Question

Without actual division, show that each of the rational numbers given below is expressible as a
repeating decimal :
(1) 23/24 (2) 79/30 (3) 100/9 (4) 205/27​

Answer 1

Given :- Without actual division, show that each of the rational numbers given below is expressible as a repeating decimal :

(1) 23/24 (2) 79/30 (3) 100/9 (4) 205/27

Concept used :-

• we have to check prime factors of denominators of given fraction .
• if Prime factor are 2, or 5 , or 2 and 5 both . Than the given fraction is a terminating decimal expansion .
• if prime factors are other than 2 or 5 , than the given fraction is a non - terminating decimal expansion and this expression have repeating decimal .

Solution :-

So, checking Prime factors of denominators of given fractions we get,

(1) 23/24

Prime factors of 24 = 2 * 2 * 2 * 3

As , we have 1 prime factor other than 2 or 5 which is 3.

Therefore, the given expression is a repeating decimal .

(2) 79/30

Prime factors of 30 = 2 * 3 * 5

As , we have 1 prime factor other than 2 or 5 which is 3.

Therefore, the given expression is a repeating decimal .

(3) 100/9

Prime factors of 9 = 3 * 3

As , we have only prime factor is 3.

Therefore, the given expression is a repeating decimal .

(4) 205/27

Prime factors of 27 = 3 * 3 * 3.

As , we have only prime factor is 3.

Therefore, the given expression is a repeating decimal .

Learn more :-

57.63 : 3 is the. Place

https://brainly.in/question/23514539

Product of digit in the tenths place and thousandths place of 15.246

https://brainly.in/question/23804815

Answer 2

$\displaystyle\huge\red{\underline{\underline{Solution}}}$

TO PROVE

Without actual division to show that each of the rational numbers given below is expressible as a repeating decimal :

$\displaystyle \sf{ (1) \: \: \frac{23}{24} \: }$

$\displaystyle \sf{ (2) \: \: \frac{79}{30} \: }$

$\displaystyle \sf{ (3) \: \: \frac{100}{9} \: }$

$\displaystyle \sf{ (4) \: \: \frac{205}{27} \: }$

CONCEPT TO BE IMPLEMENTED

$\displaystyle \sf{\: Fraction \: = \: \frac{Numerator}{Denominator} \: }$

A fraction is said to have terminating decimal expansion if the 2 and 5 are the only prime factors of the denominator

Otherwise the fraction is said to have non terminating decimal expansion ( Repeating Decimal Expansion )

EVALUATION

CHECKING FOR OPTION (1)

$\displaystyle \sf{ (1) \: \: \frac{23}{24} \: }$

Here denominator = 24

Now

$\sf{24 = 2 \times 2 \times 2 \times 3} \:$

Since 3 is a prime number present in prime

factorisation of 24

Hence this fraction has repeating decimal expansion

CHECKING FOR OPTION (2)

$\displaystyle \sf{ (2) \: \: \frac{79}{30} \: }$

Here denominator = 30

Now

$\sf{30 = 2 \times 3 \times 5 \: }$

Since 3 is a prime number present in prime

factorisation of 30

Hence this fraction has repeating decimal expansion

CHECKING FOR OPTION (3)

$\displaystyle \sf{ (3) \: \: \frac{100}{9} \: }$

Here the denominator = 9

Now

$\sf{9 = 3 \times 3 \: }$

Since 3 is a prime number present in prime

factorisation of 9

Hence this fraction has repeating decimal expansion

CHECKING FOR OPTION (4)

$\displaystyle \sf{ (4) \: \: \frac{205}{27} \: }$

Here the denominator = 27

Now

$\sf{27 = 3 \times 3 \times 3 \: }$

Since 3 is a prime number present in prime

factorisation of 27

Hence this fraction has repeating decimal expansion

━━━━━━━━━━━━━━━━

LEARN MORE FROM BRAINLY

Out of the following which are proper fractional numbers

(i)3/2(ii)2/5(iii)1/7(iv)8/3

https://brainly.in/question/4865271