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# 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?

Last updated date: 17th Jul 2024
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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.

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.