Proportions
https://edboost.org/index.php/
enSimple Conversions (Proportions)
https://edboost.org/index.php/node/534
<span>Simple Conversions (Proportions)</span>
<div class="field field--name-body field--type-text-with-summary field--label-hidden field-item"><div class="tex2jax_process"><p>Many word problems ask students to convert units. Sometimes they explicitly ask how many inches are in 5 feet. But, the "conversion" part of the question often does not seem that obvious.</p><p><strong>However, anytime you have a problem that provides one unit and asks for answers in a different unit, you will need to do a conversion</strong> (unless you are adding a dimension, such as going from inches to square inches, in those cases, the conversion is part of the actual calculation, e.g., length in inches times width in inches, gives area in square inches, because length is one dimensional and area is two dimensional).</p><p>This lesson will focus on problems in which conversion is the main operation, but these strategies also work when you have to do other calculations (usually, the conversion is one of the last calculations you do before marking an answer).</p><p>We find that using proportions is one of the simplest ways to do conversions. And, we firmly believe that students should label their proportions as they write them. Labeling can seem silly, or like a waste of time, but labels help prevent you from making careless errors!</p><p>Let's start with a simple problem:</p><p><em>Example</em>: </p><p>You are buying a sofa for your living room. You measure the space and you want a sofa that is 7 feet long. But, when you get to the store, they show the dimensions of the sofas in inches. How many inches do you want your sofa to be? (1 foot = 12 inches).</p><p> </p><p>The first thing you want to do, is set up a proportion. On one side of the proportion, you put the ratio that you know (the conversion rate). </p><p>$\dfrac{1\text{ foot}}{12 \text{ inches}} = $</p><p>On the other side of the proportion, you match up the measurement you know (7 feet), and the one you want to find out (write it in as a variable).</p><p>Notice how we label everything. Your labels should match <em><strong>horizontally</strong></em>. When your labels match horizontally, you know that you have it set up properly.</p><p>$\dfrac{1\text{ foot}}{12 \text{ inches}} = \dfrac{7\text{ feet}}{x \text{ inches}}$</p><p>Remember, in a true proportion, the top left times the bottom right equals the top right times the bottom left. Thus, you can cross multiply to solve for $x$.</p><p>$$\eqalign{\dfrac{1\text{ foot}}{12 \text{ inches}} &= \dfrac{7\text{ feet}}{x \text{ inches}}\\1 \times x &= 7 \times 12\\ x &= 84}$$</p><p>You are looking for a sofa that is 84 inches long.</p><p> </p><p>All conversion problems actually are this simple. But some seem harder when you read them. Also, while some provide the conversion, others will expect you to know the conversion (e.g., how many inches are in a foot).</p><p><em>Example</em>: </p><p>Jonathon exercised for 2 hours. How many minutes did he exercise for?</p><p> </p><p>Again, the first thing you want to do, is set up a proportion. On one side of the proportion, you put the ratio that you know (the conversion rate). Do you know how many minutes are in an hour? 60.</p><p>$\dfrac{1\text{ hour}}{60 \text{ minutes}} = $</p><p><em>Note: it does not matter if you put hours or minutes on top. Just make sure, whatever label is on top in this ratio, the same label is on top in the next.</em></p><p>On the other side of the proportion, you match up the measurement you know (2 hours), and the one you want to find out (write it in as a variable).</p><p>Notice how we label everything. Your labels should match <em><strong>horizontally</strong></em>. When your labels match horizontally, you know that you have it set up properly.</p><p>$\dfrac{1\text{ hour}}{60 \text{ minutes}} = \dfrac{2\text{ hours}}{x \text{ minutes}}$</p><p>Now, just cross multiply to solve for $x$.</p><p>$$\eqalign{\dfrac{1\text{ hour}}{60 \text{ minutes}} &= \dfrac{2\text{ hours}}{x \text{ minutes}}\\1 \times x &= 2 \times 60\\ x &= 120}$$</p><p>Jonathon exercised for 120 minutes.</p><p> </p><p>Some conversion problems contain extra information, that you don't really need, and that can confuse you.</p><p><em>Example</em>: </p><p>April rode her bike at a speed of 15 miles per hour. She wants to tell her friend in France how many kilometers per hour she rides. What is April's approximate speed in kilometers? (1 mile $\approx$ 1.6 kilometers)</p><p> </p><p>This question can seem a little tricky, because it's talking about miles per hour and kilometers per hour, and those seem like ratios already. In fact, they ARE ratios already ($\dfrac{miles}{hour}$). But you don't need to include "hour" in your calculation. You're trying to talk about your riding speed, in miles or kilometers, per hour. The hour is constant in both measurements. So you can ignore it. All you have to do is convert miles to kilometers. </p><p>So, as usual, the first thing you do is write the conversion rate as a fraction (use your labels!).