500 soldiers in a fort had enough food for 30 days. after 6 days, some soldiers were sent to another fort and thus the food lasted 32 more days. how many soldiers left the fort?

Best   There's enough food for 500 soldiers for 30 days. Say that each soldier has a unit of food a day, so there are 500 units for 1 day and 500*30=15000 units over the 30 days (remember this value).

500*6=3000 units are used by the time some soldiers leave, leaving a total of 15000-3000=12000 units.
Now we know that x units are used a day, and there are 32 more days left (because it tells us). You can make the following formula with this:

Number of soldier*days the food will last=units of food
x*32=12000

Then just rearrange this:

x=12000/32=375 soldiers left in the fort. So 125 soldier had left.