NCERT Solutions Class 9 Mathematics Number System Exercise 1.3

Exercise 1.3

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Question : 1:Write the following in decimal form and say what kind of decimal expansion each has:

(i) 36/100       (ii) 1/11       (iii) 4            (iv) 3/13          (v) 2/11             (vi) 329/400

Answer :

(i) 36/100 = 0.36

So, the decimal expansion of 36/100 is terminating.

(ii) 1/11 = 0.090909….. = 0.09

Thus, the decimal expansion of 1/11 is non-terminating repeating.

(iii) 4 = 4 + 1/8 = 33/8 = 4.125

Thus, the decimal expansion of 4 is terminating.

(iv) 3/13 = 0.230769230769…. = 0.230769

Thus, the decimal expansion of 3/13 is non-terminating repeating.

(v) 2/11 = 0.181818………. = 0.18

Thus, the decimal expansion of 2/11 is non-terminating repeating.

(vi) 329/400 = 0.8225

Thus, the decimal expansion of is terminating.

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Question : 2:You know that 1/7 = 0.142857 can you predict what the decimal expansion of 1/7, 3/7, 4/7, 5/7, 6/7. Are, without actually doing the long division? if so how?

Answer :

Given, 1/7 = 0.142857

Now,

2/7 = 2 * (1/7) = 2 * 0.142857 = 0.285714

3/7 = 3 * (1/7) = 3 * 0.142857 = 0.428571

4/7 = 4 * (1/7) = 4 * 0.142857 = 0.571428

5/7 = 5 * (1/7) = 5 * 0.142857 = 0.714285

6/7 = 6 * (1/7) = 6 * 0.142857 = 0.857142

Thus, without actually doing the long division we can predict the decimal expansions of the

above given rational numbers.

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Question : 3:Express the following in the form p/q where p and q are integers and q ≠ 0.

(i) 0.6                                 (ii) 0.47                        (iii) 0.001

Answer :

(i) Let x = 0.6

⇒ x = 0.66666…….                  ……….1

Multiply equation 1 by 10 on both sides, we get

      10x = 6.6666….

⇒ 10x = 6 + 0.6666……

⇒ 10x = 6 + x                    [From equation 1]

⇒ 10x – x = 6

⇒ 9x = 6

⇒ x = 6/9

⇒ x = 2/3

(pi) Let x = 0.47

⇒ x = 0.477777…….                  ……….1

Multiply equation 1 by 10 on both sides, we get

      10x = 4.7777….                  …….2

Multiply equation 2 by 10 on both sides, we get

⇒ 100x = 47 + 0.7777……

⇒ 100x = 43 + 4 + 0.7777……

⇒ 100x = 43 + 4.7777……

⇒ 100x = 43 + 10x                    [From equation 2]

⇒ 100x – 10x = 43

⇒ 90x = 43

⇒ x = 43/90

(iii) Let x = 0.001

⇒ x = 0.001001001…….                  ……….1

Multiply equation 1 by 1000 on both sides, we get

      1000x = 1 + 0. 001001

⇒ 1000x = 1 + x                        [From equation 1]

⇒ 1000x – x = 1

⇒ 999x = 1

⇒ x = 1/999

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Question : 4:Express 0.99999 . . . in the form p/q. Are you surprised by your answer? With your teacher and classmates discuss why the answer makes sense.

Answer :

Let x = 0. 99999…….                  ……….1

Multiply equation 1 by 10 on both sides, we get

      10x = 9. 99999….

⇒ 10x = 9 + 0. 99999…..

⇒ 10x = 9 + x                    [From equation 1]

⇒ 10x – x = 9

⇒ 9x = 9

⇒ x = 9/9

⇒ x = 1

The answer makes sense as 0.99999…. is very close to 1. That’s why we can say that

1. 99999… = 1

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Question : 5:What can 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.

Answer :

Since, the number of entries in the repeating block of digits is less than the divisor. In 1/17 the

divisor is 17.

So, the maximum number of digits in the repeating block is 16.

To perform the long division, we have

1/17 = 0.588235294117647

Thus, there are 16 digits in the repeating block in the decimal expansion of 1/17.

Hence, our answer is verified.

 

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Question : 6:Look at 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?

Answer :

Let some examples are:

2/5 = 0.4,            1/10 = 0.1,             3/2 = 1.5,                   7/8 = 0.875

The denominator of all the rational numbers are in the form 2^m * 5ⁿ where m and n are

integers.