Ratio, Part II of II

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1 Lesson 4 Ratio, Part II of II The prolific 20 th Century Christian author G.K. Chesterton ( ) wrote in Orthodoxy that Proportion cannot be a drift: it is either an accident or a design. What Chesterton means is that, for example, when we look at the features of a face, this humpback whale s, the various parts (eyes, head, mouth, blowhole, etc.) have an amazingly complex arrangement of proportions relative to each other. This arrangement makes a humpback a humbpack and not something else. That this whale s offspring will have features of similar proportions is no accident, nor will this arrangement of features drift over time from being designed to being accidental, or vice-versa. Proportion is not a drift, and in nature is always evidence of its Designer. And proportion in art, architecture, engineering, etc. is evidence of a human designer, male or female, created in His image (Genesis 1:26-27). Review: Shormann Algebra 1, Lessons 5, 6, 28, 72 Rules and Definitions Rules No new rules Definitions - proportional: having a constant ratio. - analogy: resemblance in some particulars between things otherwise unlike; similar. 4A Proportion and the Christian Adventure Each winter, humpback whales migrate from the North Pacific to Hawaii. I was fortunate enough to have an up-close encounter with one of these massive and beautiful creatures, and captured the photo above. But how do we know it s a humpback, and not some other cetacean? Well, there are obvious physical differences, but we could also make some measurements. For example, if I measured the distance from the whale s eye

2 to its blowhole, and then measured the distance from it s eye to the tip of its rostrum (snout), the ratio of those distances should be fairly constant from one humpback to the next. In other words, these features should be proportional to each other. If you want to, take a few minutes and try to draw this humpback whale on a piece of paper. If you do that, you will quickly find how important propotions are to drawing a whale that looks like a humpback, and not a puffer fish or some other creature! In Calculus: A Liberal Art, W.M. Priestly writes: The notion of a proportion - an equality of ratios - manifests itself in mathematics as measurement, in rhetoric as analogy, and in music as harmony. Proportion is, of course, closely tied to beauty in all the classical arts. As you can see, proportion, this idea of equal ratios, is much more than just a mathematical concept! But this is what we would expect in a world created by a God who is both one and many, and with Whom relationship matters. For Christians, we should seek to explain the obvious differences between things, but also see how they relate. Many times, the world seeks to only explain the differences between things. For example, many government education systems seek to train students to separate facts and opinions. They treat facts as things that can be tested and proven, and opinions as beliefs held by individuals and not necessarily true. However, not all facts can be proven (like Euclid s axioms and postulates, Lesson 10), and it is perfectly reasonable to believe in a fact. As Christians, we should seek to understand the obvious differences in facts and opinions, but we should also seek to understand how they relate, which brings us back to proportion. The idea of proportion should encourage us to find the real harmony that exists between things likes facts and opinions. Proportion also helps us see the inherent design in the world, helping us make the commonsense connection between the Creator and his creation. Here s one final thought on proportion. As a human designed in God s image, you are filled with incredible purpose for this world and eternity. The time you spend in this math course or elsewhere will be a lot more fun for you if you think of it not as a horrible chore to trudge through with much complaining. The biblical way to think of it is as an adventure (James 1:2-3). Quoting Chesterton in Orthodoxy again, he said Man must have just enough faith in himself to have adventures, and just enough doubt of himself to enjoy them. 4B Word Problems and Proportion Before we look at proportion word problems, let s review some basic F of a equals b word problems.

