Chapter 2: Graphing & Geometry Section 2.4: Ratios, Rates & Proportions Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates

Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates Ratios and Rates A ratio is a comparison between two similar quantities. The can be written as a fraction (e.g. 5/4), with a colon separator (5:4), or with the word “to” separating the numbers (5 to 4). Common uses of ratios include schools describing their student to teacher ratio, such as 20:1. Ratios that include units of measure, such as 35 mi/hr or $3.65/gallon, are called rates. A unit rate is a rate with a denominator of 1. Speed is a generalized unit rate comparing distance to time. When comparing miles to hours, we use mph for the abbreviation. Mileage is a rate comparing distance to volume. When comparing miles to gallons of fuel used, we use mpg for the abbreviation. Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates

Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates Ratios and Rates Example: Sarah travelled 250 miles in 5 hours, and used 12 gallons of gas on the trip. Find the average speed and mileage for her trip. Round to the nearest tenth, as necessary. For the speed: 250 miles/5 hours = 250/5 mi/hr = 50 mph For the mileage: 250 miles/12 gallons = 250/12 mi/gal = 20.8 mpg Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates

Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates Unit Pricing The unit price of an item (or items) is the price per desired unit of measure. They are typically used to compare two or more quantities of the same item to see which is the better deal. They are typically stated in dollars per unit, rounded to the nearest thousandth. If the unit prices are all less than $1, we state them in cents per unit, rounded to the tenth of a cent. Example: Terry needs 24 ounces of black beans for his famous bean salad. He sees the 12-oz cans are $1.29 apiece and the 8-oz cans cost 79¢ each. Find the unit price of each can (rounded to the tenth of a cent) and decide if he should buy two large cans or three small cans. For the larger cans: $1.29/12 oz = $0.1075/oz ≈ 10.8¢/oz For the smaller cans: 79¢/8 oz = 9.875¢/oz ≈ 9.9¢/oz With a lower unit price, the smaller cans are a better deal. Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates

Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates Proportions A proportion is a statement that two ratios (or rates) are equal. With the introduction of an equals sign, we can also solve for a missing measure, provided we have the other three measures. Example: Solve for x. 9 15 = 𝑥 35 Cross multiplying to get 15x = 315 x = 21 Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates

Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates Similar Triangles Similar triangles are triangles whose angles have the same measure, but their sides have different lengths. The triangles will look identical, but one will be smaller than the other. If two triangles are similar, their side lengths are proportional to each other. Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates

Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates Similar Triangles Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates

Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates Similar Triangles Example: Let's say you are 5.5 feet tall, and, at a certain time of day, you find your shadow to be 10 feet long. At the same time of day, you measure the shadow of a tree to be 38 feet long. How tall is the tree? Round to the nearest foot. Do you see the similar triangles? Set up a proportion that compares the heights to the shadow lengths. 𝑦𝑜𝑢𝑟 ℎ𝑒𝑖𝑔ℎ𝑡 𝑡𝑟𝑒𝑒 ℎ𝑒𝑖𝑔ℎ𝑡 = 𝑦𝑜𝑢𝑟 𝑠ℎ𝑎𝑑𝑜𝑤 𝑡𝑟𝑒𝑒 𝑠ℎ𝑎𝑑𝑜𝑤 5.5 𝑓𝑡 𝑥 = 10 𝑓𝑡 38 𝑓𝑡 Tree height = (38 ft)(5.5 ft)/(10 ft) Tree Height = 20.9 ft To the nearest foot, the tree is 21 ft tall. x 38 ft 5.5 ft 10 ft Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates