What is the minimum and maximum root of the equation

$$
\frac{3 x+5.03}{4}=\frac{2 x-3}{1.75}
$$

if we consider that the decimal numbers involved are the result of rounding?
I. Solution: The root of the equation, considering the decimal numbers as accurate for now and using the usual computational method, rounded to three decimal places, is:

$$
x=\frac{12+5.03 \cdot 1.75}{8-3 \cdot 1.75} \approx 7.565
$$

If we take into account that the decimal data were created by rounding, then any number that rounds to these values when rounded to two decimal places can be used, i.e., any $b$ and $c$ such that

$$
5.025 \leqq b \leqq 5.035, \quad \text { and } \quad 1.745 \leqq c \leqq 1.755
$$

Then the root is:

$$
x=\frac{12+b c}{8-3 c}
$$

Among fractions with the same positive denominator, the one with the largest numerator is the largest, and the one with the smallest numerator is the smallest. Among fractions with the same numerator, if, as in our case, the numerator and all possible denominators are positive, the one with the smallest denominator is the largest, and the one with the largest denominator is the smallest. In our case, both the numerator and the denominator are variable, so we need to determine the possible maximum and minimum values of $x$ by considering both selection criteria simultaneously. Increasing $c$ increases the numerator of $x$ because $b$ is positive, and decreases the denominator, thus increasing $x$ itself. And it clearly reaches its maximum value when both $c$ and $b$ take their maximum values. Therefore,

$$
x_{\max }=\frac{12+5.035 \cdot 1.755}{8-3 \cdot 1.755} \approx 7.618
$$

rounded to three decimal places.

Similarly, decreasing $c$ decreases the numerator (because $b$ is positive) and increases the denominator, thus decreasing $x$, and the minimum value is obtained when both $c$ and $b$ are at their minimum values:

$$
x_{\min }=\frac{12+5.025 \cdot 1.745}{8-3 \cdot 1.745} \approx 7.512
$$

rounded to three decimal places. (According to the rounding rule, this fraction should be rounded down as a standalone number, but we provide tighter bounds, i.e., a higher lower bound.) Therefore,

$$
7.512 \leqq x \leqq 7.618
$$

Denes Lajos (Budapest, Rakoczi F. g. II. o. t.)

Remarks. I. When we rounded the value of $x_{\max }$ down and the value of $x_{\min }$ up, we somewhat satisfied the rounding rule "by customary law" with the following addendum: if the number to be rounded is a finite decimal, the last digit is 5 (or an integer, and the last non-zero digit is 5), and this is the only digit to be omitted, - then we choose the rounding direction that makes the last remaining digit even. Indeed, adhering to this agreement (1) would have < in place of all four $\leqq$ signs, and thus $x$ would even less likely reach the values 7.619 and 7.511, which could happen if (1) held with $\leqq$. - However, we point out that the rounding rule itself implements an idea, that the rounded value should be as close as possible to the actual value; the addendum mentioned here is a customary agreement for the rare exceptional "tie" case, when the actual value is equally distant from the values obtained by rounding up and down. People dealing with calculations could have agreed on rounding to an odd digit in such indeterminate cases as well.

2. Many solutions show the author's incomplete understanding of "long" but still finite and infinite decimal fractions. For example: "5.03 could only have come from the following numbers: 5.025, 5.026, ..., 5.033, 5.034." According to this, only one digit could have been omitted after the last retained digit. Or: "The smallest value of 5.03 is 5.025, the largest value is 5.0349." This solution considers the omission of two digits, but according to the wording, it would round 5.034901 to 5.04, and even the smaller 5.03490001 to an infinite, periodic decimal. - Among the numbers that round to 5.03 to two decimal places according to the rounding rule - if we exclude 5.025 and 5.035 as mentioned above - there is neither a smallest nor a largest! That is, we can state a smaller number than any alleged smallest such number, which is still greater than 5.025, and a larger number than any alleged largest such number, which is still less than 5.035, - and for this reason, we never need to look for an infinite decimal, as there is always a number with the stated property among the "sufficiently long" finite decimals.
3. We also point out again that many people interpret the words "at least" and "at most" contrary to their general usage, and this occurs in various regions of the country. According to one solution from Miskolc, "x is at least 8.59 and at most 7.34," and according to one from Budapest, "at least 7.673 and at most 7.458." We clarify that by "at least 7.512, at most 7.618," we mean that the number in question cannot be less than 7.512 and cannot be greater than 7.618. This interpretation - although not related - corresponds to the usual upward direction of the $Y$-axis in the coordinate system: the lowest allowed value of the function in question is 7.512, and the highest is 7.618. (It is another matter that the use of "at least" and "at most" to express this fact is redundant, and from a mathematical point of view, the terms "first" and "last" are also not standard. Similar mathematical terms are the lower and upper bound, or the lower and upper limit.)

II. Solution: Considering the decimal numbers as accurate, we solve the equation graphically by plotting the linear functions on both sides and reading the abscissa of the intersection point $M$ of their graphs, the lines I and II.

![](https://cdn.mathpix.com/cropped/2024_05_02_c7cbd030883f8df368c7g-2.jpg?height=526&width=643&top_left_y=276&top_left_x=730)

However, taking into account the rounding, we need to think of infinitely many lines on each side. Writing the equation in the form

$$
\frac{3}{4}