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Payments by Area People living in an apartment building decide to buy the building. They will put their money together in such a way that each will pay an amount that is proportional to the size of their apartment. For example, a man living in an apartment that occupies one fifth of the floor area of all apartments will pay one fifth of the total price of the building. Circle Correct or Incorrect for each of the following statements. Statement Correct / Incorrect A person living in the largest apartment will pay more money for each square metre of his Correct / Incorrect apartment than the person living in the smallest apartment. If we know the areas of two apartments and the price of one of them we can calculate the price of Correct / Incorrect the second. If we know the price of the building and how much each owner will pay, then the total area of Correct / Incorrect all apartments can be calculated. If the total price of the building were reduced by Correct / Incorrect 10%, each of the owners would pay 10% less. There are three apartments in the building. The largest, apartment 1, has a total area of 95m2. Apartments 2 and 3 have areas of 85m2 and 70m2 respectively. The selling price for the building is 300 000 €. How much should the owner of apartment 2 pay? Show your work.

Rock Concert For a rock concert a rectangular field of 100 m by 50 m was reserved for the audience. The concert was completely sold out, and the field was full of fans standing. Which one of the following is likely to be the best estimate of the total number of people attending the concert? 2 000 5 000 20 000 50 000 100 000 1 000 000 Can you find a trick to calculate the number of people attending an event with a determinate area?


Improve your key competencies The first person to make the connection between math and music was Pythagoras of Samos, a famous philosopher and cult leader who lived most of the time in southern Italy in 5th century BC. Among his claims to fame is the oldest known proof of what we call the "Pythagorean Theorem". If you have never heard of this guy, he is one of western civilizations strangest, but most influential thinkers. For Pythagoras, ratios were everything. He believed every value could be expressed as a fraction (he was wrong, but that is a whole different story). He also is the first to believe in the idea that mathematics is everywhere. One bit of evidence of underlying rational numbers was in Greek music. At the time, music was not as complicated as it is today. The Greek octave had a mere five notes. Pythagoras pointed out that each note was a fraction of a string. Lets say you had a string that played an A. The next note is 4/5 the length (or 5/4 the frequency) which is approximately a C. The rest of the octave has the fractions 3/4 (approximately D), 2/3 (approximately E), and 3/5 (approximately F), before you run into 1/2 which is the octave A. Many of the ancient Greek harps (kitharas) had six strings corresponding to these notes. (Kitharas, like all things preindustrial, were hand made and string lengths and count were never standardized, but six strings based on these simple ratios were probably popular choices.) Also, for you music experts, note that the scale is a "minor" scale, which we associate today with sounding sad or tragic. Pythagoras was excited by the idea that these ratios were made up of the numbers 1,2,3,4, and 5, and that there were five planets that moved along similar ratios and that all this meant something. Pythagoras imagined a "music of the spheres" that was created by the universe.

So, how did we get the 12 notes scale out of these six notes? There are 12 tones in a octave. Basically, some unknown follower of Pythagoras tried applying these ratios to the other notes on the scale. For example, B is the result of the 2/3 ratio note (E) applied to itself. 2/3 * 2/3 = 4/9. There was a problem, however, if you performed this transformation a second and a third time. The 12 tone octave created by starting with an A was different than the 12 tone octave created when you started with a different A. Two harps tuned to different keys would sound out of tune with one another. Also, music written in one scale could not be transposed easily into another because it would sound quite different. The solution was created around the time of Bach. Using an irrational number to fix music based on ratios, Pythagoras probably rolled over in his grave.

A complete and very verbose explanation of the ratios can be found at emfaq/harmony/pyth4.html


Music and Pisa  
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