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

Studdy Solution

STEP 1

Assumptions1. We have5 quarts of a30% antifreeze solution. . We want to obtain a40% antifreeze solution.
3. We are adding pure antifreeze to the existing solution.
4. The percentage of antifreeze in a solution is calculated by the volume of antifreeze divided by the total volume of the solution.

STEP 2

Let's denote the amount of pure antifreeze to be added as x (in quarts). The total volume of the new solution will be5 + x quarts.

STEP 3

The amount of antifreeze in the initial solution is30% of5 quarts. We can calculate this by multiplying the volume of the solution by the percentage of antifreeze.
$Initial\, antifreeze =5\, quarts \\times30\%$

STEP 4

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

STEP 5

Calculate the initial amount of antifreeze.
$Initial\, antifreeze =5\, quarts \\times0.3 =1.5\, quarts$

STEP 6

The total amount of antifreeze in the new solution will be the initial amount of antifreeze plus the amount of pure antifreeze added. $Total\, antifreeze = Initial\, antifreeze + x$

STEP 7

Substitute the calculated initial antifreeze into the equation.
$Total\, antifreeze =1.5\, quarts + x$

STEP 8

The percentage of antifreeze in the new solution is40%. We can set up an equation using the definition of percentage the volume of antifreeze divided by the total volume of the solution.
$40\% = Total\, antifreeze / Total\, volume$

STEP 9

Convert the percentage to a decimal value.
$40\% =.4$ $.4 = Total\, antifreeze / (5\, quarts + x)$

STEP 10

Substitute the equation for total antifreeze from7 into the equation.
$0.4 = (.5\, quarts + x) / (5\, quarts + x)$

STEP 11

olve the equation for x. First, multiply both sides by (5 + x) to get rid of the fraction.
$0.4(5 + x) =.5\, quarts + x$

STEP 12

istribute on the left side of the equation.
$2 +0.4x =.5\, quarts + x$

STEP 13

Subtract0.x from both sides of the equation.
$2 =.5\, quarts +0.6x$

STEP 14

Subtract. quarts from both sides of the equation.
$0.\, quarts =0.6x$

STEP 15

Divide both sides by0. to solve for x.
$x =0.5\, quarts /0.$

STEP 16

Calculate the value of x.
$x =0.5\, quarts /0.6 =0.83\, quarts$ So, approximately0.83 quarts of pure antifreeze must be added to the5 quarts of a30% antifreeze solution to obtain a40% antifreeze solution.

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