Step 1: Assume variable/s The first task in solving mixture problems as in any other
algebra word problems is to correctly relate the variables assumed to the problem given. Generally the variable will denote that which is required to find. Suppose the given problem reads as follows:
"A grocer mixed grape juice which costs 2.25 dollars per gallon with cranberry juice which costs 1.75 dollars per gallon. How many gallons of each should be used to make 200 gallons of cranberry/grape juice which will cost 2.10 dollars per gallon?"
As we need to find the quantities of each type of juice used in the mixture, we assume as follows:
Let the volume of Grape Juice mixed = x gallons and volume of Cranberry Juice mixed = y gallons.
Step 2: Forming the equation/s If we have used one variable in our initial assumption, then we will form one equation, often on the basis of amount of substance or value. When there are two variables assumed, then we form a system of two
linear equations.
For our problem on juice mixture, we form two equations, one on total volume and the other on value.
Sum of the quantities of two juices used = Quantity of the Mixture got
x + y = 200 (First equation on Quantity)
Sum of the values of two juices used = Value of the Mixture got.
We find a value using the formula, Value of an item = rate x weight or quantity of the item.
2.25x + 1.75y = 200 $\times$ 2.10 ( Second equation on value)
You may also make a table to understand the situation clearer.
Item
 Volume

Rate

Value = Rate x Volume

Grape Juice  x gal 
2.25 
2.25 x

Cranberry Juice  y gal

1.75 
1.75 y

Mixture  200 gal  2.10  200 * 2.10

Step 3: Solve the equations formed Using any method you have already learned.
Solving the two equations given in step 2, we get,
Quantity of Grape Juice in the mixture = 140 gallons
quantity of Cranberry Juice in the mixture = 60 gallons.