# Question 5ad8d

Feb 10, 2017

Solution given in a lot of detail so that you can see where everything come from. Used first principles method.
$A = 2 \frac{1}{2} \text{ gallons}$
$B = 2 \frac{1}{2} \text{ gallons}$

#### Explanation:

Note that % is like a unit of measurement and it is worth $\frac{1}{100}$
10% is the same as $10 \times \frac{1}{100} = \frac{10}{100}$

50% is the same as $50 \times \frac{1}{100} = \frac{50}{100}$

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$\textcolor{b l u e}{\text{Considering the different parts}}$

Given that the blend is 5 gallons

Let the amount of drink $\textcolor{red}{\text{A be } x}$
Then the amount of drink $\textcolor{g r e e n}{\text{B is } 5 - x}$

The amount of the juice in the blend is 30% so it is 30% of 5 gallons
$\frac{30}{100} \times 5 = \frac{30}{20} = \frac{3}{2}$ gallons of pure juice

$\textcolor{red}{\text{The amount of juice from A is 10% of } x \to \frac{10}{100} x}$
$\textcolor{g r e e n}{\text{The amount of juice from B is 50% of } \left(5 - x\right) \to \frac{50}{100} \left(5 - x\right)}$
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$\textcolor{b l u e}{\text{Putting all together to determine the value of A}}$

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

$\text{ "10/100x" "+" "50/100(5-x)color(white)(.)=" } \frac{30}{100} \times 5$

$\text{ "0.1x" "+" "2.5-0.5xcolor(white)(.)=" } 1.5$

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

$\text{ "x" "+" "25-5xcolor(white)(.)=" } 15$

$\text{ } - 4 x + 25 = 15$

Subtract 15 from both sides

$\text{ } - 4 x + 10 = 0$

Add $4 x$ to both sides

$\text{ } 10 = 4 x$

Divide both sides by 4

$x = \frac{10}{4} = \frac{5}{2} = 2 \frac{1}{2}$ gallons = A
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$\textcolor{b l u e}{\text{Determine the amount of B}}$

$B = 5 - A \text{ "->" } B = 5 - 2 \frac{1}{2} = 2 \frac{1}{2}$
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$\textcolor{b l u e}{\text{Check}}$

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

RHS is $\frac{3}{2}$ gallons of pure juice content.

Consider just the LHS:

$\left(\frac{10}{100} \times \frac{5}{2}\right) + \left(\frac{50}{100} \times \frac{5}{2}\right) = \frac{3}{2}$

LHS = RHS so correct