Inserting Numbers Between Two Rational Numbers

 

An infinite number of rational numbers can be inserted between any two rational numbers (there are infinitely many numbers between two rational numbers). However, under certain conditions, it is possible to insert a finite number of rational numbers between them.

To insert numbers that satisfy specific conditions between two fractions:

1) The denominators of the two fractions are equalized at the least common multiple (LCM) of the denominators.

2) An expansion or simplification process is carried out to write the numbers satisfying the desired conditions between these fractions.

 

Example:

 

Given that the denominator of the number exactly in the middle of the fractions \( \frac{2}{3} \text{ and } \frac{3}{4} \) is 72, let’s find the numerator of this number.

Let’s expand the given fractions so that their denominators become 72:

\[\begin{array}{l l } \displaystyle\frac{2}{3} = \displaystyle\frac{2}{3} \cdot \displaystyle\frac{24}{24}= \displaystyle\frac{48}{72}\\
\displaystyle\frac{3}{4}=\displaystyle\frac{3}{4} \cdot\displaystyle \frac{18}{18} = \displaystyle\frac{54}{72} \quad \text{holds.}\\
\end{array}\]

Since the desired fraction is in the exact middle of the given fractions, the numerator of this fraction must be the mid-point of the numbers 48 and 54, which is:
$$\frac{48+54}{2}=51$$

Therefore, the desired fraction is \(\frac{51}{72}\).

As a second method, if we want to find the number exactly halfway between the fractions \( \displaystyle\frac{2}{3} \text{ and } \displaystyle\frac{3}{4} \), this number is the arithmetic mean of the given fractions. Therefore, the arithmetic mean of \(\displaystyle\frac{2}{3} \text{ and } \displaystyle\frac{3}{4}\) is:

$$ \frac{\left(\frac{2}{3} + \frac{3}{4}\right)}{2} = \frac{17}{24} $$

Since the denominator of the fraction is required to be 72, expanding the found fraction by 3 yields the number \(\displaystyle\frac{51}{72}\). Accordingly, the numerator of the desired fraction is found to be 51.

 

Example:

 

Let’s find the minimum possible value for the numerator of the largest fraction when at most 7 fractions are inserted between the fractions \(\displaystyle\frac{1}{3} \text{ and } \displaystyle\frac{3}{5}\).

 

 

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