Author: Math Expert

Categories
Math for Grownups

Preakness Math: What Are the Odds?

This morning at 6:00 a.m., I gave my 10 year old $25 and sent her off to the racetrack.

It’s Preakness weekend here in Baltimore, and just a few blocks from my house, Pimlico Racetrack will host the 136th Annual Preakness Stakes.  It’s a half-day at school today, and one of my daughter’s friends is totally into horse racing.  Besides, she has adult supervision.

But I probably should have taught her a little bit about odds.

Preakness is the second leg of the Triple Crown, and all eyes will be on Animal Kingdom, the 20-1 long shot who won the Kentucky Derby in early May.  Animal Kingdom isn’t a long shot anymore.  At the Preakness, his odds are  2-1.

Where do these numbers come from?  Betting on horse racing is a popularity contest.  In other words, the payoff depends on the bets themselves.  People like a winner, and so they tend to bet on the favored horse.

That means that the favored horse will have the best odds — and the lowest payoff.  And that’s why Animal Kingdom was such a great win at the Derby.  Odds like 20-1 suggest that the horse is not expected to win.  It’s the unusual event that every better wants in on.

But while it’s easy to see which horse is favored to win, it’s a little tougher to figure out the payoff.  Here’s a quick look at how it will work at the Preakness.  To make things simple, we’ll just consider a $2 bet on Animal Kingdom, who currently has 2-1 odds.  (Those odds will change as we get closer to the race.)

First multiply the amount of the bet by the first number in the odds ratio:

$2 x 2 = $4

Easy enough, right?  Now divide by the second number in the odds ratio:

$4 ÷ 1 = $4

And finally, add the amount you bet, and that’s your payoff:

$4 + $2 = $6

So if you place a $2 bet on Animal Kingdom with 2-1 odds — and he wins — you’ll get a $6 payoff.

Clearly, things get a little more complicated with different odds.  So let’s look at another example.  What if you wanted to place a $2 bet on Dialed In, another Preakness contender? This horse currently has 9-2 odds.

$2 x 9 = $18

$18 ÷ 2 = $9

$9 + $2 = $11

So placing the same $2 on a horse with 9-2 odds, means a bigger payoff ($11), if the horse wins.  How come?

Higher odds have lower payoffs.  A long-shot (like Animal Kingdom in the Kentucky Derby) has lower odds, so if they do win, the payoff can be pretty big.

Problem is, it’s not likely that a horse with low odds will win the race.  And that’s why Animal Kingdom’s win in the Derby was such a big deal.  Still, horses with higher odds have won the Kentucky Derby.  In 1913, Donerail won with 91.45-1 odds!

Of course, the more you bet, the more you’ll win — if your horse wins.  Take a look at a $150 bet on Dialed In at 9-2 odds:

$150 x 9 = $1,350

$1,350 ÷ 2 = $675

$675 + $150 = $825

Not a bad take. Still it’s a gamble, and that’s why plenty of people lose. To learn more, visit the Preakness website, which includes a great tutorial on betting the horses.

What do you think Animal Kingdom’s chances of winning the Preakness are?  Do we have another Triple Crown winner on our hands?  Give us your odds in the comment section.

Categories
Personal Finance Work

Is Your Boss Ripping You Off?

In last Friday’s Open Thread discussion, Gretchen posted this question:

My husband’s company does not provide health insurance for me and the kids, which is a $12,000 value. In his field, there is a salary scale based on education, number of years experience, geography, etc. The salary scale assumes that the employer provides health insurance for the family. His salary is currently at 79% of the scale, and his employer wants to eventually get him up to 100%. But that doesn’t include the insurance, so it won’t really be at 100% and is not now really at 79%. But I can’t figure out which way to do the math so he can show them the actual percentage. They’re saying he’s at 79 percent. I’m saying it’s lower because they aren’t accounting for that $12K.

All of that boils down to this: What percent of the salary scale is Gretchen’s husband actually making, given that he, and not his employer, pays the $12,000 bill for insurance? There are two steps to this problem:

1. Find the actual salary that is at 100% of the scale.

2. Find the actual percent of Gretchen’s husband’s salary, minus the cost of insurance.

I’m going to tell you up front that we’re going to use a proportion here.  What is  proportions?  A proportion is two equal ratios.  So, if you have two fractions with an equal sign between them, you have a proportion.

And how did I know to use a proportion?  Well, my big clue was that we’re working with percents.  Percent means “per one hundred,” and per one hundred means “out of one hundred,” which just means, “put the percent value over 100.” In other words:

79% = 79/100

The tricky part is figuring out what the proportions should be.

Step 1:

salary/x = 79/100,

where “salary” is Gretchen’s husband’s salary, and x is the top salary on the scale.

That’s because the company assumes that your husband’s salary is 79% of the scale. (Notice this: “salary” and “79″ are in the numerators — or top values of the fractions.)

