Solving proportions free essay sample

Algebra A comparison of two numbers Is referred to as a ratio, slmllar to tractions that can be reduced to lowest terms and then converted into a ratio of integers. Ratios allow one to compare sizes of two quantities and unit measurements. Any statement expressing the equality of two ratios is known as a proportion, which Is used In numerous formulas in todays real world settings and applications. Using proportions Is an effective way to find solutions by using the extreme means property or cross-multiplying. Extreme means property is simply the end result of the product f the extremes equaling the products of the means. Cross-multiplying is a short cut In proportions providing it Is a faster way to solutions rather than multiplying each side of the rational expression equation by the LCD_ Applications of rational expressions Involving formulas Include finding the equation of a line, distance, rate, time, uniform motion, and work problems. We will write a custom essay sample on Solving proportions or any similar topic specifically for you Do Not WasteYour Time HIRE WRITER Only 13.90 / page Proportions are used on a daily basis without even one realizing it by comparing measurements, unit pricing, driving distances, and calculating populations and wildlife on a dally basis to find a solution. For example, I will be using the extreme means property to estimate bear population In Keweenaw Penlnsuld. I was asked to solve problem #56, on page 437 of Elementary and intermediate algebra, (Dugopolski, M. , 2012) which states, that conservationists captured, tagged, and released 50 bears. over a one-year period, a random sample of 100 bears included only 2 tagged bears in Keweenaw Peninsula. To calculate the proportion, it will allow me to expect the ratio of bears that were originally tagged to the whole population Is equal to the ratio of the returning bears otaling 100 but only 2 tagged bears to the size of the sample. The variable b for bears Is applied, then followed by cross-multiplying the extremes and means to the proper set up of the proportion to find the solution. The two ratios are as follows: 50/ b = the originally tagged bears to the whole population and 2/100 = the recaptured bears to the sample size. I ne means are 2 ana D ana tne extremes are 50 ana 1 50 2 b 100 Correct setup of proportion. 5,000 = 2b cross multiply the means (2*b) and the extremes (50*100) followed by division of 2. 2,500 Answer after division was carried out. z 500 The estimated number of bears in Keweenaw Peninsula. Continuing onto the second assignment involving proportions, the following equation must be solved for y. Since there are single fractions (also referred to as ratios) on both sides of the equation, the extreme means property will be used again in this proportion. Written as an equation; solving for y. Cross multiplying was done. † l) = -3x(x+ 3) 4y- 1 +4 = -3x +3 +3Distribute 4 on left side and 3 on the right side. y = -3x -3+ 1 Add 1 to both sides. 4y = 2x -5 Last step, 4 is divided on both sides. 4 lope of -3/4. Linear equation in the form of y = mx + b and with a Taking notice that the slope of -3/4, is the same number as the number on the right hand side of the previous equation. I must continue trying another method but still use the extreme means property and try another method to see if I get a different solution. This may be an extraneous solution that I may come upon considering if the solution does not satisfy the rational expression. y-1 left. Original equation. Distribute (x-3) on both sides and multiply. Cancel out common factors which eliminates denominator on 0 Isolate y, 1 Is a aaea to Dotn sloes. ancel common factors. y=-3x-1 Equation complete and simplified. For this equation, I could have multiplied the LCD to both sides, but I found the extreme means property was an efficient shortcut. Cross- multiplying allowed me to eliminate the fractions and have the same ending result. We can now consider this an extraneous solution because the number showing as the solution but causes zero (O) in the denominator. As rational expressions can be tricky when there is a variable involved in the denominator so caution must be adhered. The use of proportions is everyday life and real world settings and applications are used without one even realizing it. While proportions can determine a solution whether it be driving distance, estimated population count, unit measuring, gas mileage, or to estimate an average time for a Job to be completed, it is a necessary tool that is used in many ways. The ratios that build the proportion can be easily solved by cross- multiplying the extremes and means in a fast and effective way. The wildlife can be assured that their tags will be calculated with an accurate solution for ny conservationist inquiring about a certain species. So the next time you find yourself comparing two quantities, deciding the average time for a specific Job, or determining how many miles you can go on a half of tank of gas on your next road trip, remember you are actually calculating proportions! [References:] Dugopolski, M. (2012). Elementary and intermediate algebra (4th ed. ). New York, NY: McGraw-Hill Publishing. Retrieved from http:// ashford. ebooks. edu (2009). Word problems involving proportions [Video]. Retrieved from http://www. youtube. com/watch

