You know that 1/7 = 0.142587. Can you predict what the decimal expansions of 2/7, 3/7, 4/7, 5/7, 6/7 are, without actually doing the long division? If so, how?
[Hint : Study the remainders while finding the value of 1/7 carefully.]
Solution:
We have the decimal expansion of 1/7 = 0.142587
Let's find the decimal expansions of 2/7, 3/7, 4/7, 5/7, 6/7.
To proceed with this, let's observe the pattern of remainders and the digits of the quotient in the long division of 1/7 as shown below.
As we can see, 1/7 is a non-terminating recurring decimal. We can use this to find the decimal expansion of 2/7, 3/7, 4/7, 5/7, 6/7.
To write the decimal expansion for:
i) 2/7
2/7 = 2 × (1/7)
= 2 × 0.142857
= 0.285714
Also, we observe that we get 2 as a remainder after the second step in the above division.
Hence, we start writing the quotient after the second decimal place and we get 2/7 = 0.285714
Hence, 2/7 = 0.285714
ii) 3/7
3/7 = 3 × (1/7)
= 3 × 0.142857
= 0.428571
Also, we observe that we get 3 as a remainder after the first step in the above division.
Hence, we start writing the quotient after the first decimal place and we get 3/7 = 0.428571
Hence, 3/7 = 0.428571
iii) 4/7
4/7 = 4 × (1/7)
= 4 × 0.142857
= 0.571428
Also, we observe that we get 4 as a remainder after the fourth step in the above division.
Hence, we start writing the quotient after the fourth decimal place and we get 4/7 = 0.571428
Hence, 4/7 = 0.571428
iv) 5/7
5/7 = 5 × (1/7)
= 5 × 0.142857
= 0.714285
Also, we observe that we get 5 as a remainder after the fifth step in the above division.
Hence, we start writing the quotient after the fifth decimal place and we get 5/7 = 0.714285
Hence, 5/7 = 0.714285
v) 6/7
6/7 = 6 × (1/7)
= 6 × 0.142857
= 0.857142
Also, we observe that we get 6 as a remainder after the third step in the above division.
Hence, we start writing the quotient after the third decimal place and we get 6/7 = 0.857142
Hence, 6/7 = 0.857142
☛ Check: Class 9 NCERT Solutions Maths Chapter 1
Video Solution:
You know that 1/7 = 0.142587. Can you predict what the decimal expansions of 2/7, 3/7, 4/7, 5/7, 6/7 are, without actually doing the long division? If so, how? [Hint : Study the remainders while finding the value of 1/7 carefully.]
NCERT Solutions Class 9 Maths Chapter 1 Exercise 1.3 Question 2:
Summary:
Given the decimal expansion of 1/7 = 0.142587, the decimal expansions of 2/7, 3/7, 4/7, 5/7, and 6/7 are 0.285714, 0.428571, 0.571428, 0.714285, and 0.857142 respectively.
☛ Related Questions:
- Express the following in the form of p/q, where p and q are integers and q ≠ 0. i) 0.6 ii) 0.47 iii) 0.001
- Express 0.99999 .... in the form of p/q. Are you surprised with your answer? With your teacher and classmates discuss why the answer makes sense?
- What can be the maximum number of digits be in the repeating block of digits in the decimal expansion of 1/17? Perform the division to check your answer.