Definitions Deconstructed

Fractional Failure

S. G. Lacey

Definition:

1: A small part or amount of something.

2: A division or portion of a whole number. [REF]

Deconstruction:

Pulling into the convenience store, you instinctively check your gas gauge. It’s well below 1/8th of a tank, the last marking on the display, with the needle now sitting in the ominous, but unquantifiable, red zone. Should probably fill up.

Aligning with the nearest pump, you drop the jalopy in park, and contemplate your options. With two maxed out credit cards in your wallet, racking up 25% interest rates, cash is the only option. Thumbing through a weathered wallet, you find a $10, a $5, and two $1’s, all crumpled. 17/20th of a Jackson, a good start.

Now, it’s time to raid the center console. This foray yields a menagerie of round metal discs, several of them stuck together with spilled soda. Or melted candy. Placing these coins in sorted stacks on the faded vinyl dashboard allows for ease of counting.

9 quarters, 13 dimes, 8 nickels, and 5 pennies.

The exact amount you need to refuel, at the even $3.00 price posted on the big board when you rolled in.

It’s only now, as you put the last few drops of gasoline into the tank, logging exactly 7 gallons, your lucky number, that you realize the math is somehow wrong. Looking up at the massive, illuminated sign, which displays the price for petrol, including the cheapest 87 octane grade, a hidden value at the end of the digits reveals itself.

9/10th, in horizontal fractional form, each numeral less than half the height of the main price denomination, thus easy to miss. This secret 0.9 cent fee per gallon is apparently the cause on your overpour. Time to start scrounging for more pennies under the passenger seat. Or a single dime, 1/10th of a dollar, which would completely solve the payment problem.

This anecdote aside, fractions rarely creep into everyday life in these modern times. As our world has gotten more computerized, basic math functions have become a ubiquitous, rapid, precise, and hidden element of many everyday tasks. However, back in the Middle Ages, before cell phones, calculators, and even chalk, math was very simplified, and highly transactional.

Thus, the need for fractions.

Tallying the total bushels of potatoes going into winter cellar storage. Dividing hectares of farmland into peasant’s individual plots. Butchering a whole cow into sides, each with numerous different cuts of meat.

The root word of fraction is the Latin “fractio”, which literally means “to break”. A fitting term, when fairly sharing a loaf of bread at the dinner table, or chopping wood into equitable burn piles.

Though the actual term was not invented yet, the concept of fractions, like many elements of mathematics, can be traced to ancient Egyptians. Way back in the 2nd millennium BC, these individuals utilized fractional counting, with a base 10 numerical system, and different hieroglyphic images representing each decimal place.

As shown in the graphic below, the oval eye shape was used to denote fractions, with the numerator value always being a one. This simple system worked, but had limitations in what fractional values could be expressed, and was cumbersome for arithmetic operations.

Over the years, there have been countless ways to express fractions, motivated by ease of writing, simplification of calculation, electronic capability, or purely artistic expression. Take the simple fraction 1/6th. This can be represented in the following equivalent ways: 0.167, two bagels in a dozen, 1:6, one-sixth, 2 inches out of a foot, and numerous more obscure forms.

Another scary element of fractions are the long, unique words associated with this branch of mathematics. Quotient. Numerator. Reciprocal. Denominator. Other than adding common to the last word, these terms are rarely used in literature, let alone daily conversation.

For the average 4th grader, fractions combine the already terrifying math elements of times tables and rapid counting, with an entirely new, entirely foreign, numerical nomenclature. The multitude of ways in which the same two numbers of a fraction can expressed, in written, electronic, and digital forms, doesn’t help simplify the topic.

The greatest gaff, and clearest ineptitude, of American citizens interaction with fractions occurred in one of our society’s most popular realms. Fast food. Apparently, even the thought of a greasy meal can cloud one’s algebraic judgement.

In 1972, McDonald’s launched their quarter pounder, a burger whose beef patty size is built into the name. This Mickey D’s iconic menu offering remains relevant to this day.

