## MP Board Class 9th Maths Solutions Chapter 1 Number Systems Ex 1.3

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

- \(\frac{36}{100}\)
- \(\frac{1}{11}\)
- 4\(\frac{1}{8}\)
- \(\frac{3}{13}\)
- \(\frac{2}{11}\)
- \(\frac{329}{400}\)

Solution:

1. \(\frac{36}{100}\)

\(\frac{36}{100}\) = 0.36

The decimal expansion is terminating.

2. \(\frac{1}{11}\)

The decimal expansion is non-terminating repeating.

3. 4\(\frac{1}{8}\)

The decimal expansion is terminating.

4. \(\frac{3}{13}\)

The decimal expansion is non-terminating repeating.

5. \(\frac{2}{11}\)

The decimal expansion is non-terminating repeating.

6. \(\frac{329}{400}\)

The decimal expansion is terminating.

Question 2.

You know that \(\frac{1}{7}\) = \(\overline { 0.142857 } \). Can vou 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.]

Solution:

\(\frac{1}{7}\) = \(\overline { 0.142857 } \)

Question 3.

Expressthe following in the form \(\frac{p}{q}\), wherep and q are integers and q ≠ 0.

- \(0 . \overline{6}\)
- \(0 . \overline{oo1}\)

Solution:

1. \(0 . \overline{6}\)

Let x = \(0 . \overline{6}\) …(i)

10x = \(6 . \overline{6}\)

[Multiplying (i) by 10 on both sides] …(ii)

Subtracting (i) from (ii). we get

9x = 6

x = \(\frac{6}{9}\) = \(\frac{2}{3}\)

∴ \(6 . \overline{6}\) = \(\frac{2}{3}\)

2. \(0 . \overline{oo1}\)

1000x = \(1 . \overline{001}\)

[Multiplying (i) by 1000] …(ii)

Subtracting (i) from (ii), we get

999x = 1

x = \(\frac{1}{999}\)

∴ \(0 . \overline{001}\) = \(\frac{1}{999}\)

Question 4.

Express 0.99999….. in the form \(\frac{p}{q}\). Are you surprised by vour answer? With your teacher and classmates, discuss why the answer make sense.

Solution:

0.99999 = \(0 . \overline{9}\)

Let x = 0.9 …(i)

10x = \(9 . \overline{9}\)

[Multiplying (i) by 10] …(ii)

Subtracting (i) from (ii), we get

9x = 9

x = \(\frac{9}{9}\)

∴ \(9 . \overline{9}\) = 1

Question 5.

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.

Solution:

The maximum number of digits in the repeating block of digits in the decimal expansion \(\frac{1}{17}\) can be 16.

0. 05882352941176470588235294117647….

By Long Division, the number of digits in the repeating block of digits in the decimal expansion of = \(\frac{1}{17}\) = 16

∴ The answer is verified.

Question 6.

Look at several examples of rational numbers in the form \(\frac{p}{q}\) (q ≠ 0),where p and q are integers with no common factors other than 1 and having terminating decimal representation (expansions). Can you guess what property q must satisfy?

Solution:

Examples:

The property that q must satisfy is that the prime factorisation of q have only powers of 2 or powers of 5 or both.

Question 7.

Write three numbers whose decimal expansions are non – terminating non – recurring.

Solution:

0. 01001000100001……..,

0. 20200220002200002…….,

0. 003000300003

Question 8.

Find three different irrational numbers between the rational numbers \(\frac{5}{7}\) and \(\frac{9}{11}\).

Solution:

Irrational numbers between \(\frac{5}{7}\) and \(\frac{9}{11}\)

\(\frac{5}{7}\) = 0.71 and \(\frac{9}{11}\) = 0.81

Three irrational numbers between \(\frac{5}{7}\) and \(\frac{9}{11}\) are

0. 7201001000…

0. 7301001000…

0. 7401001000…

Question 9.

Classify the following numbers as rational or irrational:

- \(\sqrt{23}\)
- \(\sqrt{225}\)
- 0. 3796
- 7. 478478…
- 1. 101001000100001…

Solution:

- \(\sqrt{23}\) is an irrational number
- \(\sqrt{225}\) = 15, a rational number
- 0. 3796 is a rational number
- 7. 478478….. is an irrational number
- 1. 101001000100001… is an irrational number