Overview:
Number statements that contain proportions are statements that fractions are equivalent. It can easily be shown that 2/4 = 3/6. If the proportions contain variables, they can be solved like any other number statement. The solution may just take more steps.
What Are Rates, Ratios, and Proportion?
The words rate, ratio, and proportion all refer to a relationship that can be expressed as a fraction. Suppose that a car is traveling at the rate of speed of 70 miles per hour. In order to find out how far that car would travel in 45 minutes, an equation would need to be written to compare equal rates. That proportion would be x/70 = 45/60. It can be solved several different ways, and each way will give the same answer. In the first method, 60x = 70(45), or 60x = 3150, or x = 3150/60 = 52.5. In the second method, 45/60 can be simplified to ¾, so that x/70 = ¾. Then 4x = 210, so x = 210/4 or 52.5. The third method is a good way to check if the proportion chosen is correct. The fraction 45/60 is also equivalent to the fraction 75/100, also known as the decimal .75, and .75 ∙70 is also equal to 52.5.
What Is the Means-Extremes Property?
The Means-Extremes Property is a shortcut to finding equivalent proportions. A proportion such as 75/100 = 45/60 can also be written as 75:100 = 45:60, as the ratio of 75 to 100 equals the ratio of 45 to 60. The numbers 75 and 60 can be thought of as the outside numbers (the extremes) and 100 and 45 as the inside numbers (the means). According to the Means-Extremes Property, the product of the outside numbers and the product of the inside numbers are equal. The product of 75 and 60 is 4500 and the product of 100 and 45 is 4500. Similarly, the ratio of x:70 = 45/60, or 60x = 70(45) = 3150.
What Are Similar Figures?
Similar figures have corresponding sides in the same ratio to another. Imagine one right triangle with sides that measure 50, 120, and 130. Suppose there is another right triangle with sides that measure 25, 60, and 65. The sides are in the same ratio so that 25/50 equals 60/120 equals 65/130. This property can be used to find the length of sides if some of the corresponding sides are known by comparing ratios.
What Is Proportional Thinking?
Proportional thinking is a way to use ratios to compare numbers. It can be used as a way to estimate answers, then find the exact answer. Suppose that the distance between two cities is about 150 miles. Traveling on the freeway at 60 miles per hour, how many hours will it take to get there? Estimating the time, it would take 2 hours to go 120 miles, and another half hour to go 30 miles more, so it would take about 2 ½ hours to travel 150 miles. One could also set up a proportion that 60 miles/1 hr = 150 miles/h hr. Using the Means-Extremes Property 60h = 150, or h = 150/60, so h = 2.5.
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