Mixed tyres Width
 

HI , I need to know if there's any benefits/guidelines for having different tyre width on the front and the back ..

You can do it if you want. I think the general consensus is it's not really necessary for most of us, but there are no rules against it.

Can you tell us what you have in mind??

Funny one. Because it works both ways depending on what you want.
The rear gets more weight, so bigger is better in terms of taking the bumps and while you tend to see and dodge things the front may hit, the rear often smacks whatever is in the way.

The rear transmits the power, so smaller is better in terms of power transmission. So for a sprint - smaller is better.

In the end most folks use the same size. The rear will wear faster.

The Conti Attack/Force combo is F/R specific. The front is 22 and the rear 24 (well, at least they were as of a couple years ago - haven't bought them in a while). Read over their marketing and take it with a grain of salt.

I ride a 24 front and 26 rear on my Venge and it suits me well. The smaller tire on the front compliments the wheel very nicely as it gives it an aero profile.

I did a couple of gravel events last year and I saw a couple of people with wider tires on the front to my surprise. After chatting with a guy about his choice he explained that the wider front allows him to hit larger gravel, divots and objects and not have the front end give. I have no first hand experience with this set up.

Some people (like those that obsess over their tyre pressures) will run a lower tyre pressure on the front as it has less weight on it that the rear. Why not run a narrower tyre on the front and keep the front and rear pressure the same instead?

I use a 23mm F and 25mm R. I don't like the 25mm on the front as I have narrow rims and I feel that the larger tyre on the front balloons too much which makes the front too squirrelly when turning which counteracts the extra grip that a 25mm is supposed to provide.

Most theory points to it being somewhat sensible to go wider in back for road.
There's more load on the rear tire, which a tire with more air volume and travel should handle better, both in rolling behavior and wear life. Aero effects are is also more significant at the front wheel, so potential aero penalties of wider are smaller in back.
Although, it doesn't seem to matter much either way, and staying the same front and back tends to be more stylish.

A lot of MTB folks prefer the opposite, wider in front.

well nothing in particular , i just saw some youtube videos and some tyres set which are made with different width , so i wanted to understand the science behind that ..

For road bikes , i see that most people agree that effect is not that much to consider , and that it's better to go wider on the back as it process more weight and to benefit from narrower /more aero dynamic in front ..

I think for on the road, it is of very limited value. If you are off-roading it, may make more sense to mix it up.

Many road cyclists (myself included) like to "rotate" our tires, when the rear is worn we move the front to the rear and put the new one on the front. That makes using the same width tires more important.

But as I stated, there is no harm in mixing it. YMMV.

Another small question about rear tyre , my rear tyre indeed just start to crack , i need to replace it , but i can't find same brand , front tyre is still in very good shape , not sure if it's better to buy new set , or just buy one tyre for the rear that's different brand/pattern , etc ..
I feel it was a food idea to always buy spare tyre when buying a tyre-set for such circumstances

If it's just a crack in the rubber/tread, you may be able to continue to use it. You can mix/match brands though, if need be.

This is why I stopped buying Attack/Force pairs.

I also like to rotate my tires, so I'll likely keep using 25mm for both on the road.

For off-road, I currently run 30mm, which is about the limit in the back, but next time I need tires I may try out a bigger (33 or 35) size up front for better gravel cornering.

Of course, this presents a problem when switching from 23mm to 25mm...
My next switch is going to 25mm front, 23mm rear, simply because I wanted to try 25mm tubeless, and I still have a good 23mm on the front.

Fairly common approach for all loose surface riding. Although the exact reasons vary. Some simply refer to grip and roll. Grippier front for better cornering. Better roll for the more heavily loaded rear where a skid is more controllable.

Bigger up front for the better cornering traction and less fatigue. Bigger in the back for the better puncture resistance. So... bigger all round.

Bigger all round to fluster other nearby cyclists.