which can still be simplified further (because 9 divides 936 as well as 2475):

x=104/275 or p=104, q=275

as the final solution.

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