Question

How do you solve mixture problems using a system of equations?

Answer 1

Hint: Mixture problems are ones where two different solutions are mixed together resulting in a new final solution. We generally use a table to solve mixture equations and plot the different variables as per the criteria given to arrive at a rational conclusion.

Complete step-by-step answer:
A set of simultaneous equations, also known as a system of equations or an equation system, is a finite set of equations for which common solutions are sought.
Mixture equation is generally solved by using the following table format:

	Amount	Part	Total
Item 1
Item 2
Final

The first column is for the amount of each item we have. The second column is labelled “part” which can be generally in form or percentage or any other criteria. To arrive at the total, we simplify having to multiply the “Amount” and “Part”. We can get an equation by adding the amount and/or total columns that will help us solve the problems. These problems can have either one or two variables.
Let us understand with the help of an example:
A chemist has $$70$$mL of a $$50\mathrm{\%}$$ methane solution. How much of the $$80\mathrm{\%}$$ solution must she add so the final solution is $$60\mathrm{\%}$$ methane?
Step 1: Construct the mixture table with the given data. We start with $$70$$, but don′ t know how much we add, that is $$x$$. The part is the percentages, $$0.5$$ for start, $$0.8$$ for add.

	Qty(ml)	%	Total
Start	$$70$$	$$0.5$$
Add	$$x$$	$$0.8$$
Final

Step 2: Add the values to arrive at the final column. The percentage for this quantity is $$0.6$$ because we want the final solution to be $$60\mathrm{\%}$$ methane.

	Qty(ml)	%	Total
Start	$$70$$	$$0.5$$
Add	x	$$0.8$$
Final	$$70+x$$	$$0.6$$

Step 3: Multiply the quantity with % to arrive at the total column.

	Qty(ml)	%	Total
Start	$$70$$	$$0.5$$	$$35$$
Add	$$x$$	$$0.8$$	$$0.8x$$
Final	$$70+x$$	$$0.6$$	$$42+0.6x$$

Step 4: Now we can construct an equation:
$$35+0.8x=42+0.6x$$
Subtracting by $$0.6$$on both the sides,
$$35+0.2x=42$$
Further simplifying
$$0.2x=42-35$$
$$0.2x=7$$
$$x={\displaystyle \frac{7}{0.2}}$$
$$x=35$$
Hence, we can conclude that a $$35$$ ml solution will be required.
So, the correct answer is “$$35$$ ml”.

Note: Mixture problems can be solved easily by drawing a table and then applying simple addition, subtraction, multiplication and division rules. There may be more than one unknown variable which can be found as per the above method. Mixture problems are used practically in a variety of fields for solving real life problems for example concentration of acid in a solution, finding value of product in staggered price structure, deciding discount policy for product based on given margin etc.