{"id":7165,"date":"2017-05-08T23:56:34","date_gmt":"2017-05-08T23:56:34","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/prealgebra\/?post_type=chapter&p=7165"},"modified":"2019-05-27T05:13:22","modified_gmt":"2019-05-27T05:13:22","slug":"ratios-and-rate","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/monroecc-prealgebra\/chapter\/ratios-and-rate\/","title":{"raw":"Introduction to Writing Ratios and Calculating Rates","rendered":"Introduction to Writing Ratios and Calculating Rates"},"content":{"raw":"

What you'll learn to do: Write ratios as fractions and calculate unit rates<\/h2>\r\n[caption id=\"attachment_13339\" align=\"aligncenter\" width=\"1000\"]\"Several<\/a> How can you compare prices on packages of different sizes?[\/caption]\r\n\r\nWhen you go grocery shopping, do you sometimes find it difficult to compare prices when you're choosing between packages of different sizes or quantities of an item? For example, imagine you want to buy a block of cheese. One brand is\u00a0$4.99 for a 16-ounce brick. Another brand is on sale for $6.99 for a 24-ounce brick. Which one is the better deal? To make sure you get the most for your money, you'll need to figure out the price of cheese per ounce so that you can compare equal quantities. In this section, we'll explore ratios and rates, which will help you calculate unit rates and unit prices.\r\n\r\nBefore you get started, take this readiness quiz.\r\n
\r\n

readiness quiz<\/h3>\r\n1)\r\n\r\n[ohm_question]146014[\/ohm_question]\r\n\r\nIf you missed this problem, review this video.\r\n\r\nhttps:\/\/youtu.be\/_2Wk7jXf3Ok\r\n\r\n2)\r\n\r\nDivide: [latex]2.76\\div 11.5[\/latex]\r\n\r\nSolution: [latex]0.24[\/latex]\r\n\r\n3)\r\n\r\n[ohm_question]146100[\/ohm_question]\r\n\r\nIf you missed this problem, review the video below.\r\n\r\nhttps:\/\/youtu.be\/zw7WdhQnXHw\r\n\r\n<\/div>\r\n

<\/h1>\r\n

Translate Phrases to Expressions with Fractions<\/h1>\r\nHave you noticed that the examples in this section used the comparison words ratio of, to, per, in, for, on<\/em>, and from<\/em>? When you translate phrases that include these words, you should think either ratio or rate. If the units measure the same quantity (length, time, etc.), you have a ratio. If the units are different, you have a rate. In both cases, you write a fraction.\r\n
\r\n

Example<\/h3>\r\nTranslate the word phrase into an algebraic expression:\r\n\u24d0 [latex]427[\/latex] miles per [latex]h[\/latex] hours\r\n\u24d1 [latex]x[\/latex] students to [latex]3[\/latex] teachers\r\n\u24d2 [latex]y[\/latex] dollars for [latex]18[\/latex] hours\r\n\r\nSolution\r\n\r\n\r\n\r\n\r\n\r\n
\u24d0<\/td>\r\n<\/td>\r\n<\/tr>\r\n
<\/td>\r\n[latex]\\text{427 miles per }h\\text{ hours}[\/latex]<\/td>\r\n<\/tr>\r\n
Write as a rate.<\/td>\r\n[latex]\\Large\\frac{\\text{427 miles }}{h\\text{ hours}}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\r\n\r\n\r\n\r\n\r\n
\u24d1<\/td>\r\n<\/td>\r\n<\/tr>\r\n
<\/td>\r\n[latex]x\\text{ students to 3 teachers}[\/latex]<\/td>\r\n<\/tr>\r\n
Write as a rate.<\/td>\r\n[latex]\\Large\\frac{x\\text{ students}}{\\text{3 teachers}}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\r\n\r\n\r\n\r\n\r\n
\u24d2<\/td>\r\n<\/td>\r\n<\/tr>\r\n
<\/td>\r\n[latex]y\\text{ dollars for 18 hours}[\/latex]<\/td>\r\n<\/tr>\r\n
Write as a rate.<\/td>\r\n[latex]\\Large\\frac{y\\text{ dollars}}{\\text{18 hours}}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n \r\n\r\n<\/div>\r\n \r\n
\r\n

TRY IT<\/h3>\r\nTranslate the word phrase into an algebraic expression.\r\n\r\n\u24d0 [latex]689[\/latex] miles per [latex]h[\/latex] hours \u24d1 [latex]y[\/latex] parents to [latex]22[\/latex] students \u24d2 [latex]d[\/latex] dollars for [latex]9[\/latex] minutes\r\n\r\n\u24d0 [latex]689[\/latex] mi\/h<\/em> hours\r\n\u24d1 y<\/em> parents\/[latex]22[\/latex] students\r\n\u24d2 $d<\/em>\/[latex]9[\/latex] min\r\n\r\nTranslate the word phrase into an algebraic expression.\r\n\u24d0 [latex]m[\/latex] miles per [latex]9[\/latex] hours \u24d1 [latex]x[\/latex] students to [latex]8[\/latex] buses \u24d2 [latex]y[\/latex] dollars for [latex]40[\/latex] hours\r\n\r\n\u24d0 m<\/em> mi\/[latex]9[\/latex] h\r\n\u24d1 x<\/em> students\/[latex]8[\/latex] buses\r\n\u24d2 $y<\/em>\/[latex]40[\/latex] h\r\n\r\n<\/div>\r\n \r\n\r\n
\r\n\r\n

Applications of Ratios<\/h2>\r\nOne real-world application of ratios that affects many people involves measuring cholesterol in blood. The ratio of total cholesterol to HDL cholesterol is one way doctors assess a person's overall health. A ratio of less than [latex]5[\/latex] to [latex]1[\/latex] is considered good.\r\n
\r\n

