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

Jul 5, 2018

$\frac{2}{3}$ or $0. \setminus \overline{6}$ cups.

## $\setminus \textcolor{s e a g r e e n}{\setminus \textrm{N e w a n s w e r}}$

• You have $30$ cups. 8% of those cups are grapefruit juice. To
calculate: $\frac{8}{100} \times 30 = 2.4$ cups are juice.

Remember that 92% of this punch is non-juice. $\frac{92}{100} \times 30 = 27.6$ cups of non-juice.

• But now we want to increase the volume, by adding juice cups, until we have 10% juice. Let's make the amount added $x$.

Thus, $30 + x$ cups will be our new total punch volume.

• We will still have 27.6 cups of punch that is NOT juice! But in our new volume, that will only be 90% of the punch.
$\frac{27.6}{\left(30 + x\right)} = \frac{90}{100}$
$\frac{90}{100} \left(30 + x\right) = 27.6$
$0.90 \left(30 + x\right) = 27.6$
• Let us solve for $x$ to find the amount of grapefruit juice added.
$\frac{27.6}{0.90} = 30 + x$
$x = \frac{27.6}{0.90} - 30$
$x \setminus \approx 0.66666 \ldots$ or $x = 0. \setminus \overline{6}$
• This is also known as $\frac{2}{3}$.
Therefore, you add two-thirds of a cup to the punch, and you will have 10% juice.

Checking results:

Remember that we had $2.4$ cups of juice when we started. Now we have $\frac{2}{3}$ cup more of juice.

• Let's divide to calculate how much of our new amount is juice.
$\frac{2.4 + \frac{2}{3}}{30 + \frac{2}{3}} = 0.1$
• To get percentage, multiply by 100%.
0.1xx100%=10%\color(green)(√)

## \color(red)(\text(Old answer)

$30$ cups. 8% of those 30 cups is juice.
This means $\frac{8}{100} \times 30 \setminus \textrm{\cup s} = 2.4$ cups worth of juice.

You want 10% of those 30 cups to be juice, though.
That means $\frac{10}{100} \times 30 \setminus \textrm{\cup s} = 3$ cups worth of juice.

To get from 2.4 cups to 3 (8% to 10%), we need to subtract:
$3 - 2.4 = 0.6$ cups.

We need 0.6 cups of juice to get 10% grapefruit juice punch.

Another method:
10%-8%=2%
So we need a 2% increase to hit 10% grapefruit juice.

Let's multiply:
$\frac{2}{100} \times 30 \setminus \textrm{\cup s} = 0.6$ cups of juice that need to be added.

Jul 5, 2018

$\frac{2}{3}$ a cup of 100% juice should be added.

Much better to stick to fractions as rounding errors occur when using decimal.

#### Explanation:

Lets consider the information we have been given:

Let the volume of the 8% juice be color(white)("dd")j_8 = 30" cups"

Let the volume of the 100% juice be ${j}_{100} \text{ cups "larr" unknown}$

Let the blend volume of 10% juice be ${j}_{10} \text{ cups "larr"unknown}$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Using the above information consider volume:

${j}_{8} + {j}_{100} = {j}_{10} \leftarrow \text{ in cups}$

$30 + {j}_{100} = {j}_{10} \leftarrow \text{ by volume in cups.....Equation(1)}$

Using the above information consider percentage:

[8%xxj_8]+[100%xxj_100]=[10%xxj_10]" "...Equation(2)

but from $E q u a t i o n \left(1\right) \textcolor{w h i t e}{\text{dd}} {j}_{10} = 30 + {j}_{100}$
Also we know that ${j}_{8} = 30$ so we substitute this into $E q n \left(2\right)$

[color(white)(2/2)8%xx30color(white)("d")]+[color(white)(2/2)100%xxj_100color(white)("d")]=[color(white)(.)10%xx(30+j_100)]

$\textcolor{w h i t e}{\text{ddd")[240/100]color(white)("dddd")+color(white)("ddd")[(100j_100)/100]color(white)("ddd}} = \left[\frac{300}{100} + \frac{10 {j}_{100}}{100}\right]$

Multiply both sides by 100

$240 + 100 {j}_{100} = 300 + 10 {j}_{100}$

$100 {j}_{100} - 10 {j}_{100} = 300 - 240$

$90 {j}_{100} = 60$

${j}_{100} = \frac{60}{90} = \frac{6}{9} = \frac{2}{3} \text{ cups}$

Note that $\frac{2}{3}$ is a bite more that $0.6$