Unfortunately, QuickBooks cannot automatically apply the payment with the invoice in the import operation. Even though the payments are imported, they are just listed and still need to be received, applied, and deposited.
So, to apply your payments to your invoices after importing, go to the Receive Payments option in QuickBooks, and hit previous to go through each of the payments there and apply them to the appropriate invoice. Note that the invoice number for the payment will be shown in the memo field of the payment check.
Add to favorites