Jharkhand Board JAC Class 9 Maths Solutions Chapter 1 Number Systems Ex 1.3 Textbook Exercise Questions and Answers.

## JAC Board Class 9th Maths Solutions Chapter 1 Number Systems Exercise 1.3

Question 1.

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

(i) \(\frac{36}{100}\)

(ii) \(\frac{1}{11}\)

(iii) \(4 \frac{1}{8}\)

(iv) \(\frac{3}{13}\)

(v) \(\frac{2}{11}\)

(vi) \(\frac{329}{400}\)

Answer:

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

(ii) \(\frac{1}{11}\) = 0.09090909… = \(0 . \overline{09}\) (Non-terminating and repeating)

(iii) \(4 \frac{1}{8}\) = \(\frac{33}{8}\) =4.125 (Terminating)

(iv) \(\frac{3}{13}\) = 0.230769230769… = \(0 . \overline{230769}\) (Non-terminating and repeating)

(v) \(\frac{2}{11}\) = 0.181818181818… = \(0 . \overline{18}\) (Non-terminating and repeating)

(vi) \(\frac{329}{400}\) = 0.8225 (Terminating)

Question 2.

You know that \(\frac{1}{7}\) = 0.142857. Can you predict what the decimal expansion of \(\frac{2}{7}, \frac{3}{7}, \frac{4}{7}, \frac{5}{7}, \frac{6}{7}, \frac{3}{7}, \frac{4}{7}, \frac{5}{7}, \frac{6}{7}\) are without actually doing the long division? If so, how?

[Hint: Study the remainders while finding the value of \(\frac{1}{7}\) carefully.]

Answer:

Yes, We can do this by:

\(\frac{2}{7}=2 \times \frac{1}{7}=2 \times 0 . \overline{142857}=0 . \overline{285714}\)

\(\frac{3}{7}=3 \times \frac{1}{7}=3 \times 0 . \overline{142857}=0 . \overline{428571}\)

\(\frac{4}{7}=4 \times \frac{1}{7}=4 \times 0 . \overline{142857}=0 . \overline{571428}\)

\(\frac{5}{7}=5 \times \frac{1}{7}=5 \times 0 . \overline{142857}=0 . \overline{714285}\)

\(\frac{6}{7}=6 \times \frac{1}{7}=6 \times 0 . \overline{142857}=0 . \overline{857142}\)

Question 3.

Express the following in the form p/q, where p and q are integers and q ≠ 0.

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

(ii) \(0 . 4 \overline{7}\)

(iii) \(0 . \overline{001}\)

Answer:

(i) \(0 . \overline{6}\) = 0.666…

Let x = 0.666…

∴ 10x = 6.66…

∴ 10x = 6 + 0.66…

∴ 10x = 6 + 0.666…

∴ 10x = 6 + x

∴ 9x = 6

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

(ii) \(0 . 4 \overline{7}\) = 0.4777… = \(\frac{4}{10}\) + \(\frac{0.777}{10}\)

Let x = 0.777…

∴ 10x = 7.77…

∴ 10x = 7.777…

∴ 10x = 7 + 0.777…

∴ 10x = 7 + x

∴ x = \(\frac{7}{9}\)

\(0 . 4 \overline{7}\) = \(\frac{4}{10}\) + \(\frac{0.777}{10}\) = \(\frac{4}{10}\) + \(\frac{7}{90}\)

= \(\frac{36}{90}\) + \(\frac{7}{90}\) = \(\frac{43}{90}\)

(iii) \(0 . \overline{001}\) =0.001001…

Let x = 0.001001…

∴ 1000x = 1.001….

∴ 1000x = 1.001001…

∴ 1000x = 1 + 0.001001 …

∴ l000x= 1 + X

∴ 999x = 1

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

Question 4.

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

Answer:

Let x = 0.9999…

10x = 9.999…

∴ 10x = 9.9999…

∴ 10x = 9 + 0.9999…

∴ 10x = 9 + x

∴ 9x = 9

x = 1

The difference between 1 and 0.999999 is 0.000001 which is negligible. Thus, 0.999… is too much near 1. Therefore, the answer 1 can be justified.

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:

\(\frac{1}{17}\) = 0.05882352941176470588…

= \(0 . \overline{0588235294117647}\)

There are 16 digits in the repeating block of the decimal expansion of \(\frac{1}{17}\)

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 representations (expansions). Can you guess what property q must satisfy?

Answer:

We observe that when q is 2, 4, 5, 8, 10,… then the decimal expansion is terminating. For example:

\(\frac{1}{2}\) = 0.5, denominator q = 2^{1}

\(\frac{4}{5}\) = 0.8, denominator q = 5^{1}

We can observe that terminating decimal may be obtained in the situation where prime factorisation of the denominator of the given fractions has the power of 2 only or 5 only or both.

Question 7.

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

Answer:

Three numbers whose decimal expansions are non-terminating non-recurring are:

(i) 0.303003000300003…

(ii) 0.505005000500005…

(iii) 0.7207200720007200007200000…

Question 8.

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

Answer:

\(\frac{5}{7}\) = \(0 . \overline{714285}\)

\(\frac{9}{11}\) = \(0 . \overline{81}\)

Three different irrational numbers are:

0.73073007300073000073…

0.75075007500075000075…

0.76076007600076000076…

Question 9.

Classify the following numbers as rational or irrational:

(i) \(\sqrt{23}\)

(ii) \(\sqrt{225}\)

(iii) 0.3796

(iv) 7.478478…

(v) 1.101001000100001…

Answer:

(i) \(\sqrt{23}\) = 4.79583152331…

Since the decimal expansion is non-terminating and non-recurring therefore, it is an irrational number.

(ii) \(\sqrt{225}\) = 15 = \(\frac{15}{1}\)

The number is rational number as it can represented in \(\frac{p}{q}\) form where p, q ∈ Z and q ≠ 0.

(iii) 0.3796

Since the decimal expansion is terminating therefore, it is a rational number.

(iv) 7.478478… = \(7 . \overline{478}\)

Since, this decimal expansion is non-terminating recurring, therefore, it is a rational number.

(v) 1.101001000100001…

Since the decimal expansion is non-terminating and non-repeating, therefore, it is an irrational number.