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enWhy Is It Important to Study Math?
https://www.quickanddirtytips.com/education/math/why-is-it-important-to-study-math
<p><img alt="Playground" class="qdt-wrap-left" height="300" src="https://www.quickanddirtytips.com/sites/default/files/styles/insert_medium/public/images/9226/playground.jpg?itok=hxDSiIOL" width="448" />Today is a very special episode of the Math Dude. To begin with, it’s episode 300. And because we humans have 10 fingers, we love to give special meaning to multiples of 10. But while that’s fun, it’s not the big news of the day or what makes this episode special to me. The big news is that this 300th episode is my last. Between my day job as a physics and astronomy professor and my day-and-night job of being “Dad” to an awesome and bustling 3-year-old, my free time for Math Dude duties has dwindled. And although I will surely miss all of you math fans, after seven years on the job, it's time to say goodbye.</p>
<p>But before I go, I have one more thing to say—and I think it’s the most important thing I’ve ever said on the show. It’s not something that I would (or even could) have said when I wrote the first episode seven years ago, because I wasn’t yet a father and so I wasn’t yet watching somebody discover the world for the first time. So please take a few minutes and listen, because I think this is something that everybody who has kids or might have kids or works with kids or might work with kids should know.</p>
<p>Here it is: <em>Math is a playground … so play!</em> Allow me to explain.</p>
<h2>Math Is a Playground</h2>
<p>A few days ago, I was at the park with my daughter watching her play. She’s at a very adventurous age and is constantly testing out every possible pathway to the top of what she has dubbed the “mermaid castle.” As she stretched her relatively tiny legs from rung-to-rung over what comparatively looked like a gaping chasm, I squiggled and squirmed as I struggled to keep myself from jumping up and lifting her over what I perceived to be a great danger. But she was careful, she didn’t fall, and she learned a bit about the world.</p>
<p>To be sure, playgrounds can be dangerous. So why do we let our kids play on them? Because playgrounds exercise their bodies. And not just in the sense of improving cardiovascular health or building strong bones and muscles. Those are all lovely side-effects, but what playgrounds do is provide kids with a relatively safe way to learn about using their bodies to navigate the world—how to balance, how to get from here-to-there, what to do when you get stuck. In other words, how to solve problems in the physical world.</p>
<p>As I was watching my daughter, I realized that math too is a playground. But it’s not a playground for our bodies, it’s a playground for our minds. In a way I’ve always known this to be true, but I’d never thought about it quite like this. And the thing is that this is pretty much the opposite of the way kids are commonly talked to and taught about math (and many of the sciences, too). All too often we’re taught that math is a tool—and only a tool—that we need to master in order to complete some boring but purportedly important task. We drill and drill our kids with arithmetic or factoring problems, but we never allow them to explore. And we never really allow them to play.</p>
<hr />
<h2>So Play!</h2>
<p>But play is exactly what kids do best. It’s how they learn. And it’s what we need to allow them to do—both with their bodies on the physical playground and with their minds on the mathematical one. Many adults struggled with math as kids. Why? In many cases, it’s because their parents struggled with math. And why was that? Because their parents struggled with math. And so on. Which means it’s time to break the cycle.</p>
<p>Your child, or grandchild, or friend’s child, or student, or whoever it may be, does not need to struggle just because you did. Please don’t let your past struggles determine your child’s future struggles. Don’t say things like “I’m bad at math” or “I’m just not a math person” when talking to kids. When they hear such words they just hear that “math = pain,” and it gives them the easy excuse that “I too am bad at math.” But they aren’t bad at math. They’re bad at drills. And drills aren’t math. Math is the playground.</p>
<p>So let them play. And just like my daughter figuring out how to span that jungle gym chasm, let them work to get better at playing. Because even play requires work. Before children walk, they fall. But when our kids fall taking those initial shaky steps, we help them to their feet and we encourage them to keep trying. Eventually those steps lead them to the playground and to chasms like the one my daughter crossed a few days ago. So why is math any different? Why do we treat the mental playground differently than the physical one? I don’t honestly know, but I know that we shouldn’t. And I believe that if we change our attitude and encourage our children to play—both physically and mentally—we’ll help create the greatest generation of creative and critical thinkers the world has ever seen. Which is something we could use right about now.</p>
<p>That leaves us with one final but very important question: What does playing with math actually look like? What is the mathematical playground? The good news is it’s simply the world around you. Math is everywhere if you just stop and look. You can explore the patterns on sea shells. Play with tiles and shapes. Categorize objects in bizarre and creative ways. Do puzzles. Do origami. Study the patterns in music. Study the structure of trees. Paint. Knit. Create secret codes. Program computers. Be creative. And check out the many amazing resources on the web such as <a href="http://naturalmath.com" target="_blank">http://naturalmath.com</a> and <a href="http://gdaymath.com" target="_blank">http://gdaymath.com</a> that are there to help you, your kids, your grandkids, your students, and everybody else learn how to play with math. Because math is a playground…so go play!</p>
<h2>Wrap Up</h2>
<p>OK, that's all the math we have time for.</p>
<p>Thanks again to everybody who has tuned in over the past seven years to hear what I’ve had to say. It’s been a lot of fun and I’ve learned a ton along the way. Hopefully you’ve gotten as much out of it as I have (although I doubt that’s possible). If you want to follow along with me on my future adventures in life, you can always find me on <a href="http://twitter.com/jasonmarshall" target="_blank">Twitter</a>.</p>
<p>For the last time, this is Jason Marshall with <a href="https://www.quickanddirtytips.com/math-dude">The Math Dude’s Quick and Dirty Tips to Make Math Easier</a>. Thanks for reading, math fans.</p>
<p><em><a href="https://www.shutterstock.com/image-photo/modern-children-playground-park-576726937?src=zWws2cqbj-_rYl8rqNZ_IA-1-31" target="_blank">Playground image</a> from Shutterstock.</em></p>Fri, 24 Feb 2017 23:20:46 -0500Fri, 24 Feb 2017 23:20:46 -0500https://www.quickanddirtytips.com/education/math/why-is-it-important-to-study-mathHow Are Distances and Absolute Values Related?
https://www.quickanddirtytips.com/education/math/how-are-distances-and-absolute-values-related
<p> </p>
<p>In the <a href="http://mathdude.quickanddirtytips.com/what-are-absolute-values.aspx" target="_blank">last article</a>, we learned exactly what <a href="http://mathdude.quickanddirtytips.com/what-are-absolute-values.aspx" target="_blank">absolute values</a> are and how you can find the absolute value of a number. In today’s article, we’re going to put this knowledge to work and learn about the very practical skill of using absolute values to find distances between numbers and places.</p>
<h2>Review: What are Absolute Values?</h2>
<p>As we talked about last time, the quick and dirty way to think about absolute values is that the absolute value of a number simply tells you how far away it is from zero on the number line. For example, since the numbers 5 and –5 are both 5 steps away from zero on the number line, they both have the same absolute value of 5.</p>
<h2>What is Distance?</h2>
<p>Does this idea that the absolute value of a number tells you how many steps away it is from zero on the number line remind you of anything in the real world…perhaps the idea of distance? The connection here is actually pretty straightforward, but let’s take a minute to look at an example that will drive home the relationship between absolute values in math and distances between objects in the real world.</p>
<p>As you know, the distance between two trees in your backyard is just a number that tells you how far apart the trees are. If you draw a <a href="http://mathdude.quickanddirtytips.com/what-are-1d-2d-and-3d-coordinates.aspx" target="_blank">coordinate system</a> in your backyard (which is really just a number line) and set one of the trees at the origin of your coordinate system (the location marked zero), then the distance to the other tree is the absolute value of its location in your coordinate system. For example, if one tree is at the location marked 0 and the other tree is 7 steps away in whatever direction you choose to be the positive direction, then the distance to the tree is |7| = 7. If the second tree is instead at –7 in in your coordinate system (in the opposite direction), its distance to the first tree is |–7| = 7. In other words, independent of direction, the second tree is always 7 steps away.</p>
<h2>How to Find the Distance Between Positive and Negative Numbers</h2>
<p>So we know that the absolute value of a point on the number line (or the absolute value of the coordinate of a tree in your backyard) tells you the distance between that point (or tree) and the number zero at origin of your coordinate system. But how do we find the distance between any two numbers? In other words, what if the first tree in our example wasn’t located at the origin of the coordinate system? What if one tree is at 2 and the other is at –5? How do you find the distance between them in that case?</p>
<p>Let’s start by realizing that this problem with trees is the same as the problem of figuring out the distance between the numbers 2 and –5. We know that the distances from 2 to 0 and from –5 to 0 are each given by the absolute values |2| = 2 and |–5| = 5. And if we think about where 2 and –5 sit on the number line, we can see that the distance between these two numbers is equal to the distance from –5 to 0 plus the distance from 0 to 2. In other words, the distance between –5 and 2 is equal to |–5| + |2| = 5 + 2 = 7.</p>
<p>But is that always true? Is the distance between any pair of numbers always equal to the sum of their absolute values? For example, what if we want to find the distance between the numbers 2 and 5 instead of 2 and –5. Can we just add the distance from 0 to 2 to the distance from 0 to 5 to get |2| + |5| = 7. Is 7 the distance between 2 and 5? No! As you can easily see by looking at a number line, the distance from 2 to 5 isn’t 7…it’s 3!</p>
<h2>How to Find the Distance Between Any Two Numbers</h2>
<p>So what went wrong? Well, since the absolute value of a single number is its distance from zero, the sum of the absolute values of two numbers is not the distance between them, it’s the sum of each number’s distance from zero. When one number is positive and the other is negative, this is the same thing as the distance between the numbers. But it doesn’t work when both numbers are either positive or negative.</p>