To solve this proportion, we need to plug in Gretchen’s husband’s salary and then solve for x. In order to make this easy to explain, I’m going to assume that his salary is $100,000.

substitute:   {$100,000}/x = 79/100 cross multiply:   {$100,000*100} = 79x simplify:     {$10,000,000} = 79x solve for x:     $126,582 = x

So if his salary is $100,000, the top salary on the scale is $126,582.

Step 2:

{$100,000-12,000}/{126,582} = p/100,

where p is the actual percent of the scale.

Let’s look carefully at this proportion: The first ratio is just the salary minus the cost of insurance, over the max salary in the scale.  (That’s what we found in step 1.)  The second ratio is just like the second ratio in step 1, except that we don’t know what the percent is.

Now, pay close attention to this.  Check the top numbers to be sure they match. We want to know the actual percent of the scale that Gretchen’s husband is making — and that’s what’s represented in the top number of each ration.

Check the bottom numbers to be sure they match.  Do they?  Why yes!  Yes they do!  That’s because $126,582 is 100% of the salary scale.

(Unlike my 10-year-old daughter’s outfits, math is very matchy-matchy.  Knowing that will help you organize your problems and check to see if they’re set up properly.)

Now all we need to do is solve for p.

simplify:    {$88,000}/{126,582} = p/100 cross multiply:     {$88,000*100} = {126,582p} simplify:       {$8,800,000} = {126,582p} solve for p:       69.5 = p

So what does this mean? If Gretchen’s husband makes $100,000 a year and is paying $12,000 for insurance, he’s earning 69.6% of the salary scale.

If you made it this far, you get a gold star!  Pat yourself on the back, and take the rest of the day off.  This is a complex problem that depends on an understanding of proportions and how to solve for a variable in an algebraic equation.

Never fear!  I’ll unravel some of these mysteries in later blog posts.  And of course, if you have a question, ask it in the comments section!

Categories
Math for Grownups

Twitter math

OMGoodness!  Two posts within the hour!

I can’t resist sharing this terrific video.  If you’re as addicted to Twitter as I am — or as I was two weeks ago, just not getting it — take a look.  This guy is funny, and, wow, can he draw!

Categories
Home Math for Grownups Work

Math at the Permit Office (or Contractors Ain’t No Dummies)

If you’ve ever been to your city’s or county’s permit office, you can probably imagine how frustrated I was yesterday at around noon.  All I wanted was a demolition and construction permit for our newest renovation project.

Image courtesy of Ross Crawford

In my floral skirt and gold flip-flops (I had painted my toenails an hour earlier), I felt just a teensy bit out of place, among the blue-jeaned, unshaven contractors, who brandished rolled up blueprints and wore cell phones and tape measures clipped to their belts.. Still, I had my hand-sketched scale drawing and photos of the house as it looks now. As long as I could get the form filled out correctly, I was good to go.

But this was my second trip downtown in search of approval for our reno plans, and I was determined to get out of there with a permit this time.  That meant I was prepared to stay all afternoon — and go up to the counter as many times as I was asked to do.

This dude was in trouble. No scale drawing and no clue how to make one.

On my sixth visit to the counter — after completing the form three different times and calling my contractor once to clarify some measurements–I realized that I was just about home free.  The attendant asked me to add some notes to my paperwork, while she helped the next person in line.

This dude was in trouble.  No scale drawing and no clue how to make one.  The attendant gave him a quick lesson, along with a blank piece of paper and a scale ruler.  But it was clear that this guy was in for a long, long afternoon.

So, what is a scale drawing, and why is it important?

A scale drawing shows an object to scale. (Duh, right?) In other words, all of the measurements in a scale drawing are proportionate to the measurements of the actual object.  But making a scale drawing doesn’t burn up too many brain cells.  That’s because of three simple tools.

  1. Graph paper.  Each square on a piece of graph paper is 1/4 inches wide and tall.  So, if you define your scale as 1/4 inch = 1 foot, 10′ will be 10 boxes .
  2. Scale rulers: These are great if you don’t have graph paper, and you can use the same scale: 1/4 inch = 1 foot.  (See the picture above.)
  3. Computer programs: These translate your measurements into scale drawings for you.  But if you’re like me, it’s hard to visualize how to input the correct measurements.  I prefer to just make a drawing by hand.

Scale drawings are useful in lots of situations, but I’ve found them most helpful in home improvements and gardening.  (A quick sketch of my flowerbeds keeps me from overcrowding my begonias.)  And apparently, the city permits office wants to see them, too!

(Wondering if I got my permit? I can proudly say, yes!  And I know for a fact that my scale drawing helped.)

What math have you used in home improvements?  Is there a time when math got in the way of a home improvement project?

Categories
Math for Grownups

Open Thread Friday: What’s Your Math Question?

So when does math make you crazy in your everyday life?  Are there situations that make your hands sweat?

