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 valuePractice 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