Question

How much $90$ percent vinegar does you have to add to a gallon of $5$ percent vinegar to get $20$ percent vinegar?

Answer 1

Hint: In this problem we need to calculate the amount of vinegar to be added to meet the required concentration of the vinegar according to the problem. In the problem they have given that there is a one gallon with $5$ percent vinegar. So, we will treat it as the initial amount of vinegar we have. Now we need to add some amount of vinegar to the existing vinegar. So, we will assume the amount of vinegar to be added is $x$ gallons. Now we will calculate the concentration of the vinegar added to the existing vinegar that is $90$ percentage in the amount of vinegar added. After adding the vinegar, we will calculate the concentration of the vinegar by adding both the values and equate it to the $20$ percent of the final vinegar amount to get the required solution.

Complete step by step solution:
Given that one gallon has $5$ percent vinegar.
The initial vinegar concentration we have is given by
$\begin{align}
  & i=5\%\text{ of }1 \\
 & \Rightarrow i=\dfrac{5}{100}\times 1 \\
 & \therefore i=\dfrac{1}{20} \\
\end{align}$
Let $x$ gallons of vinegar are added to the initial vinegar, then the final amount of vinegar is $x+1$.
The concentration of the vinegar which is added to initial vinegar is $90$ percent and the value is
$\begin{align}
  & j=90\%\text{ of }x \\
 & \Rightarrow j=\dfrac{90}{100}\times x \\
 & \therefore j=\dfrac{9x}{10} \\
\end{align}$
Now the final concentration of the vinegar is calculated by adding the both the concentration, then we will get
$\begin{align}
  & f=\dfrac{1}{20}+\dfrac{9x}{10} \\
 & \therefore f=\dfrac{18x+1}{20} \\
\end{align}$
But in the problem, they have mentioned that the final concentration of the vinegar is $20$ percent. So, equating the both the values, then we will get
$\begin{align}
  & \dfrac{18x+1}{10}=20\%\text{ of }\left( x+1 \right) \\
 & \Rightarrow \dfrac{18x+1}{20}=\dfrac{20}{100}\left( x+1 \right) \\
\end{align}$
Simplifying the above equation, then we will get
$\begin{align}
  & 18x+1=\dfrac{20\times 20}{100}\left( x+1 \right) \\
 & \Rightarrow 18x+1=4x+4 \\
 & \Rightarrow 14x=3 \\
 & \therefore x=\dfrac{3}{14} \\
\end{align}$
Hence, we need $\dfrac{3}{14}$ gallons.

Note: This problem comes with a confusing language. So please read the question two or three times to get a better clarity on the question. After that many students may make mistakes at the concentration of vinegar after mixing. After mixing vinegar the quantity also changes so we need to consider the modified quantity.