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Chapter 41: Problem 1190

How many quarts of pure alcohol must be added to 40 quarts of a mixture that is \(35 \%\) alcohol to make a mixture that will be \(48 \%\) alcohol?

Short Answer

Expert verified
Approximately 10 quarts of pure alcohol must be added to the 40-quart mixture to make a mixture that is \(48 \%\) alcohol.

Step by step solution

01

Set up two equations representing the alcohol in each mixture

Let x be the amount of pure alcohol that needs to be added to the 40-quart mixture. The 40-quart mixture is \(35 \%\) alcohol. So, the current amount of alcohol in the mixture is \(0.35(40)\). The total amount of alcohol in the final mixture will be the sum of alcohol in the mixture and the pure alcohol added, which is \(0.35(40) + x\). Since we want the mixture to be \(48 \%\) alcohol, we can represent the total amount of alcohol as a percentage of the final mixture: \(0.48(40 + x) = 0.35(40) + x\)
02

Solve the equation for x

Now we need to solve the equation we set up in step 1: \(0.48(40 + x) = 0.35(40) + x\) Expand the equation: \(19.2 + 0.48x = 14 + x\) Subtract 14 from both sides: \(5.2 + 0.48x = x\) Subtract 0.48x from both sides: \(5.2 = 0.52x\) Now, divide both sides by 0.52 to find x: \(x \approx 10\)
03

Interpret the result

The value of x is approximately 10 quarts. This means that 10 quarts of pure alcohol must be added to the 40-quart mixture to make a mixture that is \(48 \%\) alcohol.

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