# Adjusting the Relative Percentages of a Whole

by Thetom
 P: 56 Hi, how can I work out the relationship between percentages of a whole, when one of those percentages changes? I'm rubbish at maths and can't really explain myself properly, but I'll do my best. Here's the problem: I want to be able to workout the percentage of component colours in any mix of colour, and then change the amount of one component but keep the ratio between the other colours the same. I need to do this a hundred times, but here is just one as an example: I have a Purple (P) that I have mixed using... 30% (W)hite 55% (C)yan 11% (M)agenta 4% (Y)ellow I've tried to express this mathematically (and probably unconventionally) as... P = 30%W + 55%C + 11%M + 4%Y Now I need to mix a new colour (D) by decreasing the amount of white by half while keeping the relation between the other colours the same. I then need to find the percentages of these component colours to allow me to mix the new colour. Sooo... (30%W - 50% = 15%W) D = 15%W + ?%C + ?%M + ?%Y C = ?% M = ?% Y = ?% How can I find out the percentages of CMY once I have decrease W by a certain factor (in this case 50%)? It's important that the ratio between the other colours doesn't change, just their ratio to white. I've tried to explain my problem as clearly as I can, but I'm rubbish at maths. If it's not clear what I mean, please let me know. I can go into more detail, or include some diagrams which will allow me to explain it more clearly. Any help with this practical problem would be very, very.... helpful :)
 P: 15,294 Very strange colour palette you're using. CMYW? That's a new one. Anyway, 30% (W)hite 55% (C)yan 11% (M)agenta 4% (Y)ellow If W=15, then C = 55 + 55/70*15 M = 11 + 11/70*15 Y = 4 + 4/70*15 More generally, Given, A + B + C + D = 100 If A' = A-x then B' = B + B/(B+C+D)*(x) C' = C + C/(B+C+D)*(x) D' = D + D/(B+C+D)*(x) Example: W=35,C=27,M=22,Y=16 x = 25 W' = 10 C' = 27 + 27/(65)*25 M' = 22 + 22/(65)*25 Y' = 16 + 16/(65)*25 That could probably be boiled down further...
P: 56
 Quote by DaveC426913 Very strange colour palette you're using. CMYW? That's a new one.
Black and white simply adjust the value of a colour. Value is how light or dark a colour is. Adding white increases the value. Printers (for example) use CMYB, and the black decreases the value of the colour to make it darker. A printer doesn't use white ink though, it just uses the white from the paper. Really the palette is CMY + BW. I'm eventually gonna be using a pallet of CMY & RBG & BW . It's for mixing polymer clays.

Here's an example of the colours I've mixed using CMYW if your interested. The transition of colour at the bottom of the photo is the sum of the component colours at any given point along the event line . I'm trying to formulate a new way of blending clays. I can explain in more detail of your interested? I'm finding it hard to work out the maths part though.

 Quote by DaveC426913 If W=15, then C = 55 + 55/70*15 M = 11 + 11/70*15 Y = 4 + 4/70*15
Brilliant! Thanks, I just need to work that out, and make sure I understand what has happened so I can do it myself. I think I get the relationship...

Each component (CMY) has added to it... its own value divided by 70 (the old remaining percentage) times 15 (the new percentage of W). I think that explains it!?

I find it hard to express this stuff in English, let alone mathematically. The only way I've been able to do it so far is visually, by drawing out a graph on squared paper and counting the squares when I changed a value. Takes far to long doing that.

Thanks again. I'll try applying it to other problems to see if I got it.

P: 56

## Adjusting the Relative Percentages of a Whole

 Quote by DaveC426913 More generally, Given, A + B + C + D = 100 If A' = A-x then B' = B + B/(B+C+D)*(x) C' = C + C/(B+C+D)*(x) D' = D + D/(B+C+D)*(x)
Yes! I think I understand that. I'll try applying it and see what happens...
P: 15,294
 Quote by Thetom Each component (CMY) has added to it... its own value divided by 70 (the old remaining percentage) times 15 (the new percentage of W). I think that explains it!?
No, times W minus 15. See my example, where I made a change that is not 50%.

Do a few examples on paper, using round numbers to ensure it's working. I'd hate for you to apply it to the clay and have it come out all wrong.
P: 15,294
 Quote by Thetom Here's an example of the colours I've mixed using CMYW if your interested. The transition of colour at the bottom of the photo is the sum of the component colours at any given point along the event line . I'm trying to formulate a new way of blending clays. I can explain in more detail of your interested? I'm finding it hard to work out the maths part though.
This is fascinating. Can you walk me through it?
P: 56
 Quote by DaveC426913 This is fascinating. Can you walk me through it?
I would love too :) Give me a minuet to get some examlpes uploaded that explain what is going on.

I think I'm abit confused with this equation, actually.

I'll run through what I just did..