<p>[[AdMiddle]Okay, but what’s the right way to calculate the distance between any pair of numbers? The quick and dirty tip is that the distance between any pair of numbers is given by the absolute value of their difference. To see what this means, let’s go back to finding the distance between the numbers 2 and 5. The absolute value of their difference is |5–2| = |–3| = 3…which, as you can check by looking at a number line, is exactly the distance between 2 and 5! And, as you can also check, this works for <em>any </em>pair of numbers. The absolute value of the difference of two numbers, two coordinates on a map, or two locations of trees in your backyard, always tells you the distance between them.</p>
<h2>Number of the Week</h2>
<p>Before we finish up, it’s time for this week’s featured number selected from the various numbers of the day posted to the <a href="http://facebook.com/TheMathDude" target="_blank">Math Dude’s Facebook page</a> and to QDT’s new blog, <a href="http://blog.quickanddirtytips.com/2011/09/20/how-to-tell-how-fast-someone-is-running/" target="_blank">The Quick and Dirty</a>. This week’s number is actually a conversion that you can use to figure out how fast a wide receiver in the football game you’re watching is running as he jets toward the end zone.</p>
<p>The trick is to know that a speed of 10 yards per second is about the same as 20 miles per hour. So if it takes a player one second to run 10 yards, he must be running at about 20 miles per hour. If it takes two seconds to cover those 10 yards, his speed must be only 10 miles per hour. That’s all there is to it! Now, the next time you’re watching a football game, you can impress your friends by telling them how fast everybody is running. It’s sure to give you a whole new perspective on what’s taking place on the field!</p>
<h2>Wrap Up</h2>
<p>Okay, that’s all for today. Remember to become a fan of the Math Dude on <a href="http://facebook.com/TheMathDude" target="_blank">Facebook</a> where you’ll find a new number of the day and math puzzle posted each and every weekday. And if you’re on Twitter, please <a href="http://twitter.com/jasonmarshall" target="_blank">follow me</a> there too. Finally, if you have math questions, feel free to send them my way via <a href="http://facebook.com/TheMathDude" target="_blank">Facebook</a>, <a href="http://twitter.com/jasonmarshall" target="_blank">Twitter</a>, or by email at <a href="mailto:mathdude@quickanddirtytips.com">mathdude@quickanddirtytips.com</a>.</p>
<p>Until next time, this is Jason Marshall with <em>The Math Dude’s Quick and Dirty Tips to Make Math Easier</em>. Thanks for reading, math fans!</p>
Sat, 18 Feb 2017 02:34:38 -0500Sat, 18 Feb 2017 02:34:38 -0500https://www.quickanddirtytips.com/education/math/how-are-distances-and-absolute-values-relatedHow to Use Math to Fly Rockets to Space
https://www.quickanddirtytips.com/education/math/how-to-use-math-to-fly-rockets-to-space
<p><img alt="Space Shuttle" class="qdt-wrap-left" height="224" src="https://www.quickanddirtytips.com/sites/default/files/styles/insert_small/public/images/9055/edu_shuttle_launch_sts-43.jpg?itok=JRotYeJa" width="169" />… 5 … 4 … 3 … 2 … 1 … and liftoff!</p>
<p>Ever since I was a kid, I’ve loved rockets and everything about flying to space. So the sound of the countdown leading up to a rocket launch is music to my ears. Of course, the sound that follows the countdown is anything but musical because rockets are really loud … but they’re also beautiful. And they’re marvelous machines that will soon be playing an increasingly crucial role in our day-to-day lives as we begin the journey towards becoming a truly space-faring species. And to top it all of, they’re machines powered by math (and, of course, a bunch of physics and fuel).</p>
<p>What's the math that powers rockets? How does it help us get them to space? And how do we use that math to put a satellite or person in orbit around the Earth? Let's find out.</p>
<h2>The Mathematics of Getting to Space</h2>
<p>When people think about going to space, they usually think about going up. And that’s certainly true, but it’s only part of the story. It’s sort of hard to define exactly where the atmosphere ends and outer space begins (since the atmosphere gradually falls off as you go up in altitude), but one popular choice is the so-called “Karman line” at a height of 100 km (or around 62 miles) above sea level. A lot of people are surprised to find that space begins only 100 km up … since that’s really not that far. But the problem with getting there is that it’s “uphill” the whole way, which means you have to fight gravity the whole way.</p>
<p class="qdt-pull-quote-right">A rocket traveling at 8 km/s completes one orbit every 90 minutes.</p>
<p>But getting up that high is only half the battle of getting into orbit around the Earth. Because if you fly a spacecraft 100 km straight up and then turn off the engines, it will simply come right back down to the ground (this is called a sub-orbital flight). If your goal is to get a satellite into orbit around the Earth or to deliver a person to the International Space Station, the rocket doesn’t just need to get into space, it needs to stay there. And that means it needs to end up flying sideways really, <em>really</em> fast—around 8 km/s or almost 18,000 miles per hour!</p>
<p>How fast is that? Well, a rocket or satellite traveling at 8 km/s completes one orbit every 90 minutes. Which is amazingly fast considering it takes 5 hours to fly across the United States in an airplane. For comparison, a rocket in orbit crosses the US in about 10 minutes.</p>
<h2>The Mathematics of Orbiting the Earth</h2>
<p>But why does a rocket or satellite or space station need to be moving sideways so fast to stay in orbit? The answer is mainly geometry (and a healthy dose of physics). As you know, the Earth is roughly spherical. While it’s possible to go around the Earth (or anything else) in an elliptical orbit (which looks like a squashed circle), we’re going to think about the simple case of a perfectly circular orbit. If you think about it, you’ll see that a rocket going around the spherical Earth in a circular orbit some height above the ground will stay at that height above the ground the entire orbit. This is kind of obvious, but it really is the key to understanding the mathematics of being in orbit.</p>
<p class="qdt-pull-quote-left">Orbits come down to geometry and traveling sideways really fast.</p>
<p>To see how this works, imagine standing at the edge of a tall cliff overlooking the ocean. If you drop a ball, the ball will fall straight down into the water. If you throw the ball with a bit of sideways speed, the ball will travel in a parabolic arc and land a bit further away from the cliff. Now imagine throwing the ball harder and harder with more sideways speed. Each increase in horizontal speed means the ball lands in the water farther from the cliff than before. If you throw the ball hard enough (and we’re talking <em>really</em> hard), something weird happens: the amount the ball falls towards the Earth is exactly matched by the amount the spherical Earth curves away from the ball. The net result is that the height of the ball above the water doesn't change—and it will just keep going and going.</p>
<p>Keep in mind that even though it doesn't hit the ground, the ball is actually falling towards the Earth the whole time—it simply never gets closer to the ground since its curved trajectory matches the curvature of the Earth. In other words, the ball is in orbit. As I said, orbits come down to geometry and traveling sideways really fast. Of course, you can’t actually get a ball into orbit by throwing it off a cliff like this since air molecules in Earth’s atmosphere will slow it down and eventually make it fall to the ground. Which is exactly why rockets also have to travel upwards into space before they can orbit the Earth.</p>
<h2>The Rocket Equation</h2>
<p>Now that we know what it means to get a satellite into orbit, let’s think about how we get it there. In other words, let’s think about what determines how big a rocket needs to be to lift a satellite into space and get it moving sideways fast enough to orbit the Earth. To begin with, let’s contemplate what we have to do to put a person (and their toothbrush) or a satellite into orbit. The answer is that we need to attach a rocket underneath this payload that has enough fuel and power to lift the required mass into orbit. But, the rocket we just attached to the payload also has some mass (mostly its fuel), which means we need another rocket under the first that has enough fuel and power to lift it. But, this second rocket we just attached also has some mass (again, mostly its fuel), so we once again need another rocket to lift it! And on, and on, and on. Even if a rocket's payload is small, it needs a lot of fuel to lift it … and it needs fuel to lift the fuel … and so on. As I said earlier, space is only 100 km away, but it’s 100 km <em>straight up …</em> which makes it hard to get to.</p>
<p class="qdt-pull-quote-right">Space is only 100 km away, but it’s 100 km straight up … which makes it hard to get to.</p>
<p>There’s an equation that summarizes this whole situation and tells us roughly how much fuel is needed to lift a given amount of mass into orbit by a particular rocket. It’s called, logically, the rocket equation. We’re not going to go into all the details of this equation, but the gist is that it tells engineers how to calculate the speed gained by a rocket as it burns its fuel. In particular, the equation says that the speed increase is proportional to the logarithm of the initial mass of the rocket (including the rocket itself, the payload, and all of its fuel) divided by the final mass of the rocket (once all the fuel is burned). This ultimately tells us that adding more and more fuel to a rocket offers diminishing returns in terms of speed gained since, as we’ve seen, all of that fuel requires even more fuel. Which is exactly why rockets have to be such enormous, magnificent, and beautiful machines.</p>
<h2>Wrap Up</h2>
<p>Okay, that's all the rocket math we have time for today.</p>
<p>For more fun with numbers and math, please check out my book, <em><a href="http://us.macmillan.com/themathdudesquickanddirtyguidetoalgebra/" target="_blank">The Math Dude’s Quick and Dirty Guide to Algebra</a></em>. Also, remember to become a fan of The Math Dude on <a href="https://www.facebook.com/TheMathDude" target="_blank">Facebook</a> and to follow me on <a href="http://twitter.com/jasonmarshall" target="_blank">Twitter</a>.</p>
<p>Until next time, this is Jason Marshall with <a href="https://www.quickanddirtytips.com/math-dude">The Math Dude’s Quick and Dirty Tips to Make Math Easier</a>. Thanks for reading, math fans!</p>
<p><em><a href="https://www.nasa.gov/audience/forstudents/k-4/stories/nasa-knows/what-is-the-space-shuttle-k4.html" target="_blank">Rocket image</a> courtesy NASA.</em></p>Sat, 11 Feb 2017 00:27:15 -0500Sat, 11 Feb 2017 00:27:15 -0500https://www.quickanddirtytips.com/education/math/how-to-use-math-to-fly-rockets-to-spacePolygon Puzzle: How Many Degrees Are in a Polygon?
