Egyptian addition and subtraction used essentially the same methods as we use today. The really interesting part of Egyptian arithmetic was their multiplication. The Egyptians regarded division as the inverse operation of multiplication, and once we see how multiplication is done, it is simple to see how division would have been achieved.
Egyptian multiplication used the method of duplation and mediation. This involved reducing the multiplication problem to a problem in addition. We take the two given numbers, and successively divide one of them by 2, discarding any remainders, until 1 is reached. We mark the odd numbers obtained from the division. At each step, the other number is multiplied by 2, and the result written next to the division step. The required product of the two numbers is then found as the sum of the multiples positioned next to the marked values.
The following applet illustrates this process.