Question #5ad8d

1 Answer
Feb 10, 2017

Answer:

Solution given in a lot of detail so that you can see where everything come from. Used first principles method.
#A=2 1/2 " gallons"#
#B=2 1/2" gallons"#

Explanation:

Note that % is like a unit of measurement and it is worth #1/100#
10% is the same as #10xx1/100 = 10/100#

50% is the same as #50xx1/100=50/100#

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Considering the different parts")#

Given that the blend is 5 gallons

Let the amount of drink #color(red)("A be "x)#
Then the amount of drink #color(green)("B is "5-x)#

The amount of the juice in the blend is 30% so it is 30% of 5 gallons
#30/100xx5 = 30/20 = 3/2# gallons of pure juice

#color(red)("The amount of juice from A is 10% of "x->10/100x)#
#color(green)("The amount of juice from B is 50% of "(5-x) ->50/100(5-x))#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Putting all together to determine the value of A")#

#(10%" of A")+(50%" of B")" "=30% "of 5 gallons"#

#" "10/100x" "+" "50/100(5-x)color(white)(.)=" "30/100xx5#

#" "0.1x" "+" "2.5-0.5xcolor(white)(.)=" "1.5#

Don't like decimals so lets get rid of them.
Multiply everything by 10 giving:

#" "x" "+" "25-5xcolor(white)(.)=" "15#

#" "-4x+25=15#

Subtract 15 from both sides

#" "-4x+10=0#

Add #4x# to both sides

#" "10=4x#

Divide both sides by 4

#x=10/4 = 5/2 = 2 1/2# gallons = A
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Determine the amount of B")#

#B=5-A" "->" "B=5-2 1/2 = 2 1/2#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Check")#

#10%A+50%B=30%5#

RHS is #3/2# gallons of pure juice content.

Consider just the LHS:

#(10/100xx5/2)+(50/100xx5/2)= 3/2#

LHS = RHS so correct