Thank you for your post.
A Summary field totals/counts/averages/etc. records across an entire database or a subset of records. Since you want to total "precinct portions", a Summary field type is necessary.
Pull down the File menu and select "Manage -> Database..." Click on the Fields tab (if not already selected) and create a new field TOTAL of type Summary, which is a Total of your voter count field. Click OK and return to your database. At the bottom of the layout, you will probably see TOTAL with a total number for the found set of records.
Pull down the View menu and select Layout mode. Pull down the Layouts menu and select "New Layout/Report...". You are now creating a new layout. In the first dialog box, give the layout/report a name, select "List/Report", and click Next. On the next screen, select the option "Report with grouped data", check the option "Include Subtotals" (and also select Include grand totals if you want a final total) and click Next. In the next window, select the fields you want to display in the layout. and click Next. In the next window, select the Precinct (not the precinct portion) and click Next. In the next window, sort the records by Precinct and click Next. In the next window, select the summary field TOTAL and be sure to click the button "Add Subtotal" so it appears in the bottom portion of the dialog box. Click Next. In the next window, select "Default" and click Next. Click Next a few more times and click Finish. You should now see a report similar to what you want to display with each precinct totaled.
If you need clarification for any of the above steps, please let me know.