Question

Question #5ad8d

Answer 1

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))`
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`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
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`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