Proportionality (mathematics)

In mathematics, two sequences of numbers, often experimental data, are proportional or directly proportional if their corresponding elements have a constant ratio. The ratio is called coefficient of proportionality (or proportionality constant) and its reciprocal is known as constant of normalization (or normalizing constant). Two sequences are inversely proportional if corresponding elements have a constant product, also called the coefficient of proportionality.

This definition is commonly extended to related varying quantities, which are often called variables. This meaning of variable is not the common meaning of the term in mathematics (see variable (mathematics)); these two different concepts share the same name for historical reasons.

Two functions $f(x)$ and $g(x)$ are proportional if their ratio ${\textstyle {\frac {f(x)}{g(x)}}}$ is a constant function.

If several pairs of variables share the same direct proportionality constant, the equation expressing the equality of these ratios is called a proportion, e.g., $.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}a/b = x/y = ⋯ = k$ (for details see Ratio). Proportionality is closely related to linearity.

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Let's talk about proportionality. All this means, if we say that two variables are proportional, it just means that they are constant multiples of each other. So if we say that y is equal to 3 times x, we can say that y is-- and we can be a little bit more specific-- we could say y is directly proportional. y is directly proportional to x. And of course, we could divide both sides of this equation by 3, and we would get 1/3 y is equal to x. So once again, x is a constant multiple of y. We could write this as x is equal to 1/3 y. So we could also write that x is directly proportional to y. If y is directly proportional to x, then x is also directly proportional to y, by a different proportionality constant. So this term right here, this is the constant of proportionality between y and x, or between x and y. This is the constant of proportionality. So in general, if I say that a is directly proportional-- sometimes you'll just see it written as a is proportional-- a is directly proportional to b. In general, from this statement, we could write that a is equal to some constant times b, where this constant is equivalent to that constant or that constant. It's equal to some number. And you could also say that b is directly proportional to a, where if you divide both sides of this equation by k, you would get b is equal to 1/k a. This is direct proportionality. Now another thing you might see-- and we'll see it later in this video-- is being inversely proportional. So let me draw a little line here. So let me give you a few examples of inverse proportionality. Let's say that y is equal to 5 times 1/x. Notice what's going on here. In this situation, y is proportional to the inverse of x, right? Or you could say that y is inversely proportional-- I'll abbreviate it-- proportional to x. It is directly proportional to the inverse of x. So here is our proportionality constant, but it is directly proportional to the inverse of x. Here's another example, and I'm just playing with letters here. I could write a is equal to 10/b. Once again, a is inversely proportional to b. You could rewrite this as a is equal to 10 times 1/b. a is proportional to the inverse of b, or it's inversely proportional to b. So now that we have that out of the way, let's do some examples using our newly-found knowledge of direct proportionality and inverse proportionality. So here I have a problem. It says that x is proportional to y. So that means that x is equal to some constant times y. And just so you're familiar with other notation that you might see when you're talking about proportionality. Other ways to write this statement here. x is proportional to y. You could write it like this. x is equal to some constant times y. Or you could write it as x is proportional to y. That little, I don't know, left facing fish looking thing means proportional. This and this mean the same thing. This means that x is equal to some constant times y. Or sometimes you'll even see it x squiggly line y. x is equal to some constant times y. This, this, and this, and that are all equivalent statements. But this is the most useful because you say, oh, there's some constant, maybe we can solve for that constant. Then the next statement here says, x is proportional to y and inversely proportional to z. So that second statement says that x is equal to k. It's going to be a different constant, not necessarily the same constant. So let me put a different-- well, I'll call it k2, I'll call this k1-- is equal to some other constant-- not necessarily some other, but I'm guessing it's going to be some other constant-- times the inverse of z, times 1/z. That's what inversely proportional to z means. And other ways we could write this is, x is proportional to the inverse of z. x is inversely proportional to z, or x is inversely proportional to z. These things in orange are all equivalent. I just wanted to make you familiar with it in case you ever see it. And then they tell us, OK, if x is proportional to y, inversely proportional to z-- we wrote that already-- and x is equal to 2 when y is equal to 10. So let's do that. x is equal to 2 when y is equal to 10. So x is equal to 2 when k1 is multiplied by y. That's what that statement tells us, and y is 10 when k1 is multiplied by 10. x is 2 when y is 10. I just put those numbers in there. And then we can actually use this now to solve for k1. But before that, let's see what this next statement tells us. It says x is 2 when y is 10 and z is 25. So x is 2 when k2 is multiplied by 1/z or 1/25. So that's what that second statement tells us. So they say find x when y is equal to 8 and z is equal to 35. So essentially what they're asking us, hey, why don't you use this information that we gave you in the green and the red, solve for the different proportionality constants, and then use that to write, I guess, the specific equations for the proportionality or the inverse proportionality, and then solve for y or z. Let's just do it. So here, up here in green, let me rewrite it over here. We know that x is 2 when k1 is multiplied by y where y is 10. Divide both sides of this equation by 10. You get 2 over 10 is 1/5. You get k1 is equal to 