A decorator can buy pink paint from two manufacturers.

- Paint A is made up from red and white paint in the ratio $1:3$
- Paint B is made up from red and white paint in the ratio $1:7$

He can mix the paints to produce a different shade of
pink.

If Paint A and Paint B come in same size cans, what is the least number he would need of each type in order to produce pink paint containing red and white in the following ratios:

- $1:4$
- $1:5$
- $1:6$

Another decorator buys pink paint from two different
manufacturers:

- Paint C is made up from red and white paint in the ratio $1:4$
- Paint D is made up from red and white paint in the ratio $1:9$

What is the least number he would need of each type in order to produce pink paint containing red and white in the following ratios:

- $1:5$
- $1:6$
- $1:7$
- $1:8$

Is it always possible to combine two paints made up in the
ratios $1:x$ and
$1:y$ and turn them into
paint made up in the ratio $1:z$ ? (where $x < z < y$)

Experiment with a few more examples.

Can you describe an efficient way of doing this?

Mixing More Paints is a follow-up question to this one.