Practice Problem #1
Suppose you have a jug of 120 ounces of juice that is 20% pure apple juice and the rest is water. You feel this is a bit too sweet, so you want to add water to the juice so that your new juice is 15% pure apple juice. How much water should you add?
Solution:
1.) Your unknown quantity is how much water you should add to your juice. We will represent this with the variable w.
2.) We want to set up an equation in w with the information given.
The initial jug of juice is 120 ounces, and it contains 20% pure apple juice. We can figure out how many ounces of pure apple juice are in the jug by finding 20% of 120 ounces. We convert 20% to decimal form by multiplying by 1 / 100 to get 0.20. Then we multiply 120 by 0.20. Thus, we have 120 * 0.20 = 24 ounces.
We are adding w ounces of water to our initial juice, so our new juice will contain 120 + w ounces, and 24 of those ounces will still be pure apple juice. We want this new juice to be 15% pure apple juice, so we want 24 to be 15% of 120 + w. Putting this into an equation, we have 0.15( 120 + w ) = 24. This is our equation.
3.) Now we will solve our equation using algebra.
4.) We see that w = 40 ounces, so we should add 40 ounces of water to our jug of juice in order to have a juice that is 15% pure apple juice.
Practice Problem #2
Assume you have 15 gallons of a 15% oil solution. How many gallons of a 30% oil solution need to be added to make a 25% oil solution?
Solution:
1.) You're unknown quantity is how many gallons of the 30% oil solution you should add to your 15% oil solution. We will represent this with the variable s.
2.) We want to set up an equation in s with the information given.
Our initial solution is 15 gallons of 15% oil solution. To find the number of gallons of oil our initial solution contains, we convert 15% to decimal form (as previously demonstrated) to get 0.15. Then we multiply by the 15 gallons. Our initial solution contains 15 * 0.15 = 2.25 gallons of oil. The solution we are adding is s gallons of 30% oil solution, so this solution contains 0.30s gallons of oil.
When we mix these two solutions together, we have s + 15 gallons of mixture, and we want 25% of that to be oil, so we want our mixture to contain 0.25( s + 15) gallons of oil.
Putting this all together to create an equation, we recognize that the amount of oil in the 15% solution (2.25 gallons) plus the amount of oil in the added 30% solution (0.30s gallons) will be equal to the amount of oil in the desired final mixture (0.25(s +15) gallons). In equation form, we have 2.25 + 0.30s = 0.25( s + 15).
3.) Now we solve the equation we found in step 2 using algebra.
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4.) We see that s = 30 gallons, so we should add 30 gallons of our 30% oil solution to our 15 gallons of our 15% oil solution to obtain a 25% oil solution.
Practice Problem #3
Assume you have 500 milliliters of saline solution (salt and water) that is 4% salt. How much water must evaporate to get a saline solution that is 8% salt?
Solution:
1.) Our unknown quantity is how much water must evaporate to get a saline solution with 8% salt. We will represent this with the variable x.
2.) Our original solution is 500 milliliters of saline solution that is 4% salt. Thus, we convert 4% to decimal form and multiply by 500 to find that it contains 500 * 0.04 = 20 milliliters of salt. When x milliliters of water are evaporated, we will have 500 - x milliliters of saline solution, which will still contain 20 milliliters of salt. We want those 20 milliliters to be 8% of our 500 - x milliliters of saline solution. In equation form, we have 0.08( 500 - x ) = 20.
3.) Now we use algebra to solve our equation.
4.) We see that x = 250 milliliters, so 250 milliliters of water must evaporate from our original saline solution to obtain a saline solution that is 8% salt.
Lesson Summary
To solve mixture problems in algebra, we follow these four steps.
1.) Represent the unknown quantity with a variable.
2.) Create an equation involving that variable.
3.) Solve the equation for the variable.
4.) Answer the question posed in the problem.
The practice problems in this lesson have given us the chance to become more comfortable with these steps. The more you practice these types of problems, the easier they will become, and you will familiarize yourself with the pattern of the solving process. Just remember, practice makes perfect!