On selected Fridays, I’ll host an open thread where you can ask your questions or share your specific frustrations with everyday math.  And if you see a question from someone else that you can answer? Go for it!  I’ll select questions for future posts here at Math For Grownups.  In those posts, I’ll show you easy ways to get around these frustrations.

So let ‘er rip in the comments section.  Ask about fractions or grocery store math or the best way to place a bet on the ponies. Just remember that we’re not here to do your homework for you.  And leave your calculus, trig and diff eq questions for another blog.

Whatcha got?

Categories
Math for Grownups Math for Parents Math for Teachers Math for Writers

Math Secret #1: There’s More than One Way to Skin a Math Problem

The more I talk to people about math, the more I hear this refrain: “I don’t like math, because math problems have only one answer.”

Peshaw!

Okay, so it’s not such a crazy idea.  Most math problems do have one answer (as long as we agree with some basic premises, like that we’re working in base ten).  But math can be a very creative pursuit — and I’m not talking about knot theory or fractals or any of those other advanced math concepts.

I have a friend who is crazy good at doing mental math.  She can split the bill at a table of 15 — even when each person had a completely different meal and everyone shared four appetizers — without a calculator, smart phone or pencil and paper!  This amazed me, so I asked her how she does it.  And what I discovered was pretty surprising. She approaches these simple arithmetic problems in ways that I never would have thought of.  She subtracts to solve addition problems, divides to multiply.  And estimation? Boy howdy, does the girl estimate.  In other words, she gets creative.

(She also has a pretty darned good understanding of how numbers work together, which is probably the biggest reason she can accomplish these feats of restaurant arithmetic.)

While there may be one absolutely, without-a-doubt, perfectly correct answer to “How much do I owe the waiter?” there are dozens of ways to get to that answer.  Problem is, your fourth grade math teacher probably didn’t want to hear about your creative approach.

See, when we learn math as kids, we’re focused on computation through algorithms.  (In case you’re not familiar with the word, algorithms are step-by-step procedures designed to get you to the answer.)  You did drill after drill of multiplication, long division, finding the LCM (Least Common Multiple) and converting percents to fractions.  But nobody ever asked you, “How would you do it in your head?”

The good news is that now you’re all grown up.  There’s not a single teacher who is looking over your shoulder to see if you lined up your decimal points and carried the 2.  You can chart your own path!  And when people are given this freedom, they often find really interesting ways to solve problems.

Don’t believe me?  Try this out: Add 73 and 38 in your head.  How did you do it?  Now pose the question to someone else.  Did they do something different?  If not, ask someone else.  I will guarantee that among your friends and family, you’ll find at least three different ways of approaching this addition problem.

So, let’s do this experiment here.  In the comments section, post how you solved 73 + 38 without a calculator or paper and pencil.  Then come back later to see if someone else had a different approach.  If you’re feeling really bold, post this question as your Facebook status, then report the results in the comments section.

And while you’re at Facebook, be sure to visit and like the Math For Grownups Facebook fan page!

Categories
Math for Grownups Math for Parents Math for Teachers Math for Writers

My Math Story

The biggest fights my father and I had were about math.  I kid you not.

The year was 1984.  I was a junior in high school, taking Algebra II.  Radicals were kicking my scrawny, little butt.

(Remember radicals?  They look like this: sqrt{24}. In Algebra II, you mostly learned to simplify them, as well as add, subtract, multiply and divide with them.)

My father wanted to help, and he had the patience of Job.  But he was not great at accepting that I didn’t understand.  And I wasn’t great at controlling my emotions.  I hollered, cried and probably threw things.  Somehow, I got the impression that my dad thought I couldn’t do math, and I did what any strong-willed girl will: I dug in my heels.

That’s when I started drinking coffee, actually.  I was so determined to show my dad–and my Algebra II teacher, Mr. Gardner–that I got up at 4:30 a.m., sat in my dad’s easy chair with a cup of coffee and a stack of sharpened pencils, and did problem after problem after problem.

I did every single radicals problem in the textbook.  And then I did them again. I took what Mr. Gardner and my dad taught me and figured the darned things out.  It took time, but I was determined not to give up.

Why on earth would I do this?  Well, I’m stubborn, for one.  But probably the biggest reason is Mrs. Ivey.  She was my geometry teacher the year before, and she changed my perspective about math.  You see, before then, I knew I couldn’t do math.  Mrs. Ivey convinced me that I was wrong.

She and my father are the reasons I majored in math.  I found out I’m a math teacher, not a mathematician. (Sometimes we’re one or the other.)  I’m fascinated by the ways people choose to do math, not by complex computations or proofs.

Math geeks aren’t always born.  Sometimes a teacher inspires us.  Sometimes we’re dragged kicking and screaming. And sometimes we just learn to deal with math–because we have to.

What’s your math story? Share it in the comments section!