Find the best tutors and institutes for Class 10 Tuition
Q3(iii):
Express the following in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$.
(iii) $0.\overline{001}$
Solution :
Step 1: Initial Setup & Variable Assignment
Let the given repeating decimal be assigned to a variable $x$. The vinculum (the bar over the digits) indicates the specific block of digits that repeats infinitely.
$x = 0.\overline{001}$
Expanding the repeating block, we can write this as an infinite decimal:
$x = 0.001001001... \quad \text{--- (Equation 1)}$
Step 2: Determining the Multiplier [Base-10 Shift]
To convert a recurring decimal into a rational fraction of the form $\frac{p}{q}$, we must eliminate the infinitely repeating fractional part. [Per the properties of repeating decimals], we achieve this by shifting the decimal point to the right by exactly the number of digits in the repeating block.
The repeating block is "001", which consists of exactly $3$ digits. Therefore, we multiply both sides of Equation 1 by $10^3$ (which is $1000$).
$1000 \cdot x = 1000 \cdot (0.001001001...)$
$1000x = 1.001001001... \quad \text{--- (Equation 2)}$
Step 3: Algebraic Subtraction & Elimination of the Infinite Series
We now have two equations where the fractional parts (the digits after the decimal point) are identical. By subtracting Equation 1 from Equation 2, the infinite repeating sequence will cancel out completely [By the Subtraction Property of Equality].
Mathematically, this is executed as:
$1000x - x = (1.001001001...) - (0.001001001...)$
$999x = 1$
Step 4: Isolation of the Variable
Divide both sides of the equation by $999$ to isolate $x$ [By the Division Property of Equality]:
$x = \frac{1}{999}$
Step 5: Verification of Rational Conditions
The problem requires the final answer to be in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$.
- $p = 1$, which is an integer ($1 \in \mathbb{Z}$).
- $q = 999$, which is an integer ($999 \in \mathbb{Z}$).
- The denominator $q = 999 \neq 0$.
All conditions for a rational number are strictly satisfied. Furthermore, the greatest common divisor $\text{GCD}(1, 999) = 1$, meaning the fraction is in its simplest form.
Final Solution: The repeating decimal $0.\overline{001}$ expressed in the form $\frac{p}{q}$ is $\frac{1}{999}$.
More Questions from Class 9 Mathematics Number Systems EXERCISE 1.3
- Q1(i): Write the following in decimal form and say what kind of decimal expansion each has : (i) $\frac{36}{100}$
- Q1(ii): Write the following in decimal form and say what kind of decimal expansion each has : (ii) $\frac{1}{11}$
- Q1(iii): Write the following in decimal form and say what kind of decimal expansion each has : (iii) $4\frac{1}{8}$
- Q1(iv): Write the following in decimal form and say what kind of decimal expansion each has : (iv) $\frac{3}{13}$
- Q1(v): Write the following in decimal form and say what kind of decimal expansion each has : (v) $\frac{2}{11}$
- Q1(vi): Write the following in decimal form and say what kind of decimal expansion each has : (vi) $\frac{329}{400}$
- Q2: You know that $\frac{1}{7} = 0.\overline{142857}$. Can you predict what the decimal expansions of $\frac{2}{7}, \frac{3}{7}, \frac{4}{7}, \frac{5}{7}, \frac{6}{7}$ are, without actually doing the long division? If so, how? [Hint : Study the remainders while finding the value of $\frac{1}{7}$ carefully.]
- Q3(i): Express the following in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. (i) $0.\overline{6}$
- Q3(ii): Express the following in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. (ii) $0.4\overline{7}$
- Q4: Express $0.99999 ....$ in the form $\frac{p}{q}$. Are you surprised by your answer? With your teacher and classmates discuss why the answer makes sense.
- Q5: What can the maximum number of digits be in the repeating block of digits in the decimal expansion of $\frac{1}{17}$? Perform the division to check your answer.
