Question

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?

Answer 1

`2/3` or `0.\bar(6)` cups.

Explanation:

`\color(seagreen)(\text(New answer))`

• You have `30` cups. `8%` of those cups are grapefruit juice. To
  calculate: `8/100xx30=2.4` cups are juice.

  Remember that 92% of this punch is non-juice. `92/100xx30=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.
  `27.6/((30+x))=90/100`
  `90/100(30+x)=27.6`
  `0.90(30+x)=27.6`
• Let us solve for `x` to find the amount of grapefruit juice added.
  `27.6/0.90=30+x`
  `x=27.6/0.90-30`
  `x\approx0.66666...` or `x=0.\bar(6)`
• This is also known as `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 `2/3` cup more of juice.

• Let's divide to calculate how much of our new amount is juice.
  `(2.4+2/3)/(30+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 `8/100xx30\text( cups)=2.4` cups worth of juice.

You want `10%` of those 30 cups to be juice, though.
That means `10/100xx30\text( cups)=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:
`2/100xx30\text( cups)=0.6` cups of juice that need to be added.

Answer 2

`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" cups "larr" unknown"`

Let the blend volume of 10% juice be `j_10" cups "larr"unknown"`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Using the above information consider volume:

`j_8+j_100=j_10 larr" in cups"`

`30+j_100=j_10 larr" 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 `Equation(1)color(white)("dd") j_10=30+j_100`
Also we know that `j_8=30` so we substitute this into `Eqn(2)`

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

`color(white)("ddd")[240/100]color(white)("dddd")+color(white)("ddd")[(100j_100)/100]color(white)("ddd")=[300/100+(10j_100)/100]`

Multiply both sides by 100

`240+100j_100=300+10j_100`

`100j_100-10j_100=300-240`

`90j_100=60`

`j_100=60/90 = 6/9=2/3" cups"`

Note that `2/3` is a bite more that `0.6`