Check If a Given Rational No. Is Terminating or Not

Q. Question 7
$\text{Express the following in the form}$$\frac{p}{q}$, $\text{where p and q are integers and}q\ne 0$
(i) 0.2
(ii) 0.888…
(iii) $5.\overline{2}$
(iv) $0\overline{.001}$
(v)0.2555…
(vi)$0.1\overline{34}$
(vii) 0.00323232...
(viii) 0.404040…

Q.

Without actual division, show that each of the following rational numbers is a nonterminating repeating decimal.

$\left(i\right)\frac{11}{(2^3\times 3)}\left(ii\right)\frac{73}{(2^2\times 3^3\times 5)}\left(iii\right)\frac{129}{(2^2\times 5^3\times 7^2)}\left(iv\right)\frac{9}{35}\left(v\right)\frac{77}{210}\left(vi\right)\frac{32}{147}\left(vii\right)\frac{29}{343}\left(viii\right)\frac{64}{455}$

Q.

Question 2
Write down the decimal expansions of
(i) $\frac{13}{3125}$
(ii) $\frac{17}{8}$
(iii) $\frac{15}{1600}$
(iv) $\frac{23}{2^3{5}^2}$
(v) $\frac{6}{15}$
(vi) $\frac{35}{50}$

Q. Question 2
Write down the decimal expansions of those rational numbers in Question 1 above which have terminating decimal expansions.

Q.

Without actual division, show that each of the following rational numbers is a terminating decimal. Express each in decimal form.

$\left(i\right)\frac{23}{(2^3\times 5^2)}\left(ii\right)\frac{24}{125}\left(iii\right)\frac{171}{800}\left(iv\right)\frac{15}{1600}\left(v\right)\frac{17}{320}\left(vi\right)\frac{19}{3125}$

Q.

Write down the decimal expansions of : $\frac{15}{1600}$

Q.

Which of the following rational numbers is expressible as a terminating decimal ?

$\left(a\right)\frac{124}{165}\left(b\right)\frac{131}{30}\left(c\right)\frac{2027}{625}\left(d\right)\frac{1625}{462}$

Q.

Write down the decimal expansions of : $\frac{35}{50}$

Q.

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

Q.

Question 2
You know that $\frac{1}{7}=0.\overline{142587}$ .Can you predict the decimal expansion of $\frac{2}{7},\frac{3}{7},\frac{4}{7},\frac{5}{7},\frac{6}{7}$ without actually doing the long division? If so, how?

Q.

Write down the decimal expansions of : $\frac{23}{2^3{.5}^2}$

Q.

$3.7777...$when expressed in the form of $\frac{p}{q}\text{is}........$

Q.

The $43.12345678$ real numbers have decimal expansions. Decide whether it is rational or not. If it is rational, and of the form, p q what can you say about the prime factors of $\mathrm{q}$.

Q.

Write down the decimal expansions of : $\frac{17}{8}$

Q.

Write down the decimal expansions of : $\frac{6}{15}$

Q.

Question 3 (iii)
Express $0.\overline{001}$ in the form $\frac{p}{q}$, where p and q are integers and $q\ne 0$.

Q.

Write down the decimal expansions of : $\frac{13}{1325}$

Q.

If the rational number $\frac{a}{b}$ has a terminating decimal expansion, what is the condition to be satisfied by $b$?

Q. Assertion (A): $\frac{29}{9261}$ will have a non-terminating, non-repeating decimal expansion.
Reason (R): Let $a=\frac{p}{q}$ be a rational number such that $p$ and $q$ are co-prime and the prime factorisation of $q$ is of the form $2^n\times 5^m$, where $n$ and $m$ are non-negative integers. Then $a$ has a decimal expansion, which is non-terminating.

1. (A) and (R) are true and (R) is the correct explanation of (A)
2. Both (A) and (R) are false
3. (A) is true and (R) is false
4. (A) is false and (R) is the correct explanation of (A)

Q. The decimal expansion of pi is:

1. A whole number
2. Terminating
3. Non-terminating but repeating
4. Non-terminating, non-recurring

Q. Write down the decimal expansions of
(i) $\frac{13}{3125}$
(ii) $\frac{17}{8}$
(iii) $\frac{15}{1600}$

Q. Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion:
(i) $\frac{13}{3125}$ (ii) $\frac{17}{8}$ (iii) $\frac{64}{455}$ (iv) $\frac{15}{1600}$ (v) $\frac{29}{343}$

Q.

You know that $\frac{1}{11}=0.\overline{09}$ . Can you predict what the decimal expansions of $\frac{2}{11},\frac{3}{11},\frac{4}{11},\frac{5}{11},\frac{6}{11}$ are, without actually doing the long division? If so, how?

Q.

Question 1 (iv)
Without actually performing the long division, state whether $\frac{15}{1600}$ will have a terminating decimal expansion or a non-terminating repeating decimal expansion.

Q. Which of the following fraction(s) is/are non terminating?

1. $\frac{1}{2}$
2. $\frac{1}{5}$
3. $\frac{1}{6}$
4. $\frac{1}{3}$

Q. Let $x=\frac{p}{q}$ be a rational number, such that the prime factorisation of q is not of the form $2^m{5}^n$, where n, m are non-negative integers. Then, x has a decimal expansion which is:

1. Non-terminating repeating (recurring)
2. Real Integer
3. None of these
4. Terminating non-repeating (non-recurring)

Q.

Question 134

The volume of the earth is approximately $7.67\times {10}^{-7}$ times the volume of the Sun. Express this figure in usual form.

Q. Write down the decimal expansions of
(i) $\frac{13}{3125}$
(ii) $\frac{17}{8}$
(iii) $\frac{15}{1600}$

Q.

Question 1 (iii)
Without actually performing the long division, state whether $\frac{64}{455}$ will have a terminating decimal expansion or a non-terminating repeating decimal expansion.

Q.

Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a non-terminating decimal expansion : $\frac{23}{2^3{.5}^2}$