A bar over a sequence of digits in a decimal indicates that

Thanks for explaining this and for link too ! you rock

I did this:

Factor out 10^2. You get 10^2 (100-1) (.0012)

12/100 = .12. Therefore 12/1000 = .012, therefore 12/10000 = .0012

10^2 (99) * (12/10000)

10^2 = 100 and 100^2 = 10000

Cancel out the 100 with the 10^2

Left with 99(12/100).

.99*12 = 11.88. That should be the answer I dont think you listed the choices.

Sorry, this was my first post, not familiar with rules. Thanks for the clarity - I would never have considered factoring out that 10^2. Did you do that all in 2 minutes? Seems like it would take too long

Hey von
Unless you are running a competition against a number crunching super machine. 10^2 can be ignored when compared to 10^4. Since its 1 percent.
Knocking 10^2 off. 10^4 * 0.0012 = 12 approx But remember use inequality for safety -
11.9< answer < 12.0 if the choices are close.

(10^4-10^2)x(0.0012)
= 10^2(10^2-1)x(0.0012)
= (10^2-1) x 0.12
= 99 x 0.12 ----> eliminate all choices except E

(10^4 - 10 ^2) (.00(12)) --- 12 within () is recurring decimal .
=10^2(10^2 -1) (.00(12))
=10^2 x 99 x .00(12)
=.(12) x 99 (Since 12/99 = .121212..... )
= 12

Answer E

Hi thompstc,

You can solve this question either way. Just that by distribution, we can arrive at the answer easily.
(10^4−10^2)∗0.0012 = 9,900 * (12/9,900) = 12

Correct Option: E

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.

10^4 = 10*10*10*10 = 10,000
10^2 = 10*10 = 100

Therefore 10^4−10^2 = 10,000 − 100 = 9900

Now, I just multiplied 9900*12 = 118,800
Finally add back the 4 decimal places of the 0.0012 = 11.8800 which is more or less 12, Hence E.

Note : Rule to convert mixed recurring decimal to fraction : In the numerator write the entire given number formed by the (recurring and non - recurring parts) and subtract from it the part of the decimal that is not recurring. In the denominator, write as many nines as the number of digits recurring and then place next to it as many zeros as there are digits without recurring in the given decimal.

Converting the mixed recurring decimal (0.0012) to fraction we get;
\(\frac{(12 - 00)}{9900} = \frac{12}{9900} = \frac{4}{3300}\)

(\(10^4\) -\(10^2\))(0.0012) can be written as = \((10^4 -10^2)* \frac{4}{3300}\)
\(10^2(10^2 - 1)* \frac{4}{3300}\)= \(100 * (100 - 1)* \frac{4}{3300}\)
\(99 * \frac{4}{33}\) \(= 3 * 4 = 12\). Answer E...

_________________
PS: Consider giving Kudos if my post helped you in some way

(10⁴ - 10²)(0.00121212...) = 10⁴(0.00121212...) - 10²(0.00121212...)
= 10,000(0.00121212...) - 100(0.00121212...)
= 12.121212... - 0.121212...
= 12 (since the decimal parts, in blue, are identical, they cancel out]

Answer: E

Bunuel i dont get after this 10,000*0.00(12) why do you get 12.(12) it should 12

10,000*0.00(12) = 12 isnt it

A bar over (brackets around in our case) a sequence of digits in a decimal indicates that the sequence repeats indefinitely.

So, 0.00(12) is 0.001212121212121212...

10,000*0.00(12) = 10,000*0.001212121212121212... = 12.12121212.... = 12.(12)

It is helpful to understand the following about repeating decimals:

When ONE digit is repeated, a denominator of 9 is implied:
\(0.\overline{5} = \frac{5}{9}\)
When TWO digits are repeated, a denominator of 99 is implied:
\(0.\overline{27} = \frac{27}{99} = \frac{3}{11}\)
When THREE digits are repeated, a denominator of 999 is implied:
\(0.\overline{006} = \frac{006}{999} = \frac{2}{333}\)



\((10^4 -10^2)(0.00\overline{12}) = (10^2)(10^2-1)(0.\overline{12})(10^{-2}) = (10^2)(10^{-2})(10^2-1)(0.\overline{12}) = (1)(99)(\frac{12}{99}) = 12\)

Can anyone please help me? why is B wrong?
(10^4-10^2)(0.00121212....)

(10^4-10^2) = 10^2 = 100

and then 100 * 0.00121212 = 0.121212....

can anyone tell what am i missing ? thanks in advance.

The red portion is incorrect.
\(10^4-10^2 = 10000-100 = 9900\)
Algebraically:
\(10^4-10^2 = 10^2(10^2-1) = 100(99) = 9900\)

­Blunel, what do you think about my way of solving this problem? 

Factoring out 10^2 first

(10^2)(10^2-1)(0.00(12)) 

This gives us 

(0.(12))(99) 

99 approximated is 100 

So (0.(12))(99)  is somewhere around 12.