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Two cans of unstirred pink paint

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.