On Reduction, 11 of 11

     Problem Set: Elementary Reduction

     #1. Find in lowest terms:

            6/5 <+> 8/3    =  ?

           11/3 <+> 11/4   =  ?

           3/11 <+> 4/11   =  ?

            6/7 <-> 9/5    =  ?

     #2. 3<+>7=? , 52<+>117=? , 70<+>126=? , 2<+>2=?

     #3. 5<->15=? , 21<->70=? , 3<->5=? , 28<->44=?

    #4. a+b=c implies ac<+>bc = ab. Use this to create three reductions involving integers only. (E.g. 1+2=3, so 3<+>6=2.)

     #5. Prove: (a+b)(c<+>(bc/a))  =  (a<+>b)(c+(bc/a))

     #6. Assuming that their work rates add, how long would Alice and Bob, working together, take to trim the hedges, if separately:

     a) Alice takes 45 minutes, and Bob takes 36 minutes.

     b) Alice takes 17 minutes, and Bob takes infinite time.           

     c) Alice takes 23 minutes, and Bob takes no time at all.

     #7. Two ships cross the same ocean, starting at the same time and heading for each other's home port. Let F be the time that the first ship took to go from port A to port B, let S be the time that the second ship took to go from port B to port A, and let M be the time it took for them to meet in mid-ocean.

     What is M, if:

     a) T = 60  and  F = 84

     b) T = 60  and  F =  ∞   (This ship is stuck in port.)

     c) T = 60  and  F =  0   (This ship is a starship.)

     #8. You drive a mile at velocity v₁, and then another mile at velocity v₂; on average you go at velocity V.

     a) What is V if  v₁ = 55  and  v₂ = 66 ?

     b) What is v₂ if V = 70 and v₁ = 60 ?

     #9. Two racecars, when driving towards each other from a distance of a mile, pass each other in 12 seconds; when the faster car pursues the slower car from one mile behind, it takes 60 seconds to catch up. How quickly does each car run a mile?

     #10. Resistances R₁ and R₂ are wired in parallel, and their combined resistance is R_12P.

     a) What is  R_12P  if  R₁ =  90  and  R₂  =  10 ?  

     b) What is  R_12P  if  R₁ =   2  and  R₂  =  ∞ ?     (A break.)

     c) What is  R_12P  if  R₁ = 100  and  R₂  =  0  ?    (A short.)

     d) What is  R₁   if  R_12P = 20  and  R₂  =  45?

     #11. Graph:  y  =  2 <+> x

     #12. Graph:  y  =  2 <-> x

    

     #13.

      *     *                     

      |\   /|      a) What is z when x=680  and  y=920?       

      | \ / |                   

    x |  *  | y    b) What is x when z=546  and  y=2310?      

      | /|\ |              

      |/ |z\|                    

      *--*--*                  

     #14. Solve:     a)   7x<+>2 = 5

                      b)   4x<->4 = 5x<+>8

     #15. Solve:           33x <+> 42y  =  110

                      48x <+> 21y  =   80

     #16. Solve:     x<+>y  =  12

                      x<->y  =  84

     #17. Solve:           x² <+> 6x <+> 150  =  ∞.

     #18. Solve:           x + y  =  144

                      x <+> y  =   35

     #19. Solve:           (3 <+> x) + x  = 8

     #20. (x<+>1)³   =   ?  (Expand using <+> only, no +)

     Elementary Reduction Answer Key

     #1.  24/29, 11/7, 12/77, 18/11

     #2.  21/10,  36,  45,  1

     #3.  15/2,  30,  15/2,  77

     #4. This is up to the student.

     #5.(a+b)(c<+>(bc/a)) = (a+b)(a<+>b)(c/a) = ab(c/a) = bc

        (a<+>b)(c+(bc/a)) = (a<+>b)(a+b)(c/a) = ab(c/a) = bc

     #6.  a: 45 <+> 36 = 20 minutes

          b: 17 <+> 1/0  = 17 minutes

          c: 23 <+> 0  =  0 minutes

     #7.  a) M = 35

          b) M = 60

          c) M =  0

     #8.  a) 60

          b) 84

     #9. 20 seconds and 30 seconds.

     #10.  R_12P  =   9

           R_12P  =   2

           R_12P  =   0

           R₁   =  36

     #11.

                                  .*     |

                                  .*     |

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               _. _  _ ._  _  _  _ *_  _ |_2_  _  _  _  _  _  _

                                   *     |        .      .

                                   *     | .

                                   *     .                      

                                 -2*   . |

                                   *  .  |

                                   * .   |

     #12.

                                   |     *  .

                                   |     *   .

                                   |     *     .

                                   |     *        .     

               _  _  _  _  _  _  _2|_  _ * _  _  _  _ ._  _

                .      .           |     *               

                              .    |     * 

                                   .     *         _        

                                   | .   *2

                                   |   . *

                                   |    .*

                                   |    .*

     #13. a) 391

          b) 715

     #14. a) x =  (5<->2)/7      =  -10/21

          b) x =  (8<+>4)/(4<->5)  =  2/15

     #15.  X = 7,  y = 5

     #16. x  =  2(12<+>84)  =  21

          y  =  2(12<->84)  =  28

     #17. x = -15 or -10

     #18. x = 84 ,  y = 60

     #19. x = 6 or -4

     #20.  x³ <+>  x²/3 <+> x/3 <+> 1