The Curious Warehouse Manager

When we solve purely mathematical problems, we get exact results. However, when we use mathematics to solve problems in the world around us, we rarely achieve absolute precision in the answer. Approximation is sometimes the result of a simplification of the real situation in our minds. Sometimes the input data are approximated (e.g. we can only measure lengths or time with limited accuracy) or an absolutely exact result is realistically unattainable and must be rounded off.

Rounding to a given number of significant digits is often used in practice (and in the following problems). We round a positive real number \(r\) to \(n\) significant digits as follows:

• We express \(r\) in the form \(a\cdot 10^b\), where \(a\in\mathbb{R}\),\(a\in\left\langle 1,10 \right)\) and \(b\in\mathbb{Z}\), and then we round the number \(a\) to \(n-1\) decimal places according to the standard rules for rounding.
• E.g. numbers \(r=31{.}258\,16\) and \(s=0{.}023 \,123\,6\) we round to four valid digits as follows: \[ \begin{aligned} r &= 31{.}258\,16 = 3{.}125\,816 \cdot 10^1 \quad \doteq\quad 3{.}126 \cdot 10^1 = 31{.}26 \\ s &= 0{.}023 \,123\,6 = 2{.}312\,36 \cdot 10^{-2} \quad \doteq\quad 2{.}312 \cdot 10^{-2} = 0{.}023\,12. \end{aligned} \]

Notably, rounding input data can have surprising consequences for the accuracy of the result, for example when solving equations, as we will see in the following series of problems.

Solution. First, let’s solve the problem without rounding. Let \(x\) be the price per package of Ixodine and \(y\) be the price per package of Nopolio. The information in the assignment leads to a system of two linear equations with two unknowns \[ \begin{alignat*}{2} 597x &\,+& 386 y &= 401{,}950 \\ 86x &\,+& 19 y &= 39{,}600 \end{alignat*} \] whose solution gives us the real purchase price of a package of Ixodin vaccine \(350\,\text{CZK}\) and of a package of Nopolio vaccine \(500\,\text{CZK}\).

After rounding the coefficients to one significant digit, we solve the system \[ \begin{alignat*}{2} 600x' &\,+ & 400 y' &= 400\,000 \\ 90x' &\,+ & 20 y' &= 40\,000. \end{alignat*} \] The solution is the pair \(x'=\frac{1{,}000}{3}\doteq333\) and \(y'=500\). Now we have the actual price of the medicine and the estimate of the price by the warehause manager. Let’s also calculate the relative error in the price of the medicine due to rounding. Relative error is the absolute error (absolute value of price difference) divided by the actual price per package. We summarize the results in the table:

Solution. We will solve the problem in the same way as before, this time we will denote \(x\) the price of one package of Antiflu and \(y\) the price of one package of Contradift. The real prices are the solution of the system \[ \begin{alignat*}{2} 504x &\,+\,& 81 y &= 198{,}900 \\ 98x &\,+\,& 18 y &= 40{,}700 \end{alignat*} \] where we get \(x=250\) and \(y=900\).

When rounding the coefficients, we solve the system \[ \begin{alignat*}{2} 500x' &\,+\,& 80 y' &= 200{,}000 \\ 100x' &\,+\,& 20 y' &= 40{,}000, \end{alignat*} \] whose solution is \(x'=400\) and \(y'=0\). From the solution of the warehouse manager, it seems that that the second vaccine was delivered to the warehouse free of charge, while it is actually almost four times more expensive than the first one. We calculate the relative error and enter all the values in the table again:

Solution. Let \(p_1\), \(p_2\) (or \(q_1\), \(q_2\)) be the lines given by the equations of the system with unrounded coefficients in Exercise 1 (or Exercise 2), namely \[ \begin{align*} p_1 &\colon 597x + 386 y = 401{,}950 \\ p_2 &\colon 86x + 19 y = 39{,}600 \\[2mm] q_1 &\colon 504x + 81 y = 198{,}900 \\ q_2 &\colon 98x + 18 y = 40{,}700. \end{align*} \] Let us denote the lines given by the corresponding equations with rounded coefficients by \(p'_1\), \(p'_2\), \(q'_1\) and \(q'_2\) and further denote the points \(P\in p_1\cap p_2\), \(P'\in p'_1\cap p'_2\), \(Q\in q_1\cap q_2\) and \(Q'\in q'_1\cap q'_2\). A graphical representation of the pair of systems for each problem separately is shown in following figure.

By comparing the two graphical representations, it can be seen that in the case of Exercise 2, the pair of lines \(q_1\) and \(q_2\) are almost parallel. When rounding the coefficients of the equation, the position of the lines relative to the coordinate system generally changes and the position of the intersection changes as well. The change in the position of the intersection is much greater for lines that are almost parallel. The figure also shows why the second coordinate of the intercept (i.e., the price of the vaccine Contradift) will be much more affected by rounding in the second problem. Because of the slope of the lines \(q_1\) and \(q_2\), a small change in the \(x\) coordinate of the intersection would mean a large change in its \(y\) coordinate.

Literature

• Biermann K., Grötschel M., Lutz-Westphal B. (2010). Besser als Mathe: Moderne angewandte Mathematik aus dem MATHEON zum Mitmachen. Berlin: Vieweg+Teubner.