Question 43·Easy·Ratios, Rates, Proportional Relationships, and Units

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A decorator mixes white paint and blue paint in a $3{:}2$ ratio by volume to create a custom shade. If exactly $6$ gallons of the custom paint are needed, how many gallons of blue paint should the decorator use?

A part-to-part ratio compares two ingredients, while a given total counts both together. Add the ratio parts to make the whole before finding the requested ingredient's fraction. Multiply that fraction by the total, using Desmos for the arithmetic. Check which ingredient the question asks for: using the first ratio number may give the other ingredient instead.

Hints

Hint 1
A part-to-part ratio matches the colors in the order named: $3$ equal parts white and $2$ equal parts blue. How many parts make up the whole mixture?
Hint 2
A part-to-whole fraction puts the requested color's parts on top and all the mixture's parts on the bottom. What fraction of the mixture is blue?

Step-by-step

Find blue's fraction of the whole
Step 1
Find blue's share
The ratio gives $3$ parts white and $2$ parts blue, so the whole mixture has $3+2$ equal parts. Blue gets $2$ of those parts:
$\frac{\text{blue paint}}{\text{all paint}}=\frac{2}{3+2}=\frac25.$
Using $\frac23$ here would compare blue with white, not with all the paint.
Step 2
Apply that share to the total
The $6$ gallons are the whole mixture, so multiply by blue's fraction to get blue gallons:
$\text{blue gallons}=6\left(\frac25\right).$
Step 3
Calculate the blue paint needed
Type $6\left(\frac25\right)$ in Desmos. It displays $2.4$ , so the decorator should use $2.4$ gallons of blue paint. Choice B.

Question 43·Easy·Ratios, Rates, Proportional Relationships, and Units

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