https://www.quickanddirtytips.com/education/math/polygon-puzzle-how-many-degrees-are-in-a-polygon
<p><img alt="Soccer Ball" class="qdt-wrap-left" height="224" src="https://www.quickanddirtytips.com/sites/default/files/styles/insert_small/public/images/6745/hexagons-and-pentagons-in-a-soccer-ball.jpg?itok=bDrgi8ue" width="224" />We recently talked about <a href="/node/6442">why the three interior angles of a triangle must always add up to 180º</a>. And at various points in the past, we've noted that the quartet of 90º “right" angles in a square must mean that the interior angles of a square add up to 360º. But we've never talked about what happens when we toss more sides into the mix.</p>
<p>In other words, we've never talked about how to figure out the total number of degrees in a pentagon. Or a hexagon...or an octagon. Or any other polygon, for that matter! And just as importantly, we’ve never dealt with whether or not there’s some clever way to figure all of this out without resorting to making <a href="/node/6269">measurements </a>with a protractor.</p>
<p>Until now, that is - because these are exactly the questions we’ll be talking about today as we dive into a delectably delicious polygon puzzler.</p>
<p class="qdt-adv-rte">Sponsor: Visit <a href="http://x.co/4trSA" target="_blank">GoDaddy.com</a> to get your $2.95 .COM domain. Some limitations apply, see website for details.</p>
<h2>Review: Interior Angles of Polygons</h2>
<p>Our big goal for today is to figure out exactly how the interior angles of polygons change as the number of sides in the shape increases.</p>
<p>As a super quick review, a polygon is any shape made up of three or more connecting sides that you can draw on a flat sheet of paper. For a more thorough look at the definition of a polygon, check out <a href="/node/6621">the episode on that topic</a>.</p>
<p>The angles formed in the interior of a polygon where pairs of sides intersect are called "interior angles." As noted earlier, we've talked about using a clever trick to prove that a <a href="/node/6442">triangle's interior angles</a> always add up to 180º. And we’ve seen that the four right interior angles of a square (or any rectangle) must add up to 360º.</p>
<p>Which might lead you to wonder...</p>
<h2>How Many Degrees Are In a Pentagon?</h2>
<p>What happens when the number of sides increases beyond four? In other words, <a href="/node/6678">what’s the sum</a> of the interior angles of a pentagon, a hexagon, an octagon, or any other polygon?</p>
<p class="qdt-pull-quote-right">Interior angles get larger as the number of sides increases.</p>
<p>Let’s start by taking a look at the 5-sided regular polygon (meaning, its sides and angles are all equally sized), better known as a pentagon. If you sketch a pentagon, you’ll immediately see that its interior angles are all greater than 90º. So the first thing we can conclude is that the interior angles of a polygon get larger as the number of sides increases. But by how much?</p>
<p>At this point, I encourage you to stop for a minute and see if you can figure out how you might go about answering this question. If you’re having trouble getting started, think about the fact that <a href="/node/4391">you can draw</a> a diagonal line across a 360º square to break it up into a pair of 180º triangles.</p>
<hr />
<p>Hmm, very interesting. Now take a minute or two and see what you can come up with.</p>
<p><em>Any ideas yet?</em>...</p>
<p><em>Isn’t this fun?</em>...</p>
<p><em>OK, ready for the answer?</em>...</p>
<p><img alt="Pentagon" class="qdt-wrap-right" height="214" src="https://www.quickanddirtytips.com/sites/default/files/styles/insert_small/public/images/6745/MD227-01.png?itok=yxDCAhB-" width="224" />Hopefully, you noticed that a 5-sided regular pentagon can be broken up into t<a href="/node/4508">hree triangles</a>. To see this, draw a line from the top corner of your pentagon down to each of its opposing lower corners. Once you draw this picture, you’ll see that a pentagon is indeed composed of three triangles. And from that you should see that a pentagon must contain 3 x 180º = 540º.</p>
<p>So if the interior angles of a 3-sided triangle add up to 180º, those of a 4-sided square add up to 360º, and those of a 5-sided pentagon add up to 540º. It looks like we keep adding <a href="/node/6442">180º </a>for each additional side. Does this trend hold up as we move to polygons with more and more sides?</p>
<h2>How Many Degrees In a Hexagon, Octagon, Or Any Polygon?</h2>
<p>To see, let’s make the next logical leap and move from thinking about a 5-sided pentagon to a 6-sided hexagon. Again, I encourage you to take a few minutes to think about what we’ve done so far, and see if you can puzzle out the total number of <a href="/node/5586">degrees </a>contained in the interior angles of a hexagon.</p>
<p><em>Do you see the trick?</em>...</p>
<p><em>Same idea as before...</em></p>
<p><em>Got it?</em>...</p>
<p>To answer this question, draw yourself a lovely regular hexagon, and then draw three lines from one of its corners to the three opposing corners. When you do this, you should see that a hexagon can be broken up into four triangles. And, therefore, you should see that a hexagon must contain 4 x 180º = 720º.</p>
<p><img alt="Hexagon" class="qdt-wrap-right" height="195" src="https://www.quickanddirtytips.com/sites/default/files/styles/insert_small/public/images/6745/MD227-02.png?itok=I6FPLSH0" width="224" />Do you see <a href="/node/6724">a trend</a> yet? If you really wanted to, you could continue this game for the 7-sided heptagon, the 8-sided octagon, and so on. In fact, you can do it for any and every polygon.</p>
<p>When you do that, you’ll find that an <em>n</em>-sided polygon (where <em>n</em> simply represents the number of sides in the polygon) can be broken down into <em>n</em>–2 triangles. Which means that the interior angles of an <em>n</em>-sided polygon will always add up to (<em>n</em> – 2) x 180º.</p>
<p>Which, lo and behold, is the answer to today’s great polygon puzzler!</p>
<h2>Wrap Up</h2>
<p>OK, that’s all the math we have time for today.</p>
<p><img alt="" class="qdt-wrap-left" height="224" src="https://www.quickanddirtytips.com/sites/default/files/styles/insert_small/public/images/6745/mathdude.png?itok=f5Ue47AD" width="82" /></p>
<p>For more fun with math, please check out my book, <a href="http://us.macmillan.com/themathdudesquickanddirtyguidetoalgebra/" target="_blank"><em>The Math Dude’s Quick and Dirty Guide to Algebra</em></a>. And remember to become a fan of The Math Dude on <a href="https://www.facebook.com/TheMathDude" target="_blank">Facebook</a>, where you’ll find lots of great math posted throughout the week. If you’re on <a href="https://twitter.com/jasonmarshall" target="_blank">Twitter</a>, please follow me there, too.</p>
<p>Until next time, this is Jason Marshall with <a href="https://www.quickanddirtytips.com/math-dude" target="_blank">The Math Dude’s Quick and Dirty Tips to Make Math Easier</a>. Thanks for reading, math fans!</p>
<p><em><a href="http://www.shutterstock.com/pic-90097009/stock-photo-black-and-white-soccer-ball-or-football-graphic-white-background.html?src=pp-photo-54648064-LWUnHTl9Vpjr7InG3LJGVw-5" target="_blank">Soccer ball image</a> courtesy of Shutterstock.</em></p>Fri, 03 Feb 2017 23:52:46 -0500Fri, 03 Feb 2017 23:52:46 -0500https://www.quickanddirtytips.com/education/math/polygon-puzzle-how-many-degrees-are-in-a-polygonThe Simple Math Behind Crunching the Sizes of Crowds
https://www.quickanddirtytips.com/education/math/the-simple-math-behind-crunching-the-sizes-of-crowds
<p><img alt="Inauguration Crowd" class="qdt-wrap-left" height="149" src="https://www.quickanddirtytips.com/sites/default/files/styles/insert_small/public/images/9004/crowd-at-presidential-inauguration.jpg?itok=ml9bdFI3" width="224" />There are over 7 billion people on Earth today, and occasionally a bunch of them decide to converge for one reason or another. We’re talking thousands, tens of thousands, hundreds of thousands, or even millions of people in the same place at the same time. As those of us in the United States (and no doubt around the world) have recently witnessed, such events include things like U.S. presidential inaugurations and political marches.</p>
<p>In case you haven’t heard, there’s been a bit of a fracas brewing over the exact sizes of crowds at certain events held in the U.S. this past week. While I don’t want to get into the political aspects of these issues (nor the very reasonable questions about why we’re having these sideshow conversations at all), I feel that it’s important to note that estimating crowd sizes is a solved problem that’s actually pretty straightforward. And it’s relevant to us math fans because it’s really nothing more than a simple exercise in basic math.</p>
<p>So, how do crowd estimate experts estimate crowds? Let’s find out.</p>
<h2>Counting Crowds</h2>
<p>The most reliable method for estimating the size of a crowd is to actually count the size of the crowd. In truth, I’d say this method provides a <em>measurement</em> of the crowd size rather than an <em>estimate</em> since it’s just a matter of counting up people—there’s no other math involved. The beauty of this method is that the uncertainty in your measurement should be extremely low, which means you can be confident about the count’s reliability. Such a direct count is easy to do when crowd sizes are relatively small or when people have to pass through doors or turnstiles, but it’s hard (or even impossible) to do when crowds are large and spread out.</p>
<p class="qdt-pull-quote-right">The most reliable method for estimating the size of a crowd is to actually count the size of the crowd.</p>
