Firstly, I will simply give the ratios that are the answers to the given questions before I give the general formula for finding the number of pots of each paint:
Paints A and B:
1:4 : 3 pots of A and 2 pots of B.
1:5 : 1 pot of A and 2 pots of B.
1:6 : 1 pot of A and 6 pots of B.
Paints C and D
1:5 : 2 pots of A and 1 pots of B.
1:6 : 3 pots of A and 4 pots of B.
1:7 : 1 pots of A and 3 pots of B.
1:8 : 1 pot of A and 8 pots of B.
It is possible to find a ratio of paint 1:z with 2 pots of paint with ratios 1:x and 1:y where x<z<y. To show this, I will use simultaneous equations:
To start off, we first need to know the fact that the mixing of 2 paints essentially gives us the average of the two ratios. Eg; mixing 1:2 and 1:5 gives us 1:4, which is ((â…“)+(â…™))/2.
So, let:
The number of pots of paint A = a,
The number of pots of paint B = b,
The ratio of paint A = 1:x,
The ratio of paint B = 1:y,
And, the target ratio = 1:z.
Since we know that the mixing of paints A and B will give us the mean of the two ratios:
(a(1/(x+1)) + b(1/(y+1)))/a+b = 1/(z+1)
(I am using x+1, y+1, and z+1 rather than x, y, and z because both terms of a ratio add up to make a whole. Eg; 1:3 means that something is divided up into ¼ and ¾, not ⅓ and 3/3)
Now, we split the equation above into 2 parts, one being the numerator of both sides of the equation and the other being the denominator:
a(1/(x+1)) + b(1/(y+1)) = 1
=> (a(y+1) + b(x+1))/(x+1)(y+1) = 1
=> a(y+1) + b(x+1) = (x+1)(y+1)
a + b = z+1
=> a = (z+1) - b
Substituting 2) into 1),
((z+1) - b)(y+1) + b(x+1) = (x+1)(y+1)
=> (z+1)(y+1) - b(y+1) + b(x+1) = (x+1)(y+1)
=> b((x+1)-(y+1)) = (x+1)(y+1)-(z+1)(y+1)
=> b(x-y) = (y+1)((x+1)-(z+1))
=> b = ((y+1)(x-z))/(x-y)
Therefore, a = (z+1) - ((y+1)(x-z))/(x-y)
Now, let’s go back to the answers I gave above and see if my general solution works:
Paint A and B, target: 1:4 :
x = 3, y = 7, z = 4
a = 5 - (8*(-1))/-4
= 3
b = (8*-1)/-4 = 2
Answer = 3 pots of A and 2 pots of B.
However, this general solution does give answers in fractions or in unsimplified form,
Eg: Paint C and D, target: 1:7
x = 4, y = 9, z = 7
a = 8 - (10*-3)/-5
= 6
b = (7+1) - 6 = 2
Answer = 6 pots of A and 2 pots of B.
But this is not the least number of pots that can be used, so we simplify the answer to 3 pots of A and 1 pot of B.