</p><p>$\dfrac{1\text{ mile}}{1.6 \text{ kilometers}} = $</p><p>On the other side of the proportion, you match up the measurement you know (15 miles), and the one you want to find out (write it in as a variable).</p><p>$\dfrac{1\text{ mile}}{1.6 \text{ kilometers}} = \dfrac{15\text{ miles}}{x \text{ kilometers}}$</p><p>Now, just cross multiply to solve for $x$.</p><p>$$\eqalign{\dfrac{1\text{ mile}}{1.6 \text{ kilometers}} &= \dfrac{15\text{ miles}}{x \text{ kilometers}}\\1 \times x &= 1.6 \times 15\\ x &= 24}$$</p><p>April rides about 24 kilometers per hour.</p><p> </p><p>So, whenever you are faced with a conversion -- either by itself in a word problem or as part of another word problem -- isolate the conversion rate, set up a proportion, use labels, make sure the labels line up horizontally, cross multiply, and you'll have your answer.</p></div></div>
<span><span>edboost</span></span>
<span><time datetime="2024-05-31T14:52:42-07:00" title="Friday, May 31, 2024 - 14:52">Fri, 05/31/2024 - 14:52</time>
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<h4><i class="icon-bookmark"></i> Practice Problems:</h4>
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<li><article data-history-node-id="420" class="node node-type-math-practice-problems node-view-mode-default">
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<a href="https://edboost.org/index.php/node/420" rel="bookmark"><span>Proportions: Measurement Conversions</span>
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<div class="field field--name-body field--type-text-with-summary field--label-hidden field-item"><div class="tex2jax_process"><p>Use proportions to perform the conversions:<br> </p><p>1. If one pound equals 16 ounces, how many ounces are in 3.25 pounds?</p><p>2. If one foot equals 12 inches, how many inches are in 1.5 feet?</p><p>3. If one cup equals 8 fluid ounces, how many fluid ounces are in 2.75 cups?</p><p>4. If one pound equals 16 ounces, how many pounds is an object weighing 48 ounces?</p><p>5. If one foot equals 12 inches, a stick measuring 88 inches is how many feet long?</p><p>6. If one yard equals 3 feet, how many feet are in 6.75 yards?</p><p>7. How many minutes are in 3.5 hours?</p><p>8. How many hours long is a movie that lasts 129 minutes?</p><p>9. If James runs 900 meters a day, how many kilometers does he run in a week? (1 kilometer=1000 meters)</p><p>10. Imani made a poster that was 70 centimeters long. How many meters long was it? (1 meter = 100 centimeters)</p><p>11. Abraham left school at 3:45. He arrived home at 6:10. How many hours did it take him to get home?</p><p>12. Josiah swam a 100 meter freestyle race in 1.2 minutes. How many seconds did it take him to swim the race?</p></div></div>
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<h4><i class="icon-bookmark"></i> Answer Key:</h4>
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<li><a href="https://edboost.org/index.php/node/135" hreflang="en">Proportions: Measurement Conversions</a></li>
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<h4><i class="icon-bookmark"></i> Skill:</h4>
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<li><a href="https://edboost.org/index.php/taxonomy/term/42" hreflang="en">Proportions</a></li>
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Fri, 31 May 2024 21:52:42 +0000edboost534 at https://edboost.orgScale measurements
https://edboost.org/index.php/node/533
<span>Scale measurements</span>
<div class="field field--name-body field--type-text-with-summary field--label-hidden field-item"><div class="tex2jax_process"><p>Anytime you are given a scale drawing (or the scale for something) in a word problem, you can be fairly sure that you are going to have to use a proportion.</p><p>What is a scale drawing?</p><p><strong>A scale drawing (or a scale figure) is a representation of something that is drawn proportionally.</strong> A scale drawing isn't a little map that we draw on a napkin or a post-it note. Think of it more like something that an architect or designer would use. Scales are also used in maps. The scale tells you what fraction of the size of the real item the map, or drawing, or figure is. So, on a map, one inch might represent 10 miles. On a blueprint, one centimeter might represent 20 feet. In a scale picture of a cell in your biology book, one inch might represent $\dfrac{1}{100}$ of a millimeter$ (in this case, the microscopic cell is scaled up so that you can see it).</p><p>A scale is basically a conversion ratio. So, like when you do measurement conversions, you'll use proportions to solve scale problems. </p><p><strong>Like in conversions, you start with the ratio that you know is true (the scale), use labels, fill in the other piece of information that you know, and solve for the final piece of information.</strong></p><p>Let's try one. </p><p><em>Example</em>: </p><p>Katya uses graph paper to draw a scale model of her apartment. In her model, one square on the graph paper represents 2 feet in the apartment. If her room is 6 squares long on the graph paper, how long is her room?