“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 “4x.”
Then, we add 3 to the answer, so 4x + 3. We then triple
the 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 = 24/3
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.
Thus, x = 1.25
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