Rounding Off

TASK. In a certain multi-million dollar company, Division Managers are required to submit monthly detail and summary expense reports on which the amounts are rounded, for ease of reading, to the closest $1,000. One month, a Division Manager's detail report shows $1,000 for printing and $1,000 for copying. In the summary report, the total for "printing and copying" is listed as $3,000. When questioned about it by the Vice President, he claims that the discrepancy is merely round-off error. In subsequent months, the Vice President notices that such round-off errors seem to happen often on this Division Manager's reports. Before the Vice President asks that the Division Manager re-create the reports without rounding, she wants to know how often this should happen.

COMMENTARY. We are often quoted rounded numbers that do not then turn out to be quite exact. Even a bank's approximate computational program for principal and interest can eventually drift far enough off the actual payment for the difference to be important. In any problem, we have to be concerned about which numbers are exact and about the accuracy of those that are not.

People don't often realize how huge the consequences of rounding numbers can be. Suppose, for example, that a company's board of directors has received a report indicating that each of the machines manufactured by their company will take up 2% of the freight capacity of their cargo planes, and the board wants to know how many machines can be shipped on each plane. In our standard notation, 2% represents a number somewhere between 1.5% and 2.5%. Solving the problem with each of these two exact percentages yields answers that are quite different. Using 1.5%, the board
will find that the plane can hold 100% ÷ (1.5%/machine) x 66 machines; but by using 2.5%, the board will find that the plane can hold 100% ÷ (2.5%/machine) = 40 machines. So, in truth, all the board can say is that the answer is between 40 and 66 machines! Clearly, the report has not supplied accurate enough information, especially if the profitability of the shipment depends strongly on the number of machines that can be shipped.

If, on the other hand, the report had indicated that the board could assume another decimal place of accuracy, by stating that each machine accounted for 2.0% of the plane's capacity, then, with rounding, the board can be sure that the exact portion is somewhere between 1.95% and 2.05%. Using these exact percentages, the board can conclude that the plane can hold between 48 and 51 machines. One decimal place of additional accuracy in the reported data reduced the uncertainty in the answer from 26 machines to 3.

This problem is important for another reason as well, for its solution introduces a useful mathematical connection: the notion of geometric probability, where the range of options (technically, the "sample space") is represented by a geometric figure so that the probability of certain events correspond to the areas of certain portions of that figure. Geometric probability enables us to use our knowledge of the area (or length or volume) of geometric figures to compute probabilities.

MATHEMATICAL ANALYSIS. Fundamental to an understanding of geometric probability is the idea that on a portion of a line, probability is proportional to length, and on a region in a plane, probability is proportional to area. For example, suppose that in Figure 1, the areas of regions A, B, and C are 2, 1, and 3 respectively, for a total area of 6. Then a point picked at random from these regions would have probability of 2/6, 1/6, and 3/6 of being in regions A, B, and C respectively.

Figure 1: An area model for probability

Note that the boundaries of the regions are not significant in the calculations because they have no area. Ideally (as opposed to in a physical model) these boundaries are lines with no thickness. Thus, the probability that a point from this rectangle will lie exactly on one of these boundaries, rather than close to a boundary, is zero.

In order to answer the question at hand, it must be stated more mathematically: Given a pair of numbers that both round to 1, and assuming that all such pairs are equally likely, find the probability that their sum rounds to 2. This assumption may or may not be reasonable in a particular business and would require some knowledge of typical expenses and some non-mathematical judgment.

A number that rounds to 1 is somewhere between .5 and 1.5. These numbers may be represented by a line segment, shown as the shaded portion of the number line in Figure 2.

Figure 2: A linear representation of numbers that round to 1

To state this a bit more formally, a number x will be rounded to 1 if .5 < x < 1.5. (Again, we can ignore the boundaries, .5 and 1.5, because the probability that a number will be exactly on the boundary is zero.) Suppose y also rounds to 1, so that .5 < y < 1.5. If we consider a coordinate plane with points (x, y), these two inequalities determine a square of side 1. This square (Figure 3) represents all pairs of numbers where both could be rounded to 1. For example, point A represents (.8, .6), B represents (1.1, 1.1), and C represents (1.3, 1.4).

Figure 3: An area representation for 1 + 1

What can we say about x + y for points inside the square? Most of the time, x + y will round to 2, but sometimes it will round to 3, and sometimes it will round to 1. Note that the components of A add to 1.4, which rounds to 1; the components of B add to 2.3, which rounds to 2; and the components of C add to 2.7, which rounds to 3.

The probability that 1 + 1 rounds to 1 is the fraction of the square containing pairs that, when added, round to 1. Now, x + y rounds to 1 if x + y < 1.5, which will occur for points below the line x + y = 1.5. Similarly, x + y rounds to 3 for points above the line x + y = 2.5. These conditions each cut off a triangular corner of the square (shown as the darker shaded regions in Figure 4).

Figure 4: An area representation for 1 + 1, with rounding boundaries

     The legs of these right triangles are each of length 1/2, so they each have area 1/8. Thus, the probability that 1 + 1 = 3 is 1/8, and the probability that 1 + 1 = 1 is also 1/8. Finally the probability that 1 + 1 = 2 is 3/4, the remaining fraction of the square.

EXTENSIONS. What's the probability that 1 x 1 = 2? This requires calculating the portion of the square that satisfies xy > 1.5 (Figure 5). Is this bigger or smaller than 1/8, calculated as the area of the upper triangle in Figure 4? A comparison of Figures 4 and 5 shows remarkable similarity. What is the precise relationship between the line x + y = 2.5 and the curve xy = 1.5? Solving the first equation for y and substituting into the second yields x(x ­ 2.5) = 1.5, a quadratic which simplifies to ­x2 + 2.5x ­1.5 = 0 or 2x2 ­5x + 3 = 0. This second equation factors easily as (2x ­ 3)(x ­ 1) = 0, yielding solutions x = 1.5 and x = 1. These solutions imply that the line x + y = 2.5 and the curve xy = 1.5 intersect the square at the same points. By the concavity of the curve xy = 1.5, the curve must lie below the line inside the square. So the answer should be a little bigger than 1/8 = .125.

Figure 5: An area representation for 1 x 1, with rounding boundaries

Calculus allows us to calculate the shaded area as precisely:

ƒ₁^1.5 (1.5 ­ 1.5/x) dx = .75 ­ 1.5 ln 1.5 .142.

Similarly, if x = .6 and y = .7, then xy = .42 < .5, which would round to 0. The probability that xy rounds to 0 is .5 ln 2 ­ .25 .097.

What about 1/1? It rounds to 0 with probability .0625, to 1 with probability .75, to 2 with probability .175, and to 3 with probability .0125. These calculations require only geometry, no calculus.