Now that you have had the opportunity to enjoy a well-earned day's rest, let's switch the format of our meeting. I offer some solutions before I pose new problems. Variety is an intellectual stimulant (as it is a leading cause for divorce in the US...but then, that's in a different context). Please refer to the last two posts on this blog, as it's generally a good policy to know the questions tha the answers relate to!
Question: What would the amount and the dilution be of hyroxyidiotic acid be in bottle #3 at the end of this process? This is a question that can be answered intuitively, without much math at all (phew). In Question #3, we were left with 1 gallon of 20% hydroxyidiotic acid in bottle #3. In Question #4, we added 1 gallon of 10% of hydroxyidiotic acid to bottle #3. In sum, bottle #3 now contains 1 gallon of 20% acid, and 1 gallon of 10% acid, or 2 gallons of 15% hydroxyidiotic acid. If we were to have done the math, we might have used this calculation: (.50) (.20) +(.50) (.10) = .15 , or 15%.
How many 5-letter combinations can you make of the name OBAMA, where none of those combinations started with an "O"? The computation is simple (but with a twist): for the first letter slot you have 4 possible letters; for the second, 4; for the third, 3: for the fourth, 2; and for the fifth, 1 --- by multiplying 4x4x3x2x1, we get 96 arrangements of letters that do not begin with the letter "O". But what about the arrangements that are redundant because they have two "A"s in them....Hmmm....perhaps you need a bit more time.
Since every arrangement of these letters will contain two letter "A"s, and it doesn't matter which A is being used, half of our arrangements will be redundant. What we must then do is eliminate half of the arrangements in order to eliminate these redundancies. If we divide 96 in half, the result will be 48 arrangements, which is our answer.
Now for a second surprise...there is no problem for today. There is merely a wonderful exercise. Take at least 30 minutes at some time time today, and sample some of the audio/video clips below for mind expansion purposes. Scroll down to EXERCISE YOUR MIND 6: MEDITATION, STIMULATION AND RELAXATION. Enjoy the experience!