NCERT Solutions for Class 9 Maths Exercise 1.1 Question 3
Understanding the Question 🧐
This question asks us to find five rational numbers between two given fractions, &&\frac{3}{5}&& and &&\frac{4}{5}&&. The method is very similar to finding numbers between integers. The key is to create equivalent fractions with a larger denominator to make space for new numbers in between. These ncert solutions will guide you through the process.
Find five rational numbers between &&\frac{3}{5}&& and &&\frac{4}{5}&&.
Step-by-Step Solution 📝
We will follow the same &&(n+1)&& method we used for integers. This method works perfectly for fractions as well.
Step 1: Check the denominators and determine the multiplier
The given numbers are &&\frac{3}{5}&& and &&\frac{4}{5}&&. We can see that the denominators are already the same, which is great!
We need to find &&5&& rational numbers, so &&n = 5&&. We will use the multiplier &&n+1 = 5+1 = 6&&.
Step 2: Create equivalent fractions
Now, we multiply the numerator and the denominator of each fraction by our multiplier, &&6&&.
- For &&\frac{3}{5}&&:
&&\frac{3}{5} = \frac{3 \times 6}{5 \times 6} = \frac{18}{30}&& - For &&\frac{4}{5}&&:
&&\frac{4}{5} = \frac{4 \times 6}{5 \times 6} = \frac{24}{30}&&
Our task has now become finding five rational numbers between &&\frac{18}{30}&& and &&\frac{24}{30}&&.
Step 3: List the rational numbers
We can now easily list the fractions between our two new fractions by counting up the numerator.
The five rational numbers are: &&\frac{19}{30}, \frac{20}{30}, \frac{21}{30}, \frac{22}{30}, \frac{23}{30}&&
Step 4 (Optional but Recommended): Simplify the fractions
It’s good practice to simplify any fractions that can be reduced.
- &&\frac{20}{30} = \frac{2}{3}&&
- &&\frac{21}{30} = \frac{7}{10}&&
- &&\frac{22}{30} = \frac{11}{15}&&
So, the simplified list is: &&\frac{19}{30}, \frac{2}{3}, \frac{7}{10}, \frac{11}{15}, \frac{23}{30}&&.
Conclusion and Key Points ✅
Five rational numbers between &&\frac{3}{5}&& and &&\frac{4}{5}&& are &&\frac{19}{30}, \frac{20}{30}, \frac{21}{30}, \frac{22}{30},&& and &&\frac{23}{30}&&. The key was to make the denominators larger by multiplying both the numerator and denominator by &&6&& (which is &&5+1&&), allowing us to easily find the numbers in between.
Multiplying by &&10&& is often easier and quicker. You can multiply the numerator and denominator of &&\frac{3}{5}&& and &&\frac{4}{5}&& by &&10&& to get &&\frac{30}{50}&& and &&\frac{40}{50}&&. Then you can easily pick any five numbers, such as &&\frac{31}{50}, \frac{32}{50}, \frac{33}{50}, \frac{34}{50}, \frac{35}{50}&&. This is also a correct answer!
- First, ensure the fractions have a common denominator. (In this case, they already did).
- To find &&n&& numbers, multiply the numerator and denominator by &&n+1&& (or any number larger than &&n&&).
- List the new fractions by incrementing the numerator.
- Always simplify the final fractions if possible.
FAQ
Q: How is this question different from finding rational numbers between two integers?
A: The method is exactly the same. The only difference is that you start with fractions instead of integers. Since the denominators were already the same (both &&5&&), the first step of creating a common denominator was already done for you.
Q: Why do we multiply by 6/6 in this specific problem?
A: We need to find five rational numbers, so we let &&n = 5&&. The &&(n+1)&& method suggests multiplying by &&(5+1)&&, which is &&6&&. By multiplying the numerator and denominator by &&6&&, we create enough ‘space’ between the numerators to easily pick five rational numbers.
Q: Can I multiply by 10/10 instead of 6/6?
A: Yes, absolutely. Multiplying by &&\frac{10}{10}&& would change &&\frac{3}{5}&& to &&\frac{30}{50}&& and &&\frac{4}{5}&& to &&\frac{40}{50}&&. You could then easily pick numbers like &&\frac{31}{50}, \frac{32}{50},&& etc. This is another set of correct answers.
Q: How many rational numbers are there between 3/5 and 4/5?
A: Just like with integers, there are infinitely many rational numbers between any two distinct fractions like &&\frac{3}{5}&& and &&\frac{4}{5}&&. We are only asked to find five of them.
Further Reading
For more information on Number Systems, you can refer to the official NCERT textbook or visit the NCERT website.
