Phone is plugged in to a fax machine and the phone rings too many times before answering machine picks up so the fax machine takes over. How can I reduce the number of rings so that the answering machine picks up and not the fax?

It seems that you're sharing your home phone line with your Lexmark fax machine. You can adjust the "pick-up" ring from the printer operator panel by pressing the Setup button > Fax Setup > Ringing and Answering > Pick Up On > select your preferred number of rings before the fax answers.
Please note:
If you have an answering machine connected to the same line as the fax, set the answering machine to answer incoming calls before the printer does. Increase the number of rings by a factor of 3 over the answering machine. For example, if the answering machine is set to pick up on 4 rings, set the printer to pick up on 7 rings.

If you have a voicemail feature in the same line as the fax, you may subscribe from your phone service provider for a distinctive ring pattern feature. You can program your printer to have a distinctive ring pattern and phone number for incoming faxes.

Should you still need assistance, please leave a comment to my response and I'll get back to you. Should this resolves your problem, your rating will be greatly appreciated.

I'm going out on a limb here, and guessing it's a Canon imageClass MF4150... If not, please disregard this solution.

The advanced manual can be viewed / downloaded / printed here, directly from Canon:

http://gdlp01.c-wss.com/gds/0900006459/imageCLASS_MF4150_Advanced_Guide_EN.pdf

To stop it from interfering with the telephone conversation:

Issue: Fax interrupts telephone conversations in answering machine mode (AnsMode)
Solution: Setting the Receive Mode - ICMF 4150

Setting the Receive Mode

1. Press <Additional Functions>.
   
   
   
2. Use <> or <> to select [5 FAX SETTINGS], then press <OK>.
   
   
   
3. Use <> or <> to select [3 RX SETTINGS], then press <OK>.
   
   
   
4. Use <> or <> to select [3 RX MODE], then press <OK>.
   
   
   
5. Use <> or <> to select the desired receive mode, then press <OK>.
   - For [AnsMode], make sure an answering machine is connected to the machine.
   If [AnsMode], [FAX/TEL], or [DRPD] is selected, you can select some settings as below:
   
   [AnsMode]
   --- [AnsMode CHK TIME] (select time from 10 to 30 seconds for the answering machine to start playing outgoing message after receiving a call)
   --- [FAX DETECT TIME] (select the duration from 10 to 70 seconds for the fax to detect)
   
   [FAX/TEL]
   --- [RING START TIME] (select the time from 0 to 30 seconds for the phone to start ringing after receiving a call)
   --- [F/T RING TIME] (select the duration of ringing from 10 to 45 seconds)
   --- [F/T SWITCH ACTION] (select the action the machine will take after the ringing ends, Receive or Disconnect)
   
   [DRPD]
   --- [NORMAL RING]
   --- [DOUBLE RING]
   --- [SHORT-SHORT-LONG]
   --- [SHORT-LONG-SHORT]
   --- [OTHER RING TYPE]
   
6. Press <Stop/Reset> to return to the standby display.

The fax is set to answer on the 4th ring. Change it to answer on say... 6 rings.It cannot differentiate the call before it's picked up ... it just answers at at set number of rings

You need to set your phone to pick up later than the fax machine. set it to pick up on the 4th or 5th ring, set the fax to pick up at 2 or 3. Better yet, get a distintive ring # set up by your phone co. then the fax will only recognize that ring.

I’m happy to assist further over the phone at https://www.6ya.com/expert/joe_8b8c2cd6ce148309

Set rings to answer
You can select the number of rings that will occur before the fax answers an incoming call. If you are using your machine as both a telephone and a fax machine, we suggest you set the ring count to at least 4 to give you time to answer.
TIP: HP 1050 Fax: The Toll Saver function for the Digital Message Center interacts with Rings to Answer. See Toll Saver on page 40. If you have the HP 1050 Fax and have Toll Saver turned on, set the Rings to Answer to 2.
1. Press Menu repeatedly until BASIC FAX displays and press Start/Enter.
Rings to Answer will display.
2. Press Start/Enter.
3. Enter the number of rings you want (2-7) and press Start/Enter.
The HP Fax rings the set number of times and then answers. It then produces a different sounding ring to alert you to which type of call is incoming.
• The additional ring may occur once for a fax call
• The additional ring occurs 5 times for a voice call. After the 5 additional rings, the answering machine will answer.

Set your answering machine to pick up after four rings . This will give the fax machine enough time to take the call if it senses that it's an incoming fax. Setting the answering machine to pick up after four rings would give the fax machine enough time to accept the fax.
hope this was helpful .

I have a HP Officejet J6480 All-In-One and needed the answering machine and fax to be on at the same time.  I called the help line at HP and they said I needed another phone line into the home but I tried this and it worked. 
First I got a different phone number for the fax machine - they do this for no charge at my phone company. This is not another physical line into the home - it is just another phone number with a different ring - it rings 2 fast rings for the fax machine instead of the phones regular one ring. I set the answering machine built into the phone to pick up after 4 rings and I set the fax machine to pick up on 2 rings. Be sure to set it so the fax pick up number of rings are less than the answering machine pick up number of rings.  It worked for me. Hope this works for you too.

If this is on a network your network tech needs to set up your machine through one of the routers.
If the machine was pluged into the phone socet direct, there would be no problem. The phone will ring constantly, until you plug the phone to the machine.
If the machine does not answer the phone, then the problem maybe in the machines confguration for auto answer, or a modem board not working.

But on a network, the computer is trying to find the signal, but can't because it cannot find the machines signal.
This problem is in the network side. Check with your network specialist.

Set your machine to MANUAL, this will allow you to answer every call yourself, then if it is a fax, the machine will take over. Sounds like you have it in fax/tel mode now, which allows for the fax machine to answer every call then if not a fax, double rings to you.

There is a help menu on this machine. To print this out, just press the help button. To change the fax/tel ring, 1st have in mind how you want your fax machine to work. Do you have an a TAD connected? Do you want the fax machine to take priority over tel calls? The fax machine seizes the line after the set number of rings, that means any other device connected to other phone outlets, will not answer after the fax has answered. Have this in mind when you set the ring time. The TAD will have it's own ring time too.