# How do you express 0.0001/0.04020 as a decimal?

Feb 11, 2016

$\frac{1}{402}$

#### Explanation:

Take $\frac{0.0001}{0.04020}$ and multiply top and bottom by 10000.
$\frac{0.0001 \times 10000}{0.04020 \times 10000} .$
Use the "move the decimal" rule. ie. $3.345 \times 100 = 334.5$ to get:
$\frac{1}{402.}$ This is the answer in fraction form.

If the goal was to covert the decimal directly to fractions and then solve, in $0.0001$, the $1$ is in the ten thousandth column, making it the fraction $\frac{1}{10000}$ and the 2 in 0.0402 is also in the ten thousandth column so $0.0402 = \frac{402}{10000}$.

$\frac{0.0001}{0.04020} = \frac{\frac{1}{10000}}{\frac{402}{10000}} = \frac{1}{10000} \div \frac{402}{10000}$
$= \frac{1}{10000} \times \frac{10000}{402} = \frac{1}{402}$.

Feb 11, 2016

Multiply numerator and denominator by ${10}^{4}$ to get $\frac{1}{402}$, then long divide to get:

$\frac{1}{402} = 0.0 \overline{0} 2487562189054726368159203980099 \overline{5}$

#### Explanation:

To calculate $\frac{0.0001}{0.04020}$ first multiply both numerator and denominator by ${10}^{4}$ to get $\frac{1}{402}$

Assuming we want a decimal expansion of the quotient, let's use long division.

First write out the multiples of $402$ we will use:

$0 : \textcolor{w h i t e}{X X 000} 0$
$1 : \textcolor{w h i t e}{X X 0} 402$
$2 : \textcolor{w h i t e}{X X 0} 804$
$3 : \textcolor{w h i t e}{X X} 1206$
$4 : \textcolor{w h i t e}{X X} 1608$
$5 : \textcolor{w h i t e}{X X} 2010$
$6 : \textcolor{w h i t e}{X X} 2412$
$7 : \textcolor{w h i t e}{X X} 2814$
$8 : \textcolor{w h i t e}{X X} 3216$
$9 : \textcolor{w h i t e}{X X} 3618$

Then our long division starts:

Write the dividend $1.000$ under the bar and the divisor $402$ to the left. Since $402$ is somewhat less than $1$, there are several zeros for the quotient before it 'gets going'. Once we have brought down 3 $0$'s from the dividend our initial running remainder is $1000$ and the first non-zero digit of the quotient is $\textcolor{b l u e}{2}$ resulting in $2 \times 402 = 804$ to be subtracted from the remainder to yield the next remainder.

Bring down another $0$ from the dividend alongside the remainder $196$ to give $1960$ and choose the next digit $\textcolor{b l u e}{4}$ for the quotient, etc.

Notice that with the running remainder having arrived at $10$ we are essentially back to dividing $\frac{1}{402}$ again - that is we have found the recurring decimal expansion:

$\frac{1}{402} = 0.0 \overline{0} 2487562189054726368159203980099 \overline{5}$

Feb 11, 2016

I want to capitalize on George C. answer and give my version of $\frac{1}{402}$!!!

#### Explanation:

have a look:

Feb 11, 2016

Just for fun I thought I would add a solution as well. I am going to considerably limit the number of decimal places!!

$\textcolor{b l u e}{\frac{0.0001}{0.04020} \text{ "~=" } 0.00024}$

#### Explanation:

Given:$\text{ } \frac{0.0001}{0.04020}$

$\textcolor{p u r p \le}{\text{Making them into more mentally manageable numbers}}$$\textcolor{p u r p \le}{\text{and apply a correction at the end!}}$

Multiply the numerator by ${10}^{7}$ giving: 1000 so the correction is$\times {10}^{- 7}$

so $\frac{0.0001}{0.04020} \text{ "=" } \frac{1000}{0.0402} \times {10}^{- 7}$

Multiply the denominator by ${10}^{4}$ in the form of

$\frac{1}{0.0402} \times \frac{1}{10} ^ 4 \to \frac{1}{402}$ so the correction for this bit is $\times {10}^{4}$

Putting this all together gives:

$\frac{1000}{402} \times \left({10}^{4 - 7}\right) \text{ "=" } \frac{1000}{402} \textcolor{g r e e n}{\times {10}^{- 3}}$
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

$\textcolor{b l u e}{\text{Step 1}}$
" "underline(" ")
Write as:$\text{ } 402 | 1000$

Consider just the hundreds: $10 \div 4 = 2 + \text{Remainder}$
Do not worry about the remainder!

" "underline(" 2 ")
Now write:$\text{ } 402 | 1000$
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Step 2}}$

$2 \times 402 = \textcolor{b r o w n}{804}$

" "underline(" 2 ")
Now write:$\text{ } 402 | 1000$
$\text{ } \textcolor{b r o w n}{\underline{804 -}}$
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Step 3}}$

subtract the 804 from the 1000
" "underline(" 2 ")
$\text{ } 402 | 1000$
$\text{ } \textcolor{b r o w n}{\underline{804 -}}$
$\text{ } 196$
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Step 5}}$

402 > 196 so put a decimal place to the right of the 2 and put a
$\textcolor{red}{0}$ to the right of 196

" "underline(" 2"color(red)(.)" " )
$\text{ } 402 | 1000$
$\text{ } \underline{804 -}$
$\text{ } 196 \textcolor{red}{0}$

$402 \times 5 = 2010 > 1960$ so too big
$402 \times 4 = \textcolor{m a \ge n t a}{1608} < 1960$ so we pick this one

so $1960 \div 402 = \textcolor{g r e e n}{4} + \text{Remainder}$

So now we write:

" "underline(" "2"."color(green)(4)" " )
$\text{ } 402 | 1000$
$\text{ } \underline{804 -}$
$\text{ } 1960$
$\text{ } \underline{\textcolor{m a \ge n t a}{1608 -}}$

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Step 6}}$

" "underline(" "2"."color(green)(4)" " )
$\text{ } 402 | 1000$
$\text{ } \underline{804 -}$
$\text{ } 1960$
$\text{ } \underline{1608 -}$
$\text{ } 352$

352 < 402 so put $\textcolor{red}{0}$ to the right of 352 and we repeat step 5. This cycle go on for ever if the number is irrational!
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

So far we have 2.4. Applying the correction this becomes:

$2.4 \textcolor{g r e e n}{\times {10}^{- 3}} \text{ "->" "2.4/1000" "=" } 0.00024$

$\frac{0.0001}{0.04020} \text{ "~=" } 0.00024$

Look at the beginning to see where $\textcolor{g r e e n}{\times {10}^{- 3}}$ comes from.