Answer:
8.34 lb/gal to 3 sig figs
Step-by-step explanation:
We are given two conversions:
(1lb =453.59) and (1 gal= 3.785 L)
Lets start with the raw data and calculate a density that is expressed as grams/ml.
Density = (10.0 grams)/(9.98 ml)
Density = 1.002 g/ml
To convert this into pounds per gallon (lbs/gal) we can use the given conversion factors. It is important to keep the units in the calculations to guide us on how the conversions can be properly made.
(1lb =453.59 g)
(1 gal= 3.785 L)
We want to convert from (g/ml) to (lbs/gal).
Lets rearrange 1lb =453.59 g into a conversion factor by dividing both sides by 453.59g:
1lb =453.59 g
1lb/453.59 g = 1
Note that this is equal to 1. Math tells us we can multiple 1 times anything and it will be a valid number. The difference is that we are actually changing the value to be consistent with the new unit. It is still a valid number, but only with the new unit.
I chose to divide both sides by the grams value, since I can now multiply that times the metric density (g/ml) is the following manner:
(1.002 g/ml)*(1 lb)/(453.59 g)
Note that the grams cancel, leaving just lbs and ml:
(1.002 /ml)*(1lb/453.59 )
Density = 0.002205 lb/ml
Note: [One may also use (453.59 g)/(1 lb) as a conversion factor, but to get grams to cancel, the density of (g/ml) must be divided by the factor in order for the units to cancel:
(1.002 g/ml)/((453.59g/1 lb)
I don't like division. It is harder to visualize the approximate result and I prefer to take a guess to help avoid a calculation error. It is easier, for me, to mentally multiply to an approximate result so that a blatant error might be detected. I always arrange conversion factors so that they can be multiplied (personal preference).
Now that we have Density = 0.002205 lb/ml, we need to change ml to gallons.
Use the second conversion of 1 gal= 3.785 L. Our goal is to make this a conversion factor that will allow us(me) to cancel out the ml unit and replace it with gallons.
1 gal= 3.785 L
First, let's convert the L to ml:
1000 ml = 1 L
1000 ml/L = 1
1 gal= (3.785 L)*(1000 ml/L )
1 gal = 3785 ml
In this case I will divide both sides by 1 gal to get the desired conversion factor:
1 gal = 3785 ml
1 = (3785 ml)/(1 gal)
This arrange it allows us to multiply the (0.002205 lb/ml) density result from above so that the ml units cancel and gallons is added:
Density = 0.002205 lb/ml
Density = (0.002205 lb/ml)*((3785 ml)/(1 gal))
Density = 8.34 lb/gal to 3 sig figs
[Note: An online conversion utility returns the value of 10.04 lbs/gal. I don't know if the conversion factors are different here, but be careful with the math.]