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1. For the most part, no. As long as you don't have upwards of ten cards or more (If it ends up that way, you may have issues), it won't hurt your score as long as you keep making the payments on time. But something between 5-8 cards total with differing credit limits is average among U.S. consumers. Once you hit 8+ cards though, your available credit to income ratio (the amount of credit you have vs. your annual income) will start going too high, and your score will start to fall.
2. It again depends... if you have somewhere between 5-8 cards total, then you should leave the cards open with small ($10-20) balances on them, or use them but only for small purchases you intend to pay off every month. Closing accounts negatively affects your score because it shows the credit bureaus that you're not handling your debt properly. But, if you do have 8+ cards out there, then you would want to close your newest account firsts, because closing the oldest accounts hurts your score the most, so closing those new cards might be okay in the long run, if you have lots of cards. If you don't have that many cards yet, then leave them open for now.
The best strategy to use if you're serious about paying it off is that when you do transfer the balance to a 0% offer card, take the card and put it in a desk drawer or safe deposit box. Then instead of making minimum payments as the bills come, add as much as you can afford to the bill. And since your card is locked in your desk, you won't be tempted to buy with it. And then try and pay cash or only use a debit card straight out of your checking account so that you don't rack up more debt unless it's absolutely neccessary.
Just make sure those payments are made ON TIME. Because no strategy will work if you make just one late payment since all your rates will rise to the delinquency APR, usually over 20%. On time is crucial.
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