<p>In such cases, we have to rely upon our math and reasoning skills. In particular, we have to change tactics from performing a count to performing an estimation. As with any estimate, there will be uncertainty in the final tally since the whole process relies upon a set of assumptions which each are accompanied by some uncertainty. But the beauty of math and statistics is that it provides us with a framework within which we can not only estimate the size of a crowd, but also estimate the quality of our estimate. Which means we can accurately calculate the range of the crowd size which, with very high probability, contains the actual number of people.</p>
<h2>Low-Tech Crowd Size Estimates</h2>
<p>For medium to large crowds spread out over medium to large areas, your best-bet low-tech solution for obtaining accurate crowd counts is to use the method developed by UC Berkeley professor Herbert Jacobs in the 1960s. While observing crowds of protestors gathering in the plaza below his office window, Jacobs realized that he could take advantage of a geometrical/architectural feature of the plaza to come up with crowd size estimates. Since the concrete of the plaza was poured in a grid pattern, Jacobs started by performing an accurate count of the number of people in a few typical grids, and then obtained his total tally by multiplying this average count per grid by the total number of grids. As I said, this isn’t exactly rocket science.</p>
<p>We can get a bit fancier and obtain a more complete understanding of crowd sizes by incorporating uncertainty. For example, imagine you are in charge of counting a crowd. The first thing you do is find a vantage point that gives you a good overview of the scene. Even if you’re not so lucky as to have a grid pattern poured in concrete to work with, you can still mentally divide up the scene into some sort of grid. Suppose you count the people in several squares and conclude that the average square contains between 20 and 25 people (since the density of all crowds does naturally vary from place-to-place). To come up with your estimate of the crowd size, you can then multiply the low and high estimates of the number of people per square by the total number of squares. If there are 300 squares, you would conclude that there are between 20•300=6,000 and 25•300=7,500 people in total.</p>
<hr />
<h2>Modern High-Tech Crowd Size Estimates</h2>
<p>With really big events such as presidential inaugurations or political marches, modern technology offers some assistance to the crowd estimating experts of the world. Today you can launch a balloon or drone-borne camera system to hover over an event and take high resolution images of the crowd. But even with these technological advances, the best crowd estimates still ultimately come down to measuring the average density of people in a representative region of the crowd and then multiplying this value by the total area over which those people are spread out.</p>
<p class="qdt-pull-quote-left">Modern technology offers some assistance to the crowd estimating experts of the world.</p>
<p>There are a few additional modern twists on the simple grid pattern technique in use today. For example, computers (and the humans that write the software they are running) are pretty good at looking at images and classifying different regions of those images based upon their densities. So instead of breaking up an image into a regular grid pattern, a computer can figure out the different amoeba-shaped regions in an image that share similar densities. Once you have such a map of the different regions, you can turn it into a more precise crowd count by calculating the actual density of humans in each region and performing the arithmetic problem just as before. The idea is the same, but the technique does provide more precise results.</p>
<h2>The Certainty and Uncertainty of Crowd Sizes</h2>
<p>Even though we have the mathematical know-how to reliably estimate crowd sizes, you’ll still find that different groups can come up with very different estimates. Most of the time these differences don’t point to a problem with the math but are instead traceable to the biases of the person (or people) performing the estimate. For example, police, media, and event organizers often report different numbers. Why? Well, it’s fairly obvious that different groups might benefit by having larger or smaller numbers reported. Event organizers typically want to bolster their message, so they might be inclined to be “generous” with the assumptions they make in calculating their values.</p>
<p>While it’s good to keep all of this in mind, the real takeaway message here is that calculating crowd sizes isn’t all that complicated. The bottom line is that pictures don’t lie … and with the help of a bit of simple math, they provide all the information needed to accurately estimate crowd sizes.</p>
<h2>Wrap Up</h2>
<p>Okay, that’s all the math we have time for today.</p>
<p>For more fun with math, please check out my book,<em><a href="http://us.macmillan.com/themathdudesquickanddirtyguidetoalgebra/" target="_blank">The Math Dude’s Quick and Dirty Guide to Algebra</a></em>. Also, remember to become a fan of The Math Dude on <a href="https://www.facebook.com/TheMathDude" target="_blank">Facebook</a> and to follow me on <a href="http://twitter.com/jasonmarshall" target="_blank">Twitter</a>.</p>
<p>Until next time, this is Jason Marshall with <a href="https://www.quickanddirtytips.com/math-dude">The Math Dude’s Quick and Dirty Tips to Make Math Easier</a>. Thanks for reading, math fans!</p>
<p><em><a href="https://www.shutterstock.com/image-photo/washington-dc-january-21crowd-approximately-750-126121280?src=jEJsUl_HvfiDOGZi7OLsqQ-1-59" target="_blank">Crowd image</a> from Shutterstock.</em></p>Fri, 27 Jan 2017 23:18:15 -0500Fri, 27 Jan 2017 23:18:15 -0500https://www.quickanddirtytips.com/education/math/the-simple-math-behind-crunching-the-sizes-of-crowdsMath Tips for Smart Shopping
https://www.quickanddirtytips.com/education/math/math-tips-for-smart-shopping
<p><img alt="Calculator in a Shopping Cart" class="qdt-wrap-left" height="224" src="https://www.quickanddirtytips.com/sites/default/files/styles/insert_small/public/images/6711/calculator-in-a-shopping-cart.jpg?itok=Q_YbNM18" width="189" />I recently ran across a 2012 article from <em>The Atlantic</em> called <a href="http://www.theatlantic.com/business/archive/2012/07/the-11-ways-that-consumers-are-hopeless-at-math/259479/" target="_blank">The 11 Ways That Consumers Are Hopeless at Math</a>. The title of this article hooked me, and as I began reading I found that there are indeed a few ways in which consumers misunderstand math - and pay the price as a result.</p>
<p>But I also found that most of the so-called “math tricks" that people get caught up in are really better described as number-based psychological hacks, which marketers use to extract every last penny from us that they can.</p>
<p>So it's not so much that consumers are <a href="/node/6036">hopeless at math</a> as they are susceptible to being tricked. Which is precisely what a <a href="/node/6625">savvy shopper</a> knows how to avoid.</p>
<p>What are some of these mathematical misunderstandings that you should be aware of? And what are some of the most common number-based psychological hacks? Those are exactly the questions we’ll be looking at today, as we finish up the year with <a href="/node/1397">a resolution</a> to become even smarter shoppers in the new year.</p>
<p class="qdt-adv-rte">Sponsor: This episode is brought to you by NatureBox. Discover smarter snacking with a new NatureBox each month. Get your first box FREE when you go to <a href="http://naturebox.com/qdt" target="_blank">naturebox.com/qdt</a>.</p>
<h2>How Much Bang For Your Buck?</h2>
<p>The article I mentioned from <a href="http://www.theatlantic.com/business/archive/2012/07/the-11-ways-that-consumers-are-hopeless-at-math/259479/" target="_blank"><em>The Atlantic</em></a> begins with an anecdote that nicely points out one of the biggest flaws in the way the average consumer shops. Namely, that when it comes to <a href="/node/1521">pricing and deals</a>, most people go with their gut instead of taking a few seconds to <a href="/node/6412">think things through</a>.</p>
<p>Here's the story: Imagine you walk into a <a href="/node/3153">coffee shop</a>, take a look at the day’s specials, and see a sign that says, “Today only, your choice—get 33% more coffee for the regular price, or pay 33% less for the regular amount of coffee!” If you were presented with these two options, which would you choose?</p>
<p>In truth, choosing the best deal isn't always just a question of numbers. For example, if you really wanted more than your regular amount of coffee that day, then the extra coffee option would be a fine choice. But that’s not really what I’m talking about here, so let’s rephrase the question a bit to <a href="/node/3175">focus on the math</a>.</p>
<p>The real question is this: Which option is the better deal in terms of dollars spent per ounce of coffee? After all, that’s what we’re really talking about when we speak of being a savvy shopper—<a href="/node/1615">getting the most bang for your buck</a>.</p>
<p class="qdt-pull-quote-right"><span>Most people's gut instinct is that the two deals are about equally as good.</span></p>
<p>Most people’s gut instinct is that the two deals—33% more coffee for the same price or the same amount of coffee for 33% less money—are equally as good. After all, they both have the same 33% in them. But <a href="/node/5502">let’s do the math</a> to see if this assumption is really true.</p>
