How do you express 0.0001/0.04020 as a decimal?

4 Answers
Feb 11, 2016

1/402

Explanation:

Take 0.0001/0.04020 and multiply top and bottom by 10000.
{0.0001 xx 10000}/{0.04020 xx 10000}.
Use the "move the decimal" rule. ie. 3.345 xx 100=334.5 to get:
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 1/10000 and the 2 in 0.0402 is also in the ten thousandth column so 0.0402=402/10000.

0.0001/0.04020= {1/10000}/{402/10000} =1/10000-:402/10000
=1/10000 xx 10000/402 =1/402.

Feb 11, 2016

Multiply numerator and denominator by 10^4 to get 1/402, then long divide to get:

1/402 = 0.0bar(0)2487562189054726368159203980099bar(5)

Explanation:

To calculate 0.0001 / 0.04020 first multiply both numerator and denominator by 10^4 to get 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:color(white)(XX000)0
1:color(white)(XX0)402
2:color(white)(XX0)804
3:color(white)(XX)1206
4:color(white)(XX)1608
5:color(white)(XX)2010
6:color(white)(XX)2412
7:color(white)(XX)2814
8:color(white)(XX)3216
9:color(white)(XX)3618

Then our long division starts:
enter image source here

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 color(blue)(2) resulting in 2 xx 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 color(blue)(4) for the quotient, etc.

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

1/402 = 0.0bar(0)2487562189054726368159203980099bar(5)

Feb 11, 2016

I want to capitalize on George C. answer and give my version of 1/402!!!

Explanation:

have a look:
enter image source here

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!!

color(blue)( 0.0001/(0.04020)" "~=" "0.00024)

Explanation:

Given:" " 0.0001/(0.04020)

color(purple)("Making them into more mentally manageable numbers")color(purple)("and apply a correction at the end!")

Multiply the numerator by 10^7 giving: 1000 so the correction isxx10^(-7)

so 0.0001/(0.04020)" "=" "1000/0.0402xx10^(-7)

Multiply the denominator by 10^4 in the form of

1/0.0402xx1/10^4 ->1/402 so the correction for this bit is xx10^4

Putting this all together gives:

1000/402 xx (10^(4-7))" "=" "1000/402color(green)(xx10^(-3))
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

color(blue)("Step 1")
" "underline(" ")
Write as:" " 402|1000

Consider just the hundreds: 10-:4=2+"Remainder"
Do not worry about the remainder!

" "underline(" 2 ")
Now write:" " 402|1000
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
color(blue)("Step 2")

2xx402=color(brown)(804)

" "underline(" 2 ")
Now write:" " 402|1000
" "color(brown)(underline(804 -))
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
color(blue)("Step 3")

subtract the 804 from the 1000
" "underline(" 2 ")
" " 402|1000
" "color(brown)(underline(804 -))
" "196
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~
color(blue)("Step 5")

402 > 196 so put a decimal place to the right of the 2 and put a
color(red)(0) to the right of 196

" "underline(" 2"color(red)(.)" " )
" " 402|1000
" "underline(804 -)
" "196color(red)(0)

402xx5=2010 >1960 so too big
402xx4=color(magenta)(1608)<1960 so we pick this one

so 1960-:402=color(green)(4) +"Remainder"

So now we write:

" "underline(" "2"."color(green)(4)" " )
" " 402|1000
" "underline(804 -)
" "1960
" "underline(color(magenta)(1608-))

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
color(blue)("Step 6")

" "underline(" "2"."color(green)(4)" " )
" " 402|1000
" "underline(804 -)
" "1960
" "underline(1608-)
" "352

352 < 402 so put color(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 color(green)(xx10^(-3))" "->" "2.4/1000" "=" "0.00024

0.0001/(0.04020)" "~=" "0.00024

Look at the beginning to see where color(green)(xx10^(-3)) comes from.