When a certain number is increased by five, the result is six times the same number decreased by ten. What is the number?

Translate the problem into an equation.
"A certain number is increased by five" can be written as $x + 5$.
"Six times the same number decreased by ten" can be written as $6(x - 10)$.
The problem states that these two expressions are equal.

$$x + 5 = 6(x - 10)$$

Subtract $x$ from both sides:

$$5 = 5x - 60$$

Add 60 to both sides:

$$5 + 60 = 5x$$ $$65 = 5x$$

$$\frac{65}{5} = x$$ $$x = 13$$

The number is $\boxed{\text{13}}$.

3. If $\frac{3}{5}$ of a numbers is 4 more than $\frac{1}{2}$ the number, what is the number? Ans 40

Translate the problem into an equation.
"$\frac{3}{5}$ of a number" can be written as $\frac{3}{5}x$.
"4 more than $\frac{1}{2}$ the number" can be written as $\frac{1}{2}x + 4$.
The problem states that these two expressions are equal.

$$\frac{3}{5}x = \frac{1}{2}x + 4$$

$$10 \cdot \left(\frac{3}{5}x\right) = 10 \cdot \left(\frac{1}{2}x + 4\right)$$ $$6x = 5x + 40$$

Subtract $5x$ from both sides:

$$6x - 5x = 40$$ $$x = 40$$

The number is $\boxed{\text{40}}$. This matches the given answer.

4. I am thinking of a certain number. If I multiply by 6, add -18 to the product and multiply by one - third of the result, I get -2. What number am I thinking of?

Translate the problem into an equation.
"Multiply by 6": $6x$
"Add -18 to the product": $6x - 18$
"Multiply by one-third of the result": $\frac{1}{3}(6x - 18)$
"I get -2": means the expression equals -2.

$$\frac{1}{3}(6x - 18) = -2$$

$$\frac{1}{3}(6x) - \frac{1}{3}(18) = -2$$ $$2x - 6 = -2$$

Add 6 to both sides of the equation:

$$2x = -2 + 6$$ $$2x = 4$$

$$x = \frac{4}{2}$$ $$x = 2$$

The number is $\boxed{\text{2}}$.