Math
posted by Anonymous on .
Explain which method you would use to determine whether the following proportion is true or false. Show how you know that it is true or false.19/6=4/27

I like this question! It's simple, but should make you think, and there are many ways to approach it. I'd hate to deprive you of the thinking experience, but here are some questions for you.
What does it mean for a statement about numbers being equal to be "true" or "false"?
If numbers are equal, is one bigger than the other?
Consider any fraction of the form a/b, like 13/17 or 184/63. How can you know really quickly whether that fraction is less than, equal to, or greater than 1?
Is 3/4 greater or less than 1? What about 4/4? or 5/4?
Now go look at your two fractions again. 
Well, 3/4 is less then one, 4/4 is equivalent to 1 and 5/4 is greater then one....right?

Right! If it's bigger than one, the number on top is larger.
Now, is 19/6 greater or less than one?
Is 4/27 greater or less than one?
So without calculating anything, you can show that the two fractions cannot be equal. 
Yes! 19/6 is greater then one and 4/27 is less then one!
Thank you! 
Glad I could help. :)

Jim, one more question....
In a case of 4/15 and 5/16 which would be bigger? 
Good question! Really good question, since you're thinking of the more general case.
You can't use the same trick there; I'm afraid you'll have to calculate.
The standard way, that will always work, is to find a common denominator of the two fractions, so that you transform both to have the same bottom number. You can always do that by multiplying top and bottom of each by the bottom of the other. Exampl:
4/15  we want to multiply top and bottom by 16:
4 / 15 = (4 * 16) /(15 * 16) = 64 / 240
5/16  we want to multiply top and bottom by 15
5 / 16 = (5 * 15) / (16 * 15) = 75 / 240
Note: We haven't _changed_ the numbers; just the way we write them: 2/4 is exactly the same as 1/2, and 64/240 is still exactly the same as 4/15.
So now we have the question which is bigger: 64 / 240 or 75/240?
Since they're over the same denominator, i's the same wquestion as which is bigger: 64 or 75? Obviously 75.
So 75/240 is bigger than 64/240
So 5/16 is bigger than 4/15.
(If you want a quick check, pull out a calculator, divide 5 by 16 and see that the decimal is bigger than dividing 4 by 15.) 
You are very smart! I don't know why I didn't think of that. That makes the most sense.

The reason you didn't think of that is just that you haven't had enough practice yet. :)
Keep plugging at it, and you'll be doing these things without even having to think about them! 
Thank you!