"Simplification of an answer" question (involves fractions)

[jescobedo]

Greetings all,

I am new to the forum -- I am a few years out of college studying for the FE exam coming up in October. Algebra is beautiful! We see it all the time, but unfortunately for me, I have forgotten a few concepts and I am humbly asking for your help.

My apologies if there are similar threads -- I would have searched for this algebra concept, however, I wasn't too sure what to search for.

My question is as follows:
Upon finding the area of a Circular Segment, I have simplified my answer down to:

Area = (49/2) * [ (5pi/6) - (1/2) ]

HOWEVER, the answer has been further simplified to:

Area = (49/12) * (5pi - 3)

Now if I multiply 6 * [ (5pi/6) - (1/2) ], I can get [ 5pi - 3 ], however it appears the reciprocal of 6 was multiplied to (49/2) to get (49/12)

Forgive me for my ignorance, but I can't remember for the life of me what that concept is or called!

How do you simplify Area = (49/2) * [ (5pi/6) - (1/2) ] to get Area = (49/12) * (5pi - 3) ??

Thank you!

-JEE

 

[wjm11]

Area = (49/2) * [ (5pi/6) - (1/2) ]

HOWEVER, the answer has been further simplified to:

Area = (49/12) * (5pi - 3)

Now if I multiply 6 * [ (5pi/6) - (1/2) ], I can get [ 5pi - 3 ], however it appears the reciprocal of 6 was multiplied to (49/2) to get (49/12)

Forgive me for my ignorance, but I can't remember for the life of me what that concept is or called!

How do you simplify Area = (49/2) * [ (5pi/6) - (1/2) ] to get Area = (49/12) * (5pi - 3) ??

Here is one way to interpret the change: We can always multiply any expression by "1" and not affect the value of that expression, right? In this case, we can think of "1" as being represented by 1 = (6)(1/6).

Whenever we multiply a number by its reciprocal, we always get "1" for an answer. (Perhaps this is the principle you were trying to remember(?). So, in this problem, the (49/2) was multiplied by (1/6), and the [ (5pi/6) - (1/2) ] was multiplied by 6, (as you observed).

 

[mmm4444bot]

They multiplied the entire expression by 1 in the form of 6/6.

6/6 can be factored into 6/1 * 1/6.

Hence:

1 * [49/2 * (5Pi/6 - 1/2)]

6/6 * [49/2 * (5Pi/6 - 1/2)]

6/1 * 1/6 * [49/2 * (5Pi/6 - 1/2)]

By the Commutative Property of Multiplication, we can multiply quantities in any order:

(1/6) (49/2) (6/1) (5Pi/6 - 1/2)

As you observed, this leads to:

49/12 (5Pi - 3)

We could also "expand" this end-result by using the Distributive Property to multiply 5Pi - 3 by 49/12:

245Pi/12 - 49/4

:cool:

 

[jescobedo]

HA!

Thank you for your help. I see it now. It didn't seem so intuitive in my head at first, but what do I know?!

Sheesh.

Thanks again!!

-JEE