- Q6: Look at several examples of rational numbers in the form $\frac{p}{q}$ ($q \neq 0$), where $p$ and $q$ are integers with no common factors other than 1 and having terminating decimal representations (expansions). Can you guess what property $q$ must satisfy?
- Q7: Write three numbers whose decimal expansions are non-terminating non-recurring.
- Q8: Find three different irrational numbers between the rational numbers $\frac{5}{7}$ and $\frac{9}{11}$.
- Q9(i): Classify the following numbers as rational or irrational : (i) $\sqrt{23}$
- Q9(ii): Classify the following numbers as rational or irrational : (ii) $\sqrt{225}$
- Q9(iii): Classify the following numbers as rational or irrational : (iii) $0.3796$
- Q9(iv): Classify the following numbers as rational or irrational : (iv) $7.478478...$
- Q9(v): Classify the following numbers as rational or irrational : (v) $1.101001000100001...$
CBSE Solutions for Class 9 Mathematics Number Systems
Chapters in CBSE - Class 9 Mathematics
Top Tutors who teach Number Systems
Indirapuram Ahinsa Khand 1, Ghaziabad
3 yrs of
Exp
Online Classes
Student's Home
I am a professional french tutor with 5 years of experience in teaching languages. I can teach efficiently to all school students as well as adults, for delf-dalf, migration, jobs and other purposes. Over my years teaching, I have been know to be amiable amd flexible according to my students' needs. I always schedule a free demo session for the student to get comfortable with me my teaching techniques and vice versa. The classes I conduct are via offline and online mode both, as preferred.
Top Tutor
Talwandi Sector-B, Kota
10 yrs of
Exp
1 student
699 hrs of classes completed
1850
per hour
Online Classes
Tutor's home
My teaching exp.11 year in most reputed coaching in india. Mentor of jee top rank air-12 yatis agrawal, 19, 28 many more in top 100 in jee adavnced. Many of students got 100%tile in jee mains along with 100% marks in board exams. I also provide complete relevent study material like worksheet, practice sheet , dpps , assignment etc. Daily quizz based on assignment. Periodically test papers topic wise, minor and major as per shedule. I have helped thousands of students overcome their fear of maths.
Top Tutor
Mahmoorganj Tulsipur, Varanasi
4 yrs of
Exp
12 students
733 hrs of classes completed
100
per hour
Online Classes
Teaching methodology is such that even weakest students can understand the concept very easily through real life analogy
Great teacher i understand subjects with better clarity now and he teaches great and it sure helped me alot
Top Tutor
Kaloor, Kochi
12 yrs of
Exp
2 students
2 hrs of classes completed
500
per hour
Online Classes
Student's Home
Tutor's home
With 12 years of teaching experience, I specialize in concept-based instruction, catering to both weaker students and advanced learners. I help struggling students grasp core concepts while providing advanced-level teaching to high achievers, enabling them to consistently reach 95% or higher in their academic performance.
Good class. Sir explained concepts well, i understood the concepts easily, and sir also cleared all my doubts.
Top Tutor
Sector 12, Noida
6 yrs of
Exp
4 students
148 hrs of classes completed
400
per hour
Online Classes
I have worked as PGT teacher in one of the best CBSE schools and all of my students gained more than 90 percent marks. I have full concept of subjects and can assist students in gaining more than 90 percent marks.
She’s a excellent teacher , her behaviour her talking style is very sweet and if you didn’t understand any topic she’ll repeat it for you as many times you want!! Satisfied and loved her teaching
Find more Tutor for Number Systems in your City
- Bangalore Mathematics Tutors
- Delhi Mathematics Tutors
- Chennai Mathematics Tutors
- Gurgaon Mathematics Tutors
- Noida Mathematics Tutors
- Hyderabad Mathematics Tutors
- Mumbai Mathematics Tutors
- Ghaziabad Mathematics Tutors
- Chandigarh Mathematics Tutors
- Pune Mathematics Tutors
- Jaipur Mathematics Tutors
- Surat Mathematics Tutors
