Wednesday, August 17, 2011

What can Batting Averages tell us?

It's the bottom of the 9th, 2 outs, bases loaded in the 7th game of the World Series. On the mound is the opposing team's left-handed pitcher trying to close out the game. As the Head Coach you have a decision to make: let your left-handed 9th batter hitting .270 for the season go up and take his hacks, or pinch hit with your young, recently called-up rookie batting .350?
The first question we need to answer before making a decision is: What do the batting average numbers mean?

Batting averages are a simple decimal that approximates the number of hits per at-bat, or more simply the probability that a batter reached first base on a hit during his previous at-bats. The equation used to calculate batting average is simple: # Hits/# At-Bats.

A batting average is written in decimal form using 3 digits after the decimal point. Avid baseball readers read these as large numbers, so .400 would be read as "four-hundred" and .283 would be read as "two eighty-three." Each individual thousandth is called a "point," so .400 would be considered 117 points higher than .283.

But not all batting averages can be read equally. Two players can have the same batting average, take .300 for example, and have very different statistics. Player 1 could have 3 hits in 10 at-bats while player 2 may have 120 hits in 400 at-bats.

So which is a more accurate description of a player's ability? Let's take a look at what happens to the players after their next at-bat.
If they were to both get a hit in the next at-bat, their averages would indicate that Player 1 is much more likely to get a hit, yet if they both made an out the numbers would swing heavily in favor of Player 2.

The key to this discrepancy lies in the number of total at-bats. With more at-bats, the denominator for the fraction becomes larger and is less affected by adding 0 or 1 to the numerator. Referring to the chart, the next at-bat for Player 1 will either increase his average by 64 points or decrease it by 27. Player 2 will see either a 2 point increase or a 1 point decrease. So batting averages are less affected with larger numbers of at-bats, and can more accurately describe a hitter's tendency over a period of time.

Now, looking back to the original question, I will add more context to the problem. In an average 162-game season a player might amass about 450 at-bats, and back-ups could see 100 at-bats. Rookies and recent call-ups (players invited to the major-league team from the minor leagues) will usually be on the team for the final 50 games of the season.

Knowing this information and having seen the chart from above, does this change your original decision for what to do? Why or why not? There is no definitive correct answer to this question, but I do ask that you use numbers to support your reasoning. Please post your decisions in the comments.

Sunday, August 14, 2011

How does GoogleMaps know how long it takes to drive somewhere?

(Picture courtesy of

Have you ever noticed that when you look up directions on GoogleMaps that along with possible routes it lists the estimated time it will take to reach the destination? Well, this is yet another example of how math from the classroom appears again in our daily lives.

We'll have to leave the topic of how it determines possible routes for another blog post, but for now let's look at how it determines how long a given route will take.

The simple formula that the site uses to calculate time uses the distance on each road divided by the approximate speed of travel on each road (speed limit). In equation form we can represent this as (Miles Traveled) /(Miles per Hour) = Hours.

However, this calculation is not the total distance divided by the average speed limit because you may be on a road with a faster speed limit for a longer distance. In the "Driving Directions" list in the left column individual distances on each road are listed step-by-step. For each step GoogleMaps will use the equation above and divide the distance traveled by the speed limit to get the estimated time. So if you drive 10 miles on a road at 60 miles per hour, you would drive for 1/6th of an hour, or 10 minutes. GoogleMaps adds all of these individual distances together to give you an estimated total travel time.

The reason it is estimated is because there are many other variables to take into account that are all random. It is nearly impossible to take into account stop signs or stop lights where no distance is being covered and time is passing. Think about when you are at a stop light - sometimes the light is green and you pass straight through, and other times it is red and you will wait for several minutes.

GoogleMaps does take traffic into account by adjusting the "expected" speed of travel to closer mirror what the driver will experience. For example: in traffic a highway that is normally a 65 mph zone would be estimated at 35 mph to accommodate for the additional traffic. So if you look up directions you may find that the website give two estimates for travel time: one based off of our formula above, and the other that takes traffic into account.

Will the time estimation be more accurate for shorter drives or longer drives? Why do you think that will be the case? Please post your comments below.

Thursday, August 11, 2011

The $7.11 Problem

This problem comes from Jim Wilson's math site from the University of Georgia (, and it is very interesting to analyze.
* Problem from Professor Doug Brumbaugh, University of Central Florida.

A guy walks into a 7-11 store and selects four items to buy. The clerk at the counter informs the gentleman that the total cost of the four items is $7.11. He was completely surprised that the cost was the same as the name of the store. The clerk informed the man that he simply multiplied the cost of each item and arrived at the total. The customer calmly informed the clerk that the items should be added and not multiplied. The clerk then added the items together and informed the customer that the total was still exactly $7.11.

What are the exact costs of each item?
So where should I begin to start solving the problem?

In looking for the 4 numbers that add and multiply to $7.11 I find that it is easy to come up with four numbers whose sum is correct, and it is much more difficult to think about numbers whose product is correct.

Thus, do find numbers that will multiply my first instinct is to factor the number. When I factored 7.11 I found that 3, 3, and 0.79 divide into it evenly. But 0.79 + 3.00 + 3.00 + 1.00 = 7.79. So I will have to manipulate the factors to accommodate for the decimals.

Decimals are nasty to work with, so I will convert everything into pennies. My sum is then multiplied by 100 to give me 711, and the product is multiplied by 100^4 to give me 711,000,000. This can be seen in the equations below:
100a + 100b + 100c + 100d = 100 (a + b + c + d) = 100 (7.11) = 711
100a * 100b * 100c * 100d = (100^4)(a * b * c * d) = (100^4)(7.11) = 711,000,000
Now when I factor 711,000,000 I can see that I am left with 79, 3, 3, 2, 2, 2, 2, 2, 2, 5, 5, 5, 5, 5, and 5. So now it's a matter of rearranging these factors so that their four products sum to 711.

