Why do this problem?
This problem follows on from Mixing Paints
and encourages students to think about ratio in new ways, explore using an interactive environment, and come up with some generalisations and proofs. The most general case will require perseverance to discover and prove.
This problem could be used in a follow-up lesson after working on Mixing Paints
, or as an extension activity for some students.
Students could start by using the interactivity to experiment, and then gradually move to pen-and-paper methods, only using the interactivity to check.
If I am mixing $1:4$ paint with $1:5$ paint, why might it be useful to start with 30 litres of paint?
If I want to mix $1:x$ paint with $1:y$ paint, what would I start with instead of 30 litres?
Coming up with a general strategy for making any ratio from any paint is a challenging extension task.
Make sure students are secure in any general strategies they came up with for Mixing Paints
before embarking on this task.