Bosco

   A speech class has $4$ freshmen and $3$ sophomores. (Everyone is distinguishable.)     

   In how many ways can they stand in line, so that all of the freshmen are standing next to each other?     

   Mentally strap the freshmen together, temporarily.     

   Now you have 4 "things" to arrange (3 sophs and 1 group).     

                  4P4  =  24 arrangements with the freshmen grouped     

   But wait.  You can group the freshmen     

                  4P4  =  24 ways to arrange only the group of freshmen     

   So, for each arrangement of the entire group, there    

   can be 24 different arrangements of the freshmen.     ==>>     24 • 24 = 576 ways to stand in line     

   _.     

   I drove to the beach at a rate of $40$ miles per hour.     

   If I had driven at a rate of $55$ miles per hour instead, then

   I would have arrived $25$ minutes earlier. How many miles did I drive?     

   Distance equals Rate times Time               D_@40  =  (40)(T)     

                                                                       D_@55  =  (55)(T – 25 / 60)     

   distance is the same both instances        (40)(T)  =  (55)(T – 0.4167)     

                                                                         40T  =  55T – 22.9167)     

                                                                      – 15T  =  – 22.9167     

                                                                             T  =  1.5278     

   D = R • T                                                     D_@40  =  (40)(1.5278)     

                                                                       D_@40  =  61.1111 miles     

   validate by comparing D_@55                        D_@55  =  (55)(1.5278 – 0.4167)     

                                                                       D_@55  =  (55)(1.1111)     

                                                                       D_@55  =  61.1105     close enough considering rounding     

   _.     

   I drove to the beach at a rate of 40 miles per hour.     

   If I had driven at a rate of 50 miles per hour instead, then     

   I would have arrived 45 minutes later earlier. How many miles did I drive?     

   Distance = Rate x Time               D₄₀  =  (40) (T)     

                                                        D₅₀  =  (50) (T – 0.75)    

    the distances are the same         40T =  50T – 37.5     

                                                        10T  =  37.5     

                                                            T  =  3.75 hr     

   D = RT            D₄₀  =  (40)(3.75)     

                                  =  150 miles     

   check answer     

   Does D₅₀ = D₄₀ ?    

            D₅₀  =  (50)(3.75 – 0.75)     

                   =  (50)(3)     

                   =  150  True     

   _.      

   All members of our painting team paint at the same rate.     

   If 20 members can paint a 2000 square foot wall in 40 minutes, then how long     

   would it take the 20 members to paint a 6000 square foot wall, in minutes?     

   Rate_{20 members}   =   2000 sqft / 40 min   =   50 sqft / min     

     6000 sqft     

   ––––––––––  =  120 min     

     50 ^sqft / _min     

   confirm answer     

   Think of the 6000 sqft wall as being three 2000-sqft walls.     

   Looking at it that way, it would take (3 • 40 min) = 120 min.     

   _.     

   At a cafeteria, Mary orders two pieces of toast and a bagel, which comes out to $3.15.     

   Gary orders a bagel and a muffin, which comes out to $3.50.     

   Larry orders a piece of toast, two bagels, and three muffins, which comes out to $8.15.     

   How many cents does one bagel cost?     

                                  T + B = 315 ¢     ==>>      T = 315 – B     

                                  B + M = 350 ¢     ==>>     M = 350 – B     

                                   T + 2B + 3M  =  815 ¢     

                                   (315 – B)  +  2B  +  [3 • (350 – B)]  =  815     

                                   315 – B + 2B + 1050 – 3B  =  815     

                                    – 2B + 1365  =  815     

                                    – 2B  =  – 550     

                                         B  =  275 ¢     

   _.     

   I have to paint one side of a wall.  The wall is 12 meters tall and 120 meters long.     

   Each gallon of paint covers 150 square feet. If a foot is approximately 0.3048 meters, then     

   what is the smallest whole number of gallons I can buy and have enough paint to cover the whole wall?     

       12 m / 0.3048 m/ft  =    39.3701 ft     

     120 m / 0.3048 m/ft  =  393.7008 ft     

                          Area  =    39.3701 ft • 393.7008 ft     

                          Area  =    15,500.04 sqft     

                          Paint  =   15,500.04 sqft / 150 sqft/gal     

                          Paint  =   103.33 gal     but you can't buy just 0.33 of a gallon     

                          Paint  =   104 gal     

   _.     

   All members of our painting team paint at the same rate.     

   If 20 members can paint a 2000 square foot wall in 40 minutes, then how long     

   would it take the 20 members to paint a 6000 square foot wall, in minutes?     

   Rate_{20 members}  =  2000 sqft / 40 min  =  50 ^sqft / _min     

     6000 sqft     

   ––––––––––  =  120 min     

     50 ^sqft / _min     

   confirm answer     

   Think of the 6000 sqft wall as being three 2000 sqft walls.     

   Looking at it that way, it would take (3 x 40 min) = 120 min.     

   _.     

   In a store window, there was a box of berries having a total weight of 200 kg.     

   The berries were 99% water, by weight.  After two days in the sun, the water     

   content of the berries was only 90%, by weight.  What was the total weight     

   of the berries after two days, in kg?     

   You're starting with 200 kg of berries, and 99% of that is water.     

   Since they're 99% water, then the other 1% is the solid stuff,     

   and 1% of 200 kg means that there are 2 kg of solid stuff.     

   Some of the water evaporates in the sun, but the solid stuff doesn't.     

   So, after the evaporation, there still remains the same 2 kg of solid stuff.     

   Since the new water content is 90%, that means that now the solid stuff is 10%.     

   So 2 kg (solid stuff) is 10% of the total weight.  (0.10) W = 2   ==>>  W = 2 / 0.10     

   Therefore, the new total weight is 20 kg.     

   _.     

   Find an angle ϴ such that 0°≤ϴ≤360° and ϴ is coterminal with 1500°.     

   1500 / 360  =  4 R 60     ==>>     theta is 60^o     

   _.