1/5. So that tells us that this first equation right here-- I'll write it in that original color-- is x is equal to 1/5 y. Now, that second statement in red, we know that 2 is equal to k2 times 1 over 25. Let's multiply both sides of this equation by 25. This cancels out. And we're left with k2-- k2 is equal to 50. k2 is equal to 50, so we can write this equation right here that x is equal to 50 times 1/z or is equal to 50/z. So we've now solved for the two constants of proportionality for these two equations, so now we can answer the second part of this question. Find x when y is equal to 8 and z is equal to 35. So the situation when y is equal to 8-- we just go right here-- x is equal to 1/5 times 8, which is equal to 8/5. So that's what x is equal to when y is equal to 8. When z is equal to 35, once again, x is equal to 50/z, is equal to 50/35. Now let's see, we can divide the numerator and the denominator here by 5, so we get 10/7. So when z is equal to 35, we get 10/7. When y is equal to 8, x is equal to 8/5. So those are our two answers, right there. Let's do another one. Ohm's Law. Ohm's Law states that current flowing in a wire is inversely proportional to resistance of the wire. So current flowing in a wire is inversely proportional to resistance of a wire. So let's use I for current. I is equal to current. And we'll use R for resistance. R is equal to resistance. And you might be wondering why I picked I, but later on when you start studying electricity and maybe you become an electrical engineer, you'll see that I is the conventional letter used for current. And I won't go into the details of why, just yet. But it tells us that current, that I, is inversely proportional to resistance. So that first statement I underlined in yellow says that current is inversely proportional. It's equal to some constant times the inverse of resistance. It's inversely proportional to resistance of the wire. If the current is 2.5 amperes when the resistance is 20 ohms. So the current is 2.5 when k is multiplied by 1 over a resistance of 20 ohms, 1/20 ohms. And I'm assuming, because they're going to keep the units in amperes and ohms later on, so we could write the units here if we wanted to, but I'll keep it simple and not write the units. They ask, find the resistance when the current is 5 amperes. So this first statement, right here. The current is 2.5 amperes when the resistance is 20 ohms. That's this equation, right here. So we can use this to solve for k. You multiply both sides of this by 20. These cancel out, and you're left with k is equal to-- what's 20 times 2.5? 20 times 2 is 40. 20 times 0.5 is 10. So it's going to be 50, 40 plus 10. So k is equal to 50. So in this situation, this equation is I is equal to 50 times 1/R or 50/R. That's our relationship, right there. And then they ask, find the resistance when the current is 5 amperes. So when our current is 5 amperes, so 5 is equal to 50 over the resistance. Let's multiply both sides of this equation by the resistance, and you get 5R is equal to 50. I just multiplied both sides of this equation by R. These canceled out, so I got 5R is equal to 50. Divide both sides of this equation by 5. We get R is equal to 50 divided by 5 is 10. So the answer is, when the current is 5 amperes, the resistance will be equal to 10 ohms. And we could write it like this, 10 ohms, just like that. Let's do one more. The intensity of light is inversely proportional to the-- now let's be careful here-- to the square of the distance between the light source and the object being illuminated. So let's just say, well, we could use I again. Let's say I is equal to intensity of light. And let's say that D-- I'm going to do it in a different color-- D is equal to the distance between the light source and object being illuminated. So that's what? D, the distance between the light source and the object being illuminated. Now what does this first sentence tell us? The intensity of the light is inversely proportional. So the intensity of the light, I, is inversely proportional. So it's going to be some proportionality constant times the inverse. But notice, the intensity of light is inversely proportional to the square of the distance. Not just to the distance. So to the square of the distance. So to distance squared. If it just said to the distance, we would just have a D here. But it says the intensity of light is inversely proportional. So 1 over the square of the distance between the light source and the object being illuminated. So that's what this first statement is going to give us, this equation right here. Now, they tell us, a light meter that is 10 meters-- so they're saying, when distance is equal to 10 meters-- a light meter that is 10 meters from a light source registers 35 lux. So when the distance is equal to 10 meters, they're telling us that the intensity-- the lux is a measure of intensity-- the intensity is equal to 35 lux. So what intensity would register 25 meters from the light source? So this first statement they gave us, that I wrote down the information in this purple color, we can use that to solve for the proportionality constant. And then once we have that constant, we can solve for I or D, given one or the other. So that statement told us, they told us, that when D is equal to 10, I is equal to 35. So I is equal to 35, so 35 is equal to K times 1 over D squared, 1 over 10 squared. Or we could say that 35 is equal to k times 1/100, multiply-- 10 squared is just 100-- multiply both sides of this equation by 100. These cancel out. And we're left with k is equal to-- what is this? 3,500. So we've solved, we've used this statement here where they gave us some values to solve for our proportionality constant. So now we know that the equation is-- the intensity is equal to 3,500-- that's k times 1 over D squared. Or you can just write over D squared. Now, they say, what intensity would it register 25 meters from the light source? So they're saying D is 25. So intensity is equal to 3,500 divided by 25 squared. Our distance is 25 meters. So the intensity is going to be equal to 3,500 divided by-- well, actually, let's just keep it simple. Let me see, well, this is divided by 25 times 25. It's 625. Let's just get the calculator out. So we get, it's going to be 3,500 divided by 25 times 25, which is 625, which is equal to 5.6. It's equal to 5.6 lux, which is the unit for light intensity. Hopefully you found that useful.

