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[tmac324]

a vintner fotifies wine that contains 10% alcohol by adding 70% alcohol solution to it. the resulting mixture has an alcohol strength of 16% and fills 1000 one liter bottles.. how many liters (L) of the wine and of the alcohol solution does he use?

 

[soroban]

Hello, tmac324!

A vintner fortifies wine that contains 10% alcohol by adding 70% alcohol solution to it.
The resulting mixture has an alcohol strength of 16% and fills 1000 one liter bottles.
How many liters (L) of the wine and of the alcohol solution does he use?

$\displaystyle \text{The final mixture is 1000 liters which is 16\% alcohol.}$
. . $\displaystyle \text{It will contain: }\:0.16 \times 1000 \:=\:\boxed{160}\text{ liters of alcohol.}$


$\displaystyle \text{He uses }x\text{ liters of wine which is 10\% alcohol.}$
. . $\displaystyle \text{This contains: }\:0.1x\text{ liters of alcohol.}$

$\displaystyle \text{He adds }100-x\text{ liters of solution which is 70\% alcohol.}$
. . $\displaystyle \text{This contains: }\:0.7(1000-x)\text{ liters of alcohol.}$

$\displaystyle \text{The final mixture will contain: }\:\boxed{0.1x + 0.7(1000-x)}\text{ liters of alcohol.}$


$\displaystyle \text{We just described the final amount of alcohol in }two\text{ ways.}$

\(\displaystyle \text{There is our equation! }\quad\hdots \quad 0.1x + 0.7(1000-x) \;=\;160\)