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QOTD #8 When the decimal point of a certain positive decimal

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QOTD #8 When the decimal point of a certain positive decimal [#permalink] New post 03 Aug 2016, 17:08
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When the decimal point of a certain positive decimal number is moved six places to the right, the resulting number is 9 times the reciprocal of the original number. What is the original number?

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[Reveal] Spoiler: OA
0.003


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Re: QOTD #7 When the decimal point of a certain positive decimal [#permalink] New post 03 Aug 2016, 17:13
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Explanation

Moving the decimal point of a positive decimal number, n, six places to the right is equivalent to multiplying n by \(10^6\). In the question, you are given that the result of such a change is 9 times the reciprocal of the original number, or \(9(\frac{1}{n})\). Therefore \(n(10^6)=9(\frac{1}{n})\). You can solve this equation for n as follows.

\(n(10^6)=9(\frac{1}{n})\)

\(n^2=\frac{9}{(10^6)}\)

\(n=0.003\).
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Re: QOTD #7 When the decimal point of a certain positive decimal [#permalink] New post 24 Aug 2016, 11:14
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sandy wrote:
Explanation

Moving the decimal point of a positive decimal number, n, six places to the right is equivalent to multiplying n by \(10^6\). In the question, you are given that the result of such a change is 9 times the reciprocal of the original number, or \(9(\frac{1}{n})\). Therefore \(n(10^6)=9(\frac{1}{n})\). You can solve this equation for n as follows.

\(n(10^6)=9(\frac{1}{n})\)

\(n^2=\frac{9}{(10^6)}\)

\(n=0.003\)


Sandy's explanation is perfect.
For anyone wondering about the last step, I might add two steps in between to get:

\(n(10^6)=9(\frac{1}{n})\)

\(n^2=\frac{9}{10^6}\)

\(n^2=\frac{3}{10^3} * \frac{3}{10^3}\)

\(n=\frac{3}{1000}\)

\(n=0.003\)
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Re: QOTD #8 When the decimal point of a certain positive decimal [#permalink] New post 09 May 2019, 20:24
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I have a confusion:
Given explanation: Moving the decimal point of a positive decimal number, n, six places to the right is equivalent to multiplying n by 10^6.
My logic: Moving the decimal point of a positive decimal number, n, six places to the right is equivalent to multiplying n by 10^-6.

Please explain.
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Re: QOTD #8 When the decimal point of a certain positive decimal [#permalink] New post 10 May 2019, 02:24
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What you said is incorrect.

Multiply a number for \(10^{-6}\) is equal to multiply the same number for \(\frac{1}{10^6}\) which is not the stem is asking you.

Ask if something is still unclear to you

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Re: QOTD #8 When the decimal point of a certain positive decimal [#permalink] New post 10 May 2019, 07:56
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Thank u for d explanation.
Suppose we have a positive decimal number: 0.123456
As per question,
When the decimal point of a positive decimal number is moved six places to the right of the decimal point, we have 123456 * 10 ^ 6.
Am I proceeding correctly?

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Re: QOTD #8 When the decimal point of a certain positive decimal [#permalink] New post 10 May 2019, 10:40
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If you do have for instance a decimal number such as 0.1 * 10^6 = 0.1 * 1,000,000 = 100,000

See the GC book for reference :)

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Re: QOTD #8 When the decimal point of a certain positive decimal [#permalink] New post 11 May 2019, 08:35
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Thank u for the clear-cut explanation.

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Re: QOTD #8 When the decimal point of a certain positive decimal [#permalink] New post 02 Jun 2020, 13:09
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sandy wrote:
When the decimal point of a certain positive decimal number is moved six places to the right, the resulting number is 9 times the reciprocal of the original number. What is the original number?

enter your value

[Reveal] Spoiler: OA
0.003


Practice Questions
Question: 21
Page: 154


A student asked me to solve this question. So here goes....

First we need to understand what must occur to move a decimal point 6 spaces to the RIGHT.

Check out these examples:
\((1.234567)(10^2) = 123.4567\). So, multiplying a number by \(10^2\) results in moving the decimal point 2 spaces to the RIGHT.
\((6.73215333)(10^3) = 6,732.15333\). So, multiplying a number by \(10^3\) results in moving the decimal point 3 spaces to the RIGHT.
\((9.865294501)(10^6) = 9,865,294.501\). So, multiplying a number by \(10^6\) results in moving the decimal point 6 spaces to the RIGHT.


