"Thinking backwards"-starting at the end of the problem and using reverse operations-can help you solve certain problems at different grade levels.

**Upper Elementary and Middle School**:

A certain number is doubled. That answer is tripled. Finally, that answer is quadrupled and the answer is 60. What is the original number?

Answer:

When we *quadruple* a number, we multiply it by 4. As division is the inverse of multiplication, 60 ÷ 4 = 15. *Tripling* a number means "multiply by 3," so 15 ÷ 3 = 5. Finally, doubling a number means "multiply by 2," thus, 5 ÷ 2 = **2.5**, or **2 ½**.

**Algebra**:

A certain number is quadrupled. 3 is added to the answer. That answer is then tripled. Finally, when that answer is cut in half, the answer is 12.

Answer:

*x. "x, quadrupled*" means "4

*x.*" Then, we add 3 to the answer, so 4

*x*+ 3. We then

*triple*the quantity "4

*x*+ 3

*"*: 3(4

*x*+ 3). Finally, we split the quantity 3(4

*x*+ 3) in

*half*in order to yield the final answer, 12, so {3(4

*x*+ 3)} /2

**= 12**.

*x*, which involves "canceling out" the numbers by using inverse operations.

*x*+ 3)} /2 = 12 (2)

*multiply both sides by 2...*

*divide both sides by 3...*

*subtract 3 from both sides, and...*

*divide both sides by 4.*

*Thus, x =*

**1.25**