Rearranging equation 1.4=(Delta e/1.9619 + Delta e) * 19.5 to solve for Delta e

[Amyt]

Hello,

I am a first time poster. The last time that I did maths was around 14 years ago at school, now I have gone back to Uni to study a Masters in Engineering Geology and I am struggling with rearranging equations.
This equation here is to help me know the change in the void ratio of a soil (Delta e).

I would like help to show me how to get to Delta e.

My equation so far is:

1.4=(Delta e/1.9619 + Delta e) * 19.5

Could someone please walk me through this?

Thank you.

 

[mmm4444bot]

\(\displaystyle \displaystyle 1.4 = \frac{\Delta e}{1.9619 + \Delta e} \cdot 19.5\)

Here's a symbolic listing of the steps; see whether you recognize the correspondence to your equation and whether the steps make sense. If not, let us know. If so, work through the symbolic example using paper and pencil; write down every step. Then, try your exercise again.

Solve for e

A = e/(B + e) ∙ C

Multiply the right-hand side

A = C∙e/(B + e)

Multiply each side by the denominator

A(B + e) = C∙e

Expand the left-hand side, using the Distributive Property

A∙B + A∙e = C∙e

We want the two e terms on the same side (so we can factor out that e) and we want the non-e term on the other side:

Subtract A∙B from each side

A∙e = C∙e - A∙B

Subtract C∙e from each side

A∙e - C∙e = -A∙B

Factor the left-hand side -- this combines like-terms A∙e and C∙e into a single term and the coefficient is A-C

(A-C)∙e = -A∙B

Divide each side by the coefficient: A - C

e = -A∙B/(A - C)

Edit: As noted by HallsofIvy in the next post, this result for e can be rewritten, if you want to eliminate that leading negative sign (multiply by -1/-1).

e = A∙B/(C - A)

If you need more help, please show what you tried on your exercise.

PS: Please try to write your decimal points on the baseline; centered dots denote multiplication. That is,

1∙4 means 1×4

1.4 means 1 + 4/10

Cheers :cool:

 

[mmm4444bot]

1.4 = (Delta e/1.9619 + Delta e) * 19.5

Note about texting ratios: In that example, the grouping symbols need to enclose the denominator, not the entire ratio:

1.4 = ∆e/(1.9619 + ∆e) * 19.5

 

[HallsofIvy]

\(\displaystyle \displaystyle 1.4 = \frac{\Delta e}{1.9619 + \Delta e} \cdot 19.5\)

Here's a symbolic listing of the steps; see whether you recognize the correspondence to your equation and whether the steps make sense. If not, let us know. If so, work through the symbolic example using paper and pencil; write down every step. Then, try your exercise again.

Solve for e

A = e/(B + e) ∙ C

Multiply the right-hand side

A = C∙e/(B + e)

Multiply each side by the denominator

A(B + e) = C∙e

Expand the left-hand side, using the Distributive Property

A∙B + A∙e = C∙e

Now we want to combine like-terms; move all e terms to one side and all non-e terms to the other side:

Subtract A∙B from each side

A∙e = C∙e - A∙B

Subtract C∙e from each side

A∙e - C∙e = A∙B

No, you should have A∙e - C∙e = -A∙B but it would probably be better to write it as C∙e - A∙e = A∙B.

Factor the left-hand side (symbolically, this combines like-terms A∙e and C∙e)

(A - C)∙e = A∙B

Divide each side by the coefficient on e: (A - C)

e = A∙B/(A - C)

If you need more help, please show what you tried on your exercise.

PS: Please try to write your decimal points on the baseline; centered dots denote multiplication. That is,

1∙4 means 1×4

1.4 means 1 + 4/10

Cheers :cool:

 

[Amyt]

Hello,

Thank you for your your quick reply.

I am not sure how you get to this step.

“Multiply the right-hand side

A = C∙e/(B + e)”

If I was to do as the method above would I not just be multiplying the top number and not the bottom, when I need to multiply both by C (or 19.5)?

I have attached a photo.

Please ignore my incorrect bracket placement.

 

[mmm4444bot]

No, you should have A∙e - C∙e = -A∙B …

Thanks for catching that copy-n-paste goof, Halls. (I ought to have used paper first.) Fixed it.

 

[mmm4444bot]

… "Multiply the right-hand side

A = C∙e/(B + e)”

… would I not just be multiplying the top number and not the bottom, when I need to multiply both by C (or 19.5)? …

Yes, you would just be multiplying the top. (I'm not sure how you got the idea that you "need to multiply both".)

When you multiply top and bottom both, you're actually multiplying by 19.5/19.5 instead of 19.5/1.

\(\displaystyle \dfrac{\Delta e}{1.9619 + \Delta e} \cdot 19.5\)


Think of the multiplication like this:

\(\displaystyle \dfrac{\Delta e}{1.9619 + \Delta e} \cdot \dfrac{19.5}{1}\)


not like this:

\(\displaystyle \dfrac{\Delta e}{1.9619 + \Delta e} \cdot \dfrac{19.5}{19.5}\)


The fraction 19.5/19.5 is actually 1.

 

[Amyt]

Aah!!! I get it
So every number is x/1 and then you just multiply it out.

That makes much more sense, and then the rest is just rearranging.

Thank you

 

[mmm4444bot]

… So every number is x/1 …

Yes! Any number may be expressed by writing it over 1. Even a fraction.

(3/5)/1

And, if it helps, any number may also be expressed by multiplying it by 1.

x = 1∙x

You can even think in terms of both, if it helps you with some situation.

x = 1∙x/1

There are situations in algebra where we often use these "hidden" factors of 1. Cheers :cool: