### Theory:

Problem $$1$$
Find the mode of the following numbers.

$$70$$, $$50$$, $$90$$, $$50$$, $$40$$, $$20$$, $$10$$, $$50$$, $$60$$, $$80$$, $$30$$, $$50$$, $$30$$

Let us now find the solution.

Step $$1$$: Arrange the numbers in the ascending order.

$$10$$, $$20$$, $$30$$, $$30$$, $$40$$, $$50$$, $$50$$, $$50$$, $$50$$, $$60$$, $$70$$, $$80$$, $$90$$

Step $$2$$: Find the number that occurs for the most number of times.

$$50$$ occurs the most number of times.

Therefore, $$50$$ is the mode of that set of data.

Problem $$2$$
Find the mode of the following numbers.

$$15$$, $$12$$, $$8$$, $$3$$, $$5$$, $$7$$, $$15$$, $$3$$, $$9$$, $$8$$, $$2$$, $$8$$, $$15$$

Let us now find the solution.

Step $$1$$: Arrange the numbers in the ascending order.

$$2$$, $$3$$, $$3$$, $$5$$, $$7$$, $$8$$, $$8$$, $$8$$, $$9$$, $$12$$, $$15$$, $$15$$, $$15$$

Step $$2$$: Find the number that occurs for the most number of times.

Both $$8$$ and $$15$$ occur thrice.

Therefore, $$8$$ and $$15$$ are the modes of that data.

In other words, the given set of data is bimodal.