How Do Gradients, Rates, and the Term 'Per' Relate to Division and Ratios?

[Miraj Kayastha]

Why does "per" in 3 miles per hour mean division?
Why are gradients and rates a ratio?

 

[Hardik Batra]

Why does "per" in 3 miles per hour mean division?

The meaning of "per" is "for each" .

Now read 3 miles "for each" hour.
And you divide the 3 miles by one hour you will get the speed of an object.

 

[256bits]

Why does "per" in 3 miles per hour mean division?
Why are gradients and rates a ratio?

Are you not familiar with the basic equation
y = mx + b
where m is the slope, or rate ( and b is the intercept )

m is the ratio of the ordinate to the abscissa for any point on the line (x,y)

If you plot y-axis as the "miles" and x-axis as the time of hours , then the slope naturally follows as miles/hour, or in English terms miles per hour.

Same thing for gradient - for a surface that has a slope, its elevation will increase y-amount for every x-amount distance.

 

[CWatters]

Why are gradients and rates a ratio?

Two "similar triangles" have the same angles but can be different sizes.

In each case

Tan{Θ} = Length of opposite side/length of adjacent side

So if interested in the angle or gradiant it makes sense to compare the ratio of the sides rather than their absolute magnitude.

 

[pmacias]

Gradient refers to the change in a quantity over a specific distance or interval. It is often represented by the symbol ∇ and can be calculated by dividing the change in the quantity by the corresponding distance or interval. This is similar to the concept of rate, which also measures the change in a quantity over time. Both gradient and rate are expressed as ratios, where the numerator represents the change in the quantity and the denominator represents the corresponding distance or time interval.

The term "per" in 3 miles per hour indicates a division because it is used to express a ratio of two quantities. In this case, it represents the ratio of distance (miles) to time (hours). The word "per" is derived from the Latin word "per" which means "by" or "through". In this context, it signifies that the quantity (miles) is being divided by the unit of time (hours).

Gradients and rates are both ratios because they compare two different quantities and express their relationship in terms of division. This allows us to measure and compare the change in a quantity over a specific distance or time interval, providing valuable information in many scientific fields such as mathematics, physics, and biology.