I like to use letters that remind me of what they stand for.
So I use:

n = the number of nickels
p = the number of pennies

$0.05n = the monetary value of the n nickels
$0.01p = the monetary value of the p pennies
Total value = $0.05n + $0.01p

If the p pennies were turned into dimes, their monetary value would be $0.10p.
If the n nickels were turned into pennies, their monetary value would be $0.01n.
Total value would be = $0.10p + $0.01n.

We set those values equal:

$0.05n + $0.01p = $0.10p + $0.01n

Solve for p in terms of n

Drop the dollar marks and multiply every term by 100 
to remove the decimals:

5n + p = 10p + n

4n = 9p

 = p

So n must be divisible by 9 in order to divide 
evenly into 4n

Then the value is

0 ≦ $0.05n + $0.01p < 2.00

0 ≦ 5n + p < 200

0 ≦ 5n + 4n/9 < 200

0 ≦ 45n + 4n < 1800

0 ≦ 49n < 1800

0 ≦ n < 36.73469388

0 ≦ n < 36

So n can equal 0, 9, 18, 27, 36

And since  = p

So p can equal , , , , 

That is, p can equal 0, 4, 8, 12, 16

So the 5 solutions are:

1. We can have 0 nickels and 0 pennies, and 0 coins. Total = $0.00

If the 0 pennies were turned into 0 dimes, their monetary value
would be 0·$0.10 = $0.00.  And,  
if the 0 nickels were turned into 0 pennies, their monetary value 
would be 0·$0.01. And the total would be $0.00. 

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2. We can have 9 nickels and 4 pennies, and 13 coins. Total = $0.49

If the 4 pennies were turned into 4 dimes, their monetary value
would be 4·$.10 = $0.40.  And,  
if the 9 nickels were turned into 9 pennies, their monetary value
would be 9·$0.01 = $0.09. And the total would be $0.49.

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3. We can have 18 nickels and 8 pennies, and 26 coins. Total = $0.98

If the 8 pennies were turned into 8 dimes, their monetary value
would be 8·$.10 = $0.80.  And,  
if the 18 nickels were turned into 18 pennies, their monetary value
would be 18·$0.01 = $0.18. And the total would be $0.98.

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4. We can have 27 nickels and 12 pennies, and 39 coins. Total = $1.47

If the 12 pennies were turned into 12 dimes, their monetary value
would be 12·$.10 = $1.20.  And,  
if the 27 nickels were turned into 27 pennies, their monetary value
would be 27·$0.01 = $0.27. And the total would be $1.47.

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5. We can have 36 nickels and 16 pennies, and 52 coins. Total = $1.96

If the 16 pennies were turned into 16 dimes, their monetary value
would be 16·$.10 = $1.60.  And,  
if the 36 nickels were turned into 36 pennies, their monetary value
would be 36·$0.01 = $0.36. And the total would be $1.96.    

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Edwin