Direct proportionality

Given an independent variable x and a dependent variable y, y is directly proportional to x^[1] if there is a positive constant k such that:

y=kx

The relation is often denoted using the symbols "∝" (not to be confused with the Greek letter alpha) or "~", with exception of Japanese texts, where "~" is reserved for intervals:

y\propto x

(or

y\sim x

)

For $x\neq 0$ the proportionality constant can be expressed as the ratio:

k={\frac {y}{x}}

It is also called the constant of variation or constant of proportionality. Given such a constant k, the proportionality relation ∝ with proportionality constant k between two sets A and B is the equivalence relation defined by $\{(a,b)\in A\times B:a=kb\}.$

A direct proportionality can also be viewed as a linear equation in two variables with a y-intercept of $0$ and a slope of k > 0, which corresponds to linear growth.

Examples

If an object travels at a constant speed, then the distance traveled is directly proportional to the time spent traveling, with the speed being the constant of proportionality.
The circumference of a circle is directly proportional to its diameter, with the constant of proportionality equal to $π$ .
On a map of a sufficiently small geographical area, drawn to scale distances, the distance between any two points on the map is directly proportional to the beeline distance between the two locations represented by those points; the constant of proportionality is the scale of the map.
The force, acting on a small object with small mass by a nearby large extended mass due to gravity, is directly proportional to the object's mass; the constant of proportionality between the force and the mass is known as gravitational acceleration.
The net force acting on an object is proportional to the acceleration of that object with respect to an inertial frame of reference. The constant of proportionality in this, Newton's second law, is the classical mass of the object.

Inverse proportionality

Two variables are inversely proportional (also called varying inversely, in inverse variation, in inverse proportion)^[2] if each of the variables is directly proportional to the multiplicative inverse (reciprocal) of the other, or equivalently if their product is a constant.^[3] It follows that the variable y is inversely proportional to the variable x if there exists a non-zero constant k such that

y={\frac {k}{x}}

or equivalently, $xy=k$ . Hence the constant "k" is the product of x and y.

The graph of two variables varying inversely on the Cartesian coordinate plane is a rectangular hyperbola. The product of the x and y values of each point on the curve equals the constant of proportionality (k). Since neither x nor y can equal zero (because k is non-zero), the graph never crosses either axis.

Direct and inverse proportion contrast as follows: in direct proportion the variables increase or decrease together. With inverse proportion, an increase in one variable is associated with a decrease in the other. For instance, in travel, a constant speed dictates a direct proportion between distance and time travelled; in contrast, for a given distance (the constant), the time of travel is inversely proportional to speed: s × t = d.

Hyperbolic coordinates

The concepts of direct and inverse proportion lead to the location of points in the Cartesian plane by hyperbolic coordinates; the two coordinates correspond to the constant of direct proportionality that specifies a point as being on a particular ray and the constant of inverse proportionality that specifies a point as being on a particular hyperbola.

Computer encoding

The Unicode characters for proportionality are the following:

U+221D ∝ PROPORTIONAL TO (∝, ∝, ∝, ∝, ∝)
U+007E ~ TILDE
U+2237 ∷ PROPORTION
U+223C ∼ TILDE OPERATOR (∼, ∼, ∼, ∼)
U+223A ∺ GEOMETRIC PROPORTION (∺)

Notes

^ Weisstein, Eric W. "Directly Proportional". MathWorld – A Wolfram Web Resource.
^ "Inverse variation". math.net. Retrieved October 31, 2021.
^ Weisstein, Eric W. "Inversely Proportional". MathWorld – A Wolfram Web Resource.

References

Ya. B. Zeldovich, I. M. Yaglom: Higher math for beginners, p. 34–35.
Brian Burrell: Merriam-Webster's Guide to Everyday Math: A Home and Business Reference. Merriam-Webster, 1998, ISBN 9780877796213, p. 85–101.
Lanius, Cynthia S.; Williams Susan E.: PROPORTIONALITY: A Unifying Theme for the Middle Grades. Mathematics Teaching in the Middle School 8.8 (2003), p. 392–396.
Seeley, Cathy; Schielack Jane F.: A Look at the Development of Ratios, Rates, and Proportionality. Mathematics Teaching in the Middle School, 13.3, 2007, p. 140–142.
Van Dooren, Wim; De Bock Dirk; Evers Marleen; Verschaffel Lieven : Students' Overuse of Proportionality on Missing-Value Problems: How Numbers May Change Solutions. Journal for Research in Mathematics Education, 40.2, 2009, p. 187–211.

This page was last edited on 4 March 2024, at 21:05

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