Manhattan GMAT Challenge Problem of the Week – 26 March 2012
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Question
(1.0002)(0.9999) – (1.0001)(0.9998) =
A. 0.00000001
B. 0.00000002
C. 0.00000004
D. 0.0001
E. 0.0002
Answer
Don’t actually multiply these numbers out! The key here is to substitute a variable for a tiny value that shows up in several places.
Specifically, let x = 0.0001. Now you can rewrite all the numbers as 1 plus or minus x or 2x.
(1.0002)(0.9999) – (1.0001)(0.9998)
= (1 + 2x)(1 – x) – (1 + x)(1 – 2x)
Now, distribute each product in the expression separately.
First product: (1 + 2x)(1 – x) = 1 + 2x – x – 
= 1 + x –
Second product: (1 + x)(1 – 2x) = 1 + x – 2x – = 1 – x –
Subtract the two products:
1 + x – – (1 – x – )
= 1 + x – – 1 + x +
= 2x
Notice how much cancels out!
Finally, substitute back in for x. The difference is 2(0.0001) = 0.0002.
It might seem odd to solve an arithmetic problem by turning it into algebra! But in this case, doing so saves you a ton of work.
Another way to tackle the problem is to estimate judiciously. Of course, if you round every number to 1 in the original expression, you get 0. But consider rounding the numbers in this way:
(1.0002)(0.9999) – (1.0001)(0.9998)
≈ (1.0002)(1.0000) – (1.0001)(0.9999) — round both of the second numbers up by 0.0001, and because the first numbers in each product are approximately equal, you’ll only have a truly small rounding error
= 1.0002 – (1.0001)(0.9999). Now, if this were 1.0002 – 1.0001, you’d get 0.0001. But you’re subtracting something even smaller (since the 1.0001 is being multiplied by a number less than 1), so the difference must be larger than 0.0001.
The correct answer is E.
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