Math problem: how many objects will fit into this glass jar

February 22, 2015 7:36 PM

I am trying to work how many 5 gram objects can fit into a 3 litre jar. I struggle with conversions of grams to litres and finding the number of objects. Can someone show me how to mathematically work it out. My mind is blank. Thanks in advance :)
posted by nassep to Education (12 answers total)

Grams are a unit of mass and liters are a unit of volume. In order to solve this problem, you need to know the density of the objects (mass/volume, or grams/liter) or the volume of each object.
posted by i_am_a_fiesta at 7:40 PM on February 22, 2015 [12 favorites]

Grams are a unit of mass and liters are a unit of volume, so there is not going to be a direct conversion. Different things of the same size weigh different amounts. What are the 5 gram objects you are trying to calculate?

edit: on preview, what fiesta said.
posted by The Architect at 7:41 PM on February 22, 2015 [1 favorite]

The missing piece of information here is the density of the objects (i.e. how much a cubic centimeter of material weighs, or vice versa, how many cubic centimeters a gram of material occupies). If we assume that the objects are as dense as water at room temperature, this becomes easy: one gram of room-temp water equals one cubic centimeter, so each object occupies 5 cubic centimeters (if our assumption about their density holds).

One liter is equal to 1000 cubic centimeters, so the jar is 3000 cubic centimeters in volume. Finally, with consistent units we can now divide 3000 by 5 to get the answer: 600 objects. (Note that this also assumes that the objects pack in perfectly with no wasted space.)

We can also use this 600 figure to scale the answer up or down if the objects' density is different from water's. If, for example, the objects have a density of (to pick a random value) 2.5 grams per cubic centimeter, the jar will hold 2.5 times as many (smaller, denser) objects as if we assumed the objects had a density of 1.0 grams per cubic centimeter (i.e. 1,500 objects). If the objects are less dense, say 0.4 grams per cubic centimeter, the jar will hold 0.4*600 = 240 (bigger, lower-density) objects.
posted by letourneau at 7:45 PM on February 22, 2015

The density of gold is about 19,300 g/l, so you could fit 11,580 pieces of gold weighing 5g each into a jar if you could pack them perfectly. The density of cork is about 250g/l, so you could fit 150 pieces of cork into a jar if you could pack them perfectly but not squish them (aka make them denser). Most things are less dense than gold and more dense than cork, so somewhere in that range probably.
posted by jeather at 7:48 PM on February 22, 2015 [2 favorites]

Don't forget the shape of the objects may prevent a perfect amount to fill the jar.

You know, it's funny, but when you say how many 5-gram objects can fit into a jar, the possible answers can be "none," if it's not very dense and hence too big, or "one," if it's just the right size to fit all by itself, or "infinite," if they're super-dense and tiny.
posted by Cool Papa Bell at 7:55 PM on February 22, 2015 [7 favorites]

I struggle with conversions of grams to litres and finding the number of objects.

Don't feel bad about struggling with this; it's simply not possible to figure this out from the information you've given here. Even if we knew the density, it would still be impossible unless the objects were basically gelatinous. If they are rigid solids then you'd need to know the shape of the jar, the shapes of the objects... it could get very complicated.
posted by jon1270 at 8:17 PM on February 22, 2015 [2 favorites]

Just to make it worst, packing problems are some of the most intractable and actual research topics.
posted by sammyo at 8:19 PM on February 22, 2015 [2 favorites]

The good news is, if the objects have a simple, uniform shape, you can look up a "packing factor" for them, and then just multiply it in.
posted by palmcorder_yajna at 9:40 PM on February 22, 2015

The objects don't happen to be nickels, do they? (Pardon the idle speculation, but nickels weigh 5.000 grams and seem like a reasonable thing to pack into a jar.)

You could pack just under 4,000 nickels into a 3-liter jar if they fit neatly in honeycombed layers.

On preview: you wrote "litre" so I am probably wrong. :)
posted by aws17576 at 10:31 PM on February 22, 2015

This is a complex problem: https://en.wikipedia.org/wiki/Sphere_packing
posted by yoyo_nyc at 2:15 AM on February 23, 2015 [1 favorite]

All the comments above are correct if you are talking about solid objects with a fixed shape made out of some unknown substance. If you ignore the shape fitting problem and answer the density question, then the problem is very simple.

Take, for example, 5 gram drops of water. 1 gram of water = 1 ml of water. 5 grams = 5 ml. 3,000/5 = 600 drops to fill the container.
posted by alms at 6:53 AM on February 23, 2015

Identical round objects like marbles will fill 52% of the volume allotted. Therefore, for marbles, find the mass per volume of glass, and multiply by .52 times the volume of the container, treating it as a uniform substance, to find how much mass the container will contain. If you mix sizes, then it depends on what the sizes are, and it gets complicated, since now smaller objects can fill in the empty space between larger objects.
posted by halhurst at 2:22 PM on February 23, 2015