Hi, Sophia,
I LOVE this type of question. With unit conversions/analysis, we can conclude something about which we have very little perspective. Sounds crazy, but bear with me, it will be fun.
The most important idea here is the nature of a conversion factor. When someone says 12 eggs in a dozen, we can write it in several ways. I'll leave off the eggs for now, since this can apply to anything - raspberries, clowns, teachers, movies, etc. It is a definition. 12 of anything can be called a "dozen," if you fell like it. I personally don't know why anyone would pick a number like "12" and give it a special name, but the custom is rampant:
5280 feet is one mile (who came up with that?)
144 is one gross (what?),
12 inches in a foot,
3 feet in a yard,
and so on.
These are all definitions of a particular number, but the idea applies to any type of conversion. It could be clowns/circus or students/class, for example.
They are equalities, so I can write them in various ways:
1 foot/12 inches,
5280 feet/mile
12 inches/1 foot
4 quarts/gallon
1 mile/5280 feet
1 gallon/4 quarts
etc.
It is important to note I can write them with the values inverted, such as 1 mile/5280 feet. Since the denominator and numerator are both equal to each other, I can write them in either format. That is very useful when it comes to problems such as these.
2 fardells=1 nooke,
4 nookes=1 yard,
and 4 yards=1 hide.
How many hides in 15 nookes?
The first three lines are all conversion factors. I can write them in a format that I choose in order to conveniently cancel units. Since the denominator and numerator are equal, the expression reduces to just 1, mathematically. That allows us to multiply that expression times anything, since it is ultimately a value of "1." The numbers change, but those changes simply reflect a change in the units, not in the actual value.
Let's start with the nookes. I want to know how many hides are in 15 of the buggers. I need to select from the conversion factors to eventually change nookes into hides. I can see from the second and third conversion factors that nookes are related to yards, and that yards are related to hides. Let's try those:
15 nookes * (1 yard/4 nookes) = 15/4 yards. I chose the conversion factor in a form that placed yards over nookes. When I multiply the 15 nookes, the unit nookes simply cancels out and we are left with just yards. I could have chosen 4 nookes/1 yard, but then I would have to divide 15 nookes by (4 nookes/1 yard), and, although simple in the case, I prefer multiplication over division because my mind can easily predict the outcome of a division as well as a multiplication.
Next, we need to convert yards into hides. Take the 2nd conversion factor and write is as (1 hide/4 yards). Then multiply it times the (15/4) yards:
(15/4) yards x (1 hide/4 yards) = 15/16 hide. The unit "yards" cancels out and all we're left with is hides. Sounds promising!
There is 15/16 hide in 15 nookes. I didn't have to envision what exactly is a nooke or a yard, or even a fardell. I just used the units to decide what steps to take in changing the number of nookes into an equivalent number of hides. I don't like doing that, since it is easy to wind up with something stupid, but don't worry about it until the end of the problem. In this case one can't envision what is happening, so just go with it.
Keeping units in calculations is an easy way to help guide your work and to check whether the answer makes sense. If your final unit is something like "presidents/square inch" then you may want to double check, especially if it is "US presidents"/square inch. But if the units have no physical meaning for you, then just be careful going through the first time.
I do conclude, however, a nooke must be a real loser if it only has 15/16 of a hide.
I know keeping units in a calculation seems like a lot of effort for nothing, but it does save time and prevents errors in the end.
I hope this helps,
Bob