Let \(n\) = the original number
This means \(\frac{1}{n}\) = the reciprocal of the original number

GIVEN: When the decimal point of a certain positive decimal number is moved six places to the right, the resulting number is 9 times the reciprocal of the original number.
We can reword this as: When \(n\) is multiplied by \(10^6\), the resulting number is 9 times \(\frac{1}{n}\)

So our equation is: \((10^6)(n) = (9)(\frac{1}{n})\)

Multiply both sides of the equation by \(n\) to get: \((10^6)(n^2) = 9\)

Divide both sides of the equation by \(10^6\) to get: \(n^2 = \frac{9}{10^6}\)

Rewrite the right-hand side as follows: \(n^2=(\frac{3}{10^3})(\frac{3}{10^3}\))

In other words, \(n^2=(\frac{3}{10^3})^2\)

This means: \(n = \frac{3}{10^3}=\frac{3}{1000}=0.003\)

Answer: \(0.003\)

Cheers,
Brent
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Re: QOTD #8 When the decimal point of a certain positive decimal [#permalink] New post 04 Sep 2020, 13:05
GreenlightTestPrep wrote:
sandy wrote:
When the decimal point of a certain positive decimal number is moved six places to the right, the resulting number is 9 times the reciprocal of the original number. What is the original number?

enter your value

[Reveal] Spoiler: OA
0.003


Practice Questions
Question: 21
Page: 154


A student asked me to solve this question. So here goes....

First we need to understand what must occur to move a decimal point 6 spaces to the RIGHT.

Check out these examples:
\((1.234567)(10^2) = 123.4567\). So, multiplying a number by \(10^2\) results in moving the decimal point 2 spaces to the RIGHT.
\((6.73215333)(10^3) = 6,732.15333\). So, multiplying a number by \(10^3\) results in moving the decimal point 3 spaces to the RIGHT.
\((9.865294501)(10^6) = 9,865,294.501\). So, multiplying a number by \(10^6\) results in moving the decimal point 6 spaces to the RIGHT.


Let \(n\) = the original number
This means \(\frac{1}{n}\) = the reciprocal of the original number

GIVEN: When the decimal point of a certain positive decimal number is moved six places to the right, the resulting number is 9 times the reciprocal of the original number.
We can reword this as: When \(n\) is multiplied by \(10^6\), the resulting number is 9 times \(\frac{1}{n}\)

So our equation is: \((10^6)(n) = (9)(\frac{1}{n})\)

Multiply both sides of the equation by \(n\) to get: \((10^6)(n^2) = 9\)

Divide both sides of the equation by \(10^6\) to get: \(n^2 = \frac{9}{10^6}\)

Rewrite the right-hand side as follows: \(n^2=(\frac{3}{10^3})(\frac{3}{10^3}\))

In other words, \(n^2=(\frac{3}{10^3})^2\)

This means: \(n = \frac{3}{10^3}=\frac{3}{1000}=0.003\)

Answer: \(0.003\)

Cheers,
Brent


Brent, could you show how this would work with plugging in numbers?

I started with N = 3

3,000,000 = 1/3 * 9

All I end up with is 3,000,000 / 3. I'm struggling to understand where I went wrong in plugging in numbers.

Thank you
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Re: QOTD #8 When the decimal point of a certain positive decimal [#permalink] New post 16 Sep 2020, 04:36
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rendition3 wrote:
Brent, could you show how this would work with plugging in numbers?

I started with N = 3

3,000,000 = 1/3 * 9

All I end up with is 3,000,000 / 3. I'm struggling to understand where I went wrong in plugging in numbers.

Thank you


We can't solve this question by plugging in numbers, since there's only one number that satisfies the given information.
This means we would have to keep testing numbers until we get the one number that satisfies all of the given information.

Let me first show you why the answer must be 0.003 (and then we'll take a look at your approach)
If we move the decimal place 6 places to the right we get: 3000
Since 0.003 = 3/1000, the reciprocal of 0.003 is 1000/3
Notice that 3000 is 9 times the reciprocal, 1000/3
That is, (9)(1000/3) = 3000
So, the correct answer must be n = 0.003

If we start testing numbers (as you have done), we'll find that all numbers (other than 0.003) won't satisfy the given information.
For example, if n = 3, then we get 3,000,000 when we move the decimal point six spaces to the right
The reciprocal of 3 is 1/3
Since 3,000,000 does NOT equal (9)(1/3), we can be certain the answer is NOT n = 3

Does that help?
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Re: QOTD #8 When the decimal point of a certain positive decimal   [#permalink] 16 Sep 2020, 04:36
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