Carl Frederickson and his Explorer friend Russell tied too many balloons onto th
ID: 3199735 • Letter: C
Question
Carl Frederickson and his Explorer friend Russell tied too many balloons onto their house. They float up and away. Up and Up and UPPP they go!! The next thing they know they are at cloud level. Then they are above the clouds. Then they are leaving the Earth’s atmosphere (thankfully Carl’s house is nice and airtight and he has extra oxygen tanks for breathing). They go past the moon, then Mars, ten Jupiter, and on past the solar system. Past several comets, and black holes and galaxies. They finally land on the planet Fluxicon. The Fluxicons greet them warmly and after staying there for a year they learn the language. The Fluxicons live pretty much like we do here on earth except for one thing. They’ve invented a new operation called flux. You can add 5 and 7 and get 12. You can flux 5 and 7 and get an answer too. The symbol for flux is @. Here is the way flux works. You write the numbers in fraction form and then multiply the numerator of each number by the other numbers denominators and add those two numbers. For example, 5 @ 2/3 ? First since 5 isn’t a fraction we make it one. 5/1@2/3. Then 5 x 3 + 2 x 1=17. So 5 @ 2/3=17. Russell likes mathematics so he is thinking about flux and is wondering about the properties that he learned about adding and multiplying. Both adding and multiplying have commutative and associative properties. For example, the associative property of multiplication says that (a×b)×c=a×(b×c) for all numbers a, b, and c. Since flux is just really adding and multiplying, Russell thinks flux might also have a commutative and associative property. But he’s not sure. a. State what the commutative property for flux would be if one exists. b. Give some examples to show that the property makes sense or give one example that shows that it fails. c. State what the associative property for flux would be if one exists. d. Give some examples to show that the property makes sense or give one example that shows it fails.
Explanation / Answer
a) let two fractions be x and y.
commutative property will be, x@y = y@x
b) let the fractions be x=2/3 and y=4/5
x@y= 2*5 +4*3=22 and y@x = 4*3 +2*5 = 22
let x= 2/9 and y= 6/7
x@y= 2*7+6*9=68 and y@x= 6*9+2*7= 68
this implies x@y = y@x so commutative property holds true.
c) let three fractions be x, y, z
associative property states that,
x@(y@z)= (x@y)@z
d) let x= 4/3 , y= 5/7 , z= 2/3
x@(y@z) = x@(5*3+2*7) = x@29 = 4/3 @ 29/1 = 4*1+29*3 = 91
(x@y)@z = (4*7+5*3)@z =43@z = 43/1 @ 2/3 = 43*3+2*1 = 131
since x@(y@z) is not equal to (x@y)@z
associative property does not hold true.