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QuestionHow many quarts of pure antifreeze are needed to change 5 quarts of a 30%30\% solution to a 40%40\% solution?

Studdy Solution

STEP 1

Assumptions1. The initial solution has5 quarts of30% antifreeze. . We are adding pure antifreeze to this solution.
3. The final solution should be40% antifreeze.
4. The volume of antifreeze in the initial solution plus the volume of pure antifreeze added equals the volume of antifreeze in the final solution.

STEP 2

First, we need to find the amount of antifreeze in the initial solution. We can do this by multiplying the total volume of the initial solution by the percentage of antifreeze in it.
Initial antifreeze=Initial volumetimesInitial percentageInitial\, antifreeze = Initial\, volume \\times Initial\, percentage

STEP 3

Now, plug in the given values for the initial volume and initial percentage to calculate the initial antifreeze.
Initial antifreeze=5 quartstimes30%Initial\, antifreeze =5\, quarts \\times30\%

STEP 4

Convert the percentage to a decimal value.
30%=0.330\% =0.3Initial antifreeze= quartstimes0.3Initial\, antifreeze =\, quarts \\times0.3

STEP 5

Calculate the initial antifreeze amount.
Initial antifreeze=5 quartstimes0.3=1.5 quartsInitial\, antifreeze =5\, quarts \\times0.3 =1.5\, quarts

STEP 6

Now, let's denote the volume of pure antifreeze to be added as xx. The final volume of the solution will be 5+x5 + x quarts, and the final volume of antifreeze will be 1.5+x1.5 + x quarts. We can set up the following equation to represent the final percentage of antifreezeFinal antifreeze percentage=Final antifreeze volumeFinal total volumeFinal\, antifreeze\, percentage = \frac{Final\, antifreeze\, volume}{Final\, total\, volume}

STEP 7

Substitute the expressions for the final antifreeze volume and the final total volume into the equation.
40%=1.5 quarts+x quarts5 quarts+x quarts40\% = \frac{1.5\, quarts + x\, quarts}{5\, quarts + x\, quarts}

STEP 8

Convert the percentage to a decimal value.
40%=0.440\% =0.40.4=1.5 quarts+x quarts5 quarts+x quarts0.4 = \frac{1.5\, quarts + x\, quarts}{5\, quarts + x\, quarts}

STEP 9

To solve for xx, first cross-multiply.
.4×(5 quarts+x quarts)=.5 quarts+x quarts.4 \times (5\, quarts + x\, quarts) =.5\, quarts + x\, quarts

STEP 10

implify the equation.
2 quarts+0.4x quarts=.5 quarts+x quarts2\, quarts +0.4x\, quarts =.5\, quarts + x\, quarts

STEP 11

Rearrange the equation to isolate xx.
0.6x quarts=0.5 quarts0.6x\, quarts =0.5\, quarts

STEP 12

olve for xx.
x=0.5 quarts0.6≈0.83 quartsx = \frac{0.5\, quarts}{0.6} \approx0.83\, quartsSo, approximately0.83 quarts of pure antifreeze must be added.

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