Question

How do you convert `4.0 \times 10^7` micrograms to pounds (lbs)?

Answer 1

`0.0881848` lbs

Explanation:

1 microgram = `2.20462xx10^-9` lbs

So amount in lbs would be:

Number of micrograms x 1 microgram expressed in lbs.

This will be:

`4.0 xx 10^7 xx 2.20462xx10^-9`

`4 xx 2.20462 = 8.81848`

`10^7 xx 10^-9 = 10^-2`

This give us:

`8.81848 xx 10^-2` lbs

`0.0881848` lbs

Answer 2

I could use all sorts of shortcuts and present a very efficient solution. This may leave you wandering where things came from. So you got the lot!

`4.0xx10^7" microns"= 8.816xx10^-2" pounds"`

Explanation:

THE CONVERSION PART IN A LOT OF DETAIL USING RATIO

Did you know you can manipulate units of measurement the same way you do the numbers. I WILL BE USING THIS!

Quote from Wikipedia: https://en.wikipedia.org/wiki/Microgram

a microgram or microgramme (μg) is a unit of mass equal to one billionth (1×10−9) of a kilogram

Using ratio: `("micrograms")/("kilograms") = 1/(10^(-9))`

.....................................................................................................
We need to change this so that we have 1 kilogram not `10^(-9)`

Note that `color(blue)(10^(-9))` is the same as `color(blue)(1/(10^9))`

So `color(blue)(10^(-9))color(red)(xx10^(9)) = 10^(color(red)(9)color(blue)(-9)) = 10^0 = 1`
.......................................................................................................
Thus we have:

`("micrograms")/("kilograms") = 1/(10^(-9))color(purple)(xx1)=1/10^(-9)color(purple)(xx(10^9)/(10^9)) = 10^9/1color(red)(..Eqn(1))`

So now we have 1 kilograms worth of micrograms

Using the conversion: 1 kilogram = 2.204 lb

`("kilograms")/("pounds")-> 1/2.204color(red)(..Eqn(2))`

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We need to end up with the ratio `("microns")/("pounds")`

So we can use the units of measurement to show us what to do with the numbers:

`("microgram")/(cancel("kilograms"))xx(cancel("kilograms"))/("pounds")=("microgram")/("pounds")`

`color(white)("dddd")darrcolor(white)("ddddddddd")darrcolor(white)("ddddddd")darr`

`color(white)("ddd")10^9/(cancel(1))color(white)("ddd")xxcolor(white)("dd")(cancel(1))/2.204 color(white)("d")=color(white)("d")10^9/2.204`

`("microgram")/("pounds")=10^9/2.204`

so for 1 microgram we have

`("microgram")/("pounds")=(10^9-:10^9)/(2.204-:10^9)= color(red)(1/(2.204xx10^(-9)))`
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`color(blue)("NOW WE HAVE THE CONVERSION WE CAN ANSWER")` `color(blue)("THE QUESTION")`

`(4.0xx10^7" micrograms")/("pounds")->color(red)(1/(2.204xx10^(-9)))xx(4.0xx10^7)/(4.0xx10^7)`

`4.0xx10^7" microgram"=(color(red)(2.204xx1/10^9))xx(4.0xx10^7)` pounds

`4.0xx10^7" microgram"= 8.816xx10^-2" pounds"`