Let and . Express both and in the form and where , and are integers and . Using the same denominator , write exactly five distinct rational numbers lying between and keeping an integer numerator. Explain why the condition is necessary to find such rational numbers between the two rational numbers and using this method.

Understand the Question To find rational numbers between two fractions, we first express them with a common denominator . If and (with ), the rational numbers between them with denominator correspond to the strictly intermediate integers between and . The number of strictly intermediate integers between and is . To find at least such numbers, the difference between the numerators must be large enough. Step 1 · Express and with a Common Denominator Satisfying Given First, express with a common denominator of Here, the difference in numerators is , which is not greater than . Multiply the numerator and denominator of both fractions by Here, , , and . Check the condition Step 2 · List Five Rational Numbers Between and Since and , we choose five integers strictly between and for the numerators: Each of these numbers lies strictly between and : Step 3 · Explain the Condition for Finding Rational Numbers For two integers and with , the integers strictly lying between them are The total number of integers in this range is To find at least distinct rational numbers of the form strictly between and , there must be at least available integer numerators: Thus, the difference between the numerators must be at least (or ) to guarantee at least distinct integers between them. Answer > , , and five rational numbers between them are . Common Mistakes Endpoint Inclusion Error: Assuming there are integers between and . Since the endpoints are excluded, there are only strictly intermediate integers. Insufficient Denominator Scaling: Multiplying by a scale factor that makes , which does not provide enough intermediate fractions.

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Question 13

Let $a = \dfrac{7}{12}$ and $b = \dfrac{5}{6}$ . Express both $a$ and $b$ in the form $\dfrac{k_1}{m}$ and $\dfrac{k_2}{m}$ where $k_1$ , $k_2$ and $m$ are integers and $k_2 - k_1 > 6$ . Using the same denominator $m$ , write exactly five distinct rational numbers lying between $a$ and $b$ keeping an integer numerator. Explain why the condition $k_2 - k_1 > n + 1$ is necessary to find $n$ such rational numbers between the two rational numbers $a$ and $b$ using this method.

Solution

Understand the Question

To find rational numbers between two fractions, we first express them with a common denominator $m$ .
If $a = \dfrac{k_1}{m}$ and $b = \dfrac{k_2}{m}$ (with $k_1 < k_2$ ), the rational numbers between them with denominator $m$ correspond to the strictly intermediate integers between $k_1$ and $k_2$ .
The number of strictly intermediate integers between $k_1$ and $k_2$ is $(k_2 - k_1 - 1)$ . To find at least $n$ such numbers, the difference between the numerators must be large enough.

Step 1 · Express $a$ and $b$ with a Common Denominator Satisfying $k_2 - k_1 > 6$

Given $a = \frac{7}{12}, \quad b = \frac{5}{6}$

First, express with a common denominator of $12$ $a = \frac{7}{12}, \quad b = \frac{5 \times 2}{6 \times 2} = \frac{10}{12}$

Here, the difference in numerators is $10 - 7 = 3$ , which is not greater than $6$ .

Multiply the numerator and denominator of both fractions by $3$ $a = \frac{7 \times 3}{12 \times 3} = \frac{21}{36}$ $b = \frac{10 \times 3}{12 \times 3} = \frac{30}{36}$

Here, $m = 36$ , $k_1 = 21$ , and $k_2 = 30$ .

Check the condition $k_2 - k_1 = 30 - 21 = 9 > 6$

Step 2 · List Five Rational Numbers Between $a$ and $b$

Since $a = \dfrac{21}{36}$ and $b = \dfrac{30}{36}$ , we choose five integers strictly between $21$ and $30$ for the numerators: $\frac{22}{36}, \quad \frac{23}{36}, \quad \frac{24}{36}, \quad \frac{25}{36}, \quad \frac{26}{36}$

Each of these numbers lies strictly between $a$ and $b$ : $\frac{21}{36} < \frac{22}{36} < \frac{23}{36} < \frac{24}{36} < \frac{25}{36} < \frac{26}{36} < \frac{30}{36}$

Step 3 · Explain the Condition for Finding $n$ Rational Numbers

For two integers $k_1$ and $k_2$ with $k_1 < k_2$ , the integers strictly lying between them are $k_1 + 1, \; k_1 + 2, \; \dots, \; k_2 - 1$

The total number of integers in this range is $(k_2 - 1) - (k_1 + 1) + 1 = k_2 - k_1 - 1$

To find at least $n$ distinct rational numbers of the form $\dfrac{k}{m}$ strictly between $\dfrac{k_1}{m}$ and $\dfrac{k_2}{m}$ , there must be at least $n$ available integer numerators: $k_2 - k_1 - 1 \ge n \implies k_2 - k_1 \ge n + 1$

Thus, the difference between the numerators must be at least $n + 1$ (or $k_2 - k_1 > n$ ) to guarantee at least $n$ distinct integers between them.

Answer

$a = \dfrac{21}{36}$ , $b = \dfrac{30}{36}$ , and five rational numbers between them are $\dfrac{22}{36}, \dfrac{23}{36}, \dfrac{24}{36}, \dfrac{25}{36}, \dfrac{26}{36}$ .

Common Mistakes

Endpoint Inclusion Error: Assuming there are $k_2 - k_1$ integers between $k_1$ and $k_2$ . Since the endpoints are excluded, there are only $(k_2 - k_1 - 1)$ strictly intermediate integers.
Insufficient Denominator Scaling: Multiplying by a scale factor that makes $k_2 - k_1 \le n$ , which does not provide enough intermediate fractions.

Question 13

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