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# August/September 2014 Newsflash: Tips & Techniques

Aug 14, 2014

"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?

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)     Write an equation that describes this problem.
b)     What is the original number?

a)   We're looking for a certain (unidentified) number, x. "x, quadrupled" means "4x." Then, we add 3 to the answer, so 4x + 3. We then triplethe quantity "4x + 3": 3(4x + 3). Finally, we split the quantity 3(4x + 3) in half in order to yield the final answer, 12, so  {3(4x+ 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 (4x + 3)} /2 = 12 (2) multiply both sides by 2...
{3(4x + 3)}/3 = 24/3divide both sides by 3...
4x + 3- 3 = 8 - 3 subtract 3 from both sides, and...
4x /4= 5 /4  divide both sides by 4.

Thus, x = 1.25

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