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How Do You Figure Out the Volume of a Solution Using Percents?

Simone is working on a chemistry experiment. She needs to make a solution that has 45% magnesium chloride. At her bench she has 55 mL of a solution that is 35% magnesium chloride. She also has a solution that is 65% magnesium chloride. What volume of the 65% solution does she need to add to the 35% solution in order to make the 45% solution?

Summary

  1. 'v' is a variable that represents the volume of 65% solution
  2. MgCl is an abbreviation for magnesium chloride
  3. The amount of MgCl is the percent of the solution (converted to a decimal) times the volume
  4. Add the amounts of MgCl in the 35% and 65% solutions and set that equal to the amount in the 45% solution
  5. Solve the equation to find v

Notes

    1. The volume of the 65% solution is not given in the problem
    2. So this is what we will be trying to solve for
    1. Our variable will represent the volume of 65% solution needed
    1. 'v' is a variable that will represent the volume of 65% solution needed
    1. Tables are a good way to organize information so it's easier to work with
    1. Tables are a good way to organize information so it's easier to work with
    1. For each solution, we have three pieces of information:
    2. The volume of the solution
    3. The percent of the solution that is magnesium chloride
    4. And the actual amount of magnesium chloride in the solution
    5. The first column of the table will have the volume of each solution
    1. MgCl is an abbreviation for magnesium chloride
    2. We'll put the actual amount of MgCl in the second column
    1. We know the percent of MgCl in each solution, so we can put these values down the left to identify each solution
    1. Now that our table is set up, we can look back at our problem and fill in the information it gives us
    1. mL is the abbreviation for milliliters
    2. Our problem tells us she has 55 mL of 35% solution
    3. Remember, milliliters are a measure of volume
    1. The problem doesn't give us the volume of the 65% solution
    2. This is what we are trying to solve for
    3. Remember, v was the variable we defined to represent the volume of the 65% solution
    1. Simone is mixing together two solutions to get a new solution
    2. So when you combine two solutions, you need to add together the volumes of each one to get the volume of the new one
    1. So we'll add the volume of the 35% solution to the volume of the 65% solution
    2. This value is 55+v
    1. 55+v is just the volumes of the 35% and 65% solutions added together
    2. This gives us the total volume of the combined solutions
    1. For each solution we know the volume and what percent of the volume is MgCl
    2. So to find the actual amount of MgCl, we can take the percent we're given of each volume
    1. We're given the percent of MgCl for each solution: 35%, 65%, and 45%
    2. And we now have the volume of each solution: 55, v, and 55+v
    3. To find the amount of MgCl in each solution, we need to multiply the percent by the volume
    1. We can't multiply using percents, so we need to convert the percent to a decimal
    2. To do this, drop the percent sign and move the decimal two places to the left
    3. So 35% becomes 0.35
    4. Now we can multiply 0.35 by 55 to get the amount of MgCl in the 35% solution
    1. 65% becomes 0.65 when we convert
    2. Multiply by v to get the amount of MgCl in the 65% solution
    3. Remember, v was the variable we defined to represent the volume of the 65% solution
    4. 45% becomes 0.45
    5. Multiply by 55+v to get the amount of MgCl in the 45% solution
    1. In order to find v, we need an equation we can solve
    1. Simone is combining the 35% and 65% solutions to make the 45% solution
    2. So the amount of MgCl in the 45% solutions will be the total amount in the 35% and 65% solutions
    3. This means we can add the two amounts together and set it equal to the amount in the 45% solution
    4. Luckily, we have all this information organized in our table!
    1. Now we have an equation that only contains v, so we can solve for it
    2. Remember, v was the variable we defined to represent the volume of 65% solution
    1. Remember, v was the variable we defined to represent the volume of 65% solution
    1. Remember, v was the variable we defined to represent the volume of 65% solution
    1. Multiply 0.35•55 on the left hand side
    2. This simplifies to 19.25
    1. On the right hand side, we can distribute the 0.45 through the parentheses
    2. Multiply 0.45 by each term inside the parentheses
    3. 0.45•55=24.75
    4. 0.45•v=0.45v
    1. We're trying to get v by itself on one side of the equation
    2. So we need to move the 19.25 to the other side
    3. Subtraction is the opposite of addition, so subtract 19.25 from both sides to get rid of it on the left
    4. This leaves us with 0.65v on the left
    5. Subtracting 19.25 from 24.75 on the right leaves us with 5.5+0.45v
    1. We want to get all our 'v' terms on one side of the equation
    2. Remember, v is the variable we are solving for, the volume of the 65% solution
    3. Since subtraction is the opposite of addition, subtract 0.45v from both sides
    4. On the left we get 0.2v
    5. The 0.45v's cancel on the right to leave 5.5
    6. Now all our v terms are on one side!
    1. But we're not done yet - we want just v by itself
    2. So we need to get rid of the 0.2
    3. Since division is the opposite of multiplication, we can divide both sides by 0.2 to undo the multiplication
    4. The 0.2's cancel on the right, leaving v by itself
    5. Dividing 5.5 by 0.2 on the right gives us 27.5
    1. Remember, v was the variable we defined to represent the volume of 65% solution
    2. So we've found that the volume of the 65% solution needed is 27.5
    1. mL is an abbreviation for milliliters