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 +)
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. R12P = 9
R12P = 2
R12P = 0
R1
= 36
#11.
.* |
.* |
. * |
. *
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_ _ ._
_ _ _ *_ _
|_2_ _
_ _ _
_ _
* |
. .
* | .
* .
-2* . |
* . |
* . |
#12.
|
* .
| *
.
| *
.
|
* .
_ _ _
_ _ _ _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
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