Seeing the success of this item, numerous other fast food chains followed suit. One of the most formidable challenges came from A&W, in the early 1980’s, when they launched a third pound burger. Offering more meat for the same price, while also beating McDonald’s sandwich in blind taste tests, this product seemed like a sure winner.

However, A&W management neglected to factor in one minor detail. Humans are terrible at fractions.

Consumer focus groups probing into the retail failure of this seemingly superior burger revealed that over half of those surveyed thought a quarter pounder was more meat than a third pounder, simply because 4 is larger than 3. Clearly, fractions are a topic which needs to be more extensively explored in primary schooling.

McDonald’s launched their own failed 3rd pounders in 2007, with the angus, then sirloin, options making brief appearances on the menu.

Amusingly, nearly 40 years after the original debacle, A&W is trying to redeemed themselves. Sort of. The recently released 3/9th pound burger includes the number 9, which is obviously larger than 4. How could this new marketing scheme not be successful?

Improper fractions are a thing. Thinking 1/4th is larger than 1/3rd is decidedly improper, but not in the proper math terminology sense.

One of the main benefits of fractions is that they allow simple representation of cumbersome numerical values.

0.86666 becomes 5/6th, or 50 seconds in a minute. 0.28125 is simply 9/32”, in furniture construction parlance. An infinite string of “4’s”, after a decimal point, can be exactly communicated as 4/9th.

A valuable use of fractions is with regards to calculating probabilities. This branch of mathematics is unfortunately undertaught in high school, but exceedingly valuable in the real world. Especially if you’re partaking in a friendly poker game, or playing board games with your family.

Fractions also abound in sports. Baseball batting averages. Monitoring progress of a race horse down the stretch. Free throw accuracy in basketball. Tallying your pins in bowling. Differentiating between running events on the track.

Time itself is often communicated in fractions. Half past. Quarter ‘til. These terms are relics of history, but make sense considering the awkward 60 dominator value for both minutes and seconds, a numerical system which was originally utilized by the Babylonians. Unfortunately, this same, marginally divisible, non-tens-based approach, is used in many other elements of life, from global DMS coordinates, to mathematical angle calculations.

One needs only to look at the convoluted measurement system for baking ingredients, to realize why people get confused with fractions.

3 teaspoons in 1 tablespoon. 16 tablespoons in 1 cup. 2 cups in 1 pint. 2 pints in 1 quart. 4 quarts in 1 gallon. This series of connected measures shows that there are 768 teaspoons in 1 gallon. Simple right?

What about the unit conversions? If you prefer metric, 1 tablespoon holds 5 milliliters. Or maybe you have a premonition for imperial units. In which case, 6 teaspoons equate to 1 ounce. This results in a 30:1 ratio between milliliters and ounces. Not the most convenient fractional value, with a repeating 3 in digital form, but a conversion every bartender has engrained in their memory banks.

For ultimate culinary precision, you can use a dash, 1/16th of a teaspoon, of a pinch, 1/8th of a teaspoon. Not surprisingly, most cooks use their eye, as opposed to a measurement instrument, to assess these tiny amounts.

Once you get your head wrapped around all these factional conversions, it’s time to differentiate between dry, using mass, and wet, using volume, conversions. Good luck. No wonder so many people are ordering take-out these days.

There is one scenario where fractional math is always precise. When a hungry group sits down around a table, looking at a steaming hot pizza, which has been sliced radially. Eventually, the meal will be reduced, from 100%, to three-fifths, to 1/10th.

That last piece is when the discussion gets real. Time to flip a coin, with a 1/2 chance of success, to determine who gets the last slice of pie.

Details:

Overview of fractions, with lots of good real-life examples. [REF]
Simple probability questions, which are nearly impossible to answer, without a basic grasp of fractions. [REF]
Ancient historical origin of fractions, complete with hieroglyphic symbols. [REF]
Background on the fractional burger battle between A&W and McDonald’s. [REF]
Amusing marketing video for the new A&W 3/9th burger. [REF]
For the confused baker, this fridge magnet can shed some light on the fractional confusion. [REF]