Example<\/h3>\r\nHector's total cholesterol is [latex]249[\/latex] mg\/dl and his HDL cholesterol is [latex]39[\/latex] mg\/dl. \u24d0 Find the ratio of his total cholesterol to his HDL cholesterol. \u24d1 Assuming that a ratio less than [latex]5[\/latex] to [latex]1[\/latex] is considered good, what would you suggest to Hector?\r\n\r\nSolution\r\n\u24d0 First, write the words that express the ratio. We want to know the ratio of Hector's total cholesterol to his HDL cholesterol.\r\n\r\n\r\n\r\n\r\n\r\n
Write as a fraction.<\/td>\r\n[latex]\\Large\\frac{\\text{total cholesterol}}{\\text{HDL cholesterol}}[\/latex]<\/td>\r\n<\/tr>\r\n
Substitute the values.<\/td>\r\n[latex]\\Large\\frac{249}{39}[\/latex]<\/td>\r\n<\/tr>\r\n
Simplify.<\/td>\r\n[latex]\\Large\\frac{83}{13}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\u24d1 Is Hector's cholesterol ratio ok? If we divide [latex]83[\/latex] by [latex]13[\/latex] we obtain approximately [latex]6.4[\/latex], so [latex]\\Large\\frac{83}{13}\\normalsize\\approx\\Large\\frac{6.4}{1}[\/latex]. Hector's cholesterol ratio is high! Hector should either lower his total cholesterol or raise his HDL cholesterol.\r\n\r\n<\/div>\r\n \r\n
\r\n

TRY IT<\/h3>\r\nFind the patient's ratio of total cholesterol to HDL cholesterol using the given information.\r\nTotal cholesterol is [latex]185[\/latex] mg\/dL and HDL cholesterol is [latex]40[\/latex] mg\/dL.\r\n\r\n[latex]\\Large\\frac{37}{8}[\/latex]\r\n\r\nFind the patient\u2019s ratio of total cholesterol to HDL cholesterol using the given information.\r\nTotal cholesterol is [latex]204[\/latex] mg\/dL and HDL cholesterol is [latex]38[\/latex] mg\/dL.\r\n\r\n[latex]\\Large\\frac{102}{19}[\/latex]\r\n\r\n<\/div>\r\n \r\n

Ratios of Two Measurements in Different Units<\/h3>\r\nTo find the ratio of two measurements, we must make sure the quantities have been measured with the same unit. If the measurements are not in the same units, we must first convert them to the same units.\r\n\r\nWe know that to simplify a fraction, we divide out common factors. Similarly in a ratio of measurements, we divide out the common unit.\r\n
\r\n

Example<\/h3>\r\nThe Americans with Disabilities Act (ADA) Guidelines for wheel chair ramps require a maximum vertical rise of [latex]1[\/latex] inch for every [latex]1[\/latex] foot of horizontal run. What is the ratio of the rise to the run?\r\n\r\nSolution\r\nIn a ratio, the measurements must be in the same units. We can change feet to inches, or inches to feet. It is usually easier to convert to the smaller unit, since this avoids introducing more fractions into the problem.\r\nWrite the words that express the ratio.\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
<\/td>\r\nRatio of the rise to the run<\/td>\r\n<\/tr>\r\n
Write the ratio as a fraction.<\/td>\r\n[latex]\\Large\\frac{\\text{rise}}{\\text{run}}[\/latex]<\/td>\r\n<\/tr>\r\n
Substitute in the given values.<\/td>\r\n[latex]\\Large\\frac{\\text{1 inch}}{\\text{1 foot}}[\/latex]<\/td>\r\n<\/tr>\r\n
Convert [latex]1[\/latex] foot to inches.<\/td>\r\n[latex]\\Large\\frac{\\text{1 inch}}{\\text{12 inches}}[\/latex]<\/td>\r\n<\/tr>\r\n
Simplify, dividing out common factors and units.<\/td>\r\n[latex]\\Large\\frac{1}{12}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nSo the ratio of rise to run is [latex]1[\/latex] to [latex]12[\/latex]. This means that the ramp should rise [latex]1[\/latex] inch for every [latex]12[\/latex] inches of horizontal run to comply with the guidelines.\r\n\r\n<\/div>\r\n \r\n
\r\n

TRY IT<\/h3>\r\n1. Find the ratio of the first length to the second length: [latex]32[\/latex] inches to [latex]1[\/latex] foot.\r\n\r\n[latex]\\Large\\frac{8}{3}[\/latex]\r\n\r\n2. Find the ratio of the first length to the second length: [latex]1[\/latex] foot to [latex]54[\/latex] inches.\r\n\r\n[latex]\\Large\\frac{2}{9}[\/latex]\r\n\r\n<\/div>","rendered":"

What you’ll learn to do: Write ratios as fractions and calculate unit rates<\/h2>\n
\"Several<\/a><\/p>\n

How can you compare prices on packages of different sizes?<\/p>\n<\/div>\n

When you go grocery shopping, do you sometimes find it difficult to compare prices when you’re choosing between packages of different sizes or quantities of an item? For example, imagine you want to buy a block of cheese. One brand is\u00a0$4.99 for a 16-ounce brick. Another brand is on sale for $6.99 for a 24-ounce brick. Which one is the better deal? To make sure you get the most for your money, you’ll need to figure out the price of cheese per ounce so that you can compare equal quantities. In this section, we’ll explore ratios and rates, which will help you calculate unit rates and unit prices.<\/p>\n

Before you get started, take this readiness quiz.<\/p>\n

\n

readiness quiz<\/h3>\n

1)<\/p>\n