Tag: decimal expansions of real numbers

Questions Related to decimal expansions of real numbers

Express $\displaystyle \frac{4}{9}$ as recurring decimal 

maths real number first forms rational numbers as recurring/terminating decimals decimal expansions of real numbers

  1. $0.\bar 5$

  2. $0.\bar 4$

  3. $0.\overline {45}$

  4. $0.\overline {54}$

Show answer
Correct Option: B
Explanation:

On dividing 4 by 9 we get

$\dfrac { 4 }{ 9 } =0.4444......$
So, correct answer is option B.

Find whether it is a terminating or a non-terminating decimal.

$3.2 \div 2.24$.

maths real number first forms rational numbers as recurring/terminating decimals decimal expansions of real numbers

  1. Terminating

  2. Non-terminating

  3. Ambiguous

  4. Data insufficient

Show answer
Correct Option: B
Explanation:

A terminating decimal is a decimal that ends. It's a decimal with a finite number of digits.

$3.2\div 2.24= 1.428571429....$.
The division does not gives end result.
Hence, it is a non-terminating decimal.

Find whether it is a terminating or a non-terminating decimal.

$0.3 \div 0.09$.

maths real number first forms rational numbers as recurring/terminating decimals decimal expansions of real numbers

  1. Terminating

  2. Non-terminating

  3. Ambiguous

  4. Data insufficient

Show answer
Correct Option: B
Explanation:

A terminating decimal is a decimal that ends. It's a decimal with a finite number of digits.

$0.3\div 0.09= 3.33333333333333...$. 
The division results in recurring factor.
Hence it is a non terminating decimal.

The rational number which can be expressed as a terminating decimal is

maths real number first forms rational numbers as recurring/terminating decimals decimal expansions of real numbers

  1. $\displaystyle \frac{1}{6}$

  2. $\displaystyle \frac{1}{12}$

  3. $\displaystyle \frac{1}{15}$

  4. $\displaystyle \frac{1}{20}$

Show answer
Correct Option: D
Explanation:

A terminating decimal is a decimal that ends. It's a decimal with a finite number of digits. 
$\displaystyle \frac {1}{20}= 0.05$ 

In the other options, the decimal does not end with a finite number of digits.

Without actually performing the long division, state whether the following rational number will have a terminating decimal expansion or non -terminating decimal expansion

$\displaystyle \frac{15}{1600}$

maths real number first forms rational numbers as recurring/terminating decimals decimal expansions of real numbers

  1.  Terminating decimal expansion

  2.  Non-terminating decimal expansion

  3. Cannot be determined

  4. None

Show answer
Correct Option: A
Explanation:

Factorize the denominator we get $1600=2 \times 2 \times 2 \times2 \times 2 \times 2 \times 5 \times 5  = 2^{6} \times 5^{2}$
so denominator is in form of $2^n \times 5^m$
Hence $\frac{15}{1600}$  is  terminating.

Which of the following is terminating decimal?
$\cfrac{23}{90}, \cfrac{111}{148}, \cfrac{29}{145}, \cfrac{1}{6}$

maths real number first forms rational numbers as recurring/terminating decimals decimal expansions of real numbers

  1. $\cfrac{111}{148}$

  2. $\cfrac{23}{90}$

  3. $\cfrac{29}{145}$

  4. $\cfrac{1}{6}$

Show answer
Correct Option: A,C
Explanation:

$\cfrac{111}{148}$  is having terminating decimal $0.75$

$\dfrac{29}{145}$ is having terminating decimal $0.2$

Decimal form of $\displaystyle \frac{3888} {1000} $ 

maths real number first forms rational numbers as recurring/terminating decimals decimal expansions of real numbers

  1. 38.88

  2. 3.888

  3. 388.8

  4. .3888

Show answer
Correct Option: B
Explanation:

correct option is B..

What is the 25th digit to the right of the decimal point in the decimal form of $\displaystyle \frac { 6 }{ 11 } $?

maths real number first forms rational numbers as recurring/terminating decimals decimal expansions of real numbers

  1. $3$

  2. $4$

  3. $5$

  4. $6$

  5. $7$

Show answer
Correct Option: C
Explanation:

$\cfrac{6}{11}$ $=$ $ 0.545454....$ 

The result is a non-terminating, non-recurring number whose odd digit is $5$ and even digit is $4$.
Thus, 25th digit from the decimal point will be $5$ (option C).

Without actually performing the long division, state whether the following rational number will have a terminating decimal expansion or non -terminating decimal expansion

$\displaystyle \frac{7}{210}$

maths real number first forms rational numbers as recurring/terminating decimals decimal expansions of real numbers

  1. Terminating decimal expansion

  2. Non -terminating decimal expansion

  3. Cannot be determined

  4. None

Show answer
Correct Option: B
Explanation:

Simplify it by dividing nominator and denominator both by 7 we get $\frac{1}{30}$
Factorize the denominator we get  $30=2 \times 3 \times 5$
Denominator has 3 also in denominator
So denominator is not in form of $2^{n} \times 5^{n}$
 Hence it is non terminating.

If $\displaystyle d=\frac { 1 }{ { 2 }^{ 3 }\times { 5 }^{ 7 } } $ is expressed as a terminating decimal, how many non zero digits will d have?

maths real number first forms rational numbers as recurring/terminating decimals decimal expansions of real numbers

  1. One

  2. Two

  3. Three

  4. Seven

  5. Ten

Show answer
Correct Option: B
Explanation:
d = $\cfrac{1}{2^3\times5^7 } $
Multiply by $\cfrac{2^4}{2^4}$  
$\Rightarrow$ d = $\cfrac{2^4}{2^3 \times 5^7 \times 2^4}$
= $\cfrac{16}{2^7 \times 5^7 }$
= $\cfrac{16}{10^7 }$
=$0.0000016 $
Hence, d will have two non-zero digits, 16, when expressed as a decimal.
Answer: B.