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## 7th grade (Illustrative Mathematics)

### Unit 2: Lesson 3

Lesson 4: Proportional relationships and equations- Identifying the constant of proportionality from equation
- Constant of proportionality from equations
- Constant of proportionality from table (with equations)
- Constant of proportionality from tables (with equations)
- Equations for proportional relationships
- Writing proportional equations from tables
- Writing proportional equations from tables

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# Identifying the constant of proportionality from equation

Understanding what a constant of proportionality is and how to identify it in an equation.

## Want to join the conversation?

- A guy who created math(33 votes)

- When Sal used the example y=kx. Does the constant of proportionality always have to be the first variable?(8 votes)
- Well, in your example,
*y = kx*, the constant of proportionality (k) is not the first variable. Anyway, no, the constant of proportionality does not have to be the first variable. You can write y = kx and kx = y and it means the same thing.

Hope this helps! :)

Edit: Also, y=xk and y=kx are the same thing(19 votes)

- This is so hard for me i cant believe i made to 8th grade!(13 votes)
- I had to do this in 5th grade. I am in an advanced math class.(8 votes)

- can the constant of proportionality be pi?(4 votes)
- Yes it can be pi! The constant of proportionality can be any real number.

For example, note that in the relationship between the circumference and the diameter of a circle (C = pi d), the constant of proportionality is pi.

Have a blessed, wonderful Christmas holiday!(10 votes)

- What is pi?

How to use pi?(6 votes)- The constant pi is the value of the ratio between the circumference of a circle to its respective diameter. Pi is used to help calculate approximate values of a circle's circumference or arc length, radius, diameter, or area when you have other known values of a circle.

Here are some of the formulas for circles (the symbol π means pi)

•*Area*= π × r^2 [Pi times radius squared]

•*Circumference*=**2**× π ×**r**[**2**times**pi**times**radius**; the diameter is twice the radius]

Circles are different from polygons, such as triangles, quadrilaterals (e.g. squares and rectangles), because they are "curvy" and thus require a somewhat different method of calculation, and is why we need pi for better accuracy, instead of just some "base times height".

In**3-D geometry**, some "solids" that have a circle as part of the structure are cylinders, cones, and spheres; so to find the surface area or volume for those, pi will be necessary.

•**Cylinder**:*volume*= π×r^2 h [**pi**times**radius squared**times**height**]

•**Cone**:*volume*=(**1/3**) × π ×**r^2**×**h**[**pi**times**radius squared**times**height**, all over**3**]

•**Sphere**:*volume*= (**4/3**) π**r^3**[**four-thirds**times**pi**times**radius cubed**](6 votes)

- I heard if a table is proportional, it will cross the origin on a graph. Is that true?(6 votes)
- yes a number is proportional if it crosess the orgin and the line is straight.(3 votes)

- Can you reverse the constant rate of proportionality? Like say y=5x then you multiply 5 and get y but can you divide y by 5 and get x?(5 votes)
- Yes. You could also say that if the constant of proportionality of y and x is 5 because of y=5x, you could also say the constant of proportionality of x and y is 1/5 because x=1/5y(3 votes)

- i do not get it?(4 votes)
- Hi! Here is some practice to help you succeed!

Find the constant of proportionality in this table. I know the table is a bit off, I apoligize for that. This is just the way Khan Academy formats it.

Scoops Cones

2 30

3 45

4 60

Please upvote this so everyone can practice. If there is any problem with the question, tell me. I am a learner too!

Edit- Please refrain from reading the comments below until you have solved the problem. If you need help I will be more than happy to answer your questions.(4 votes) - I get it that if we have equation y = kx, then k is the constant of proportionality. What if the equation is y = kx + b, where b is some other constant? Would k still be called constant of proportionality in that case?(3 votes)
- For y = kx + b with b nonzero, we would have a linear but non-proportional relationship. In this situation, k is not a proportionality constant.

However, k is the rate of change, also called the slope, whether or not b is zero.

Have a blessed, wonderful day!(2 votes)

## Video transcript

- [Instructor] When you hear
constant of proportionality, it can seem a little bit
intimidating at first. It seems very technical. But as we'll see, it's a
fairly intuitive concept, and we'll do several
examples and hopefully you'll get a lot more comfortable with it. So let's say we're trying to
make some type of baked goods, maybe it's some type of muffin, and we know that depending
on how many muffins we're trying to make, that for a given number of eggs, we always want twice as many cups of milk. So we could say cups of milk, cups of milk, that's going to be equal to two times the number of eggs. So what do you think the constant of proportionality is here, sometimes known as the
proportionality constant? Well yes, it is going to be two. This is a proportional relationship between the cups of milk
and the number of eggs. The cups of milk are always
going to be two times the number of eggs. Give me the number of eggs,
I'm going to multiply it by the constant of proportionality
to get the cups of milk. And we can see how this is
a proportional relationship a little bit clearer if we set up a table. So if we say number of eggs and if we say cups of milk and make a table here, well if you have one egg,
how many cups of milk are you gonna have? Well this right over here
would be one times two, well you're gonna have two cups of milk. If you had three eggs, well
you're just gonna multiply that by two to get your cups of milk, so you're gonna have six cups of milk. If you had 1,000,000 eggs, so
we have a very big party here, maybe we're some sort of
industrial muffin producer, well how many cups of milk? Well you put 1,000,000 in right over here, multiply it by two, you
get your cups of milk. You're going to need
2,000,000 cups of milk. And you can see that this is
a proportional relationship. To go from number of eggs to cups of milk, we indeed multiplied by two every time. That came straight from this equation. And you could also see,
look whenever you multiply your number of eggs by a certain amount, you're multiplying your cups
of milk by the same amount. If I multiply my eggs by 1,000,000, I'm multiplying my cups
of milk by 1,000,000. So this is clearly a
proportional relationship. Let's get a little bit
more practice identifying the constant of proportionality. So let's say I'll make it
a little bit more abstract, let's say I have some variable a and it is equal to five
times some variable b. What is the constant of
proportionality here? Pause this video and see
if you can figure it out. Yes, it is five. Give me a b, I'm gonna
multiply it by five, and I can figure out what a needs to be. Let's do another example. If I said that y is equal to pi times x, what is the constant
of proportionality here? Well you give me an x, I'm gonna
multiply it times a number, the number here is pi, to give you y. So our constant of
proportionality here is pi. Let's do one more. If I were to say that y is equal to 1/2 times x, what is the constant of proportionality? Pause this video. Think about it. Well once again, this is
just going to be the number that we're multiplying
by x to figure out y. So it is going to be 1/2. In general, you might sometimes
see it written like this. y is equal to k times x, where k would be some
constant that would be our constant of proportionality. You see 1/2 is equal to k here, pi is equal to k right over there. So hopefully that helps.