<p>Imagine your usual 8 oz. cup of coffee costs $2. In this case, the first option gives you about 1.33 x 8 oz. = 10.6 oz. of coffee for $2, while the second option gives you your usual 8 oz. of coffee for a price of 0.67 x $2 = $1.34. That means you pay $2 / 10.6 oz. = 18.9 cents/oz. with the first option, but only $1.34 / 8 oz. = 16.8 cents/oz. with the second.</p>
<p>So, clearly, the second option is a better deal. While it's tempting to get something "free" for the same amount you usually pay (the first option), in this case, getting the amount you actually want for less money is a better deal—especially if you don't really need that extra coffee anyway. And, as always, the math is there to back you up.</p>
<hr />
<h2>What’s the Best Deal?</h2>
<p class="qdt-pull-quote-right">People prefer to make choices between similar and easily-comparable options.</p>
<p>As I mentioned at the outset, most savvy shopping skills are really less about math and more about avoiding the number-based psychological hacks that <a href="/node/3339">marketers </a>(would love to) play on you. While perusing the news this week, I found an <a href="http://conversionxl.com/pricing-experiments-you-might-not-know-but-can-learn-from/#." target="_blank">article</a> discussing a perfect example of this kind of sneaky hackery.</p>
<p>This example was originally described in Dan Ariely's book <em>Predictably Irrational, </em>in which he talks about running across an advertisement to subscribe to the magazine <em>The Economist</em>. The advertisement lists 3 possible deals:</p>
<ol>
<li>Web-only subscription for $59/year</li>
<li>Print-only subscription for $125/year</li>
<li>Print + web subscription for $125/year</li>
</ol>
<p>If confronted with these options, which would you choose? If you’re anything like the 100 MIT students that Dan Ariely posed this question to, you’d pick the <a href="/node/6109">print + web subscription</a> for $125/year; 84% of the MIT students chose that offer, while 16% chose the cheaper web-only subscription.</p>
<p>Not surprisingly, nobody chose the middle <a href="/node/4486">print-only option</a>. After all, it’s a pretty bad deal compared to the third option, which gives you the same thing plus something extra, all for the same price. But if that middle option is such a bad deal, why did the marketers even bother to include it?</p>
<p>To answer this question, Dan Ariely removed the second option from the list and presented the two remaining options to another group of 100 MIT students. This time, with just the $59 web-only and $125 print + web subscriptions to choose from, 68% chose the cheaper web-only subscription and 32% chose the print + web subscription. Remember, when all three options were available, 84% of students chose <a href="/node/2465">the more expensive option</a> and only 16% chose the cheaper subscription.</p>
<p>So why did the marketers include that strange print-only subscription option? Because they also figured out that more people would choose the more expensive subscription if the print-only option was there.</p>
<p>What’s the math behind this? There isn’t any—this one is<a href="/node/6671"> purely psychological</a>. Sure, there are numbers involved, but all they’re really doing here is pointing out that people prefer to make choices between similar and easily-comparable options - so when they’re given the opportunity to do so, they will.</p>
<p>It <a href="/node/2719">may not be rational</a>, but it is very real. And knowing how to spot this kind of trick is a big part of<a href="/node/2580"> learning how to use math</a>—or at least numbers—to be a more savvy shopper.</p>
<h2><img alt="" class="qdt-wrap-left" height="224" src="https://www.quickanddirtytips.com/sites/default/files/styles/insert_small/public/images/6711/mathdude.png?itok=0az4mw0l" width="82" />Wrap Up</h2>
<p>Okay, that’s all the math we have time for today.</p>
<p>For more fun with math, please check out my book, <a href="http://us.macmillan.com/themathdudesquickanddirtyguidetoalgebra/" target="_blank"><em>The Math Dude’s Quick and Dirty Guide to Algebra</em></a>. And remember to become a fan of The Math Dude on <a href="https://www.facebook.com/TheMathDude" target="_blank">Facebook</a>, where you’ll find lots of great math posted throughout the week. If you’re on <a href="https://twitter.com/jasonmarshall" target="_blank">Twitter</a>, please follow me there, too.</p>
<p>Until next time, this is Jason Marshall with <a href="https://www.quickanddirtytips.com/math-dude">The Math Dude’s Quick and Dirty Tips to Make Math Easier</a>. Thanks for reading, math fans!</p>
<p><em><a href="http://www.shutterstock.com/pic-178162628/stock-photo-calculator-trolley-concept-a-calculator-in-a-shopping-trolley-cart-could-be-for-shopping-for.html?src=R1SWA3MWxN7rduQpQAowGg-1-22" target="_blank">Calculator-in-a-shopping cart image</a> courtesy of Shutterstock.</em></p>Fri, 20 Jan 2017 23:53:50 -0500Fri, 20 Jan 2017 23:53:50 -0500https://www.quickanddirtytips.com/education/math/math-tips-for-smart-shoppingHow to Measure Time Without a Stopwatch
https://www.quickanddirtytips.com/education/math/how-to-measure-time-without-a-stopwatch
<p><img alt="Burning Fuse" class="qdt-wrap-left" height="149" src="https://www.quickanddirtytips.com/sites/default/files/styles/insert_small/public/images/7372/burning-fuse.jpg?itok=cLmrdBdN" width="224" />A math puzzle a day keeps your brain saying "Yay!"</p>
<p>I know that’s not the most memorable saying in the world, but it’s definitely true—puzzles are a fantastic workout for your brain. As such, you’d be wise to try your hand at tackling at least a few different kinds of puzzles every week. And the best part of this is that mental exercise like this is fun!</p>
<p>To help you in your endeavor to start puzzling more, today we’re going to take a look at a great brain teaser that I recently ran across. This puzzle is all about time and how you can measure intervals of time in a rather unusual way: by burning bits of specially crafted string. How does it work? And what’s the big brain teaser?</p>
<p>That’s exactly what we’ll be talking about today.</p>
<h2>How Can You Measure 45 Minutes?</h2>
<p>Imagine you’ve been given several pieces of string with varying lengths and thicknesses. Not only do the lengths and widths of the pieces vary, each piece of string isn't even uniform in width along its own length. In other words, they get thicker and thinner (by different amounts and in different places) as you go from one end to the other. While all of the pieces are therefore different, they have one thing in common: if you light one end on fire, it will always take exactly 1 hour to burn through to the other side. But since each piece gets thicker and thinner as it goes, a given piece of string doesn’t necessarily burn at an even rate. By which I mean that a string doesn't necessarily burn half its length in 30 minutes—all we can say is that it burns its entire length in exactly 1 hour.</p>
<p class="qdt-pull-quote-right">Is it possible to use these pieces of string to measure a 45 minute time interval?</p>
<p>So that’s the setup. Here’s the question: Is it possible to use these pieces of string (as many as you want) to measure a 45 minute time interval? And, if it's possible, how would you do it? As with every puzzle, it’s a lot more fun if you give it a try before finding out the answer. So I encourage you to pause for a few minutes to give it a go. Then, when you’re ready, continue on for the answer.</p>
<h2>A Simpler Problem</h2>
<p>Before we solve today’s puzzle, let’s imagine a slightly simpler puzzle in which the pieces of string we’re given all burn at a uniform rate. In this case, you could solve the puzzle simply by folding a single piece of string in half, and then by folding this in half again to make creases in the string at 1/4, 1/2, and 3/4 its total length. Since this string burns at a uniform rate, all you have to do to measure 45 minutes of time is simply light one end of the string on fire and wait until it burns to the mark that's 3/4 of the way towards the other end. Easy!</p>
<p>But, sadly, the pieces of string we’re given in this puzzle are not so well behaved in that they don't burn at an even rate (since they get thicker and thinner at various points along their length). Which means we can’t simply fold a piece of string into quarters to count off time. But that doesn’t mean we’re out of luck … because we can do something much more clever!</p>
<hr />
<h2>How to Solve It — Step 1</h2>
<p class="qdt-pull-quote-left">Instead of lighting just one end of a piece of string on fire, we start by lighting both ends on fire at the same time.</p>
<p>Here’s what we need to do: Instead of lighting just one end of a piece of string on fire, we start by lighting both ends on fire at the same time. What does this do for us? Well, the uneven thickness of a piece of string means that we can't say how fast the fire will move at any given moment as it's burning from one end to the other (or the other way around). But, nonetheless, we do know that the two flames burning towards each other must come together and meet (although we’re not quite sure where) after precisely 30 minutes! Make sense?</p>
<p>That means we now have a way to measure 30 minute intervals in addition to 1 hour intervals. Of course, what we really want is to measure a time interval of 45 minutes. So we’re not done yet. What do we do now? Before answering that question, once again if you haven’t figured it out yet, I encourage you to stop for a minute and think about it before continuing on.</p>
<h2>How to Solve It — Step 2</h2>
