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Chapter 41: Chapter 41

Problem 1181

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How many gallons of a liquid that is 74 percent alcohol must be combined with 5 gallons of one that is 90 percent alcohol in order to obtain a mixture that is 84 percent alcohol?

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To obtain a mixture that is 84 percent alcohol, 3 gallons of a liquid that is 74 percent alcohol must be combined with 5 gallons of one that is 90 percent alcohol.
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TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: Create an equation involving variables

First, let's assume the required amount of 74 percent alcohol liquid is x gallons. Therefore, the total volume of the mixture is x + 5 gallons. We'll use this information to set up an equation to solve for x.

Step 2: Set up the weighted average equation with percentages

To create the mixture with 84 percent alcohol, the sum of the amounts of alcohol in the 74 percent solution and the 90 percent solution must be equal to the amount of alcohol in the final solution. So, we can form the equation: \[0.74x + 0.9(5) = 0.84(x + 5)\]

Step 3: Solve the equation for x

Now, we'll solve the equation for x step by step: 1. Distribute the percentages: \[0.74x + 4.5 = 0.84x + 4.2\] 2. Rearrange the equation to have the x terms on one side: \[0.1x = 0.3\] 3. Divide both sides of the equation by 0.1 to solve for x: \[x = 3\]

Step 4: Interpret the result

We've found the value of x, which represents the amount of the 74 percent alcohol liquid that must be mixed with 5 gallons of 90 percent alcohol liquid to achieve a mixture that is 84 percent alcohol. In this case, x = 3 gallons. So, 3 gallons of a liquid that is 74 percent alcohol must be combined with 5 gallons of one that is 90 percent alcohol in order to obtain a mixture that is 84 percent alcohol.

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