Q14. - Algebra 2

Question

One half quart of lemonade concentrate is mixed with $\mathbf{1}\frac{\mathbf{1}}{\mathbf{2}}$ quarts of water to make lemonade for 6 people. If you use the same proportions of concentrate and water, how many quarts of lemonade concentrate are needed to make lemonade for 21 people? 

Step-by-Step Solution

Answer

The quarts of lemonade concentrate that are needed to make lemonade for 21 people are $\frac{\mathbf{7}}{\mathbf{4}}$.

Step 1 ­- Description of step.

It is given that one half quart of lemonade concentrate is mixed with $1\frac{1}{2}$ quarts of water to make lemonade for 6 people.

One half is equivalent to $\frac{1}{2}$ and  $1\frac{1}{2}$ is equivalent to $\frac{3}{2}$.

The ratio in which the lemonade concentrate is mixed with water is:

$\begin{array}{c}\frac{1}{2}:\frac{3}{2}=\frac{\frac{1}{2}}{\frac{3}{2}}\\ =\frac{1}{3}\\ =1:3\end{array}$

Step 2 ­- Description of step.

Therefore, the total quarts of the solution is the sum of quarts of lemonade concentrate and quarts of water.

Therefore, it can be obtained that:

$\begin{array}{c}thetotalquartsofthesolution=\frac{1}{2}+\frac{3}{2}\\ =\frac{4}{2}\\ =2\end{array}$

Now as the total quarts of solution is served to 6 people.

That implies 2 quarts of solution is served to 6 people.

Therefore, one person will get:

 $\frac{2}{6}=\frac{1}{3}$ quarts of solution.

Step 3 ­- Description of step.

As, one person is getting $\frac{1}{3}$ quarts of solution, therefore 21 persons will be get:

$\frac{1}{3}\times 21=7$ quarts of solution.

 

Now, as the same proportion of lemonade concentrate and water is used, therefore still the ratio of lemonade concentrate and water is $1:3$.

Therefore, 7 quarts of solution is to be divided in the ratio of  $1:3$ to get the amount of lemonade concentrate and water.

Therefore, it can be obtained that:

$\begin{array}{c}lemonadeconcentrate=\frac{7}{1+3}\\ =\frac{7}{4}\end{array}$

Therefore, the lemonade concentrate is $\frac{7}{4}$ quarts.