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.

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

Regards

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?

Posted from my mobile device

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



Regards

Thank u for the clear-cut explanation.

Posted from my mobile device

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

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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