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

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A bar over a sequence of digits in a decimal indicates that [#permalink]

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19 Aug 2010, 06:10
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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

[Reveal] Spoiler:
Anyway My approach is;

9900*12/9999 = 11,88 so i chose E

is there anyway ? i did right but dont know the logic under it
[Reveal] Spoiler: OA

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Last edited by Bunuel on 17 Dec 2012, 06:51, edited 2 times in total.
Edited the question
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Re: Need a shortcut for it [#permalink]

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19 Aug 2010, 06:41
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The catch here is the pattern is repetitive .
That makes 10000*0.00121212 (in repetitive pattern ) - 100*0.00121212
That will give you 12.121212 - 0.121212 = 12

The way you tried to solved it should give (10000-100 ) * 0.00121212 = 9900*0.00121212 = 11.99
edit : However It is not the best approach to solve such problem ., mostly because multiplication will take more time .
E is right choice so your answer is write since there is no other answer which is near than 12 .
( I don't know why you divided it by 9999 )
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Re: Need a shortcut for it [#permalink]

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19 Aug 2010, 07:22
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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)? <<<<<Above "12" there is a line indicates that it repeats indefinitely. However i cannot find how i can put it there

a) 0
b) 0,12(has infinite line above 12)
c)1,2
d)10
e)12

Above "12" there is a line indicates that it is repeats indeinitely. However i cannot find how i will put it there

Anyway My approach is;

9900*12/9999 = 11,88 so i chose E

is there anyway ? i did right but dont know the logic under it

What is the value of $$(10^4-10^2)*0.00(12)$$? (You can indicate repeated part of the decimal by putting it in brackets)

You can solve it as Pinali suggests above, just open the brackets and multiply: $$(10^4-10^2)*0.00(12)=10,000*0.00(12)-100*0.00(12)=12.(12)-0.(12)=12$$, you can see that 0.(12) part is subtracted from 12.(12) which gives 12.

But you can do this problem in another way too: 0.00(12) can be written as fraction $$\frac{12}{9,900}$$ (as many 9's as numbers in repeated pattern and as many zeros after as zeros after decimal point). For more on how to convert a recurring decimal to fraction see Number Theory chapter of Math Book (link in my signature).

So $$(10^4-10^2)*0.00(12)=9,900*\frac{12}{9,900}=12$$.

Hope it help.s.
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Re: Need a shortcut for it [#permalink]

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19 Aug 2010, 07:45
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Bunuel wrote:
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)? <<<<<Above "12" there is a line indicates that it repeats indefinitely. However i cannot find how i can put it there

a) 0
b) 0,12(has infinite line above 12)
c)1,2
d)10
e)12

Above "12" there is a line indicates that it is repeats indeinitely. However i cannot find how i will put it there

Anyway My approach is;

9900*12/9999 = 11,88 so i chose E

is there anyway ? i did right but dont know the logic under it

What is the value of $$(10^4-10^2)*0.00(12)$$? (You can indicate repeated part of the decimal by putting it in brackets)

You can solve it as Pinali suggests above, just open the brackets and multiply: $$(10^4-10^2)*0.00(12)=10,000*0.00(12)-100*0.00(12)=12.(12)-0.(12)=12$$, you can see that 0.(12) part is subtracted from 12.(12) which gives 12.

But you can do this problem in another way too: 0.00(12) can be written as fraction $$\frac{12}{9,900}$$ (as many 9's as numbers in repeated pattern and as many zeros after as zeros after decimal point). For more on how to convert a recurring decimal to fraction see Number Theory chapter of Math Book (link in my signature).

So $$(10^4-10^2)*0.00(12)=9,900*\frac{12}{9,900}=12$$.

Hope it help.s.