Armor and Weapons of the Spanish Conquistadors

Armor and Weapons of the Spanish Conquistadors Christopher Columbus discovered previously unknown lands in 1492, and within 20 years the conquest of these new lands was proceeding quickly. How were the Spanish conquistadors able to do it? The Spanish armor and weapons had much to do with their success. The Swift Success of the Conquistadors The Spanish who came to settle the New World were generally not farmers and craftsmen but soldiers, adventurers, and mercenaries looking for a quick fortune. Native communities were attacked and enslaved and any treasures they may have had such as gold, silver or pearls were taken. Teams of Spanish conquistadors devastated native communities on Caribbean islands such as Cuba and Hispaniola between 1494 and 1515 or so before moving on to the mainland. The most famous conquests were those of the mighty Aztec and Inca Empires, in Central America and the Andes mountains of South America respectively. The conquistadors who took these mighty Empires down (Hernan Cortes in Mexico in 1525 and Francisco Pizarro in Peru, 1532) commanded relatively small forces: Cortes had around 600 men and Pizarro initially had about 160. These small forces were able to defeat much larger ones. At the Battle of Teocajas, Sebastian de Benalcazar had 200 Spanish and some 3,000 Caà ±ari allies: together they fought Inca General Rumià ±ahui and a force of some 50,000 warriors to a draw. Conquistador Weapons There were two sorts of Spanish conquistadors: horsemen or cavalry and foot soldiers or infantry. The cavalry would usually carry the day in the battles of the conquest. When the spoils were divided, cavalrymen received a much higher share of the treasure than foot soldiers. Some Spanish soldiers would save up and purchase a horse as a sort of investment which would pay off in future conquests. The Spanish horsemen generally had two sorts of weapons: lances and swords. Their lances were long wooden spears with iron or steel points on the ends, used to devastating effect on masses of native foot soldiers. In close combat, a rider would use his sword. Steel Spanish swords of the conquest were about three feet long and relatively narrow, sharp on both sides. The Spanish city of Toledo was known as one of the best places in the world for making arms and armor and a fine Toledo sword was a valuable weapon indeed. The finely made weapons did not pass inspection until they could bend in a half-circle and survive a full-force impact with a metal helmet. The fine Spanish steel sword was such an advantage that for some time after the conquest, it was illegal for natives to have one. Foot Soldiers Weapons Spanish foot soldiers could use a variety of weapons. Many people incorrectly think that it was firearms that doomed the New World natives, but thats not the case. Some Spanish soldiers used a harquebus, a sort of early musket. The harquebus was undeniably effective against any one opponent, but they are slow to load, heavy, and firing one is a complicated process involving the use of a wick which must be kept lit. The harquebuses were most effective for terrorizing native soldiers, who thought the Spanish could create thunder. Like the harquebus, the crossbow was a European weapon designed to defeat armored knights and too bulky and cumbersome to be of much use in the conquest against the lightly armored, quick natives. Some soldiers used crossbows, but theyre very slow to load, break or malfunction easily and their use was not terribly common, at least not after the initial phases of the conquest. Like the cavalry, Spanish foot soldiers made good use of swords. A heavily armored Spanish foot soldier could cut down dozens of native enemies in minutes with a fine Toledan blade. Conquistador Armor Spanish armor, mostly made in Toledo, was among the finest in the world. Encased from head to foot in a steel shell, Spanish conquistadors were all but invulnerable when facing native opponents. In Europe, the armored knight had dominated the battlefield for centuries and weapons such as