Walkabout wrote:
\(\frac{0.99999999}{1.0001}-\frac{0.99999991}{1.0003}=\)
(A) 10^(-8)
(B) 3*10^(-8)
(C) 3*10^(-4)
(D) 2*10^(-4)
(E) 10^(-4)
When first looking at this problem, we must consider the fact that 0.99999999/1.0001 and 0.99999991/1.0003 are both pretty nasty-looking fractions. However, this is a situation in which we can use the idea of the difference of two squares to our advantage. To make this idea a little clearer, let’s first illustrate the concept with a few easier whole numbers. For instance, let’s say we were asked:
999,999/1,001 – 9,991/103 = ?
We could rewrite this as:
(1,000,000 – 1)/1,001 – (10,000 – 9)/103
(1000 + 1)(1000 – 1)/1,001 – (100 – 3)(100 + 3)/103
(1,001)(999)/1,001 – (97)(103)/103
999 – 97 = 902
Notice how cleanly the denominators canceled out in this case. Even though the given problem has decimals, we can follow the same approach.
0.99999999/1.0001 – 0.99999991/1.0003
[(1 – 0.00000001)/1.0001] – [(1 – 0.00000009)/1.0003]
[(1 – 0.0001)(1 + 0.0001)/1.0001] – [(1 – 0.0003)(1 + 0.0003)]
When converting this using the difference of squares, we must be very careful not to make any mistakes with the number of decimal places in our values. Since 0.00000001
has 8 decimal places, the decimals in the factors of the numerator of the first set of brackets must each have 4 decimal places. Similarly, since 0.00000009 has 8 decimal places, the decimals in the factors of the numerator of the second set of brackets must each have 4 decimal places. Let’s continue to simplify.
[(1 – 0.0001)(1 + 0.0001)/1.0001] – [(1 – 0.0003)(1 + 0.0003)]
[(0.9999)(1.0001)/1.0001] – [(0.9997)(1.0003)/1.0003]
0.9999 – 0.9997
0.0002
Converting this to scientific notation to match the answer choices, we have:
2 x 10^-4
Answer is D
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