NCERT Solutions for Class 9 Maths Exercise 2.1 Question 5
Understanding the Question 🧐
Welcome to our detailed guide on NCERT Solutions for Class 9 Maths Exercise 2.1 Question 5. In this question from the Number Systems chapter, we are tasked with expressing the recurring decimal &&0.99999…&& in the form of &&p/q&&, where &&p&& and &&q&& are integers. This is a fundamental concept in understanding how recurring decimals can be represented as fractions, and our step-by-step solution will make it easy for you to grasp.
Question: Express &&0.99999…&& in the form of &&p/q&&
Step-by-Step Solution ✅
Let’s solve this problem systematically to arrive at the fraction form of &&0.99999…&& using the algebraic method commonly taught in Class 9 Maths.
- Assign a variable to the decimal: Let &&x = 0.99999…&&.
- Multiply to shift the decimal point: Since the decimal repeats after one digit, multiply both sides by 10.
&&10x = 9.99999…&& - Subtract the original equation: Subtract the first equation from the second to eliminate the repeating decimal.
&&10x – x = 9.99999… – 0.99999…&&
&&9x = 9&& - Solve for x: Divide both sides by 9.
&&x = 9/9&&
&&x = 1&& - Express in p/q form: Since &&x = 1&&, it can be written as &&1/1&&, where &&p = 1&& and &&q = 1&&.
Answer: The recurring decimal &&0.99999…&& in the form of &&p/q&& is &&1/1&&.
– Always multiply by a power of 10 based on the length of the repeating block.
– Subtracting equations eliminates the repeating part.
– The result must be in the simplest form of &&p/q&&.
Conclusion 🌟
With these NCERT Solutions for Class 9 Maths Exercise 2.1 Question 5, we’ve demonstrated how to convert the recurring decimal &&0.99999…&& into the fraction &&1/1&&. This method is essential for solving similar problems in the Number Systems chapter. If you’re looking for more NCERT solutions or practice questions, explore our website or refer to the official NCERT resources at https://ncert.nic.in.
Frequently Asked Questions (FAQs) ❓
How to express &&0.99999…&& as a fraction in NCERT Solutions for Class 9 Maths Exercise 2.1 Question 5?
Let &&x = 0.99999…&&. Multiply by 10 to get &&10x = 9.99999…&&. Subtract &&x&& from &&10x&&: &&9x = 9&&, so &&x = 1&& or &&1/1&&.
What is the concept behind converting recurring decimals to fractions?
The concept involves setting the decimal to a variable, multiplying by a power of 10 to shift the decimal, and subtracting to eliminate the repeating part, as shown in our Class 9 Maths solutions.
Why does &&0.99999…&& equal 1 in NCERT solutions?
Mathematically, when expressed as a fraction using subtraction of equations, &&0.99999…&& results in &&1&&, proving they are equal.
Are there other methods to solve recurring decimals in Class 9 Maths?
Yes, besides the algebraic method, you can use infinite geometric series, though the algebraic approach is standard in NCERT solutions for Class 9 Maths.