</p><p> </p><p>The first thing you want to do is set up a proportion. On one side of the proportion, you put the ratio that you know, which is the scale:</p><p>$\dfrac{1\text{ square (model)}}{2 \text{ feet (real)}} = $</p><p>On the other side of the proportion, you match up the information you know, and the information you want to find out (write it in as a variable).</p><p>Notice how we label everything. Your labels should match <em><strong>horizontally</strong></em>. When your labels match horizontally, you know that you have it set up properly.</p><p>$\dfrac{1\text{ square (model)}}{2 \text{ feet (real)}} = \dfrac{6\text{ squares (model)}}{x \text{ feet (real)}}$</p><p>Remember, in a true proportion, the top left times the bottom right equals the top right times the bottom left. Thus, you can cross multiply to solve for $x$.</p><p>$$\eqalign{\dfrac{1\text{ square (model)}}{2 \text{ feet (real)}} &= \dfrac{6\text{ squares (model)}}{x \text{ feet (real)}}\\1 \times x &= 2 \times 6\\ x &= 12}$$</p><p>Katya's room is 12 feet long.</p><p> </p><p>Very often in scale problems, you are given a picture. You will get at least some of the information that you need to complete the problem from the picture.</p><p><em>Example</em>: </p><p>[[{"type":"media","view_mode":"media_large","fid":"873","field_deltas":{"1":{}},"fields":{},"attributes":{"height":"208","width":"364","style":"margin: 5px auto; vertical-align: middle; display: block;","class":"media-image media-element file-media-large","data-delta":"1"}}]]</p><p>The picture above shows a new homeowner's scale drawing of her yard. She plans to plan fruit trees around the border, but wants to plant grass in the interior portion of the yard. The depth of the area she wants to plant grass in is 28 feet. What is the length of the portion of the yard she wants to plant grass in?</p><p>In this problem, we are not given the actual scale of the picture. But we are given one actual length and one drawing length, which we can use to find the scale.</p><p>Set up the ratio of measurements you know:</p><p>$\dfrac{\text{drawing}}{\text{real}}=\dfrac{7\text{ inches (drawing)}}{28 \text{ feet (real)}} = $</p><p>We can reduce this fraction to find the actual scale ration, but we don't need to. In order to set up an proportion, you just need one ratio that is true. The ratio of 7 inches in the drawing to 28 feet in real life, is true.</p><p>Now, just put in the one other dimension that you know and the one that you don't. You know that the length of the grass part of the yard is 10 inches in the drawing, but you don't know what it is in real life, so you will use a variable.</p><p>$\dfrac{7\text{ inches (drawing)}}{28 \text{ feet (real)}}= \dfrac{10\text{ inches (drawing)}}{x \text{ feet (real)}}$</p><p>Cross multiply to solve for $x$.</p><p>$$\eqalign{\dfrac{7\text{ inches (drawing)}}{28 \text{ feet (real)}}&= \dfrac{10\text{ inches (drawing)}}{x \text{ feet (real)}}\\7 \times x &= 10 \times 28\\ 7x &= 280\\ \div 7& \; \div 7\\x&=40}$$</p><p>The grass part of the yard is 40 feet long.</p><p> </p><p> </p><p>So, whenever you see a question that asks about scale, use the scale (either the info in the problem, the info from the drawing, or a combination of the two) to create a $\dfrac{\text{model}}{\text{real}}$ ratio, then create a proportion to solve for your unknown dimension.</p></div></div>
<span><span>edboost</span></span>
<span><time datetime="2024-05-31T14:51:45-07:00" title="Friday, May 31, 2024 - 14:51">Fri, 05/31/2024 - 14:51</time>
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<h4><i class="icon-bookmark"></i> Practice Problems:</h4>
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<li><article data-history-node-id="283" class="node node-type-math-practice-problems node-view-mode-default">
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<a href="https://edboost.org/index.php/node/283" rel="bookmark"><span>Scale Measurements</span>
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<div class="field field--name-body field--type-text-with-summary field--label-hidden field-item"><div class="tex2jax_process"><p>Solve the following scale measurement word problems:</p><ol><li>You are visiting Boston and looking at a tourist map. You are currently standing at Paul Revere's Historic House. You want to walk to Harvard Square. But you aren't sure if you have time. On the map they are only about 11.4 centimeters apart. If the map scale says that 1 cm is equal to 1 mile, how far will you have to walk? </li><li>You looking at a very complex treasure map. There are two possible routes to the treasure. One route is 5.5 inches long. The other is 6.2 inches long. If the maps scale says that 1 inch is equal to 50 yards, how much longer is the longer route? </li><li>You are looking at a small map. You know that it's 3600 miles from your house in California to your grandma's house in New York. If the map scale is 500 miles is equal to $\dfrac{1}{2}$ inch, how far apart will your houses be on the map? </li><li>On an emergency exit map on the wall of a building, emergency exits are 5 inches apart. If the scale of the map is 2 inches equals 14 yards, how far apart are the exits in the actual building? </li><li>You have a pattern for a dress for you. You want to make a matching dress for your doll as well. You are 52 inches tall. Your doll is 18 inches tall. If the dress for you requires 3.5 yards of fabric, how much fabric do you need for the doll's dress? </li><li>You are trying to create a mural. You have a drawing on graph paper. Each square on your graph paper is 1 square cm. The wall you want to cover is 60 meters by 60 meters. If your drawing is 12 cm tall and you want to make it fill the entire height of the wall, you need to turn each 1 square centimeter into what size square? </li></ol></div></div>