<p>OK, so what can we do to measure a 45 minute time interval instead of just 30 or 60 minute time intervals? The trick is that right after we simultaneously light that first string from both ends, we need to light another string from only one end. Then, when the two flames of the first string come together—which we've already figured out marks the 30 minute point—we need to light the other end of the now half-burned (in terms of time) second string. At this point, the two flames of the second string will start burning towards each other, and they will meet in the middle after exactly another 15 minutes pass.</p>
<p>Putting all of this together, we see that the burning ends of the second string must meet a total of 30 + 15 or 45 minutes after we started this whole process of setting strings on fire. And thus, by applying a bit of clever thinking, we have managed to measure a time interval of 45 minutes. Pretty cool, right?</p>
<h2>Wrap Up</h2>
<p>OK, that’s all the math puzzling we have time for today.</p>
<p>For more fun with math, please check out my book, <a href="http://us.macmillan.com/themathdudesquickanddirtyguidetoalgebra/" target="_blank"><em>The Math Dude’s Quick and Dirty Guide to Algebra</em></a>. And remember to become a fan of The Math Dude on <a href="https://www.facebook.com/TheMathDude" target="_blank">Facebook</a>, where you’ll find lots of great math posted throughout the week. If you’re on <a href="https://twitter.com/jasonmarshall" target="_blank">Twitter</a>, please follow me there, too.</p>
<p>Until next time, this is Jason Marshall with <a href="https://www.quickanddirtytips.com/math-dude" target="_blank">The Math Dude’s Quick and Dirty Tips to Make Math Easier</a>. Thanks for reading, math fans!</p>
<p><em><a href="http://www.shutterstock.com/pic-179409527/stock-photo-fuse-is-burning-dynamite-fuse.html?src=i2N8SMHO3qDgnCdLCoiXoQ-1-14">Burning fuse image</a> from Shutterstock.</em></p>Sat, 14 Jan 2017 02:30:21 -0500Sat, 14 Jan 2017 02:30:21 -0500https://www.quickanddirtytips.com/education/math/how-to-measure-time-without-a-stopwatch4 More FAQs About Percentages
https://www.quickanddirtytips.com/education/math/4-more-faqs-about-percentages
<p><img alt="FAQ" class="qdt-wrap-left" height="224" src="https://www.quickanddirtytips.com/sites/default/files/styles/insert_small/public/images/8957/faq.jpg?itok=Yi5f4gPX" width="224" />We here at the Math Dude ranch get numerous questions every week from math fans around the world. By far the most common questions we receive have to do with calculating percentages. In particular, how to quickly calculate percentages in your head. You know, things like: What’s 25% of $14,000? Or what’s the final price after a 33% discount on a $25 item? Or what’s the percentage increase from 30 to 40?</p>
<p>It’s not hugely surprising that this is such a popular line of questions since people in lots of different industries love to express changes in terms of percentages. So today we’re going to take a look at four of the most frequently asked questions about percentages.</p>
<h2>Holiday Puzzle Solution</h2>
<p>But before we dive into percentages, I want to fill you in on the solution to the puzzle I posed last time. Actually, it’s the puzzle <a href="https://mobile.twitter.com/RichardWiseman/status/811172770138718208" target="_blank">tweeted</a> by psychologist, magician, and guest of the show <a href="https://www.quickanddirtytips.com/education/math/how-to-win-every-bet" target="_blank">Richard Wiseman</a>. In case you’ve forgotten, here’s how it works. First, grab a calculator. Then do the following:</p>
<ul>
<li>Type your house number (i.e., your address) into a calculator.</li>
<li>Now double it.</li>
<li>Next add 5 to the result.</li>
<li>Then multiply this answer by 50.</li>
<li>Now add your age.</li>
<li>And then add 365 to the result.</li>
<li>Finally, subtract 615 from the whole thing.</li>
</ul>
<p>What do you get? If you did it right, you should see your house number and age (so long as you’re under 100 years old). Why? It’s actually fairly simple to understand with a bit of algebraic thinking. To begin, let’s call your house number “A” and your age “B”. If you follow the steps in Richard's tweet, you’ll see that the whole sequence of actions is equivalent to the algebraic expression:</p>
<p>(((((2 x A) + 5) x 50) + B) + 365) - 615</p>
<p>Which is quite a mess! How does it help us make sense of the trick? Well, if we simplify the expression a bit, we see that we can combine and arrange the terms to turn it into the equivalent expression:</p>
<p>((2A + 5) x 50) + B - 250</p>
<p>Admittedly, this isn’t much better, but if we simplify this even more we find that we can multiply and then combine terms to arrive at a much simpler equivalent expression:</p>
<p>100A + B</p>
<p>And now we’re getting somewhere. Because this expression tells us that all you're really doing is multiplying your address by 100 (which has the effect of padding the end of it with a pair of zeros) and then adding your age (which has the effect of sticking it on the end). Once you know this, you can see that all of the complicated actions were simply a distraction to keep you from noticing the simple thing happening when you weren’t looking. In other words, it's a magic trick.</p>
<hr />
<h2>Percentage Tip 1: Calculating 10%, 20%, 30%, …</h2>
<p>Math fan Amanda recently wrote to ask about calculating 10% of a dollar amount. She wrote “Every time I do this, I have to Google for a reminder and never end up working it out the same way twice. So what’s the easy way?”</p>
<p>Well Amanda, the good news is that all you need to remember to make sense of percentages is that the word “per-cent” means “per 100” (“cent” also shows up in century, centipede, and the 1 cent value of the United States penny). So 1 percent of some value is the same as the fraction 1/100 of that value. And 10 percent of some value is the same as 10/100 = 1/10 of that value.</p>
<p>As such, calculating 10% of a value is really, really easy to do in your head. All you have to do is divide the value by 10. So 10% of $90 is just $90/10 = $9. And it’s not much tougher to calculate 20% or 30% of a number. Simply calculate 10% of the number and then multiply the result by 2 or 3. So 20% of $90 is 2 x $9 = $18, and 30% of $90 is 3 x $9 = $27. You can use the same logic to find 40%, 50%, 60%, or any other similar percentage.</p>
<h2>Percentage Tip 2: Calculating 25%, 33%, 50%, 66%, and 75%</h2>
<p>Several math fans have written asking how to calculate other percentages such as 25%, 33%, 50%, 66%, and 75%. The good news is these are also easy to calculate in your head.</p>
<p>To begin with, since 25% is the same as 25/100=1/4, you can calculate 25% of any number by dividing it by 4. So 25% of $60 is just $60/4=$15. Similarly, since 33% is the same as 33/100 or approximately 1/3, you can calculate 33% of a number by dividing it by 3. And since 50% is the same as 50/100=1/2, you can calculate 50% of a number by dividing it by 2.</p>
<p>The percentages 66% and 75% are almost as easy. Since 66% is the same as 66/100 or roughly 2/3, you can calculate 66% of some number by dividing it by 3 and then multiplying the result by 2 (or the other way around if it's easier). And since 75% is the same as the fraction 75/100=3/4, you can calculate 75% of a number by dividing it by 4 and then multiplying the result by 3.</p>
<h2>Percentage Tip 3: Calculating Fractional Percentages</h2>
<p>Math fan Amirah wrote: “I have always had trouble calculating percentages quickly in my head; I can do it slowly on paper but it’s not the most efficient method. Do you have any tips for calculating decimal percentages quickly? For example, what’s 0.001% of $1,000,000?”</p>
<hr />
<p>Fortunately for Amirah, calculating percentages that are a fraction of 1% isn’t all that difficult once you understand how to calculate 10% of a number. This is most easily seen by working out Amirah’s example of finding 0.001% of $1,000,000.</p>
<p>The trick is that for each factor of 10 smaller percentage, we divide the number by yet another factor of 10. Remember that to calculate 10% of a number we divide by 10. And to calculate 1% of a number we divide by 100. To calculate smaller percentages, we just continue this trend. So to calculate 0.1% we divide by 1,000; to find 0.01% we divide by 10,000; and to find 0.001% we divide by 100,000. So 0.001% of $1,000,000 is $1,000,000/100,000=$10.</p>
<h2>Percentage Tip 4: Calculating Percentage Increase</h2>
<p>Finally for today, math fan Matthew recently wrote: “In the course of my work, I read sentences such as ‘Some company delivered another strong quarter with sales of $9.4 billion, or an increase of 6%.’ What’s a quick and dirty way to ballpark what the original number was?”</p>
<p>The trick that helps me here is to translate this into a very simple equation in my head. The equation says that 1.06 (which is the same as 106%) times some number is equal to $9.4 billion. The question is what’s the number? If you think about it, you’ll see that all we have to do to find out is divide $9.4 billion by 1.06. In general, you can always solve a problem like this to find the initial number by dividing the final number by the percentage increase or decrease (represented as a decimal number). Once you know the trick and where it comes from, you should never need to write the equation down again.</p>
<h2>Wrap Up </h2>
<p>Okay, that’s all the math we have time for today.</p>
<p>For more fun with math, please check out my book,<em><a href="http://us.macmillan.com/themathdudesquickanddirtyguidetoalgebra/" target="_blank">The Math Dude’s Quick and Dirty Guide to Algebra</a></em>. Also, remember to become a fan of The Math Dude on <a href="https://www.facebook.com/TheMathDude" target="_blank">Facebook</a> and to follow me on <a href="http://twitter.com/jasonmarshall" target="_blank">Twitter</a>.</p>
