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That advice is given when someone is trying to MAXIMIZE their scores, like right before applying for a mortgage or a car loan.
You do want a mix of credit ... three revolving (credit card accounts) ... and one installment (car/mortgage) is ideal, but if you don't have it, there are ways to get it.
As for the history ... there's little you can do about it now. I've tried, unsuccessfully, to have late payments deleted by prior creditors and have never been successful. So I stopped focusing on that and focused on paying all my bills on time each month. As the credit lines age, the late payments will age off. As for carrying a balance versus reporting zero, it doesn't matter. What matters is that monthly payment being noted in your credit report (and it reports as paid on time, even if you don't carry a balance that month).
Just avoid paying interest ... especially while rebuilding.
Actually its 9% of a single cards CL or less to let report for best Fico, you will have to find the actual percentage that gives you the best Fico score.
Utilization of credit (of which % util of revolving is the major component) and payment history are separate scoring categories.
Payment history is the highest weighted, but util of credit is the second highest.
Yes, maintaining or improving util scoring is still important for those who have marred payment history.