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<h4><i class="icon-bookmark"></i> Answer Key:</h4>
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<li><a href="https://edboost.org/index.php/node/103" hreflang="en">Scale Measurements</a></li>
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<h4><i class="icon-bookmark"></i> Skill:</h4>
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<li><a href="https://edboost.org/index.php/taxonomy/term/42" hreflang="en">Proportions</a></li>
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Fri, 31 May 2024 21:51:45 +0000edboost533 at https://edboost.orgProportions from Word Problems (using Ratios)
https://edboost.org/index.php/node/532
<span>Proportions from Word Problems (using Ratios)</span>
<div class="field field--name-body field--type-text-with-summary field--label-hidden field-item"><div class="tex2jax_process"><p><strong>Ratios show how numbers are related. When you put two ratios together in a proportion, you show that two sets of numbers are related in the same way.</strong></p><p>Whenever you have a proportion (see <a href="4563">Proportion</a> lesson), or can create a proportion, you can solve by cross-multiplying, which is a great way to solve word problems using rates, scale, ratios (see Ratios lesson), and any other kinds of numbers that change proportionally. Hint, if you see the word "proportionally" in a word problem, you will definitely want to use a proportion to solve it!</p><p>Just as you have to be careful about order when setting up a ratio, when you set up a proportion, the trickiest part is setting it up. Always write labels in your fractions. Labels should match across the proportion (so labels on top should be the same and labels on the bottom should be the same). If your labels are not the same, something is wrong! Either you need to flip one of the ratios, OR, you need to do a calculation to get the proper item in the fraction. </p><p><em>Example</em>:</p><p><em>The teacher student ratio at school is 1:19. If there are 29 teachers, how many students are there?</em></p><p>$$\eqalign{\dfrac{1 \text{ teacher}}{19 \text{ Students}}&=\dfrac{29 \text{ teachers}}{x \text{ students}}\qquad&&\text{Set up a proportion. Make sure labels match}\\\dfrac{1}{19}\searrow\dfrac{29}{x} \text{ and }\dfrac{1}{19}\swarrow\dfrac{29}{x}&= 1\times x = 19 \times 29 \qquad && \text{Cross multiply for a new equation}\\x&=551\qquad&&\text{Solve the equation}}$$</p><p>There are 551 students in the school.</p><p>Sometimes, problems seem more complicated. The easiest way to start is to find your "base ratio" -- the one where you know both numbers. Then, just set it equal to another ratio, which will include your unknown value. </p><p><em>Example</em>:</p><p><em>A building has 3,500 square feet of surface that needs to be painted. If 2 gallons of paint will cover 500 square feet, what is the least number of whole gallons that must be purchased to have enough paint to apply one coat to the surface?</em></p><p>$$\eqalign{\dfrac{2\text{gallons}}{500\text{ square feet}}& \qquad && \text{Find the fraction you can fill in the top and bottom for}\\\dfrac{2 \text{ gallons}}{500 \text{ square feet}}&=\dfrac{x \text{ gallons}}{3500 \text{ square feet}}\qquad&&\text{Set equal to fraction with variable. Match labels.}\\\dfrac{2}{500}\searrow\dfrac{x}{3500} \text{ and } \dfrac{2}{500}\swarrow\dfrac{x}{3500}&= 2\times 3500 = x \times 500 \qquad && \text{Cross multiply for a new equation}\\7000&=500x\qquad&&\text{Solve the equation by dividing each side by 500}\\\div 500 \quad & \quad \div 500 \\x&=7\qquad &&\text{You need 7 gallons of paint}}$$</p><p>Finally, although all of the problems that you need to set up a proportion are often included in the problem. Sometimes they aren't. Sometimes you have to do some math just to set up your proportion (see <a href="4579">Creating Ratios</a> lesson for more). Sometimes that math means finding a value that is not given (but is implied) in the problem. Other times, it means converting to a different unit (feet to inches, seconds to minutes, etc.)</p><p> <em>Example</em>:</p><p><em>Researchers worry about how much advertising students see when they are watching TV. The average children's television show contains 21 minutes of program time and 9 minutes of commercial time. A new children's network argues that children should watch 4 hours of television a day. If they do watch 4 hours of television a day, how many minutes of commercials will that be?</em></p><p>$$\eqalign{\dfrac{21\text{ minutes of program}}{9 \text{ minutes of commercial}}& \qquad && \text{Find the fraction you know the top and bottom for}\\\dfrac{21 \text{ minutes of program time}}{9\text{minutes of commercial time}}&=\dfrac{x \text{ minutes of commercial time}}{4 \text{ hours of television}}\qquad&&\text{Check: do the labels match?