Please show me how to solve the following ratio problem.

To make a chocolate milkshake, 3 parts chocolate syrup are mixed with 2 parts malt. To make a vanilla milkshake, 2 parts vanilla syrup are mixed with 1 part malt. If equal parts chocolate and vanilla milkshakes are mixed, what is the proportion of malt to milkshake? In addition to the answer, please provide me with the steps taken to solve the problem.

My name is Linda, this is a 10-12 grade problem and I am the parent of a 10th grader.
Thanks for your help!

Hi Linda,

It helps me to think of a "part" as something concrete, like a cup. Thus the chocolate recipe makes 5 parts (cups) and the vanilla recipe makes 3 parts (cups). I want to mix equal amounts of the two recipes and I can do so by making 3 chocolate recipies (15 cups) and 5 vanilla recipies (15 cups). That is, the least common multiple of 5 and 3 is 15.

Thus I mix together

3 chocolate recipes 5 vanilla recipes
3x3 = 9 parts chocolate 5x2 = 10 parts vanilla
3x2 = 6 parts malt 5x1 = 5 parts malt

Thus in total there are 9 + 10 = 19 parts flavouring and 6 + 5 = 11 parts malt in a 30 part milkshake. Hence the proportion of malt to milkshake is 11 to 30.

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