fatihaysu wrote:
A bar over a sequence of digits in a decimal indicates that the sequence repeats indefinitely. What is the value of (10^4 -10^2)(0.0012)?
(A) 0
(B) 0.12
(C) 1.2
(D) 10
(E) 12
Remember that the notation [12] means there is a bar over the 12, indicating that the decimal is nonterminating.
Now, let’s start the problem by factoring out 10^2 from (10^4 – 10^2). This gives us:
(10^4 – 10^2) (0.00[12])
10^2 (10^2 – 1)(0.00[12])
We can distribute 0.00[12] with the two quantities in the parentheses. This gives us:
10^2(0.[12] - 0.00[12])
100(0.[12] - 0.00[12])
12.[12] – 0.[12] = 12
Alternate solution:The number .00[12] is the number .00121212… if we write it without the bar notation. By the distributive property, we have
(10^4 – 10^2) (.00[12]) = 10^4(.00[12]) – 10^2(.00[12]
Without the bar notation, we write this as 10^4(.00121212…) – 10^2(.00121212…)
Multiplying a number by 10^4 indicates that we move the decimal point four places to the right, giving us:
10^4(.00121212…) = 12.1212…
Similarly, multiplying a number by 10^2 indicates that we move the decimal point two places to the right, giving us:
10^2(.00121212…) = 0.1212…
Now, if we subtract the two quantities, we have
10^4(.00121212…) – 10^2(.00121212…) = 12.1212… - 0.1212… = 12 (because the .1212… gets canceled out by the subtraction).
Answer is E.
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