}}$$</p><p>There's a problem. The labels don't match in a lot of ways!</p><p>First, "minutes of commercial" should match up, so let's flip the second fraction: $$\eqalign{\dfrac{21 \text{ minutes of program}}{9\text{minutes of commercial}}&=\dfrac{x \text{ minutes of commercial}}{4 \text{ hours of television}} \rightarrow \dfrac{21 \text{ minutes of program}}{9\text{minutes of commercial}}&=\dfrac{4 \text{ hours of television}}{x \text{ minutes of commercial}}}$$</p><p>Now, "minutes of commercial" is matched up on both fractions, but "program" is next to "television" and that won't work. We know that we have 4 hours of total television time in our question, so let's leave that. Can we figure out the total time for our base ratio? There's 21 minutes of program time and 9 minutes of commercial time, so how many minutes of total television time are in our original ratio?</p><p>$21+9=30\text { minutes of total television}$</p><p>Let's rewrite our proportion: $$\eqalign{\dfrac{21 \text{ minutes of program}}{9\text{minutes of commercial}}&=\dfrac{4 \text{ hours of television}}{x \text{ minutes of commercial}} \rightarrow \dfrac{30 \text{ minutes of television}}{9\text{minutes of commercial}}&=\dfrac{4 \text{ hours of television}}{x \text{ minutes of commercial}}}$$</p><p>Now, we're close but there's still one issue. Almost all of our labels are in minutes. But we have 4 HOURS of television in our second ratio. That won't work. Let's convert that to minutes!</p><p>$4 \text{ hours} \times 60 \text { minutes} = 240 \text{ minutes}$</p><p>Plug that into the proportion and we're reading to cross multiply!</p><p>$$\eqalign{\dfrac{30 \text{ minutes of television}}{9\text{minutes of commercial}}=\dfrac{4 \text{ hours of television}}{x \text{ minutes of commercial}} \rightarrow \dfrac{30 \text{ minutes of television}}{9\text{minutes of commercial}}&=\dfrac{240 \text{ minutes of television}}{x \text{ minutes of commercial}}\\30 x&=2160\\x&=72}$$</p><p>In 4 hours of television there's 72 minutes of commercial time.</p><p>Overall, find your base ratio. Then, match it up with a new ratio, that includes the number you need to find (as a variable). Make sure labels match up (both the content and the metric). Cross multiply and use algebra to solve.</p><p>Once you get used to using proportions, they're easy! And you'll find so many uses for them!</p></div></div>
<span><span>edboost</span></span>
<span><time datetime="2024-05-31T14:50:57-07:00" title="Friday, May 31, 2024 - 14:50">Fri, 05/31/2024 - 14:50</time>
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<h4><i class="icon-bookmark"></i> Practice Problems:</h4>
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<li><article data-history-node-id="433" class="node node-type-math-practice-problems node-view-mode-default">
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<a href="https://edboost.org/index.php/node/433" rel="bookmark"><span>Using Ratios to Create Proportions</span>
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<div class="field field--name-body field--type-text-with-summary field--label-hidden field-item"><div class="tex2jax_process"><p>Create ratios and then answer the following questions using proportions:</p><ol><li>If Stephanie can make two pizzas in 15 minutes, how many pizzas can she make in 90 minutes?</li><li>If John can bike 15 miles in 25 minutes, how many minutes will it take him to bike 9 miles?</li><li>If Allison can write 3 word problems is 12 minutes, how many problems can she write in 3 hours?</li><li>If Alex can swim 4 miles in 20 minutes, how long will it take him to swim 7 miles?</li><li>If Laetitia can jog 2 miles in 30 minutes, how long will it take her to jog 5 miles?</li><li>If Lottie can cycle 6 miles in 10 minutes, how long will it take her to cyle 15 miles?</li><li>If a dozen eggs costs \$2.40, then how much do 4 eggs cost?</li><li>If a dozen eggs costs \$1.20, then how much do 7 eggs cost?</li><li>If a half-dozen eggs costs 60 cents, then how much do 3 eggs cost?</li><li>If six sodas cost \$7.00, then how much do 2 sodas cost?</li><li>If 20 pencils cost \$1.85, then then 5 pencils cost how much?</li><li>If a dozen cupcakes cost \$15, then five cupcakes cost how much?</li></ol></div></div>
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<h4><i class="icon-bookmark"></i> Answer Key:</h4>
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<li><a href="https://edboost.org/index.php/node/136" hreflang="en">Using Ratios to Create Proportions</a></li>
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<h4><i class="icon-bookmark"></i> Test Prep Practice</h4>
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<li><article data-history-node-id="708" class="node node-type-test-prep-practice node-promoted node-view-mode-default">
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<a href="https://edboost.org/index.php/node/708" rel="bookmark"><span>Pre Algebra: Using Ratios to Create Proportions </span>
