A number of parts per one hundred total parts defines a percent. This relationship can be written as a fraction with the number of parts over one hundred. It can also be written as an equivalent fraction, decimal, or as the percentage itself. (e.g. 75/100 is equivalent to ¾, and when divided equals .75 which then in turn equals 75%) The fraction or ratio becomes a decimal by being divided. To convert the decimal to percent, multiply it by one hundred and then add the percent symbol.

However, when trying to describe what one hundred and ten percent is, we find that the description above fails in that the so-called part basically exceeds the stated total.

For example, one hundred and ten parts over one hundred parts, when divided, creates the decimal one point one; wherein we make one hundred and ten percent once we’ve multiplied it by one hundred and included the percent symbol. In this way, it seems troublesome to refer to the smaller quantity in the denominator as a so-called total:

We find that improper fractions share this same seemingly-inverted premise too: a part larger than the so-called total number of parts when referring to the denominator as the “total”. Since fractions and percentages have this relationship with a stated total, it is often more useful to use the term "base" rather than total to avoid circumventing what total means.

For the purposes of this discussion though, nothing is ever going to exceed one hundred percent. So rather than using the more carefully-considered base jargon in the course of discussing percentages or denominators, we are safe to use the term "total" throughout.

But what if the parts in question need to be grouped according to some specific detail regarding density? Since density is comprised of both mass and volume, how would we decide whether we are grouping according to mass or volume? And what is the difference between percent by volume and percent by mass anyway? Since this report estimates the value of small amounts of gold, and since the value of a golden thing is often attributed its karat value, we’ll need to ultimately understand what a karat is and how it relates to percentages. First, however, let’s briefly look at the differences between percent by volume and percent by mass:

Percent by volume:
Percent by volume is the ratio of a part of a given volume over the entire volume, and is calculated when dividing the part by the total, multiplied by 100.

For example, consider two same-sized blocks of lead and Styrofoam joined together to make some sort of single unit. Let’s find the percent by volume of Styrofoam. We could notice that the block of lead is much more massive or heavy than the block of Styrofoam, but we want to compare volumes here, not masses or weights. By thinking about weight, we have gotten sidetracked by the fact that the lead block has a higher percentage of mass than the Styrofoam block. Therefore, to strictly calculate percent by volume, we simply take the one part Styrofoam volume over the two-part total volume, creating the fraction one over two, (one half), multiplied by 100. This results in fifty percent by volume Styrofoam.

With regard to this particular example, each substance happens to have the same percent by volume: fifty percent "by volume" lead and fifty percent Styrofoam. This is in contrast with the percent by mass that Styrofoam would have.

Percent by volume is often chosen instead of percent by mass when mixing two different fluids together because some sort of convenient unit for volume has already been decided upon (cups for instance).

Percent by mass:
Percent by mass is the ratio of a part of a given mass over the entire mass, and is calculated when dividing the part by the total, multiplied by 100.

Regarding a substance comprised of both Styrofoam and lead, we might see that the equal masses of lead and Styrofoam have very different sizes. However, this view preoccupies us with notions regarding volume (size). But, mass, specifically, has no volume.

Percent by mass can be thought of as percent by weight just so long as the before-mentioned difference between mass and weight is kept in mind. Percent by mass is often preferred over percent by volume when working with items such as potato chips or feathers because quantities may fluctuate. (In other words, items such as these may take up different amounts of space depending upon how they are organized)

For example, Pringles potato chips can be tightly stacked against one another, creating a greater density of chips than some other potato chips that are more loosely packed. This holds true for feathers too. A pile of loosely-packed feathers may have fewer feathers than an equal-sized pile of highly-compacted feathers. One way to assure that the correct amount of chips or feathers are in each pile is to use weight rather than size for comparison.

In some way, gold nuggets can behave a little like potato chips too. Since nuggets are generally oddly shaped, it is possible that one pile of gold could have a slightly different number of nuggets than another pile of the same size. So it is preferable to use percent by mass or weight rather than percent by volume.

The following is an illustration of switching between different materials measured in percent by weight:

Given a container filled ninety-five percent by weight gold and five percent by weight copper, imagine removing the copper so that we could replace it with something denser like silver. We should understand that when there are equal weights of silver and copper, the silver occupies less space than copper since silver is denser.

So if we hastily fill in the entire empty spot where we took out the copper, we’d end up changing the ninety-five to five percent by weight relationship, since silver is more dense than copper.

To maintain the same relationship of ninety five percent by weight gold and five percent by weight silver, we’d need to fill in the silver just to the point where we reach the same weight as the entire thing before. At that point, we’d still have a small gap that needs filling in.

To fill in that remaining gap and maintain the ninety-five to five percent by weight relationship, we’d need to start filling in this final space with both gold and silver at the same time somehow. And, we’d need to do this in a manner that maintains the desired ninety-five percent gold and five percent silver by weight relationship too.

The result: The same volume is used as before, but a heavier total weight is created, one where ninety-five percent by weight is gold and five percent by weight is silver. And we’d see that we needed to use a bit more gold this time in order to maintain the correct percent by weight relationship.

Gold value is based upon the karat which is typically stated as percent by weight. But for the previously-stated reasons, I use percent by mass instead.

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