the harquebus and crossbow were specifically designed to pierce armor and defeat them. The natives had no such weapons and therefore killed very few armored Spanish in battle. The helmet most commonly associated with the conquistadors was the morion, a heavy steel helm with a pronounced crest or comb on top and sweeping sides that came  to  points on either end. Some infantrymen preferred a salade, a full-faced helmet that looks a little like a steel ski mask. In its most basic form, it is a bullet-shaped helm with a large T in front of the eyes, nose, and mouth. A cabasset helmet was much simpler: it is a large steel cap that covers the head from the ears up: stylish ones would have an elongated dome like the pointy end of an almond. Most conquistadors wore a full set of armor which consisted of a heavy breastplate, arm and leg greaves, a metal skirt, and protection for the neck and throat called a gorget. Even parts of the body such as elbows and shoulders, which require movement, were protected by a series of overlapping plates, meaning that there were very few vulnerable spots on a fully armored conquistador. A full suit of metal armor weighed about sixty pounds and the weight was well distributed over the body, allowing it to be worn for long periods of time without causing much fatigue. It generally included even armored boots and gloves or gauntlets. Later in the conquest, as conquistadors realized that full suits of armor were overkill in the New World, some of them switched to lighter chainmail, which was just as effective. Some even abandoned metal armor entirely, wearing escuapil, a sort of padded leather or cloth armor adapted from the armor worn by Aztec warriors. Large, heavy shields were not  necessary for the conquest, although many conquistadors used a buckler, a small, round or oval shield usually of wood or metal covered with leather. Native Weapons The natives had no answer for these weapons and armor. At the time of the conquest, most native cultures in North and South America were somewhere between the Stone Age and the  Bronze Age  in terms of their weaponry. Most foot soldiers carried heavy clubs or maces, some with stone or bronze heads. Some had rudimentary stone axes or clubs with spikes coming out of the end. These weapons could batter and bruise Spanish conquistadors, but only rarely did any serious damage through the heavy armor. Aztec warriors occasionally had a  macuahuitl, a wooden sword with jagged obsidian shards set in the sides: it was a lethal weapon, but still no match for steel. The natives had some better luck with missile weapons. In South America, some cultures developed bows and arrows, although they were rarely able to pierce armor. Other cultures used a sort of sling to hurl a stone with great force. Aztec warriors used the  atlatl, a device used to hurl javelins or darts at great velocity. Native cultures wore elaborate, beautiful armor. The Aztecs had warrior societies, the most notable of which were the feared Eagle and Jaguar warriors. These men would dress in Jaguar skins or eagle feathers and were very brave warriors. The Incas wore quilted or padded armor and used shields and helmets made of wood or bronze. Native armor was generally intended to intimidate as much as protect: it was often very colorful and beautiful. Nevertheless, eagle feathers provide no protection from a steel sword and native armor was of very little use in combat with conquistadors. Analysis The conquest of the Americas proves decisively the advantage of advanced armor and weaponry in any conflict. The Aztecs and Incas numbered in the millions, yet were defeated by Spanish forces numbering in the hundreds. A heavily  armored conquistador  could slay dozens of foes in a single engagement without receiving a serious wound. Horses were another advantage that the natives could not counter. It’s inaccurate to say that the success of the Spanish conquest was solely due to superior arms and armor, however. The Spanish were greatly aided by diseases previously unknown to that part of the world. Millions died of new illnesses brought by the Spanish such as smallpox. There was also a great deal of luck involved. For example, they invaded the  Inca Empire at a time of great crisis, as a brutal civil war between brothers Huascar and Atahualpa was just ending when the Spanish arrived in 1532; and the Aztecs were widely despised by their subjects. Sources Calvert, Albert Frederick. Spanish arms and armour: being a historical and descriptive account of the Royal armoury of Madrid. London: J. Lane, 1907Hemming, John.  The Conquest of the Inca. London: Pan Books, 2004 (original 1970).Pohl, John. The Conquistador: 1492–1550. Oxford: Osprey Publishing, 2008.