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<div class="field field--name-body field--type-text-with-summary field--label-hidden field-item"><div class="tex2jax_process"><p>1. On a map of Los Angeles, a distance of 3 miles is represented by 1.5 centimeters. If San Pedro is 24 miles from downtown Los Angeles, how many centimeters away is it on the map?</p><p>(A) 36<br>(B) 12<br>(C) 15<br>(D) 30<br>(E) 39</p><p>2. A pen factory can make 3200 pens per hour. At this rate, in how many minutes can the factory make 4800 pens?</p><p>(A) 45<br>(B) 100<br>(C) 1.5<br>(D) 15<br>(E) 90</p><p>3. Sunny is remodeling his house. He has 3100 square feet of walls and ceilings that need to be painted. If a five gallon bucket of paint will cover 275 square feet, how many buckets of paint does Sunny need to apply one coat to the surface?</p><p>(A) 56<br>(B) 12<br>(C) 57<br>(D) 11<br>(E) 13</p><p>4. On a large map of the East Coast, a distance of 4 miles is represented by 80 millimeters. If Baltimore is 38 miles from Washington D.C., how many centimeters away is it on the map?</p><p> </p><p> </p><p> </p><p> </p><p> </p></div></div>
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<h4><i class="icon-bookmark"></i> Answer Key:</h4>
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<li><a href="https://edboost.org/index.php/node/671" hreflang="en">Pre Algebra: Using Ratios to Create Proportions </a></li>
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<div class="node-taxonomy-container field--name-field-skill field--type-entity-reference field--label-inline">
<h4><i class="icon-bookmark"></i> Skill:</h4>
<ul class="taxonomy-terms">
<li><a href="https://edboost.org/index.php/taxonomy/term/42" hreflang="en">Proportions</a></li>
</ul>
</div>
Fri, 31 May 2024 21:50:57 +0000edboost532 at https://edboost.orgCreating Ratios
https://edboost.org/index.php/node/531
<span>Creating Ratios</span>
<div class="field field--name-body field--type-text-with-summary field--label-hidden field-item"><div class="tex2jax_process"><p><strong>Ratios can look complicated, but they are just fractions that show the relationship between two numbers that increase and decrease together</strong>. You can write ratios with colons (6:5) or as fractions 6/5. </p><p>The only tricky thing to remember about ratios is that you have to be very careful about the order of your numbers. The way to say or write a ratio makes a huge impact on what it means. So, if you have a classroom with 1 teacher and 30 kids, it's important that you write the student:teacher ratio as 30:1. If you write the student:teacher ratio as 1:30, it means you have thirty teachers and only one student! </p><p>There are some basic facts to know about ratios:</p><ul><li><strong>Ratios show how numbers are related</strong>. When one number in a ratio goes up, the other number goes up too. When one number goes down, the other goes down too!</li><li><strong>Ratios can be written as fractions and should be reduced or simplified like fractions</strong>. So, if the student:teacher ratio in a school is 600:30, you can write it like $\dfrac{600}{30}$ and reduce it to $\dfrac{20}{1} \text{ or } 20:1$.</li><li><strong>Order matters</strong>. The first number in a ratio (or the top number, if you are writing your ratio like a fraction) is the first item stated in the ratio. </li><li><strong>Sometimes you will have to do a calculation to figure out one of the numbers in a ratio</strong> (for instance, finding a total).</li></ul><p><em>Example</em>:</p><p><br><em>Let's say that a baseball team wins 15 games in a season. They lose ten games. They do not tie any games.</em></p><p>What is the ratio of the team's wins to losses?</p><p>It often helps to start out by writing a ratio in words. That helps you know which number to put in which place in the fraction. Then substitute the words for numbers. Then reduce.</p><p>$\dfrac{\text{wins}}{\text{losses}}=\dfrac{15}{10}=\dfrac{3}{2}$</p><p>What is the ratio of the team's loses to wins?</p><p>Remember, order matters. Switches like this show why it often helps to start out by writing a ratio in words. Substitute the words for numbers. Then reduce.</p><p>$\dfrac{\text{losses}}{\text{wins}}=\dfrac{10}{15}=\dfrac{2}{3}$</p><p>What is the ratio of the team's wins to total games?</p><p>Writing out the words can help you figure out what you know and what you don't know. In this case, you were not given total games. You have to add wins and losses together. Then proceed as usual.</p><p>$\dfrac{\text{wins}}{\text{total games}}=\dfrac{15}{15+10}=\dfrac{15}{25}=\dfrac{3}{5}$</p><p>Note, here, the ratio also helps you make a little percentage. They won $\dfrac{3}{5}$ of their games or 60% of their games. </p><p>Overall, ratios are basic (and are usually just the first step in a more complicated question. Just make sure to pay attention and be meticulous about matching up items and numbers (you may have to figure out numbers that are not given in the problem!).</p></div></div>
<span><span>edboost</span></span>
<span><time datetime="2024-05-31T14:48:48-07:00" title="Friday, May 31, 2024 - 14:48">Fri, 05/31/2024 - 14:48</time>
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<div class="node-taxonomy-container field--name-field-practice-problems field--type-entity-reference field--label-above">
