## Tuesday, November 5, 2013

### 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 R12P.
a) What is  R12P  if  R1 =  90  and  R2  =  10 ?
b) What is  R12P  if  R1 =   2  and  R2  =  ∞ ?     (A break.)
c) What is  R12P  if  R1 = 100  and  R2  =  0  ?    (A short.)
d) What is  R1   if  R12P = 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 +)

#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.  R12P  =   9
R12P  =   2
R12P  =   0
R1   =  36

#11.
.*     |
.*     |
. *     |
.    *     |
_. _  _ ._  _  _  _ *_  _ |_2_  _  _  _  _  _  _
*     |        .      .
*     | .
*     .
-2*   . |
*  .  |
* .   |

#12.
|     *  .
|     *   .
|     *     .
|     *        .
_  _  _  _  _  _  _2|_  _ * _  _  _  _ ._  _
.      .           |     *
.    |     *
.     *         _
| .   *2
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#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