This one goes out to all the people who hated word problems. Yes, you do use this stuff in real life. Or at least I do. But I can’t remember how to set this one up.
I’m trying to figure out which is the better deal. One vendor sells glycerin by weight, and the other by volume. I know the density of glycerin is 1.26g/cm^3.
So which is the better deal?
16 ounces (by weight, not volume) for $3.92
or
500mL for $6.59
Help? (While an answer would be fantastic, if you can tell me HOW to do it, then I can figure out the same thing for the propylene glycol…)
Ahhh, both glycerin and propylene glycol (antifreeze) are liquids. Liquids are not compressible therefore there weight per volume will be constant. Unless you mean their concentration or purity.
You have a measurement with weight and a measurement by volume. You need to decide whether you want to convert the weight to volume or convert the volume to weight. It could go either way.
I like the idea of converting the weight to volume. We have 16 ounces (1 pound) in the $3.92 bottle. In order to translate that to volume, we need density. But the density given is in metric. So the first step is to convert 1 pound in to grams. Google says 1 pound equals 453.592 grams.* So now we have 453.592 grams for $3.92 together with the fact that the density is 1.26 grams per cm3.
The tricky part here is figuring out whether we should multiply those numbers or divide them. So we’ll use a method called dimensional analysis. Basically, you make sure the units cancel.
Make a fraction with 453.592 grams on top and $3.92 on the bottom. You could have done it the other way, putting dollars on top and grams on the bottom. It doesn’t matter but you have to pick one. I picked grams on top. Now we put another fraction next to it so we can multiply the two fractions and something will cancel. Our second fraction will either be 1.26 grams on top and 1 cm3 on the bottom, or vice versa. If we put grams on the top in both fractions then we’ll be multiplying grams times grams, which isn’t what we want. Put grams on the bottom and now we have grams on top in one of the fractions and grams on the bottom in the other fraction so the grams cancel.
453.592 grams 1 cm3
____________ x _______
$3.92 1.26 grams
Multiply 453.592 times 1 and mulitply 3.92 by 1.26 then divide the top number by the bottom number to simplify the fraction and you get 91.835115 cm3 /
Now look at the other bottle, which is 500 mL for 6.59. Notice that 1 ML and 1 cm3 are the same thing. So it's really 500 cm3 on top and 6.59 dollars on the bottom. Divide those two numbers to simplify the fraction and you get 506.59 cm3 /
Now ask yourself which is better. Would you rather get 92 cm3 for a dollar or 507 cm3 for a dollar? The second is better, by about a factor of 5.
We’re ignoring the fact that weight and mass are not exactly the same thing. Converting pounds to grams requires an assumption about what gravity field the pounds were measured in. Most places on Earth, you’d get very similar measurements, but it’s not precisely the same everywhere. And if you did the measurement on the Moon your answer would be way off, by about a factor of 6.
I usually follow dimensional analysis pretty well, but in this case I’m not getting the same numbers, which is where I confused myself.
When I convert oz to grams and divide by density. That gives me mL (or cc, or cm^3) that I can then divide the price by to get cost per mL
3.92/((16*28.35)/1.26) = 0.010888…
compared to the one sold in mL
6.59/500 = 0.01318
And that’s damn near identical, not a fivefold difference. Plus previous experience tells me that 16 ounces of something that’s even vaguely in the same universe of density as water isn’t going to be less than 100mL. So I think there’s a problem with your DA somewhere. I’m just not sure where.
Or what am I doing wrong?
Could be. But they’re the only ones doing it. (For a definition of “only” that includes the half dozen vendors I’ve checked; not an exhaustive list, to be sure.) It’d be nice if they sold in volume listed numbers and converted them they own dangselves to weight for filling if that’s how their equipment works.
Obligatory Gimli Glider reference. Long story short: Everyone measures fuel by volume (gallons/liters) except aircraft people, who measure it by weight…
16 [del]oz[/del] * 28.3495231 [del]g[/del] * 1 [del]cm³[/del] * 1 mL = 359.993944127 mL
1 [del]oz[/del] 1.26 [del]g[/del] 1 [del]cm³[/del]
$3.92 / 360 mL = $0.01089 per mL
$6.59 / 500 mL = $0.01318 per mL
The fancy pants multiplication is how my chemistry teacher taught us to do dimensional analysis. You know that you want to end with mL on top, since it’s your answer. That tells you which direction to put all the other fractions. Then you multiply the top and divide by the bottom, like you always do when multiplying fractions.
Seems like you already did the math, WhyNot, so mine can help you to check your numbers. (I rounded the top answer to the same number of places to the bottom one.)
Plus, it was too much trouble getting the formatting right not to post it.
Oh dear! At least my math struggles aren’t going to take down an aircraft!
I bet that this vendor is doing it by weight because that gives them smaller per bottle prices and few people take the time to figure it out on a per mL basis. People just see $3.92 and figure it’s cheaper than everyone else’s.
except for a few cases, numbers always have units. if the numbers always have the units next to them it helps the thought process. takes more space on the page but it gets the job done correctly quicker.
Only if I don’t want to know how to do it for items that aren’t on the list, like propylene glycol. I had two goals here. First, to find the answer, but more importantly to remind myself of the process. I don’t have to account for density often, but this time I did, and couldn’t remember the process. Now I know it. I’ve learned how to fish, and can feed myself.
This does make me wonder about the ratios of PG/VG that people are listing on their labels. I use 70/30 by volume…wonder if some people are using weight. VG being decidedly more dense than PG, it would make a difference. Maybe not a terribly significant one, but I wonder…
Doh! Yeah you nailed, my finger must have slipped. 500 divided by 6.59 should be 75.8725341. So now the question is would you rather get 92 cm3 for a dollar or 76 cm3 for a dollar? And 92 is better.
When I got answers off by a factor of 5, that set off a warning bell in the back of my mind. I was thinking it seemed strange that the prices would be so different from each other. I should have gone back and looked for mistakes. But the procedure was sound.
