Let and . Express both a and b in the form and where , and m are integers and 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 n + 1 is necessary to find n such rational numbers between the two rational numbers a and b using this method.

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KnowledgeBoat · Accepted Answer

Given, ⇒ and . L.C.M. of 12 and 6 is 12. Here, k2 - k1 = 10 - 7 = 3, which is not greater than 6. To make k2 - k1 > 6, multiply numerator and denominator of both fractions by a sufficiently large integer. Take m = 12 × 3 = 36 : Now, k1 = 21, k2 = 30 and m = 36. ⇒ k2 - k1 = 30 - 21 = 9, which is greater than 6. The 5 distinct rational numbers between and with integer numerators are : Why the condition k2 - k1 > n + 1 gives enough integer numerators : The integers strictly between k1 and k2 are k1 + 1, k1 + 2, …, k2 - 1. Number of such integers = k2 - k1 - 1. To pick n distinct integer numerators strictly between k1 and k2, we need : ⇒ k2 - k1 - 1 ≥ n ⇒ k2 - k1 ≥ n + 1 If we want to ensure more than enough space, we take k2 - k1 > n + 1 Hence, this condition ensures that there are enough integer numerators between k1 and k2 to write n rational numbers using the same denominator. For n = 5, we needed k2 - k1 > 6, which gave us the buffer needed to easily select 5 numbers between a and b. Hence, with and (m = 36), the 5 rational numbers between a and b are and . The condition k2 - k1 > n + 1 ensures that there are at least n integer numerators strictly between k1 and k2.

Mathematics

Whole Numbers

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