|January 4th, 2015, 07:20 PM||#1|
Joined: Feb 2014
Math Focus: algebra and the calculus
Ratios as expressed by division
This might seem like a silly question, but why do we use division to compare two quantities, i.e., a ratio? I've always taken for granted that dividing two physical quantities tells how many of one quantity there is for the other quantity, but why exactly does this work? Why don't we define some new operation that represents the comparison between two quantities rather than division?
|January 4th, 2015, 08:02 PM||#2|
Joined: Oct 2008
From: London, Ontario, Canada - The Forest City
Math Focus: Elementary mathematics and beyond
Ratios and division, though closely related, are not the same thing. More precisely, ratios
employ division to give an idea of the proportion between two or more quantities, whereas
division tells us how a certain quantity (the divisor) is in proportion to another quantity
(the dividend). Consider the following example:
A pie is divided into six equal slices. John eats two slices and Mary eats one slice.
The ratio of slices consumed (John : Mary) is 2 : 1. The portion of the pie that is consumed
is 1/2. From the ratio, if we had no other information, we can only tell that John ate twice as
much as Mary. From the fraction 1/2, we can determine that exactly half the pie was consumed,
but we can't tell who ate how much.
Note that 2 + 1 = 3, 3/6 = 1/2, etc.
Last edited by greg1313; January 4th, 2015 at 08:06 PM.
|January 4th, 2015, 11:06 PM||#3|
Joined: Oct 2013
From: Far far away
Division can be understood in two ways:
1. The intuitively understandable way i.e. 6/2 means how many 2's in 6. There are 10 apples. I want to give 2 to each person. How many person's can I give apples to? 10/2 = 5 persons.
2. The sharing concept: There are 10 apples and 5 people. How many will each person get? In this case, we want to share the 10 apples among 5 people. This is difficult (for me at least) to wrap your head around.
Look at how we read multiplication 2 * 5 = 10 is read as 2, 5's are 10 OR 2 times 5 is 10. Read this way, it implies that the first number (2 in this case) is the number of groups and the second number (5 in this case) is the number in each group.
Now division as understood in (1) above is taken to be the reverse process, finding the number of groups, each group consisting of 2 apples.
Division understood in (2) above is taken as a variant of the reverse process, finding the number in each group, such that there are 5 groups.
But remember that 2 * 5 = 5 * 2. So 2 groups of 5 each = 5 groups of 2 each = 10. So, Let 10/2 = x. This means x groups of 2 (understood the first way). But, it also means 2 groups of x (understood the second way - sharing)
When it comes to ratio, it is the second meaning (sharing) that matters.
When someone travels 50 km in 2 hours and we divide 50km/2hours and get 25km/1hour, we are sharing the distance among 2 hours and we discover that each hour gets (its share) 25 km.
We are not finding how many 2 hours are in 50 km (understood the first way).
Imagine there are 10 men and 5 women. What is the ratio of men to women? We want to 'share' the men among women. So 10/5 = 2/1. The ratio is 2 men/1 woman = 2 : 1.
This is the way I understand it. I hope I'm not wrong.
|division, expressed, fractions, ratios|
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