# JAC Class 9 Maths Solutions Chapter 1 Number Systems Ex 1.3

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) $$\frac{329}{400}$$
(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.]
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) $$0 . \overline{001}$$
(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.
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.
$$\frac{1}{17}$$ = 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?
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 = 21
$$\frac{4}{5}$$ = 0.8, denominator q = 51
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.
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 $$\frac{9}{11}$$.
$$\frac{5}{7}$$ = $$0 . \overline{714285}$$

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