### Real(ly) Numbers

If x, y and z are real numbers such that: x + y + z = 5 and xy + yz + zx = 3. What is the largest value that any of the numbers can have?

### Overturning Fracsum

Solve the system of equations to find the values of x, y and z: xy/(x+y)=1/2, yz/(y+z)=1/3, zx/(z+x)=1/7

### Bang's Theorem

If all the faces of a tetrahedron have the same perimeter then show that they are all congruent.

# A Method of Defining Coefficients in the Equations of Chemical Reactions

##### Stage: 4 and 5

Published February 1999,February 2011.

While studying the school course of chemistry we were taught two methods of defining the coefficients in the equations of chemical reactions: the method of manual selection and the method of electronic balance for the reactions of oxidation and reduction. I was surprised by the imperfection of these methods. In the work that is presented to your attention I suggest a simple method of defining the coefficients in the equations of chemical reactions with the help of a system of linear algebraic equations that describes the material balance in a chemical reaction.

For simplicity the usage of the suggested method will be demonstrated on the following chemical reactions. Let us examine the reaction of oxidation of methyl-benzol. Instead of the unknown coefficients we will put the variables $a$, $b$, $c$, $d$:
$$a \textrm{C}_6 \textrm{H}_5 \textrm{C} \textrm{H}_3 + b \textrm{O}_2 \rightarrow c \textrm{C} \textrm{O}_2 + d \textrm{H}_2 \textrm{O}$$
The number of atoms of each chemical element in the left and right sides of this equation is equal. Let us write the equations of material balance for each chemical element. \begin{array}{ll} \text{Carbon:} & 7a=c\\ \text{Hydrogen:} & 8a=2d\\ \text{Oxygen:} & 2b=2c+d \end{array} As a result we will get a system of three linear equations with four unknowns: \begin{eqnarray} 7a & = & c \\ 8a & = & 2d \\ 2b & = & 2c+d \end{eqnarray} This system has an infinite number of solutions, but we have to get the minimal natural values only. The system has the following solution: \begin{eqnarray} c & = & 7a \\ d & = & 4a \\ b & = & 9a \end{eqnarray} \begin{eqnarray} a & = & 1 \\ b & = & 9 \\ c & = & 7 \\ d & = & 4 \end{eqnarray} Thus, the chemical equation has the following coefficients:
$$\textrm{C}_6 \textrm{H}_5 \textrm{C} \textrm{H}_3 +9\textrm{O}_2 \rightarrow 7\textrm{C} \textrm{O}_2 + 4\textrm{H}_2 \textrm{O}$$
Let us examine one more example of using the suggested method for the reaction between potassiumpermanganate $\textrm{K Mn O}_4$ and iron (II) sulphate, $\textrm{FeSO}_4$, acidified with $\textrm{H}_2\textrm{SO}_4$.
$$a\textrm{KMnO}_4+b\textrm{FeSO}_4+c\textrm{H}_2\textrm{SO}_4\rightarrow d\textrm{MnSO}_4+e\textrm{Fe}_2(\textrm{SO}_4)_3+f\textrm{K}_2\textrm{SO}_4 +g\textrm{H}_2\textrm{O}$$
We have obtained the following system: \begin{eqnarray} a & = & 2f \\ a & = & d \\ 4a+4b+4c & = & 4d+12e+4 \\ f & = & g \\ b & = & 2e \\ b+c & = & d+3e+f \\ 2c & = & 2g \end{eqnarray} \begin{eqnarray} a & = & \frac{1}{4}g \\ b & = & \frac{5}{4}g \\ c & = & g \\ d & = & \frac{1}{4}g \\ e & = & \frac{5}{8}g \\ f & = & \frac{1}{8}g \end{eqnarray} \begin{eqnarray} a & = & 2 \\ b & = & 10 \\ c & = & 8 \\ d & = & 2 \\ e & = & 5 \\ f & = & 1 \\ g &= & 8 \end{eqnarray} $$2\textrm{KMnO}_4+10\textrm{FeSO_4}+8\textrm{H}_2\textrm{SO}_4\rightarrow 2\textrm{MnSO}_4+5\textrm{Fe}_2( \textrm{SO}_4)_3+\textrm{K}_2\textrm{SO}_4 +8\textrm{H}_2\textrm{O}$$ Thus, the suggested method is both easy-to-use and it can be used for quick definition of coefficients for complex equations of chemical reactions.