This problem follows on from Mixing Paints .

A decorator can buy pink paint from two manufacturers.

• Paint A is made up from red and white paint in the ratio $1:4$
• Paint B is made up from red and white paint in the ratio $1:5$

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

What is the least number she would need of each type in order to produce pink paint containing red and white in the following ratios:
• $2:9$
• $3:14$
• $10:43$
Another decorator buys pink paint from two different manufacturers:
• Paint C is made up from red and white paint in the ratio $1:3$
• Paint D is made up from red and white paint in the ratio $1:7$
What is the least number she would need of each type in order to produce pink paint containing red and white in the following ratios:
• $2:9$
• $3:14$
• $10:43$

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 $a:b$ ? Experiment with a few more examples.

Can you describe an efficient way of doing this?