"Balancing the checkbook" means carefully noting the amount of each check you've written, and subtracting it from the amount of money you have in the account. Many people write checks in a hurry without making a note of it, and sometimes they accidentally write a check when there is no money left in the account. So every week or so, you should take time to calculate how much money is in the account.
Then, at the end of the month when you get your bank statement, you should compare all their numbers (checks written, balance left in the account, etc) with those in your checkbook.
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