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Warm-up: Monty Hall • Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door (it remains unopened) and the host, who knows what's behind the doors, opens a door with a goat. He then gives you the option of switching to the other unopened door. Is it to your advantage to switch doors?
Ratios And Egyptian Fractions!
Time, Speed, and Distance • The basic formula for these types of problems is the following:distance = speed * time • Most problems of this type are based off of this principle. Of course, not all are as simple.
Example 1: It’s actually easy! • Arthur drives for one hour at 60 mph and then drives one hour at 40 mph. What is his average speed for the entire trip?
Example 2: Not the same as 1 • Becky drives the first half of a 100 mile trip at 60 mph and then drives the second half at 40 mph. What is her average speed for the entire trip? • Hint: Speed = distance/time
Example 3: Lots of variables • Andy’s lawn has twice as much area as Beth’s lawn and three times as much area as Carlos’ lawn. Carlos’ lawn mower cuts half as fast as Beth’s mower and one third as fast as Andy’s mower. They all start to mow their lawns at the same time. Who will finish first?
Example 4: Equations, equations • Henry rows a boat down a river at constant speed from point A to point B in 15 minutes. He rows back upstream at the same speed from point B to point A in 20 minutes. How long would it take a stick floating on the water to travel from point A to point B?
Of Interest: Egyptian Fractions • An Egyptian fraction is the sum of distinct unit fractions. • Interestingly, every positive rational number can be expressed by an Egyptian fraction.
Some Egyptian Fractions • One type of Egyptian fraction: • The general case:
Example 5: A little algebra • Prove the general case of splitting an Egyptian fraction. • Use the general case to write as the sum of two unit fractions.
More Egyptian Fractions • So far we’ve looked at fractions with numerator 2 and odd denominator. • What about fractions with numerator 1? • Obviously, they can be written as the sum of two equal unit fractions. • The general case for these follows. Of course, since the fractions are not distinct this is not an Egyptian fraction…yet.
Even More Egyptian Fractions • Unit fractions can also be written as the sum of two distinct unit fractions. • Write and prove the general case in terms of n.
Now, about Ratios • Sometimes there are problems that give you relationships between certain values. Are they inversely proportional or directly proportional? • Sometimes the hardest part about these problems is writing equations.
Example 6: Chickens lay eggs • It takes 6 chickens 10 days to lay 20 eggs. How many chickens would it take to lay 70 eggs in 3 days?
Example 7: We’ll start with acid • If two liters of a 20% acid solution are mixed with 8 liters of a 50% acid solution, what is the concentration of the resulting solution?
Example 8: Concentrate… • Lori likes extra-strength fruit punch that is 120% the strength of normal fruit punch. Tom likes weak fruit punch that is 75% the strength of normal fruit punch. How many milliliters of water should Tom add to three liters of Lori’s punch to produce his desired punch strength?
