Why is it that when I run water from my sink, it gurgles and causes water to come up from my tub drain

The sink ties into the tub drain. This problem means that you have a clog in the tub drain.

Best solution for this issue is to locate the bathtub trap and remove it. You might find that the clog is in the trap. If the clog is not in the trap, then you need to snake the line going away from the trap.

Alternatively, you can try to snake the line from the overflow tube at the tub. See the picture here for reference:
Ask Me Help Desk - View Single Post - Bathtub p trap

I have a 1916 3 bedroom house with old plumbing. One bathroom upstairs. The water pressure is low. My toilet sometimes flushes and other times partially flushes. My bathroom sink was clogged and I used liquid plumber to unclog it. It with appeared to work but now when I run the sink water comes up from the tub drain. The kitchen sink seems to be fine. What would be a ballpark figure on a complete re-plumbing job.

Thank you very much. I had a feeling it was a clog but I thought it was just the sink. I guess in 1916 that was how they did plumbing.

Can you reach the tub trap from below or is it boxed in by a ceiling? You can rod down the tub overflow plate once you remove the plate. Unclog Tub Drain

We can't give an estimate, a good plumber would take at least several hours on site to make an estimate. Call 3 plumbing contractors for bids. Sounds like you have both a supply and a drain issue. The drain issue could be resolved by having a pro rod all of your lines with a sewer auger. Replacing a drain line is rarely needed but with a 100 year old home anything can go bad.

Bob's right we can't give you a estimate without any knowledge of how much material and labor it will take. We can ,however, help with the other problems.

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The water pressure is low

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This is causes by iron oxide build up in the old galvanized piping which cuts down on the volume. The solution would be to repipe with plastic. ( I had the same problem.)

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when I run the sink water comes up from the tub drain.

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You have a blockage downstream from your bath tub. The solution would be to snake from the tub overflow ,(see image). Unscrew and remove the overflow plate,(The one with the drain lever) and pull out the tub stopper. You now have a clear shot at the trap. The way you were going in through the drain it ran you square into a tee that you can't get around. Now feed the cable( 1/4" best or 3/3" with a small tip. You can rent a Ridgid K50 sink cable.) About 2' down you will run into the bend of the trap. Crank and push at the same time to work around it. Your blockage will be found in the lateral pipe going to the stack. You shouldn't have to put out more then 10 or 12 feet. I have found tub and shower clogs to be mostly hair and the worm should auger into it and pull it back. After you snake pour a few pans of boiling water down the drain. This will loosen the grease and flush the mess out into the main.

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My toilet sometimes flushes and other times partially flushes.

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I have one that most repair plumbers miss.. Look down at the bottom of the bowl. If there is a small hole, then that is a jet that starts the syphon action. If it's clogged the water will just swirl around and slowly go down leaving solids behind. Take your finger,(UGH! ) and run it around the inside of the opening. Over the years minerals build up and cut down on the syphon (flush) action. If it is rough or you feel build up, take a table knife and put a bend in it to get around the curve in the bowl and chip and scrap it clear. Next take a coathanger and clear out the holes around the rim. They start the swirling action. And last, check the water level in the tank. It should be 1/2" below the top of the over flow tube. And speaking of the overflow tube, Make sure the small 1/8" tube from the ballcock to the white overflow tube is connected so it discharges in it and that it's flowing when the ballcock fills. This is what raises the water level in the bowl. For a good solid flush they all have to work together. Hope this helps and thank you for rating my reply.. Tom