This question is a mixture question. These are often given in terms of liquids, but other mixtures are possible. There have been good answers for your specific problem. I present here a general approach to mixture problems.
Step 1: Analyze the problem
As with any text problem it is important to first realize what information you are given. Mixture problems contain information about mixture concentrations (how much P (part) is in the mixture), mixture amounts (can be measured in mass (e.g. kilogram), but most often in volume (e.g. liter).
- Question: Which part of the mixture does the question ask about (what is P)?
- Question: What is the total mixture?
- Question: What is unit used for the amount of the mixture.
- Question: What is unit used for the concentration of the mixture (e.g. percent, kg/liter, mol/liter).
- Determine the unit for the part from the answers to questions 3 and four.
In your example
- The part is pure orange juice (OJ)
- The whole mixture is the juice drink (Mix).
- The Mix amounts are measured in liter.
- The Mix concentrations are measured in percent.
- IA mixture concentration given in percent always means that the part and the mixture are measured in the same units. So here this means that OJ is also measured in liter.
Step 2: Set up the problem
Always set up mixture problems in a table. (What I suggest is very similar to Kathye P.'s table, but the rows and columns are switched. The advantage is that you can read off the equations needed in step 3more easily.
The general table for mixture problems:
Mxiture 1 Mixture 2 Resulting mixture
Amount of Mix <fill in this row in the units determined in question 3 above.
if the value is unknown, use a variable.>
Concentration <fill in this row in the units determined in question 4 above.
of Mix if the value is unknown, use a variable.>
Amount of P <fill in this row in the units determined under 5 above.
This is done by multiplying in each column the values of the two rows above>
In your example
Mxiture 1 (25%) Mixture 2 (10%) Resulting mixture
Amount of Mix x y 15 L
Concentration 0.25 0.1 0.18
(convert % to
Amount of OJ 0.25x 0.1y 0.18 * 15
Step 3: Set up your calculation
The two rows with the amount of the part (P) and the amount of the mix give you your equations. Just add a plus between the first two values and an equal between the second and the third value. You will get two equations with two unknowns. Multiply out any numeric values, and then proceed to solve the two equations. (If you don't know this, you need to spend some serious time reviewing it.)
In your example
from the first table row x + y = 15
from the last table row 0.25x + 0.1y = 0.18 * 15
simplified 0.25x + 0.1y = 2.7
x + y = 15
0.25x + 0.1y = 2.7
This way of setting up mixture problems will work no matter what is asked for. The most common question asks for volume of the mixtures necessary to produce a certain mixture, but other questions are possible.
This approach works for a variety of problems in amounts are not given directly but via ratios.
Common categories are
- Two people/groups or machines working at different speeds (speeds are given in some unit per hour, such as room per hour, lawn per hour, items manufactured per hour, etc.
- A vehicle traveling at two different speeds.
If you are interested in learning the general set-up so you don't have to learn different approaches for these similar problems, send me an email.