NCERT Solutions for Class 9 Maths Exercise 1.1 Question 2
Understanding the Question 🧐
The question asks us to find six rational numbers between &&3&& and &&4&&. An important concept to remember is that between any two distinct numbers, there are infinitely many rational numbers. Our goal is to find just six of them using a simple and reliable method. This guide from ncert solutions will show you how.
Find six rational numbers between &&3&& and &&4&&.
Step-by-Step Solution 📝
We will use a common method to solve this. The idea is to convert &&3&& and &&4&& into equivalent fractions with a larger common denominator, which makes it easy to pick the numbers in between.
Step 1: Determine the multiplier
We need to find &&6&& rational numbers. Let &&n = 6&&. A good practice is to use &&n+1&& as our multiplier.
So, our multiplier is &&n+1 = 6+1 = 7&&.
We will multiply and divide our numbers (&&3&& and &&4&&) by &&7&&. This is the same as multiplying by &&\frac{7}{7}&&, which is just &&1&&, so we don’t change the value of the numbers.
Step 2: Convert the numbers into equivalent fractions
Now, we convert &&3&& and &&4&& into fractions with a denominator of &&7&&.
- For the number &&3&&:
&&3 = 3 \times 1 = 3 \times \frac{7}{7} = \frac{21}{7}&& - For the number &&4&&:
&&4 = 4 \times 1 = 4 \times \frac{7}{7} = \frac{28}{7}&&
So, our problem is now to find six rational numbers between &&\frac{21}{7}&& and &&\frac{28}{7}&&.
Step 3: List the rational numbers
We can now easily list the fractions between &&\frac{21}{7}&& and &&\frac{28}{7}&& by simply increasing the numerator by &&1&&.
The six rational numbers are: &&\frac{22}{7}, \frac{23}{7}, \frac{24}{7}, \frac{25}{7}, \frac{26}{7}, \frac{27}{7}&&
Conclusion and Key Points ✅
The six rational numbers between &&3&& and &&4&& are &&\frac{22}{7}, \frac{23}{7}, \frac{24}{7}, \frac{25}{7}, \frac{26}{7},&& and &&\frac{27}{7}&&. The method involves converting the given integers into equivalent fractions with a sufficiently large common denominator, making it simple to identify the numbers in between.
The answers are not unique! You could have multiplied by any number greater than &&6&& (like &&8, 10,&& or &&100&&) and still found correct answers. For example, if you multiply by &&10&&:
- &&3 = \frac{30}{10}&&
- &&4 = \frac{40}{10}&&
- Six rational numbers could be: &&\frac{31}{10}, \frac{32}{10}, \frac{33}{10}, \frac{34}{10}, \frac{35}{10}, \frac{36}{10}&&. This is also a perfectly valid answer.
- To find &&n&& rational numbers between two numbers, it’s easiest to multiply them by &&\frac{n+1}{n+1}&&.
- First, express the integers as fractions (e.g., &&3 = \frac{3}{1}&&).
- Create equivalent fractions with the new, larger denominator.
- List the fractions between the two new fractions.
FAQ
Q: How many rational numbers exist between any two distinct rational numbers?
A: There are infinitely many rational numbers between any two distinct rational numbers. The question asks for only six of them.
Q: What is the first step to find rational numbers between two integers like 3 and 4?
A: The first step is to express the integers as rational numbers (fractions). A simple way is to write them with a denominator of &&1&&, so &&3&& becomes &&\frac{3}{1}&& and &&4&& becomes &&\frac{4}{1}&&.
Q: Why do we multiply by (n+1)/(n+1) to find ‘n’ rational numbers?
A: We multiply by &&\frac{n+1}{n+1}&& because it’s an easy way to create equivalent fractions with a larger denominator. This process creates at least ‘&&n&&’ integer gaps between the new numerators, which we can use to form the required rational numbers. Multiplying by a number like &&\frac{7}{7}&& is the same as multiplying by &&1&&, so the value of the original numbers does not change.
Q: Is the set of answers for this question unique?
A: No, the answers are not unique. Since there are infinite rational numbers between &&3&& and &&4&&, you can find many different sets of six numbers. For example, multiplying by &&\frac{10}{10}&& instead of &&\frac{7}{7}&& would give a different, but still correct, set of answers.
Q: What if I need to find 10 rational numbers between 3 and 4?
A: If you need to find &&10&& rational numbers (&&n=10&&), you would multiply &&3&& and &&4&& by &&\frac{10+1}{10+1}&&, which is &&\frac{11}{11}&&. This would give you the fractions &&\frac{33}{11}&& and &&\frac{44}{11}&&, and you could easily list &&10&& rational numbers between them.
Further Reading
For more information on Number Systems, you can refer to the official NCERT textbook or visit the NCERT website.