Thanks for explaining this and for link too ! you rock
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Re: Need a shortcut for it [#permalink]

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19 Aug 2010, 07:52
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Pinali wrote:
The catch here is the pattern is repetitive .
That makes 10000*0.00121212 (in repetitive pattern ) - 100*0.00121212
That will give you 12.121212 - 0.121212 = 12

The way you tried to solved it should give (10000-100 ) * 0.00121212 = 9900*0.00121212 = 11.99
edit : However It is not the best approach to solve such problem ., mostly because multiplication will take more time .
E is right choice so your answer is write since there is no other answer which is near than 12 .
( I don't know why you divided it by 9999 )

It is poped up in my mind, i was barely remember the rule and divide 9999 however the rule,bunuel mentioned,is we have to put 9 as much as number in the decimal different from 0.

I was trying to do that with bunuels second approach.

Thanks your way is more fast.

you got kudos from me and thanks Bunuel you are the best mate
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Re: Spent hours studying this powers problem! [#permalink]

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01 Mar 2011, 21:12
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.
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Re: Spent hours studying this powers problem! [#permalink]

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01 Mar 2011, 21:16
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
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Re: Spent hours studying this powers problem! [#permalink]

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01 Mar 2011, 22:19
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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.
von wrote:
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
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Re: Need a shortcut for it [#permalink]

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04 Mar 2011, 10:06
(10^4-10^2)*0.00121212.... >>>> (10^4-10^2)*(10^-4*12.121212...) >>>> multiplying we obtain ( 10^4*10^-4*12.121212)-(10^2-10^-4*12.121212)>>>> 10^0*12.121212 - 12.121212*10^-2 >>>>> 12.121212-0.121212= 12

I hope it's clear...
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Re: A bar over a sequence of digits in a decimal indicates that [#permalink]

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28 Aug 2014, 15:53
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(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
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Re: A bar over a sequence of digits in a decimal indicates that [#permalink]

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03 Oct 2015, 04:50
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(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

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Re: A bar over a sequence of digits in a decimal indicates that [#permalink]

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03 May 2016, 22:49
thompstc wrote:
this might seem like a silly thing to ask about, but i need some help with my thought process on this one:

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)(0.00121212121212..)

A) 0
B) 0.12 repeating
C) 1.2
D) 10
E) 12

I understand you're supposed to use the distributive property, but i'm not sure why? Following order of operations, shouldn't you simplify the parentheses first?

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
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Re: A bar over a sequence of digits in a decimal indicates that [#permalink]

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04 May 2016, 08:21
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).

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Re: A bar over a sequence of digits in a decimal indicates that [#permalink]

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10 Nov 2016, 07:08
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.
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Re: A bar over a sequence of digits in a decimal indicates that [#permalink]

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01 Apr 2017, 05:22
(10^4-10^2)(0.00(12)) [ (12) is the repeating number]
=10^2(10^2-1)(0.00(12))
=10^2(99)(0.00(12))

Let's just handle 0.00(12)
Converting it to a fraction
Let x= 0.00(12)
100x=0.(12) .........eq.1 (100 because we have to move the non-repeating part i.e.00 to the left of the decimal point. So two non- repeating digits, 100. If one digit, then 10 and so on.)

10000x= 12.(12) ..........eq.2 (Same logic as above. We already had 100 as we moved 00 to the left of the decimal point. Now we have to move 12 also to the left of the decimal point.)

Now eq.2 -eq. 1

9900x=12
x=12/9900 (Don't solve)

Now putting it back in the original equation given in the question

We get: 10^2*99*12/9900
=12
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A bar over a sequence of digits in a decimal indicates that [#permalink]

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13 Jun 2017, 23:26
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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

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...

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Re: A bar over a sequence of digits in a decimal indicates that [#permalink]

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20 Jun 2017, 06:13
I did this in an unconventional way:

10^4 - 10^2 = 9900 ---> 99 x 100

99 x 100 x 0,0012 ---> 99 x 0,12 = 11,88 thus the answer is E
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Re: A bar over a sequence of digits in a decimal indicates that [#permalink]

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28 Aug 2017, 14:44
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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

(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]

[Reveal] Spoiler:
E

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Re: A bar over a sequence of digits in a decimal indicates that   [#permalink] 28 Aug 2017, 14:44
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