Question

True or False: All rates are ratios but all ratios are not rates.

Answer 1

A ratio means a fraction or quotient $$a\div b=\frac{a}{b}$$ $$b\neq0$$

Answer 2

true

Answer 3

define rate as \( \displaystyle \lim_{x \to 0} \frac{y}{x} \)

Answer 4

If I take an orange and cut it in half, I get 2 ratios of $$\frac{1}{2}$$

Answer 5

@Directrix is that a rate?

Answer 6

Here's my current (could be wrong) thinking;

A ratio is a relationship, usually between two numbers. It may be written as a common fraction, a decimal, a % or using " : " (e.g., 1/2. .5 50%, 1:2).

A rate is a ratio where the denominator is 1. Generally, units expressed for both the numerator and the denominator, such as miles per hour, liters per km, dollars per euro, interest per year, population growth per year, sales growth (or decline) per month.

Answer 7

@skullpatrol I don't see the orange halves as a rate.

Answer 8

Correct, it is not a rate.

Answer 9

All rates are ratios but not all ratios are rates.

Answer 10

What about slope of a line that is not vertical. Is slope a rate or a ratio or both?

Answer 11

a rate is a ratio between two measurements with "different" units
so slope is a ratio, not rate

Answer 12

true

Answer 13

Don't we say that slope is the change in y over change in x? Would that not be a rate?

Answer 14

A rate is a ratio that compares the amounts of two different kinds of measurements.

Answer 15

no, just the ratio of changes....not rate

Answer 16

Slope of a line is measure of the rate of change. That what I was thinking as well @Directrix

Answer 17

It is a rate if the axises measure time and distance

Answer 18

^ ofcourse, as then we deal with different quantities in x and y axes

Answer 19

That would be the slope.

Answer 20

If the slope of a line is 3/1, then for every one unit increase in x, there is a constant three unit increase in y. Lines with slope have a constant rate of change while the slope of a curve at a point (assuming slope exists at that point) is an instantaneous rate of change of y with respect to x.

Answer 21

Simply put when you talk about rate you are implicitly using physical units, but when you talk about ratios you are using "pure" numbers.

Answer 22

Then, ratios would be viewed as "naked numbers" in that they have no units?

Answer 23

umm.... 4kg/2kg is also a ratio. (so is 4/2)
4kg/2 sec is rate.

Answer 24

Yes, sort of like the orange, but using a sphere with no units.

Answer 25

4kg/2m is also a rate, but 4m/2m is a ratio

Answer 26

I need to think more about this notion.

Answer 27

By definition: A rate is a ratio that compares the amounts of two different kinds of measurements.

Answer 28

how do you define a rate?

Answer 29

Remember they must be two DIFFERENT kinds of measurement :)

Answer 30

With different units.

Answer 31

to understand rate I usually take "birth rate" and "death rate" as examples

Answer 32

http://www.mathplanet.com/education/pre-algebra/ratios-and-percent/rates-and-ratios

Answer 33

I found something here :

http://www.differencebetween.net/science/mathematics-statistics/difference-between-rate-and-ratio

A short summary :
1.A rate refers to the frequency by which a certain event happens while a ratio refers to the relationship between the size, number, or degree of two or more things.
2.A rate is a comparison between two measurements of the same units while a ratio is the proportion of one thing to another.
3.A rate refers to the fixed quantity of two things while a ratio refers to the relationship between various things.
4.A ratio indicates the difference between things while a rate indicates the changes in their measurements or units.
5.A ratio is indicated by the quotient of one quantity divided by the other while a rate is indicated by the comparison between two things.

Answer 34

A rate is a little bit different than the ratio, it is a special ratio. It is a comparison of measurements that have different units.

Answer 35

"different" word is the catch here...

Answer 36

let y be a function of x .. rate of change of y with respect to x ... rate is dy/dx. There is an interesting article here ... http://math.stackexchange.com/questions/21199/is-dy-dx-not-a-ratio
but for some reason i never read it.

Answer 37

There are too much comments :(

Answer 38

Bottom line: "different units" is the the distinguishing factor.

Answer 39

@mathslover We need the comments to foster our thinking. After the thread activity settles, we can return and read and reflect about what has been posted.

Answer 40

I just think of "rate" as speed, example miles per hour or m/s
these are ratios. But not all ratios are speeds. example: ratios of cows/horses on a farm

Answer 41

Oh! @Directrix I meant to say that in the link ExperimentX posted : math stack exchange , there were too much comments and hence it will take time to read the complete thread.
I was not saying about this thread. sorry for my poor language that made you misunderstood my sentence.

Answer 42

Rates are to ratios like tomatoes are to fruits. Some say it is, some say it's not, but by definition, it really is one.

Answer 43

@experimentX the answers by Arturo Magidin are always epic!

Answer 44

Returning to this question two years later @Directrix I realize "what a great question!" Thanks for asking it :D