Question

How many cups of juice must be added to a 30 cup punch that is 8% grapefruit juice, to obtain a punch that is 10% grapefruit?

Answer 1

Hint: Here, we will first find the cups of juice in 30 cups. Then we will find the cups of non-juice in it. Then we will assume the amount of juice to be added to increase the percent in the punch to be a variable. We will then frame a linear equation based on the given information and simplify it further to get the required answer.

Complete step-by-step answer:
We have 30 cups from which 8% is grapefruit juice.
So, number of cups of juice \[ = 8\% \] of 30
\[ \Rightarrow \] The number of cups of juice \[ = \dfrac{8}{{100}} \times 30\]
Multiplying the terms, we get
\[ \Rightarrow \] The number of cups of juice \[ = 2.4\] Cups of juice
As, 8% is grapefruit juice so remaining 92% is non-juice.
So, the number of cups of non-juice \[ = 92\% \] of 30
\[ \Rightarrow \] The number of cups of non-juice \[ = \dfrac{{92}}{{100}} \times 30\]
Multiplying the terms, we get
\[ \Rightarrow \] The number of cups of non-juice \[ = 27.6\] Cups of non-juice
Now, as we want to increase the volume of juice, we will assume that we added \[x\] amount of juice.
$\therefore $ New total punch \[ = 30 + x\] cups
We still have \[27.6\] cups of punch that are not juice.
But in our new volume we will have only 90% of punch so,
\[\dfrac{{27.6}}{{\left( {30 + x} \right)}} = 90\% \]
Converting percentage into fraction, we get
\[ \Rightarrow \dfrac{{27.6}}{{30 + x}} = \dfrac{{90}}{{100}}\]
On cross multiplication, we get
\[ \Rightarrow 27.6 \times 100 = 90 \times \left( {30 + x} \right)\]
Multiplying the terms, we get
\[ \Rightarrow 2760 = 2700 + 90x\]
Subtracting both sides by 2700, we get
\[\begin{array}{l} \Rightarrow 2760 - 2700 = 90x\\ \Rightarrow 60 = 90x\end{array}\]
Dividing both sides by 90, we get
\[ \Rightarrow x = \dfrac{{60}}{{90}}\]
Simplifying further, we get
 \[x = \dfrac{2}{3} = 0.667\]

So, that means we have to add two-third or \[0.67\] cups of juice to the punch to make a \[10\% \] grapefruit.

Note:
Percentage is any number or ratio that can be expressed as a fraction of 100. It is a dimensionless number which has no measurement unit. The term percentage means out of a hundred. While considering a percent we take a whole which is made up of hundred equal parts. We generally use % sign and not the abbreviate ‘pct’. We have framed a linear equation here. A linear equation is defined as the equation that has the highest degree of 1 and has one solution.