# Look at the several examples of rational numbers in the form p/q (q ≠ 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?

**Solution:**

We shall look at some examples of rational numbers in the form of p/q (q ≠ 0), where their decimal representations are terminating.

2/5 = 0.4

3/100 = 0.03

27/16 = 1.6875

33/50 = 0.66

Let's observe the denominators of the above rational numbers.

2/5 = 2 / (2^{0} × 5^{1})

3/100 = 3 / (2^{2} × 5^{2})

27/16 = 27 / (2^{4} × 5^{0})

33/50 = 33 / (2^{1} × 5^{2})

We observe that the denominators of the above rational numbers are in the form of 2^{a} × 5^{b}, where a and b are whole numbers.

Hence if q is in the form 2^{a} × 5^{b} then p/q is a terminating decimal.

**☛ Check: **Class 9 Maths NCERT Solutions for Chapter 1

**Video Solution:**

## Look at the several examples of rational numbers in the form p/q (q ≠ 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?

NCERT Solutions Class 9 Maths Chapter 1 Exercise 1.3 Question 6:

**Summary:**

The property that q must satisfy to be a terminating decimal in the form of p/q (q ≠ 0), where p and q are integers with no common factors other than 1 is that q must be in the form of 2^{a} × 5^{b } where a and b are whole numbers.

**☛ Related Questions:**

- 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?
- 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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