A comparison of two numbers is referred to as a ratio, similar to fractions 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 today’s 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 of 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. 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 daily basis to find a solution.
For example, I will be using the extreme means property to estimate bear population in Keweenaw Peninsula. 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 totaling 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. The means are 2 and b and the extremes are 50 and 100. 50 = 2 b 100Correct setup of proportion. 5,000 = 2b cross multiply the means (2*b) and the extremes (50*100) 2 2followed by division of 2. 2,500 Answer after division was carried out.
x = 2, 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. y – 1 = -3 x + 3 4Written as an equation; solving for y. 4(y – 1) = -3x(x + 3)Cross multiplying was done. 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 -5Last step, 4 is divided on both sides. 4 4 y = -3 4 Linear equation in the form of y = mx + b and with a slope of -3/4. 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 = -3O riginal equation. x – 3 4 y – 1 = -3 Distribute (x-3) on both sides and multiply. x – 3 4 (x – 3)Cancel out common factors which eliminates denominator on left. y – 1 + 1 = -3x – 3 +1 To isolate y, 1 is added to both sides. Cancel common factors. y = -3 x -1 4Equation 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 (0) 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 any 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!