I have connected a 3.7v-4000mah battery to a load via a voltage booster MT3608 (3.7V to 12V) . load current is 100ma. how long does battery can work?
For this you will need to consider your batter capacity in watt-hours. So for your 3.7V 4000mAh battery, plugging into the equation mAh * V / 1000 = Wh, you get 4000mAh * 3.7V / 1000 = 14.8Wh. Now we can convert it back using 12V for our V value in the equation and get (14.8Wh * 1000) / 12V = 1233.33mAh. And with a 100mAh load that would be approximately 12.33 hours of runtime with an ideal DC/DC converter. However, you'll likely have a roughly 90% efficiency and therefore see only 12.33 * .9 = 11.09 hours of actual run time (thank you MatsK for reminding me that practicality is nearly never ideal).
I have a feeling I forgot something so others shall feel free to correct me or point out errors. I'm writing this on mobile so can't currently format it, maybe later.