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?

#\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.

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#