3 Example 4.1 a) 1/3 of 72 equals what number? b) 0.2 of what number equals 109? c) What percent of 185 equals 42.55? Recall that a ratio can be expressed as a fraction (F), decimal (D) or percent (P). Percent means per 100, so in problem solving we usually write it as P/100. We exchange F for D or P/100 in F of a equals b problems as needed. Also, F of a equals b is the same things as F times a equals b. a) F a = WN WN = what number 1/3(72) = WN 24 = WN b) 0.2 WN = 109 Divide both sides by 0.2 WN = = 545 c) P/ = P = Divide both sides by 1.85 P = = 23 Proportion word problems have 3 basic steps: 1) Set up the proportion 2) Cross multiply 3) Solve for the unknown As we saw in part 4A, proportional relationships are not just about comparing numerical quantities. One super-important thing to understand about proportion is that it reminds us that things can be distinctly different, but also related. That different things can be related is an attribute of God, and the sooner you are able to see this in mathematics, the easier all of math will be for you! Example 4.2 The model airplane was a 1/25 th scale replica. If the actual airplane was 100 feet long, how long was the model airplane? Remember that we define proportions as constant ratios, or equal ratios. So, when Step 1 says set up the proportion, that means we need to set up an equation: Step 1) The ratio of the model to the actual airplane was 1:25. We want to find out the length of the model airplane. Set up the proportion, using descriptive variables (M for model, A for actual) to help you stay organized: model actual = M A = 1 25 = M 100 Step 2) Cross multiply. This is something you should have learned in your

4 Algebra 1 course, and is just a way to simplify the algebra involved in solving for M = M M = 100(1) Step 3) Solve for the unknown by dividing both sides by M 25 = M = 4 ft There is more than one way to solve problems like this, and perhaps you also learned the F of a equals b method as well. For this problem, F would equal the fraction that was given, 1/25. So, you would say 1/25th of 100 equals? 1/ = 4 ft, the same result obtained using proportions. In general, just about any word problem that has a fraction and an unknown can be solved by setting up a proportion. It will also make it easier to understand a greater variety of problems by setting them up as proportions. You may also recall solving problems like this in pre-algebra, where you would think 25 times 4 equals 100, so 1 times 4 equals 4. Therefore, M = 4. Example 4.3 Last year, the ratio of business startups to closures in the town was 5 to 3. If 20 businesses started, how many closed? What was the total number of business startups and closures? Step 1) Set up the proportion for the first question: startups closures = S C = 5 3 = 20 C Step 2) Cross multiply: 5 3 = 20 5C = 20(3) C Step 3) Solve: C = 60 = 12 closures 5 For the 2nd question, we need to compare either startups to the total, or closures to the total. We will compare startups (5) to the total (5+3 = 8): Step 1) Set up the proportion: startups = S total T = 5 8 = 20 T

5 Step 2) Cross multiply: 5 8 = 20 T 5T = 20(8) Step 3) Solve: T = = 32 businesses Do you think it matters that we set up the ratio as S over C instead of C over S, or S over T instead of T over S? No. All that matters is that you are consistent. You can t have S over C on one side of the equation, and C over S on the other. Notice also how for the 2nd question, we weren t given a ratio of S:T, but we were able to figure out what T should be from the given information. Example 4.4 The mass (M) of iron varies directly with its volume (V). At room temperature, 7.87 g has a volume of 1 ml. What is the mass of 20 ml of iron? Round answer to 1 d.p. The italicized phrase varies directly means that as M increases, so does V (and as M decreases, so does V). You may also see the phrase directly proportional to, which is the same thing. When you see either of these in a word problem, think To solve this, I need to set up a proportion. Since its a proportion, the ratio of M to V is constant, so one way we could write the proportion is like a formula: M 1 V 1 = M 2 V 2 You have values for all but M 2, so follow the 3 steps for proportion problems. Step 1 is already done, so move on to Step 2, substituting in appropriate values, and solve: M 1 = 7.87 g V 1 = 1 ml M 2 =? V 2 = 20 ml = M 2 20 M 2 = 20(7.87) = g Did you know that the density of any substance equals its mass divided by its volume? The scientific formula for density is D = M V. Density is a difficult concept for some students to grasp, especially if their educational system has trained them to think that one thing does not depend on another. But this is why God s word should be the foundation of all our studies. His word reveals to us how one thing most definitely can depend on another. Example 4.5 In a first-class lever like a see-saw, the force (F) varies inversely with distance (d) from the pivot point, or fulcrum. On one side of the see-saw, a force of 10 N was applied at a distance of 3 m from the fulcrum. In