A = 30
B = 55
C = 11
D = 4

X = 50%
-----------------------------
A' = A-x

B' = B + B/(B+C+D) x (x)

B' = 55 + 55/ (55 + 11 + 4) x 50

B' = 55 + 55 / 70 x 50

B' = 55 + 55 / 3500

B' = 55 + 0.015714285714285714285714285714286

B' = 55.02

P: 56
 Quote by DaveC426913 No, times W minus 15. See my example, where I made a change that is not 50%.
Ok I did that now (i think) and my new answer for B' is 66.8.
 P: 56 Ok I think I got it... A' = 15 B' = 66.8 C' = 13.4 D' = 4.9 100.1 Total (due to a bit of rounding) I see where I went wrong now. I'll try some more examples on paper to test...
P: 15,294
 Quote by Thetom B' = 55 + 55 / 70 x 50 B' = 55 + 55 / 3500
You have done the order of operations wrong here. Order of operations is left to right, i.e.:

= ...+ 55 / 70 x 50
= ...+ 7.857 x 50
= ...+ 39.28
 P: 56 Am I right in thinking that X in this equation isn't the percentage that W was reduced by, but is the result of W being reduced by a certain percent? I have tried to resolve the example you gave in your first post. I think I understand the order of operations. I have included some brackets in my workings out to help show the order I did things. W=35,C=27,M=22,Y=16 x = 25 W' = 10 C' = 27 + (27 / 65) x 25 C' = 27 + 10.4 C' = 37.4 M' = 22 + (22 / 65) x 25 M' = 22 + 8.5 M' = 30.5 Y' = 16 + (16 / 65) x 25 Y' = 16 + 6.2 Y' = 22.2 Totals 100.1. This looks promasing :)
 P: 15,294 I'm working on a calculator but haven't got it working yet. http://www.davesbrain.ca/science/balancerizer.html Ironically, the calculation is working fne, it's the field entry giving me grief. I'll have to away to bed and finish it tomorrow.
 P: 56 Ok' im still totally stuck. I just don't get it :( Here's a new equation... I'll just show the workings for C. I'm not sure of the correct usage of brackets. I have just put brackets around parts that were done first. What am I doing wrong?.. C = 58, M = 10, Y = 5, W = 27 --------------------------- W' = W - 29% W' = 27 - 7.83 W' = 19.17 -------------------------- C' = C + C/ (C+M+Y) x W' C' = 58 + 58/(58+10+5) x 19.17 C' = 58 + (58/73) x 19.17 C' = 58 + (0.79 x 19.17) C' = 58 + 15.23 C' = 73.28 The problem is that when I do it for Y and M they all total greater than 100. Where am I going wrong. I'm totaly lost in the process and can't see what is happening.
P: 56
 Quote by DaveC426913 I'm working on a calculator but haven't got it working yet. http://www.davesbrain.ca/science/balancerizer.html Ironically, the calculation is working fne, it's the field entry giving me grief. I'll have to away to bed and finish it tomorrow.
Oohhh, that looks amazing! If you can make it work I'll buy it off you. That would be really, really helpful. Thanks for even trying it! :D.

I'll write up a little explination of what i'm doing for your reading pleasure...
 P: 56 Here's a fuller description of the problem I'm having. Below shows a linear transition from blue to white. At event 1 white is 0%, at event 2 white is 50% and event 3 white is 100%. And here is what I mean by a nonlinear transition. The amount of white increases exponentially (it may not be exponential, nonlinear may be a more accurate description). (note: i just made those values up, they may not correspond to the curve.) Now, finding the percentage of the component colours, when there is only two colours is easy. I can just subtract the percentage of white and the remainder is the percentage of blue. But when there are three or more colours it is much harder (for me). A liner transition of blue white and yellow (Fig.3)... But if I do the same for a nonlinear transition, it doesn't work... The curve of the white has increased the amount of yellow compared with the amount of blue. So I need a way of increasing one of the component colours while maintaining the relationship between the other colours. So referring back to fig.3 I can see that at event 2 there is 50% blue 25% yellow and 25% white. or there is twice as much blue than yellow. I know, from plotting the nonlinear curve of white, that at event 2 I want there to be only 15% white (as an example). So how to now adjust the amount of blue and yellow accordingly. This is my problem. I still havent managed to get my head around the maths of it yet. ------------------------------------------------------------------------------ So using the original values I gave in the OP, event one was: W = 30% C = 55% M = 11% Y = 4% The next event saw a decrease in white by 50% and me stumped.. W = 15% C = ? M = ? Y = ? There will be a whole series of events like this. Ideally, it should be possible to not only adjust the white in a nonlinear fashion, but also adjust the other colours, relative to each other. So the amount of white may increase at one rate, the amount of cyan increases at another rate, but the amount of yellow and magenta both increase at the same rate. I would like to be able to do this to any number of colours. Hopefully that has illustrated how all this is being applied practically - so you can see exactly what i'm doing with these numbers.
P: 15,294