# 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

**ā 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.

**ā 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.

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