I begin with 79 since it is the largest and will yield the fewest combinations. I can determine that one of my prices is a multiple of 79 cents as seen in the chart. I did not include 79 * 3 * 3 = 7.11 because then the other prices would be zero and the multiplication would fail.

From here forward the method to solve is based on combinatorics (combinations). To narrow my permutations, I can first use some logic.

At least one of the remaining numbers ends in something other than a 0 because none of the above end in 1. Thus, in order for the ones digit of our sum to be a 1, we need the digits to add to 1, 11, 21, 31. This means that at least one number will not contain a factor of 5.

For numbers 79, 158, 237, 474, and 632, this means that the six 5's would be split among two prices which would total at least 125 cents each. So I can rule out 474 and 632 as possible values because 125 + 125 = 250 and 474 +250 > 711 as well as 632 + 250 > 711.

It is at this point that I notice a pattern that gives me a hunch. My guess is that the value will be either 3.16 or 3.95. Why? Because these numbers allow for the six factors of 5 to be spread among three prices instead of two.

I begin with 3.16 so that I can examine the prices who all have factors of 5. So for my bases I have 316; 5*2; 5*2; and 5. This leaves me with the remaining factors of 3, 3, 2, 2, 5, 5, and 5. With addition limiting the maximum and minimum sizes of the prices, I can quickly come to find that the four prices are $3.16, $1.20, $1.50, and $1.25.

Share some of your strategies (successful or not) in the comments below. Are there any other solutions to be found? How about factors that aren't constrained to integer values for pennies?

Wednesday, August 10, 2011

Why are Manhole Covers Round?

Have you ever noticed that every Manhole cover you see is round? Have you ever wondered how engineers came to this decision? Did you realize that the reasoning comes down to the Geometry of Shapes?
That's right: Manhole Covers are round because Circles are the only shapes that cannot fall through themselves.

Let's examine some of the properties of shapes to see why this is true.

First, it's important to recognize that the shapes will only fall through themselves when they are rotated to be vertical. If they were lying flat then they would be covering the hole successfully!

Next, let's examine shapes with unequal sides beginning with a triangle. If the triangle had side lengths of 3, 3, and 2 feet then we could rotate it so that the side with length 2 feet was parallel to the ground. This would mean that the width of the manhole cover would now be 2 feet, with the width of the opening approaching 3 along the sides of the hole. Since 3 is greater than 2, the cover would be able to fall through the opening. Similarly for a 4-sided figure if one side is shorter or longer than the others then we will find the same result.

There is a pattern to notice from examining the previous shapes. The shortest width of each shape is compared to the longest length. But what if the covers were regular shapes with all sides and angles being the same?

Let's look at a Square with length of 1 foot on each . The shortest width of a square will be the length of a side. But how about the longest length? That can be found by measuring from the upper left-hand corner to the lower-right hand corner.

Using Pythagorean's Theorem, we know that 1^2 + 1^2 = diagonal^2. So the diagonal is square root of 2, or approximately 1.414. This means that this square could still fall through itself.

So now our focus can shift to finding a shape whose longest length and shortest width are the same. But this would mean that we would need all widths and lengths to be the same because we cannot have the longest length be shorter than the shortest width. This leads us to a circle.

A circle will always have the width of its diameter no matter which way it is rotated, so this will be the shortest width. But the longest length will also be the diameter as well because any chord will be shorter than the diameter. So the circle is unique to all polygons and shapes in that it can never fall through itself.

Want to test geometry with physical objects? Since manhole covers are heavy you should explore this idea using Tupperware containers. See how many different lid shapes will fall into the container, and how many will not - then post your comments!

Monday, August 8, 2011

Will I ever be able to Time Travel?

The faster that I travel, the less time it takes for me to arrive someplace. So when I move faster I can subtract time. So how fast will I have to go in order to arrive before I leave (i.e. When will time become a negative value)?
This question is the driving force behind the creation of Everyday Explanations. While on a bus ride home after middle school one day, my friend Kyle and I began debating each other as to whether or not this was possible.

I can't remember the conclusion we came to at the time, but I do remember learning our answer during a High School Math Class. Here's how we came to the conclusion that time traveling is impossible by simply moving faster:

Let's take a distance of 100 feet and create a chart for how long it takes to arrive based on speed of travel.

Looking at the chart we can see that as I move faster my time decreases, but at a rate proportional to my speed increase. For example, when I move at 20 ft/s it takes me 5 seconds to move 100 feet, while at 40 ft/s it will only take me 2.5 seconds. But when I move at 60 ft/s I will not arrive in 0 seconds, instead it takes me 1.67 seconds.

Notice that each column's values will multiply to a product of 100. This makes sense because we would multiply (ft/s)*s = ft, and the distance is 100 feet. So in order for time to be a negative number, we would need to have negative speed. This means that no matter how fast we

We can actually represent this truth graphically as well. Take a look at the graph of the charted points and you will notice that the time never actually reaches zero. Instead it just "flattens out" This means that no matter how fast I can move I will still always have a positive time and I will never be able to arrive before I leave.

The graph shows the concept of a "limit," which are critical to the study of Calculus. Limit means that the graph will continue to get closer to a point without ever actually reaching it. In this instance, the Limit for time as speed increases forever will be 0. Times will get smaller and smaller, but will never actually reach 0.

Using the equation from above, do your best to explain why that makes sense in the comments below.

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