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?

2 Answers
Jul 5, 2018

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.

Jul 5, 2018

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