Math Is Fun Forum

mathxyz

Member

From: Brooklyn, NY

Registered: 2024-02-24

Posts: 1,053

Constant of Proportionality k

Why is a constant of proportionality (usually the variable k) needed in variation problems?

Bob

Administrator

Registered: 2010-06-20

Posts: 10,640

Re: Constant of Proportionality k

If it's direct proportion, then, as x increases so does y whilst keeping the ratio or proportion the same.

Let's say that x = 3 when y = 5.

If you double both then x = 6 and y = 10

Treble and you get x = 9 when y = 15

With fractions you already know that 3/5 = 6/10 = 9/15

With ratio you already know that 3:5 = 6:10 = 9:15

For the first pair, (3,5), you could write x = ky and work out that k = 3/5

For the second pair, (6,10) , you could write x = jy and work out that j = 6/10 = 3/5 = k

For the third pair, (9,15) ..... I'll leave you to finish this off.

So it is the properties of equivalent fractions that lead us to the general equation x = ky and we call k the constant of proportionality.

You can develop a similar argument for inverse proportion and for all the additional proprtions involving powers etc.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

mathxyz

Member

From: Brooklyn, NY

Registered: 2024-02-24

Posts: 1,053

Re: Constant of Proportionality k

If it's direct proportion, then, as x increases so does y whilst keeping the ratio or proportion the same.

Let's say that x = 3 when y = 5.

If you double both then x = 6 and y = 10

Treble and you get x = 9 when y = 15

With fractions you already know that 3/5 = 6/10 = 9/15

With ratio you already know that 3:5 = 6:10 = 9:15

For the first pair, (3,5), you could write x = ky and work out that k = 3/5

For the second pair, (6,10) , you could write x = jy and work out that j = 6/10 = 3/5 = k

For the third pair, (9,15) ..... I'll leave you to finish this off.

So it is the properties of equivalent fractions that lead us to the general equation x = ky and we call k the constant of proportionality.

You can develop a similar argument for inverse proportion and for all the additional proprtions involving powers etc.

Bob

For the pair (9, 15), I will let x = da. 

I get d = 9/15 = 3/5.

Yes?

Bob

Administrator

Registered: 2010-06-20

Posts: 10,640

Re: Constant of Proportionality k

That's right.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

mathxyz

Member

From: Brooklyn, NY

Registered: 2024-02-24

Posts: 1,053

Re: Constant of Proportionality k

That's right.

Bob

Cooking with gas like Richie Cunningham used to say on Happy Days.