Still June, still test-free, right? Works for me. Instead, let's look at how some summer problems can help you prepare for the Math sections on The-Tests-That-Must-Not-Be Named.
I talked a little last week about real world math and every math student's eventual question: “When am I ever going to use this?” Unless you plan to live under a rock for the rest of your life, you will use math. At college alone you’ll use it to figure out the shortest route to class, to find the lowest cost textbooks, to determine the maximum number of classes you can miss and still pass, and to calculate the tip on the tab on your 21st birthday. And chances are, even if you do live under a rock, you’re still going to have to use math to figure out how to eat and drink and breathe while living with said rock pressing down on you. So here are some real world dilemmas you might encounter this summer to get you ready for math that you’ll definitely encounter this fall on The-Tests-That-Must-Not-Be-Named.
It’s time for a road trip! You and four friends (Abby, Barrett, Cayden, and Dana) are going to the city for the weekend in late June. Dana’s mom has agreed to let you all take her new convertible on one condition: only Dana and you are allowed to drive because Abby, Barrett, and Cayden are totally irresponsible. Just before you leave, Abby throws a fit and insists on sitting shotgun because her hair will get too messy if she sits in the backseat. Barrett jokes that she’s afraid her extensions will come out, which starts a 30 minute argument, and only ends when everyone agrees Abby can have the front passenger seat. That leaves three seats in the back of the bright red topless car.
Given all of these conditions, how many different ways can the five of you arrange yourselves in the five seats of the car?
Ah, a day at the lake! You’ve invited Barrett and Abby over to watch 4th of July fireworks from your dock, but they need a ride by boat. Abby lives 4 miles directly north of your lake cottage, Barrett lives 3 miles due west, and you can travel in a straight line from Abby's to Barrett's dock. The fireworks start in 55 minutes, and your boat can travel 1 mile in 5 minutes.
Can you pick up both friends and make it back to your dock in time for the start of the fireworks?
You’ve agreed to babysit for twins, Zoe and Zach, on a Monday night in mid-July because Barrett and Dana are at basketball camp and Abby and Cayden are seeing a movie. It’s going well (despite the fact that you’re wearing a tutu, superhero cape, and cowboy hat), when a summer thunderstorm rolls in and knocks out the power. The house is completely dark except for the flashlight on your cell phone, which comforts the toddlers as they huddle with you on the couch. Suddenly Zach starts screaming, “I WANT AN ORANGE POPSICLE!” and Zoe chimes in with “I WANT A PURPLE POPSICLE!” You grab your flashlight/phone to go get popsicles out of the kitchen freezer, but Zoe starts yelling, “I WANT THE LIGHT! I WANT THE LIGHT!” Sigh. You hand the children the flashlight and wander into the dark kitchen alone.
You find the brand new popsicle box, reach inside, and stop. How on earth do you pick out an orange and purple popsicle in the dark? There are 12 popsicles in the box: 4 red ones, 4 purple ones, and 4 orange ones. You blindly choose two popsicles and take them back to the flashlight to discover their color. What is the probability that one is orange AND the other is purple?
Abby, Barrett, Cayden, and you have August birthdays and you decide to throw an end-of-summer party together at your lake house. You each invite 100 different guests for a total of 400 invitees, so you’re surprised when your turnout is much smaller. Each of you decide to walk through the party and count the number of guests there that you personally invited. When you meet back up, you are happy to announce that 66 of the 100 people you invited are present. But Abby, Barrett, and Cayden should be in Summer School, because rather than count heads, they are only able to tell you the fraction of the people there that they invited: of the total attendees, 1/6 were invited by Abby, 1/8 were invited by Barrett, and 1/4 were invited by Cayden.
Can you figure out how many people were in attendance without going out and counting them all over again?
Car Combinations: 12
This is simply a Permutation question with a few specific restrictions. To solve, begin at the points of restriction and consider the limited options for those positions, and then work your way through the remaining seats remembering to reduce your options as people are seated. Start with the driver, then shotgun, then move to the three back seats: there are 2 possibilities for driver (You and Dana), 1 possibility for shotgun (Abby), 3 possibilities for backseat right (assume Dana drives; so back right is you, Barrett, or Cayden), 2 possibilities for backseat middle (say you take the back right; that leaves Barrett and Cayden for back middle), and 1 possibility for backseat left (Cayden, because Barrett took the middle). 2 x 1 x 3 x 2 x 1 = 12
Note that it doesn’t matter exactly who takes each seat, provided you meet any restrictions given, and reduce the selection group according as you move from position to position.
Lake Lineup: No
The boat must travel a right triangle to pick up Barrett and Abby. The legs of the triangle are 3 and 4, so the hypotenuse is 5 (you can solve this with the Pythagorean Theorem or memorize the 3:4:5 Pythagorean Triple). If the boat travels 1 mile every 5 minutes, it takes 20 minutes to go north 4 miles to Abby’s house, 25 minutes to travel the 5 mile hypotenuse to Barrett’s, and 15 minutes to travel the 3 miles from Barrett’s house to your house. 20 + 25 + 15 = 60 minutes
Dark Dilemma: 8/33
There are two possible selection orders that would satisfy the requirement of a single purple and a single orange popsicle: Orange then Purple, or Purple then Orange. Let’s determine the odds of each sequence:
The probability of selecting an orange popsicle with your first choice is 4/12 (4 orange popsicles out of 12 total in the box) or 1/3. The probability of then selecting a purple popsicle are 4/11 (4 purple popsicles out of 11 total popsicles left in the box; remember, you removed an orange already). Total odds of this sequence: 1/3 x 4/11 = 4/33.
Similarly, the probability of selecting a purple popsicle with your first choice is 4/12 (4 purple popsicles out of 12 total in the box) or 1/3. The probability of then selecting an orange popsicle is 4/11 (4 orange popsicles out of 11 total popsicles left in the box; remember, you removed a purple already in this scenario). Total odds of this sequence: 1/3 x 4/11 = 4/33.
Since either of those two sequences would give us what we want—one orange and one purple—we need to find the total odds of both. To do this, just add them together: 4/33 + 4/33 = 8/33.
Party Poll: 144
t = total
Your guests + Abby’s guests + Barrett’s guests + Cayden’s guests = total guests
66 + (1/6)t + (1/8)t + (1/4)t = t
(1/6)t + (1/8)t + (1/4)t = t – 66
(4/24)t + (3/24)t +(6/24)t = t – 66
(13/24)t = t – 66
13t = (t – 66)24
13t = 24t – 1584
-11t = -1584
t = 144
Photo: "7-7-07 336" courtesy of Chris