Alternator or battery?

My daughter's car wouldn't start the other day due to a dead battery. I got the car started and my battery charger/alternator tester showed the alternator was not charging. I removed the alternator, took it to the parts store and it tested fine! Thought maybe bad battery but it tested fine also. I put everything back together and put a volt meter on the battery while running and it showed 14 volts. Thought all was well.

Today I noticed that car was cranking slow. I put a voltmeter on the battery and it was just under 12 volts. I had her start the car and checked again and was only reading a little over 12 volt with the car running. Is the alternator shot? Could a bad battery/cell cause a low reading even though the car is running?

Is there a way to test the alternator without completely removing it from the car?

Try this start the engine remove the negative cable from the battery now place a volt meter on the battery cables insuring the negative cable is not touching any metal on the car. The voltages reading must be at least 14 volts if not you have a sick alternator.

Next place a charger at least a 6 amp charger on the battery for at least 1-hour, remove charger now take a reading it may read around 13v connect the negative cable on the battery, do not start engine turn on the headlights heater fans keep on for at least 5 minutes, turn off the lights and fans wait 1 minute, now take a voltage reading if the battery reads less then 12.25 volts replace the battery

Bottom line if the battery is more then 5 to 6 years old replace the battery.