Question

A tin of paint of a particular shade of color is a mixture of red, blue, and yellow. \[\dfrac{1}{4}\] of the total volume of paint is made up of red paint and the ratio of the volume of red paint to that of blue paint is 3: 2. Find the total volume of paint that will contain 84 mL of yellow paint.
(A) 120 mL
(B) 140 mL
(C) 144 mL
(D) 168 mL

Answer 1

Hint: Assume that \[x\] mL is the total volume of paint that will contain 84 mL of yellow paint. The volume of red paint is \[\dfrac{x}{4}\] mL. Now, use the information that the ratio of the volume of red paint to that of blue paint is 3: 2 and calculate the volume of blue paint. Calculate the volume of yellow paint by subtracting the volume of red paint and blue paint from the total volume. Finally, compare it with 84 mL and calculate the value of \[x\] .

Complete step by step answer:
According to the question, we are given that a tin of paint of a particular shade of color is a mixture of red, blue, and yellow.
The fraction of the total volume of paint is made up of red paint = \[\dfrac{1}{4}\] ……………………………………….(1)
The ratio of the volume of red paint to that of blue paint is 3: 2,
\[\dfrac{\text{Volume}\,\text{of}\,\text{red}\,\text{paint}}{\text{Volume}\,\text{of}\,\text{blue}\,\text{paint}}\text{=}\dfrac{\text{3}}{\text{2}}\] ………………………………………….(2)
First of all, let us assume that \[x\] mL is the total volume of paint that will contain 84 mL of yellow paint …………………………………………..(3)
Now, from equation (1) and equation (3), we get
The volume of red paint = \[\dfrac{x}{4}\] mL ……………………………………..(4)
Now, from equation (2) and equation (4), we get
\[\Rightarrow \dfrac{\dfrac{x}{4}}{\text{Volume}\,\text{of}\,\text{blue}\,\text{paint}}\text{=}\dfrac{\text{3}}{\text{2}}\]
\[\Rightarrow \] Volume of blue paint = \[\dfrac{2}{3}\times \dfrac{x}{4}\]
\[\Rightarrow \] Volume of blue paint = \[\dfrac{2x}{12}\] ……………………………………(5)
Now, from equation (4) and equation (5), we get
The volume of red paint and blue paint = \[\dfrac{x}{4}+\dfrac{2x}{12}=\dfrac{5x}{12}\] ……………………………..(6)
The total volume of the paint = \[x\] mL (from equation (3)) ………………………………………….(7)
On subtracting equation (6) from equation (7), we get
The volume of yellow paint = \[x-\dfrac{5x}{12}=\dfrac{7x}{12}\] mL ………………………………….(8)
 Now, from equation (3), we have
The volume of yellow paint = 84 mL ……………………………………………(9)
On comparing equation (8) and equation (9), we get
\[\begin{align}
  & \Rightarrow \dfrac{7x}{12}=84 \\
 & \Rightarrow x=\dfrac{84\times 12}{7} \\
 & \Rightarrow x=144 \\
\end{align}\]
Therefore, the total volume of the paint is 144 mL.

Hence, the correct option is (C).

Note:
Here, one might get the equation as $x+\dfrac{x}{4}+\dfrac{2x}{12}=84$ and get wrong results. Also, one might be careful while writing the ratio as 3:2 is Red: Blue and if they make mistakes while taking the ratio, it will not be right. In this question, the main problem faced is that one might get confused for the approach because of the complex word statement. The information is given with respect to something. Therefore, for this type of question, just assume the total volume as $x$ and then follow the steps as per information provided in the question. It will be easier to figure out the required information in mathematical terms.