<p>Until next time, this is Jason Marshall with <a href="https://www.quickanddirtytips.com/math-dude">The Math Dude’s Quick and Dirty Tips to Make Math Easier</a>. Thanks for reading, math fans!</p>
<p><em><a href="http://www.shutterstock.com/pic-143084338/stock-photo--d-illustration-of-modern-roadsign-cubes-signpost-of-faq-frequently-asked-question.html?src=i0x7_5yF-8FADdeh7D3lNg-1-2" target="_blank">FAQ image</a> from Shutterstock.</em></p>Fri, 06 Jan 2017 23:10:35 -0500Fri, 06 Jan 2017 23:10:35 -0500https://www.quickanddirtytips.com/education/math/4-more-faqs-about-percentagesThe 5 Steps of Problem Solving
https://www.quickanddirtytips.com/education/math/the-5-steps-of-problem-solving
<p><img alt="Problem Solving" class="qdt-wrap-left" height="211" src="https://www.quickanddirtytips.com/sites/default/files/styles/insert_small/public/images/5817/problem_solving.jpg?itok=UFP6a3Hi" width="224" />If the phrase "word problem" sends a shiver down your spine, you're not alone. A lot of people have trouble with so-called word problems in math. But, believe it or not, these problems usually aren't any harder to solve than non-word problems—they just look very, very different. And they require a slightly different mindset to solve.</p>
<p>Today, I'm going to tell you about my simple 5-step method that will help you solve all your math problems—including those pesky word problems. In particular, we're going to talk about how to turn a word problem into an <a href="/node/4596">algebraic equation</a> and then solve it.</p>
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<h2>A "Real World" Math Drama</h2>
<p>Today's word problem begins with a story about <a href="https://www.quickanddirtytips.com/pets/cats">cats</a> and <a href="https://www.quickanddirtytips.com/pets/dogs">dogs</a>. It goes something like this…</p>
<p><em>Like all dogs, your <a href="/node/4085">dog loves toys.</a> And you love giving them to him. Your cat, on the other hand, does not love your dog and therefore finds it amusing to hide his toys. Being quite clever, you suspect that the cat is the culprit, so you begin to monitor his favorite hiding spot: the pile of towels next to his bed. </em></p>
<p><em>But (perhaps being a little too clever for your own good) instead of constantly checking this spot, you decide that you'd like to rig up an ingenious system to automatically report to you exactly how many toys are missing.</em></p>
<p><em>The question is: How can you do this?</em></p>
<h2>Step #1: Stop and Think Before Doing Anything</h2>
<p class="qdt-pull-quote-right">The biggest mistake people make when solving problems is trying to solve them too soon.</p>
<p>The most important thing to do when faced with a problem like this is to stop working on it. Honestly, it sounds paradoxical, but the biggest mistake people make when solving problems is trying to solve them too soon. Instead, stop and think about what you need to do. Make sure you understand exactly what the question is asking and make sure you understand exactly what you are trying to solve for.</p>
<p>In our problem, we should ask ourselves: Can we actually build something that will discern the numer of hidden dog toys? Sure, all we need to do is put the cat's pile of towels on a smart scale that sends its weight to your computer. Whenever the scale senses a weight increase, it can tell your computer that another toy has been hidden. Your computer can then use some as-yet-unknown equation to figure out exactly how many toys are hidden. When that number goes above a certain limit, your computer can sound an alarm to let you know that it’s time to go fetch.</p>
<p>Now that we have a plan, it's time for the big English-to-equation translation.</p>
<h2>Step #2: English-to-Equation Translation</h2>
<p>The second step in solving word problems is turning the words into one or more mathematical <a href="/node/4542">expressions</a> or <a href="/node/4621">equations</a>. In our case, we need to figure out how to write an equation that takes the current weight on a scale and gives us back the number of dog toys hidden on it. How can we do it?</p>
<p>Well, let’s take the total weight on the scale, which we’ll call W_total, and subtract the weight of just the towels, which we’ll call W_towels. The difference between these two weights must be equal to the combined weight of all the dog toys, W_toys:</p>
<p><img alt="Weight of Dog Toys" class="image-insert_medium" height="215" src="https://www.quickanddirtytips.com/sites/default/files/styles/insert_medium/public/images/5817/weight_of_toys.png?itok=AGJlllep" width="448" /></p>
<p>But we don’t actually want to know the weight of the toys, we want to know the number of toys. How can we do that? Well, if we know the total weight of all the toys, W_toys, and we divide that by the weight of a single toy, W_toy (assuming they're all the same weight), we get the total number of toys, Ntoys:</p>
<p><img alt="Number of Dog Toys" class="image-insert_medium" height="145" src="https://www.quickanddirtytips.com/sites/default/files/styles/insert_medium/public/images/5817/number_of_toys.png?itok=hKRRGDEy" width="448" /></p>
<p>But how did we know the values of W_towels and W_toy? We must have been clever enough to measure them and write them down before we put the towels on our scale. Or, if we didn’t do that, we’d better do it now!</p>
<h2>Step #3: Solve for Whatever You’re Interested In</h2>
<p>The third step in solving our word problem—or any word problem—is to solve for the variable you’re interested in. This step will often entail going through the procedure outlined in the <a href="/node/4744">How to Solve an Equation</a> episode. Our goal is to solve for the total number of dog toys on the scale, N_toys. Combining the two equations we came up with in our English-to-equation-translation, we get just such an equation:</p>
<p>N_toys = ( W_total – W_towels ) / W_toy</p>
<p>We can now turn this abstract solution (abstract in the sense of being written in terms of a bunch of variables) into a numerical solution simply by plugging in numerical values for all the variables on the right side of the equals sign. I should say that plugging in values isn't always necessary. For example, we don’t actually have numerical values to use in our problem. Which makes sense since our equation was only intended to be used to tell our computer how to convert from total weight to total number of dog toys—we weren’t actually looking for a specific answer to a specific problem.</p>
<h2>Step #4: Make Sure You Understand the Result</h2>
<p class="qdt-pull-quote-left">Slow down and take a minute to think about your result.</p>
<p>The fourth step in the problem solving process is closely related to the first. The name of the game here is to slow down and take a minute to think about your result. Ask yourself if it makes sense. If you plugged in numbers and got a <a href="/node/1442">negative number</a>, ask yourself if you expected to get a negative number. If you got a huge number, ask yourself if you expected that.</p>
<p>The bottom line is this: Don’t declare that you’re done simply because you got an answer. The only reason you should declare that you’re done is because you understand the answer you got.</p>
<h2>Step #5: Use Your Result to Solve Other Problems</h2>
<p>The fifth and final step of the problem solving process is to use the result you’ve obtained to solve other problems. Why have I included this as a step? Because we’re talking about solving real world problems here—not just textbook problems. In the real world, many problems you solve will make you think of something related that you need to solve, too. The good news is that doing so is usually easier because you already have a solution to build upon.</p>
<p>And those are all of the steps you should go through when solving real world math problems. Of course, once you get good, you won't actually think about doing each step—they'll just happen. But while you're learning, it's best to be deliberate about things and to actually think your way through each step.</p>
<p>Good luck in all your future problem solving endeavors!</p>
<p><img alt="5 Steps of Problem Solving" class="image-insert_large" height="346" src="https://www.quickanddirtytips.com/sites/default/files/styles/insert_large/public/images/5817/5_steps_of_problem_solving.png?itok=fVga_ipJ" width="673" /></p>
<h2>Wrap Up</h2>
<p><img alt="" class="qdt-wrap-right" height="224" src="https://www.quickanddirtytips.com/sites/default/files/styles/insert_small/public/MD%2BQDG%2Bto%2BAlgebra%2BLarger_0.jpg?itok=M1PZFe4O" width="153" />Okay, that's all the math we have time for today.</p>
<p>Please be sure to check out my book <strong><a href="https://www.quickanddirtytips.com/math-dude-book" sl-processed="1"><em>The Math Dude’s Quick and Dirty Guide to Algebra</em></a></strong>. And remember to become a fan of the Math Dude on <a href="http://facebook.com/TheMathDude" sl-processed="1" target="_blank">Facebook</a> where you’ll find lots of great math posted throughout the week. If you’re on <a href="http://twitter.com/#!/jasonmarshall" sl-processed="1" target="_blank">Twitter</a>, please follow me there, too.</p>
<p>Until next time, this is Jason Marshall with <a href="http://mathdude.quickanddirtytips.com/" sl-processed="1"><strong>The Math Dude’s Quick and Dirty Tips to Make Math Easier</strong></a><strong>.</strong> Thanks for reading, math fans!</p>
<p><em><a href="http://www.shutterstock.com/pic-95641108/stock-photo-problem-solving-crossword.html?src=LGo0pFvs6LlP9KmKZxXD9Q-1-48" target="_blank">Problem solving image</a> from Shutterstock.</em></p>Fri, 30 Dec 2016 22:53:25 -0500Fri, 30 Dec 2016 22:53:25 -0500https://www.quickanddirtytips.com/education/math/the-5-steps-of-problem-solvingHow Many Gifts Are in the 12 Days of Christmas?
