Ratios and Proportions: Finding a,b,c from s-a : s-b : s-c

[zorro]

Homework Statement

Let s-a : s-b : s-c :: 1:2:3

then how do we find a:b:c from this?

Homework Equations

The Attempt at a Solution

 

[Borek]

Try to format your expression in such a way that it is unambiguous what you are looking for.

 

[zorro]

How do I do that?

 

[Borek]

You may try to use LaTeX.

https://www.physicsforums.com/showthread.php?t=8997

 

[Mark44]

Homework Statement

Let s-a : s-b : s-c :: 1:2:3

then how do we find a:b:c from this?

Rewrite this proportion as three equations. The proportion is saying is that s - b is 2 times s - a, s - c is 3 times s - a, and s - c is (3/2) times s - b.

That should give you somewhere to start.

 

[zorro]

Thanks a lot

 

[HallsofIvy]

Homework Statement

Let s-a : s-b : s-c :: 1:2:3

then how do we find a:b:c from this?

Homework Equations

The Attempt at a Solution

That's slightly confusing because it represents several proportions together.
You can analyze it as $s-a: s-b::1:2$, $s-b: s-c::2: 3$, and [itex]s-a: s-c::1: 3[itex]. Those can be written as fraction:
$$\frac{s-a}{s-b}= \frac{1}{2}$$
$$\frac{s-b}{s-c}= \frac{2}{3}$$
and
[tex]\frac{s-a}{s-c}= \frac{1}{3}

The first equation can be rewritten as 2(s- a)= s- b, 3(s-b)= 2(s- c), and 3(s-a)= s- c.

You can solve each of those for s: 2s- 2a= x- b so s= 2a- b. 3s- 3b= 2s- 2c so s= 3b- 2c. 3s- 3a= s- c so 2s= 3a- c or s= (3/2)a- (1/2)c.

Now put them back together: s= 2a- b= (2/3)a- (1/2 c, s= 2a- b= 3b- 2c. You should be able to find the relationships between a, b, and c from that.

 

[zorro]

That's slightly confusing because it represents several proportions together.
You can analyze it as $s-a: s-b::1:2$, $s-b: s-c::2: 3$, and [itex]s-a: s-c::1: 3[itex]. Those can be written as fraction:
$$\frac{s-a}{s-b}= \frac{1}{2}$$
$$\frac{s-b}{s-c}= \frac{2}{3}$$
and
[tex]\frac{s-a}{s-c}= \frac{1}{3}

The first equation can be rewritten as 2(s- a)= s- b, 3(s-b)= 2(s- c), and 3(s-a)= s- c.

You can solve each of those for s: 2s- 2a= x- b so s= 2a- b. 3s- 3b= 2s- 2c so s= 3b- 2c. 3s- 3a= s- c so 2s= 3a- c or s= (3/2)a- (1/2)c.

Now put them back together: s= 2a- b= (2/3)a- (1/2 c, s= 2a- b= 3b- 2c. You should be able to find the relationships between a, b, and c from that.

That made it more clear.
I got a:b:c :: 5:4:3
Thanks a lot Hallsofivy