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"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.

a) Write an equation that describes this problem.

b) What is the original number?

Answer:

a) We're looking for a certain (unidentified) number, *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**.

b) To find the original number, we solve for *x*, which involves "canceling out" the numbers by using inverse operations.

So, (2){3 (4*x* + 3)} /2 = 12 (2) *multiply both sides by 2...*

{3(4x + 3)}/3 = 24/3*divide both sides by 3...*

4x + 3- 3 = 8 - 3* subtract 3 from both sides, and...*

4x /4= 5 /4 *divide both sides by 4.*

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