Math Without Witches
December 30, 2003: Welcome, New Visitors
The math page of my Web site will be for non-experts (like me) who would like to be experts. Don't worry if you don't know what the above equation means; I don't know either. I just thought it looked cool.I assume some knowledge of elementary algebra, but nothing beyond that. We'll improve together in the coming months. Most of the new material, however, will be at Patrick and Debbie's Think Tank.
Exponents
To multiply two or more terms that have exponents and the same base--e.g., a5 x a4, add the exponents and keep the same base. Here the answer is a9. To divide two or more terms that have exponents and the same base, subtract the exponents and keep the same base--e.g., a7 / a3 = a4.Mathematical Expectation (from Brain Bombardment)
When you play your lottery numbers, where will you get ripped off the least in the long run? Let's begin with the daily numbers of my State lottery--the New York Numbers and the Win 4--commonly referred to as the "three-way" and the "four-way" numbers. Suppose I place a $1.00 bet that tonight's three-digit number will be 623 (my birthday is June 23). I am betting, with that same dollar, that those three digits will be in the sequence 6-2-3; i.e., what the lottery players refer to as a "straight bet." My chance of a win is 1 out of 1,000, which is 0.1%. In New York State, the payout from a $1.00 straight bet is $500. If I had all 1,000 numbers covered (000 to 999), I would get back $500 ($0.50 for every dollar spent).There is another way of looking at it (truer to real life): If you bet the same amount of money every day on that number for years and years, you should win $500 an average of 0.1% of the time. The "mathematical expectation" is still an average of 50 cents back for every dollar gambled. Some players take this matter into account by purchasing more than one ticket when their number is overdue. The odds that the number will come up that day are the same, but people are sometimes lucky.
Mixture Problems:
Pretend that I am stretching my premium coffee by mixing it with the cheap coffee. I have 3 pounds of the premium coffee and 1.5 pounds of the cheap coffee. The premium coffee costs $3.00 per pound more than the cheap coffee. Mom does not have her sales slip, but she remembers paying $27.00 for all the coffee together. How much per pound did Mom pay for the cheap coffee?Answer: Usually the hardest part of these word problems is deciding what x represents. Here let x represent the price (in dollars) per pound of the regular coffee. Then x + 3 will represent the price (in dollars) per pound of the premium coffee. Since we know how many pounds we have of each brand, we can set up the equation:
3x + 1.5(x+3) = 27
3x + 1.5x + 4.5 = 27
4.5x + 4.5 = 27
4.5x = 22.5
x = 5
Mixture Problem No. 2: Given the unit price of two different items, plus weight of final mixture, determine the proportion of the items in the mixture. Imagine that I am working at the coffee shop. One brand sells for $8.00 per pound; another brand sells for $12.00 per pound. In what proportion should I mix the two brands to make 24 pounds of a blend selling for $9.00 per pound? My blend would be as good a deal (no better, no worse) as 24 pounds of the same brands unmixed.
This one was tough. I was getting nowhere until I calculated the total price of the 24-pound blend. 24 lbs. x $9.00 per pound = $216.00. Then I realized that there was other information I could work with: 1) 24 lbs. x $8.00 per pound (cheaper coffee) = $192.00. That’s a $24.00 difference (not to be confused with the 24 lbs.).
Finally I saw the trick. I take a full 24-pound bag of the $8.00 coffee and dump the proper amount out of it. Then I replace it with an identical amount of $12.00 coffee. I arrive at the proper amount by dividing: $24.00 (value to be increased) ÷ $4.00 (price difference per pound) = 6 pounds to be increased. I must therefore remove 6 pounds of the $8.00 coffee from the bag.
The 24-pound bag of blended coffee will consist of 18 pounds of the $8.00 coffee and 6 pounds of the $12.00 coffee. Let’s check: 18 ∙ 8 = 144; 12 ∙ 6 = 72; 144 + 72 = 216.
Signed Numbers:
Simplify:3x - [3x - (-x -{2-x})]
In programming and in mathematics, the trick to working with nested parentheses is to start on the inside and work your way out. The innermost expression is (2-x). We cannot, however, just get rid of the surrounding parentheses indiscriminately. It looks as if there is no number between the minus sign and the (2. Actually there is; it's just invisible because it's a 1. So we really just have to multiply 2 x -1, resulting in a -2. In multiplying the -(1) times -x, remember that a negative times a negative makes a positive.
Now we have 3x - [3x - (-x - 2 +x)]. We want to clear out the inner parentheses. That will give us the following:
3x - [3x +x +2 -x]. Remember that when we multiply two negatives, the result is positive.
Next: 3x - 3x -x -2 +x.
Finally: -2.
When we multiplied by using the minus sign, we were really multiplying by -1.
Work Problem (Different Rates of Speed):
Here is a problem from Brain Bombardment:The multipurpose room needs to be completely decorated for a family activity. Amy can decorate it in four hours if she is working by herself. Barbara is fast; she can do it in three hours by herself. Cathy is the slowpoke; she needs six hours. How long will it take for the room to be decorated if all three persons decorate the room together? Assume that all three persons begin at the same time, are working at their usual rates of speed, and are not getting in one another's way.
Answer: The trick involves figuring out how much of the room is decorated after the first hour. If Amy finishes decorating in four hours, she has finished 1/4 of the room after the first hour. Barbara, who needs three hours, has finished 1/3 of the room after the first hour. Cathy needs six hours, so she has decorated 1/6 of the room after the first hour. To add these fractions, we use the lowest common denominator; i.e., the lowest number into which 4, 3, and 6 can be divided. That number is 12.
1/4 = 3/12, 1/3 = 4/12, and 1/6 = 2/12. 3/12 + 4/12 + 2/12 = 9/12. 9/12 is the same as 3/4. We now know that 3/4 of the room is decorated after the first hour.
Now divide 1 by 3/4, which is really the same thing as flipping the top and bottom numbers of the fraction and then multiplying. In other words 1 ÷ 3/4 is the same as 1 x 4/3, or 4/3, or 1 1/3. 1 1/3 hours = 1 hour and 20 minutes, answer.
I'm still exploring this one: selfgrowth.com. There is quite a bit of information on overall mental development. I've also liked what I'll call their "sister site": Self-Growth Newsletter.
And for people feeling inspired:
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