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This problem follows on from Mixing Paints .

Open cans of pink paints A and B

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?