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1) Which would be the lesser of two evils: closing the account and having my credit score sink, or paying the higher interest rate? I'm pretty confident that I can pay off half or more of the balance by the end of this year, and definitely all of it within two years. But money is tight, and I'd like to avoid the higher APR if I can.
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I don't think you have to close it or pay the higher interest rate. With a 700+ credit score you should be able to raise the limit on one of your other cards and balance transfer to it or apply for a new card with a limit almost high enough to take that entire balance. Have you asked how your minimum payments are calculated; for some cards the APR doesn't necessary increase the minimum payment. I know some cards do a flat rate like 2% of the balance due each month while others do 1% + all interest each month; if money is tight you may have a card that doesn't effect your minimum payment. I don't really think you have to keep the balance on that card or close the account though.
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2) How much damage to my credit report am I risking if I closed the account? I may or may not need to move into a new apartment this year and maybe need a student loan within the next one or two years.
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It's not really going to lose a whole lot unless you frequently miss payments. The more "OK" marks you have on your credit report for history the less impact one missed or delinquent payment is going to have. You may lose some points for "length of accounts" but I doubt it's going to impact you much with 2 other cards opened that are already a year old and a car loan that's more than a year old. People hurt themselves when they close down their only credit card or a credit card with no balance because they lower their total available credit limit which makes it seem like they are using a higher percentage of it or it's their only card and they are no longer receiving revolving account payment history marks on their report.
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3) What are the odds that I can talk them into keeping my rate at the current APR? I feel like I've been a good customer, and I'd prefer not to close this account. I've been satisfied with the service and perks of the card, which is why it was the only one I kept even when the balance was zero. But I've already spoken to two different representatives who weren't budging.
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Doesn't look very good if you've already talked to 2 representatives, I can't believe they are telling you to close the account down to keep your APR, and I wasn't even aware you could close an account down with a balance cause it technically can't be closed it's still going to show active on your credit report as you pay it. I guess it's some type of locked for non-use status when they do it ? I'd just get another card or ask for a limit increase on your other card and balance transfer that away.