Oh I get it, Brody has $0.20 worth of pennies and $0.20 worth of nickels. 
So he has 20 pennies and 4 nickels, 20 cents worth of each. Indeed that IS 5
times as many pennies as nickels.  Hahaha! 

And then, when you asked:

You aren't necessarily talking about Brody.  Anybody else could have:

1. what Brody had, 4 nickels and 20 pennies.
2. what somebody else could have, 2 nickels and 10 pennies. 
3. what somebody else could have, 3 nickels and 15 pennies.


I thought at first you meant that he had $0.20 worth of pennies and nickels
together!!

So here's what I posted before it occurred me what you meant:

               p = 5n
   0.01p + 0.05n = 0.20

Substitute 5n for p:

0.01(5n) + 0.05n = 0.20
   0.05n + 0.05n = 0.20
           0.10n = 0.20
               n = 0.20/0.10
               n = 2

               p = 5n
               p = 5(2)
               p = 10 

This proves there is only one way, 2 nickels and 10 pennies, not three.

If he had only 1 nickel, he'd need 15 pennies to have 20 cents.  But
15 pennies is more than 5 times the number of nickels.  So 1 nickel won't do.

If he had 3 nickels, he'd need have only 5 pennies to have $0.20.  But
5 pennies is way less than 5 times the number of nickels.  So 3 nickels won't do.

Sorry, but there is only one way, 2 nickels and 10 pennies.

Then I changed it to what is written at the top.

Edwin