Help between a Compound Interest and a Simple interest loan formula

My question if you could kindly assist me is base on the following scenario

How do we know determine the difference between a compound interest loan and a simple interest loan if both must be paid at an APR of 10% with a term of 5 years and a principal of $20000.00

If I remember well the formula for SI is principal*rate%*time and that for
CI is principal*(1+rate%)raised to the power time.

After 1 year my balance is $22000 at the above rate with any of the above formula.
If I make an annual payment of $5000.0 since no daily percentage rate was provided my balance must be $17000.0 carried over into year 2 with either SI or CI formula.

Using CI formula my new balance is $17000.0 (1+0.1) which is $18700.0 carried over to year 3 without making any payments the second year. If I made a payment of $5000.0 in year 2 then my balance is $13700.0 carried over into year 3.

Now this is where I need help

1. What is my new balance with SI carried over into year 3 without me making any payments the second year?
2. What is my balance carried over into year 3 after making $5000.0 payment?
3. What formula do we use to determine the outcome on a simple interest loan?

I know what my balance after 1 year is but I can't use this balance to compute the new interest because that would be based on the rules of compounding. This is why I am lost, correct me if I am wrong.

I'm not sure if I'm interpreting your question correctly, but this might help:
Whenever you are paying off a loan that is using a simple interest method (for example, hire purchase), the interest is always calculate based on the amount of the initial loan.
For example, a home loan of $500,000 will earn HEAPS of interest at the start of the loan, but when it gets down to the last few payments when, say, only a few grand is left to pay, not much interest at all is being generated by this loan.
Conversely, a simple interest loan of the above amount (this would never happen, but just for my example's sake) would be generating just as much interest in the last few payments as it would be at the start.
This is why hire purchase is a sucky way to buy stuff:p
So to answer your question (i think) when you do simple interest calculations after a payment or two, you will still use the normal simple interest formula, but the principal will be $20,000 still, not however much is actually left to pay at the time of calculation.
Does that help?

Really nice thread and information , thank you guys it really helped me out.

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I've never heard of a real life example of something using simple interest. I've just encountered it as a stepping stone to compound interest. Could you explain a "hire purchase" more?

Hire purchase is when you buy something in retail, but don't have to pay straight away. For example, you've probably seen those ads on TV (usually for expensive things like furniture, TVs, Hifi stuff etc) where they say things like "Buy now, pay later!" or "Pay no interest for 3 years!"
It's basically a scheme where you get the product (say a TV) straight away, and take out a loan from the shop*, which you repay later. It sounds good in theory, because you can get the TV even if you don't have enough money for it, and you can pay it off in little bits as you go, rather than having to save up a big amount for it.
But the catch is that they use simple interest for it. So for a $2000 TV, when you've paid off half of the loan, you still have to keep paying them interest as though you owe them the full amount. The interest from the loan is usually so great that it comes close to equalling the actual amount of the product. Buying a $2000 TV on hire purchase would probably end up costing you $3500-$4000 eventually.
Oh and the other catch is, if at any stage you can't make a payment, the company has the right to take the product off you, and keep all of the payments you've made so far. So you could have already paid back 90% of the loan, then you miss one payment, and you lose your TV and all the money you've already given them.
So basically hire purchase sucks, but people get sucked in by it because they get to have that fancy TV now, even though they can't afford it and don't want to have to save up for it.

Hope that answers your question.
cheers,
metal

*Usually it will actually be a separate finance company that you borrow from, which is an affiliate of the company selling you the product.