6 order to balance the see-saw, how far from the fulcrum must a 6 N force be placed on the other side? If you don t know what a see-saw is, refer to the example above. The italicized phrase varies inversely should immediately make you think To solve this, I need to set up equal products of force times distance. The product (NOT the ratio) of F and d is constant, so F 1 d 1 = F 2 d 2 You may also see the phrase inversely proportional to, which means the same thing as varies inversely. We ve already set up the inverse proportion. Now, just rearrange and evaluate for d 2 : d 2 = Fd 1 1 == 10(3) = 30 F = 5 m In thinking about how one thing compares to another, consider also that the 10 major concepts covered in Shormann Math are related. For example, setting up proportions like these is also important in other concepts, including Algebra (Lesson 6), Geometry (Lesson 10), and Measurement (Lesson 16). 4C Rate The first thing to notice about the word rate is its similarity to the word ratio. You will save yourself a lot of confusion if you simply remember that when you see the word rate, think fraction. Rate is a specific type of ratio, where usually two measurements are compared, one with a time component. For example, if you drove 60 miles in one hour, your average speed would equal 60 miles 1 hour = 60 miles per hour, or 60 mph. Rate is normally associated with speed, but it can really be for comparing any pair of measurements. For example, if you were growing strawberries, and decided to sell them for $4.99 for each quart, your rate (cost per quart) would equal $ quart = $4.99 per quart. If you understand ratio, then you can understand rate, which means you can understand calculus, too (Lessons 19-21)! After all, calculus is really just about studying rates. Using an analogy, proportion is to equal rates what calculus is to

7 changing rates. Example 4.6 If John drove 867 miles in 12 hours, what was his average speed in miles per hour? Round answer to 1 decimal place. You may use a calculator. Divide 867 by 12 to get miles per hour, or 72.3 mph. A general rule of thumb when rounding is that, if the remaining places are 5 or greater, then round up. That s why we rounded the.25 up to.3, instead of rounding down to.2. Example 4.7 Elijah bought 3 pounds of yellowfin tuna for $ At a different store, Elizabeth bought 2 pounds of yellowfin tuna for $ Who got the better deal? First, calculate the rate (cost per pound) for each, and compare: Elijah: $ = $9.25 per pound Elizabeth: $ = $9.30 per pound Elijah s price per pound is less, so he got the better deal. Practice Set 4 (subscripts tell you which lesson each problem came from) Use your best judgement as to when you should and shouldn t use a calculator W.M. Priestly wrote that the notion of proportion...manifests itself as in mathematics, in rhetoric, and in music /5 of what number equals 90? 3 4. The model airplane was a 1/40 th scale replica. If the model airplane was 2 feet long, how long was the actual airplane? 4 4. The ratio of cats to dogs in the village was 3 to 2. If there were 18 dogs in the village, what did the total number of cats and dogs equal? 5 4. The mass (M) of iron varies directly with its volume (V). At room temperature, 7.87 g has a volume of 1 ml. What is the volume of 30 g of iron? Round answer to 2 d.p For an ideal gas at room temperature, gas pressure (P) is inversely proportional to volume (V). In chemistry, this relationship is known as Boyle s Law. Initially, 1 Liter of a quantity of gas was at a pressure of 10 atm. If the gas was allowed to expand to 3 L, what would the new pressure equal? Round answer to 1 d.p If the entire volume of a 5,000,000 gallon aquarium was filtered every 3 hours, find the average filtration rate. Round answer to the nearest whole number Jeremy bought 10 pounds of broccoli for $15, and Naomi bought 9 lbs of broccoli for $14. Who got the better deal?

8 9 3. The Greek word for reasonable thought or purposeful study is Simplify the complex fraction. Remove parentheses as a first step. x 2 y Simplify Find log 2 32 Do not use a calculator Follow the pattern and find the missing value Use a calculator to find ln 2,000. Round answer to 2 decimal places Evaluate n! and T n for n = Find Simplify. 59 N log 10 N ? Simplify to eliminate all unnecessary exponents Simplify (-4) i 9 4 i Convert the following Roman numeral to the standard Hindu-Arabic system. MCCLII