Mixture problems involve combining parts, solutions, or liquids in given ratios or concentrations. The problem usually describes a change to the mixture - for example, adding a new liquid or adding more of one component already present.
Your job is to:
As you set up your equations, be consistent about what each variable represents (total amount of drink, amount of milk, amount of coffee, etc.).
Mixture problems can vary a lot, so the best preparation is practice. Try these examples.
If Rhea likes her coffee-to-milk ratio at to , and Asia likes her coffee-to-milk ratio at to , what ratio of Rhea’s to Asia’s drinks is required to make a mixture with a to ratio of coffee to milk?
A. to
B. to
C. to
D. to
E. to
Answer: B. to
To get a : coffee-to-milk ratio, you need to mix some of Asia’s drink with some of Rhea’s drink, because : lies between their preferred ratios of : and :. The question asks for the ratio of the amounts you should mix.
Begin by writing an equation. One convenient approach is to write everything in terms of the amount of milk (you could choose coffee instead, but stick to one choice).
Let:
Asia’s ratio is parts coffee to part milk. That’s total parts, so milk is of Asia’s drink.
Rhea’s ratio is parts coffee to part milk. That’s total parts, so milk is of Rhea’s drink.
The total milk in the combined mixture is:
The target ratio is parts coffee to part milk. That’s total parts, so milk is of the final mixture. The total amount of mixture is , so the milk in the final mixture is:
Set the two expressions for milk equal and solve:
So the ratio of Rhea to Asia is to .
Using weighted averages
A common method for mixture problems is the weighted average equation. (If you want a refresher, revisit the “mean, median, mode, and range” chapter.) Use weighted averages to solve the problem below.
A very hot mixture consists of three solutions, each containing a specific concentration of a melted rare metal.
Solution Rare metal concentration level A 5% B 15% C 50%
If of the mixture is made up of Solution A, is made up of Solution B, and is made up of Solution C, approximately what is the concentration of the rare metal in the mixture? Round to the nearest percent.
Answer:
Set up a weighted average. Remember:
Key information (as decimals):
Compute the mixture’s concentration:
So the concentration of the rare metal in the mixture is approximately .
As a quick check, notice that is between the smallest and largest concentrations (between and ). This is a useful fail-safe: a weighted average must fall between the minimum and maximum of the values being averaged. If your result is outside that range, something went wrong in your setup or arithmetic.
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