Question

Write down the decimal expansions of the following rational numbers by writing their denominators in the form $2^{m} * 5^{n}$, where,m,n are non-negative integers. (i) $\frac{3}{8}$ (ii) $\frac{13}{125}$ (iii) $\frac{7}{80}$ (iv) $\frac{14588}{625}$ (v) $\frac{129}{2^{2} * 5^{7}}$ [NCERT]

Answer 1

If the factors of denominator of the given rational number is of form 2^m 5ⁿ ,where n and m are non negative integers, then the decimal expansion of the rational number is terminating otherwise non terminating recurring.

SOLUTION :  

(i) Given : 3/8

3/ 2³

Here, the factors of the denominator 8 are 2³× 5^0, which is in the form 2^m 5ⁿ .

So , 23/8 has terminating  decimal expansion.

The decimal expansion of 3/ 8 :  

⅜ = 3×5³ / 2³ × 5³

= 3 × 125 / (2×5)³  

= 375 / 10³ = 375 /1000  

= 0.375  

Hence, the decimal expansion of 3/8 is 0.375.

(ii) Given : 13/125

13/5³

Here, the factors of the denominator 125 are 2^0 × 5³ which is in the form 2^m 5ⁿ .

So , 13/125 has a terminating  decimal expansion.

The decimal expansion of 13/ 125 :  

13/125 = 13 ×2³ /5³× 2³

= 13 × 8 /(5×2)³

= 104 / 10³ = 104/1000  

= 0.104  

Hence, the decimal expansion of 13/125 is 0.104.

(iii) Given : 7/80

Here, the factors of the denominator 80 are 2⁴ × 5¹, which is in the form 2^m 5ⁿ .

So , 7/80 has  a terminating decimal expansion.

The decimal expansion of 7/80 :  

7/80 = 7 × 5³ / 2⁴ × 5¹ × 5³

= 7 × 125 / (2×5)⁴

= 875/10⁴ = 875/10000

= 0.0875  

Hence, the decimal expansion of 7/80 is 0.0875 .

(iv) Given : 14588/625

Here, the factors of the denominator 625 is 2^0 × 5⁴, which is in the form 2^m 5ⁿ

So , 14588/625 has a terminating decimal expansion.

The decimal expansion of 14588/625 :  

14588/625 = 14588 × 2⁴ / 5⁴× 2⁴

= 14588 × 16 / (5×2)⁴

= 233408/ 10⁴

= 23.3408

Hence, the decimal expansion of 14588/625 is 23.3408.

(v) Given : 129/2² × 5^7  

129/2² × 5^7

Here, the factors of the denominator  are 2² × 5^7  which is in the form 2^m 5ⁿ .

So ,129/2² × 5^7  has a terminating decimal expansion.

The decimal expansion of 129/2² × 5^7 :  

129 × 2^5 /2² × 5^7 × 2^5

= 129 × 32 / (2×5)^7

= 4128 / 10000000

= 0.0004128

Hence, the decimal expansion of 129/2² × 5^7 is 0.0004128.

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Answer 2

Answer

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