<h4><i class="icon-bookmark"></i> Practice Problems:</h4>
<ul class="taxonomy-terms">
<li><article data-history-node-id="389" class="node node-type-math-practice-problems node-view-mode-default">
<h2 class="node-title">
<a href="https://edboost.org/node/389" rel="bookmark"><span>Create Ratios</span>
</a>
</h2>
<div class="node-content">
<div class="field field--name-body field--type-text-with-summary field--label-hidden field-item"><div class="tex2jax_process"><p>If there are 30 children in a classroom and there are 12 boys.</p><p>1. What is the ratio of boys to girls?</p><p>2. What is the ratio of girls to boys?</p><p>3. What is the ratio of girls to students?</p><p>4. What is the ratio of students to boys?</p><p>In a sock drawer there are 12 blue socks, 3 green socks, and 10 red socks.</p><p>5. What is the ratio of blue socks to green socks?</p><p>6. What is the ratio of red socks to blue socks?</p><p>7. What is the ratio of red socks to all socks?</p><p>8. What is the ratio of non-blue socks to all socks?</p><p>9. What is the ratio of green socks to all socks?</p><p>10. What is the ratio of all socks to blue socks?</p></div></div>
<div class="node-taxonomy-container field--name-field-answer-key field--type-entity-reference field--label-inline">
<h4><i class="icon-bookmark"></i> Answer Key:</h4>
<ul class="taxonomy-terms">
<li><a href="https://edboost.org/node/141" hreflang="en">Create Ratios</a></li>
</ul>
</div>
</div>
</article>
</li>
</ul>
</div>
<div class="node-taxonomy-container field--name-field-test-prep-practice field--type-entity-reference field--label-above">
<h4><i class="icon-bookmark"></i> Test Prep Practice</h4>
<ul class="taxonomy-terms">
<li><article data-history-node-id="730" class="node node-type-test-prep-practice node-view-mode-default">
<h2 class="node-title">
<a href="https://edboost.org/node/730" rel="bookmark"><span>Pre Algebra: Creating Ratios </span>
</a>
</h2>
<div class="node-content">
<div class="field field--name-body field--type-text-with-summary field--label-hidden field-item"><div class="tex2jax_process"><p>1. Seven of a 15 member marching band are girls and the remainder are boys. What is the ratio of boys to girls in the marching band?</p><p>(A) 7:15<br>(B) 8:15<br>(C) 7:8<br>(D) 8:7<br>(E) 7:12</p><p>2. An hour long TV program has 18 minutes of commercials. What is the ratio of commercial time to the total air time?</p><p>(A) $\dfrac{3}{7}$<br>(B) $\dfrac{10}{3}$<br>(C) $\dfrac{7}{3}$<br>(D) $\dfrac{3}{10}$<br>(E) $\dfrac{8}{3}$</p><p> </p><p>3. Richie is learning how to multiply fractions. On his last homework assignment, he got 8 questions correct and 9 questions incorrect. What is the ratio of questions he got incorrect to the total number of questions on the assignment?</p><p> </p><p> </p><p> </p><p><br> </p><p><br> </p></div></div>
<div class="node-taxonomy-container field--name-field-answer-key field--type-entity-reference field--label-inline">
<h4><i class="icon-bookmark"></i> Answer Key:</h4>
<ul class="taxonomy-terms">
<li><a href="https://edboost.org/node/691" hreflang="en">Pre Algebra: Creating Ratios</a></li>
</ul>
</div>
</div>
</article>
</li>
</ul>
</div>
<div class="node-taxonomy-container field--name-field-skill field--type-entity-reference field--label-inline">
<h4><i class="icon-bookmark"></i> Skill:</h4>
<ul class="taxonomy-terms">
<li><a href="https://edboost.org/index.php/taxonomy/term/42" hreflang="en">Proportions</a></li>
</ul>
</div>
<div class="node-taxonomy-container field--name-field-edboost-test field--type-entity-reference field--label-inline">
<h4><i class="icon-bookmark"></i> EdBoost Test:</h4>
<ul class="taxonomy-terms">
<li><a href="https://edboost.org/index.php/taxonomy/term/3" hreflang="en">Pre-Algebra</a></li>
</ul>
</div>
Fri, 31 May 2024 21:48:48 +0000edboost531 at https://edboost.orgProportions (Solving)
https://edboost.org/index.php/node/530
<span>Proportions (Solving)</span>
<div class="field field--name-body field--type-text-with-summary field--label-hidden field-item"><div class="tex2jax_process"><p>Proportions are two fractions (that sometimes represent ratios) that are equal to each other. Proportions are very helpful because you know that they are equal fractions. If you know that the fractions are equal you can find missing pieces of information by cross-multiplying. In the next lesson, we'll talk more about creating proportions, for now, let's look at how to solve them. </p><p><strong>When you have a proportion, you can cross multiply to create an algebraic equation that you can solve.</strong></p><p><strong>You can identify a proportion easily: it's two fractions joined by an equal sign.</strong></p><p><br>If you have $\dfrac{6}{7}=\dfrac{12}{14}$, you have a proportion: two fractions that are equal to each other.</p><p>This comes in handy, when you're missing either the numerator or the denominator of one of the fractions.</p><p><em>Example</em>:</p><p>$\dfrac{4}{5}=\dfrac{x}{35}\text{, what is the value of }x\text{?