Question

Vikrant’s recipe for fruit punch called for $5$ parts of water to $2$ parts of punch mix. Set up a proportion and find out how much water did he need for $8$ cups of punch mix.

Answer 1

Hint: The cup is made up of $7$ parts. Set up the ratio of water to punch mix for making a cup of fruit punch. Next, formulate the problem being asked by setting up another ratio for the second case. These two ratios will be equal. So find the amount of water required by equating these two ratios.

Complete step-by-step answer:
In this question we can consider one cup to be made up of $5 + 2 = 7$ parts. So essentially there are two units of measurement: Cups and parts, such that $1$ cup = $7$ parts.
Next we see that to make $1$ cups of fruit punch we need $5$ parts of water and $2$ cups of punch mix. Therefore, the ratio of water to punch mix for $1$ cup of punch mix is:
Water : Punch mix= $5:2$
Let us now convert $8$ cups of punch mix to the unit of parts.
$8$ cups of punch mix= $8 \times 7 = 56$ parts of punch mix.
We need to find out how much water is required for $56$ parts of punch mix. Let’s assume it will be $x$ parts. So the ratio will be $x:56$. We need to calculate $x$.
We can now set up a proportion because both these ratios should be equal
Therefore, $5:2::x:56$
$
   \Rightarrow \dfrac{5}{2} = \dfrac{x}{{56}} \\
   \Rightarrow x = \dfrac{{5 \times 56}}{2} \\
   \Rightarrow x = 140 \\
 $
So we need $140$ parts of water or $\dfrac{{140}}{7} = 20$cups of water.
Therefore, the answer is $20$ cups of water.

Note: Start by setting up the ratios of the two cases. Remember that since both ratios are meant for the same thing, that is, fruit punch, they will be in proportion to each other. Then set up the proportion and solve it to find the amount of water required. Be careful of the units while doing so. When constructing the ratios all the quantities must be in the same unit, either parts or cups.