https://www.quickanddirtytips.com/education/math/how-many-gifts-are-in-the-12-days-of-christmas
<p><img alt="Partridge In a Pear Tree" class="qdt-wrap-left" height="224" src="https://www.quickanddirtytips.com/sites/default/files/styles/insert_small/public/images/8942/partridge-in-a-pear-tree.jpg?itok=lM23elh_" width="224" /><em>Once upon a time, in a far away land, a young girl and boy were playing in the dining room of their castle when they discovered a story scrawled on the back of a painting. As legend has it, the story read …</em></p>
<p>So Christmas was weird for me last year. It all started on December 25 when my true love gave me a potted pear tree upon whose branches sat a befuddled bird. Things got even weirder the following day when I received yet another pear tree (complete with yet another bird), as well as a pair of doves. The story on the third day was similar—another pear tree, two more doves, and this time a trio of chickens. As you might imagine, I was wondering: What’s up with all the birds?</p>
<p>This sort of thing continued for more than another week. With each new day came a new gift—actually a predictable number of that new gift: 3 on the 3rd day, 4 on the 4th, 5 on the 5th, and so on—as well as a repeat of <em>all</em> the gifts given on all of the previous days. By the time we got to the 12th day, we ran out of space in the castle living room and had to move our holiday celebration to the very dining room in which this painting is hanging.</p>
<p>That evening as we sat around the table, somebody asked me how many presents my true love had given on each day? With that question we began a quest to understand the mathematics of those epic twelve days of Christmas. To the best of my recollection, dear reader, the following contains an accurate recounting of that tale of discovery.</p>
<h2>The Wonderful (and Distracting) World of Puzzles</h2>
<p>As so often happens, our journey towards counting my true love’s gifts was quickly derailed by another math puzzle. Somebody at the table mentioned that someday far into the future (in the year 2016), psychologist and magician Richard Wiseman (who was once a<a href="https://www.quickanddirtytips.com/education/math/how-to-win-every-bet" target="_blank"> guest on the famous “Math Dude” show</a>) would <a href="https://mobile.twitter.com/RichardWiseman/status/811172770138718208" target="_blank">tweet</a> the following series of instructions (which you should feel free to follow along with … you might want to go grab a calculator):</p>
<ul>
<li>Type your house number (i.e., your address) into a calculator.</li>
<li>Now double it.</li>
<li>Next add 5 to the result.</li>
<li>Then multiply this answer by 50.</li>
<li>Now add your age.</li>
<li>And then add 365 to the result.</li>
<li>Finally, subtract 615 from the whole thing.</li>
</ul>
<p>What do you get? As Wiseman points out: “Voila—your house number and age!” Needless to say, all of us at the table gave it a go, and much to our amazement we found that it does indeed work (so long as you’re less than 100 years old). Upon realizing this, roars of curiosity erupted: “Why? Huh? How can this be? I MUST KNOW NOW!”</p>
<p>So, exactly how does this puzzle from the future work? Good question, dear reader.</p>
<hr />
<p>Alas, we don’t have time for an answer right now (and my space to write on the back of this painting is limited). So I’m going to let you think about it for a bit and see if you can come up with the answer yourself (but fear not, the answer will be revealed in a future Math Dude).</p>
<p>This is also what I told my guests for dinner that evening. So instead of allowing this to distract us from our true and noble goal, we dove back into our quest to understand the numbers behind my twelve days of gifts.</p>
<h2>How Many Presents On Each Day?</h2>
<p>On the first day of Christmas, my true love gave me one gift (the bird in a pear tree). On the second day of Christmas, my true love gave me 3 gifts (a pear tree and a couple of doves). On the third day of Christmas, my true love gave me 6 gifts (a pear tree, a couple of doves, and three chickens).</p>
<p>Upon thinking this through, my dinner companions and I wondered if there was a pattern here. One of us realized that if we used little dots to represent presents, then we could arrange the dots from each successive day’s presents into successively larger triangles.</p>
<p><img alt="Triangular number construction" class="image-insert_large" height="167" src="https://www.quickanddirtytips.com/sites/default/files/styles/insert_large/public/images/8942/triangular-numbers.png?itok=mLxqe4J5" width="673" /></p>
<p>In other words, the first day’s single present is a single dot. Then if we put two dots diagonally next to that single dot, we get a triangle with three total dots representing the three total presents from the second day. Adding three more dots in a diagonal line to this three-dot triangle gives us a six-dot triangle representing the total number of presents on the third day. And so on for each successive day—the number of dots in each triangle represents the number of presents for each day. </p>
<p>The number of dots that you obtain with each successive triangle (and thus the number of gifts received on each successive day) are known in mathematical circles as “triangular numbers” since they can be used to construct triangles.</p>
<h2>The Gift of Triangular Numbers</h2>
<p>The next question was could we come up with some way to quickly figure out how many dots—or rather gifts—I had received on each day without resorting to simply counting the dots? In other words, could we figure out a simple formula to calculate triangular numbers?</p>
<p>Sure enough, some clever person at the table (the name might have been <a href="https://betterexplained.com/articles/techniques-for-adding-the-numbers-1-to-100/" target="_blank">Gauss</a>) figured out that if you wanted to know the presents received on the Nth day of Christmas (where N could be 3, 4, 5, 12, or whatever), then you could calculate it quickly by thinking about a box made up of N by N dots. If you think about it, you’ll realize that the number of gifts on the Nth day is given by the number of dots in the triangle made up of all the dots below and including those on the diagonal of the N by N square.</p>
<p><img alt="Sum of the first N integers" class="image-insert_large" height="260" src="https://www.quickanddirtytips.com/sites/default/files/styles/insert_large/public/images/8942/sum-of-first-n-integers.png?itok=Goo5PqzB" width="673" /></p>
<p>What is this number? Well, there are N•N dots in a square (that’s the area of a square), so there must must be N•N/2 dots in half a square. But that isn’t quite the triangular number we’re after because it’s missing half the dots along the diagonal of the square. How many dots is that? It’s N/2 more dots. So the total number of dots in the triangle is actually: N•N/2 + N/2. Rearranging this a bit, my dinner companions and I found that the number of presents my true love gave me on the Nth day of Christmas must be:</p>
<p>Number of Gifts on Nth Day = N • (N+1) / 2</p>
<p>So, how many presents did I receive on the 5th day of Christmas? Our newly discovered handy-dandy formula says that it must have been 5•6/2 = 15. How about the 10th day? 10•11/2 = 55 presents. And the 12th day? 12•13/2 = 78 presents!</p>
<h2>The Legend of the Twelve Days of Christmas</h2>
<p>As I said at the outset, Christmas was a little weird for me last year. And now, dear reader, you know why. My true love went a bit overboard with gift-giving and things got a little out-of-hand in the castle. But despite all of the birds and the trees (not to mention the cows, drummers, and dancers), the best gift of all was the time spent with loved-ones around the table discovering beautiful secrets hidden in the world of math…</p>
<p>At least that’s what legend says the young girl and boy discovered on the back of that painting in their castle dining room all those years ago. In truth, nobody knows for sure because, sadly, that painting was lost to history. But, thankfully, the math, the puzzles, and the song recounting those epic “12 Days of Christmas” are still with us.</p>
<p>Or at least that’s the story of this song that I’ve decided to tell my daughter this year.</p>
<h2>Wrap Up</h2>
<p>Okay, that’s all the math we have time for today.</p>
<p>For more fun with math, please check out my book, <em><a href="http://us.macmillan.com/themathdudesquickanddirtyguidetoalgebra/">The Math Dude’s Quick and Dirty Guide to Algebra</a></em>. Also, remember to become a fan of The Math Dude on <a href="https://www.facebook.com/TheMathDude">Facebook</a> and to follow me on <a href="http://twitter.com/jasonmarshall">Twitter</a>.</p>
<p>Until next time, this is Jason Marshall with <a href="https://www.quickanddirtytips.com/math-dude">The Math Dude’s Quick and Dirty Tips to Make Math Easier</a>. Thanks for reading, math fans!</p>
<p><em><a href="http://www.shutterstock.com/pic-63021532/stock-vector-a-partridge-in-a-pear-tree-christmas-bird.html?src=LZToimUxBMyVL6ArQBpjxg-1-10">”Partridge in a pear tree” image</a> from Shutterstock.</em></p>Fri, 23 Dec 2016 23:10:18 -0500Fri, 23 Dec 2016 23:10:18 -0500https://www.quickanddirtytips.com/education/math/how-many-gifts-are-in-the-12-days-of-christmas