}$</p><p>Whenever you have a proprtion you can cross multiply to create an algebraic equation you can use to solve for the variable. Cross multiply means to multiply the numerator of one fraction by the denominator of the other fraction, and then set that term equal to the term you find by multiplying the other numerator times the other denominator. Let's try it. </p><p>$$\eqalign{\dfrac{4}{5}&=\dfrac{x}{35}\\\dfrac{4}{5}&\searrow\dfrac{x}{35}\qquad&&\text{Multiply: }4\times35=140\\\dfrac{4}{5}&\swarrow\dfrac{x}{35}\qquad&&\text{Multiply: }x \times 5=5x\\140&=5x\qquad&&\text{Set the two multiplication answers equal to each other}\\\div 5 &=\div 5 \qquad&&\text{Solve for x by dividing both sides by 5}\\28&=x\qquad&&\text{You found that x=28}}$$</p><p> </p><p>Any time you have a proportion, or two equal fractions, you can cross multiply to solve for a variable. <strong>You can also use the fact that cross-products in proportions are equal if the two fractions are equal to prove that two fractions are equal!</strong></p><p><em>Example</em>:</p><p>Are the two fractions $\dfrac{6}{21}$ and $\dfrac{.4}{1.4}$ equal?</p><p>Set them up as a proportion, cross-multiply, and see. If they are equal, their cross products will be equal.</p><p>$$\eqalign{\dfrac{6}{21}&=\dfrac{.4}{1.4}\\\dfrac{6}{21}&\searrow\dfrac{.4}{1.4}\qquad&&6 \times 1.4=8.4\\\dfrac{6}{21}&\swarrow\dfrac{.4}{1.4}\qquad&&.4\times21=8.4\\8.4&=8.4\qquad&&\text{The cross products are equal so the fractions are equal}}$$</p><p>Overall, when you have two fractions that are equal, their cross-products are equal. You can use that rule to solve for missing variables or to check to see if fractions are equal. </p></div></div>
<span><span>edboost</span></span>
<span><time datetime="2024-05-31T14:46:19-07:00" title="Friday, May 31, 2024 - 14:46">Fri, 05/31/2024 - 14:46</time>
</span>
<div class="node-taxonomy-container field--name-field-practice-problems field--type-entity-reference field--label-above">
<h4><i class="icon-bookmark"></i> Practice Problems:</h4>
<ul class="taxonomy-terms">
<li><article data-history-node-id="428" class="node node-type-math-practice-problems node-view-mode-default">
<h2 class="node-title">
<a href="https://edboost.org/node/428" rel="bookmark"><span>Proportions (Solving)</span>
</a>
</h2>
<div class="node-content">
<div class="field field--name-body field--type-text-with-summary field--label-hidden field-item"><div class="tex2jax_process"><p>Solve the proportion.</p><p>1. $\dfrac{n}{12}=\dfrac{3}{4}$</p><p>2. $\dfrac{50}{20}=\dfrac{z}{16}$</p><p>3. $\dfrac{25}{3}=\dfrac{t}{51}$</p><p>4. $\dfrac{6}{c}=\dfrac{54}{99}$</p><p>5. $\dfrac{n}{14}=\dfrac{63}{84}$</p><p>6. $\dfrac{2.1}{0.9}=\dfrac{27.3}{y}$</p><p>7. $\dfrac{16.2}{67.4}=\dfrac{x}{134.8}$</p><p>8. $\dfrac{8}{a}=\dfrac{0.4}{0.62}$</p><p>9. $\dfrac{b}{1.8}=\dfrac{49.6}{14.4}$</p><p>10. $\dfrac{2}{3}=\dfrac{4}{z}$</p><p>11. $\dfrac{6}{a}=\dfrac{3}{1}$</p><p>12. $\dfrac{39}{13}=\dfrac{9}{d}$</p></div></div>
<div class="node-taxonomy-container field--name-field-answer-key field--type-entity-reference field--label-inline">
<h4><i class="icon-bookmark"></i> Answer Key:</h4>
<ul class="taxonomy-terms">
<li><a href="https://edboost.org/node/191" hreflang="en">Proportions (Solving)</a></li>
</ul>
</div>
</div>
</article>
</li>
</ul>
</div>
<div class="node-taxonomy-container field--name-field-test-prep-practice field--type-entity-reference field--label-above">
<h4><i class="icon-bookmark"></i> Test Prep Practice</h4>
<ul class="taxonomy-terms">
<li><article data-history-node-id="731" class="node node-type-test-prep-practice node-view-mode-default">
<h2 class="node-title">
<a href="https://edboost.org/node/731" rel="bookmark"><span>Pre Algebra: Proportions </span>
</a>
</h2>
<div class="node-content">
<div class="field field--name-body field--type-text-with-summary field--label-hidden field-item"><div class="tex2jax_process"><p>1. If $\dfrac{x}{16}=\dfrac{12}{y}$, then what is the value of $xy$?</p><p>(A) $\dfrac{3}{4}$<br>(B) $144$<br>(C) $192$<br>(D) $\dfrac{4}{3}$<br>(E) $224$</p><p>2. If $\dfrac{a}{8}=\dfrac{1}{16}$, then what is the value of $a$?</p><p>(A) $\dfrac{1}{2}$<br>(B) $128$<br>(C) $8$<br>(D) $16$<br>(E) $2$</p><p> </p><p>3. If $\dfrac{12}{n-2}=\dfrac{8}{2n}$, then what is the value of $n$?</p><p>(A) $-1$<br>(B) $1$<br>(C) $\dfrac{1}{2}$<br>(D) $\dfrac{-1}{2}$<br>(E) $-2$</p><p>4. The ratio of 1.8 to 3 is not equal to which of the following ratios?</p><p>(A) 18 to 30<br>(B) 3 to 10<br>(C) 6 to 10<br>(D) 9 to 15<br>(E) 0.6 to 1</p><p><br> </p></div></div>
<div class="node-taxonomy-container field--name-field-answer-key field--type-entity-reference field--label-inline">
<h4><i class="icon-bookmark"></i> Answer Key:</h4>
<ul class="taxonomy-terms">
<li><a href="https://edboost.org/node/661" hreflang="en">Pre Algebra: Proportions Practice</a></li>
</ul>
</div>
</div>
</article>
</li>
</ul>
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<div class="node-taxonomy-container field--name-field-skill field--type-entity-reference field--label-inline">
<h4><i class="icon-bookmark"></i> Skill:</h4>
<ul class="taxonomy-terms">
<li><a href="https://edboost.org/index.php/taxonomy/term/42" hreflang="en">Proportions</a></li>
</ul>
</div>
Fri, 31 May 2024 21:46:19 +0000edboost530 at https://edboost.org