NCERT Solutions for Class 10 Maths Exercise 1.4 Chapter 1 Real Numbers

Exercise 1.4  

1. 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) $\mathbf{\frac{13}{3125}}$

Ans: Given a rational number $\frac{13}{3125}$.

If the denominator of a rational number has prime factors of the form 2n  5m, where, m and n are positive integers. Then the rational number has a terminating decimal  expansion. If the denominator has factors other than 2and 5, then it has non terminating decimal expansion. 

The denominator of the given number is 3125. 

Then, factors of 3125 are 

⇒312= 5$\times$ 5$\times$ 5$\times$ 5$\times$ 5$\times$ 5 

⇒ 3125 =  55

Here, the factors of  denominator are of the form 5m . 

Therefore, $\frac{13}{3125}$  has terminating decimal expansion. 

(ii) $\mathbf{\frac{17}{8}}$

Ans: Given a rational number $\frac{13}{3125}$. 

If the denominator of a rational number has prime factors of the form 2n 5m, where, m and n are positive integers. Then the rational number has terminating decimal  expansion. If the denominator has factors other than 2 and 5, then it has non terminating decimal expansion. 

The denominator of the given number is 8 . 

Then, factors of 8 are

⇒ 8= 28 $\times $ 28 $\times $ 2 

 ⇒ 8 =23 

Here, the factors of denominator are of the form 2 n . 

Therefore,$\frac{13}{3125}$ has terminating decimal expansion. 

(iii) $\mathbf{\frac{64}{455}}$

Ans: Given a rational number $\frac{64}{455}$.

If the denominator of a rational number has prime factors of the form 2n 5m , where, m and n are positive integers. Then the rational number has a terminating decimal  expansion. If the denominator has factors other than 2and 5, then it has non terminating decimal expansion. 

The denominator of the given number is 455. 

Then, factors of 455 are 

⇒ 455 =  5 $\times$ 7 $\times$ 13 

Here, the factors of denominator are not in the form  2n 5m. The denominator has factors  other than 2and 5. 

Therefore, $\frac{64}{455}$ has non-terminating repeating decimal expansion. 

(iv) $\mathbf{\frac{15}{1600}}$

Ans: Given a rational number $\frac{15}{1600}$

If the denominator of a rational number has prime factors of the form 2n 5m, where, m and n are positive integers. Then the rational number has terminating decimal  expansion. If the denominator has factors other than 2and 5, then it has non terminating decimal expansion. 

The denominator of the given number is 1600. 

Then, factors of 1600 are 

⇒  1600 =  2$\times$ 2$\times$ 2$\times$ 2$\times$ 2$\times$ 2$\times$ 5$\times$ 5 

 ⇒ 1600 =  26 $\times$ 55

Here, the factors of denominator are of the form 2n 5m .

Therefore,$\frac{15}{1600}$ has terminating decimal expansion. 

(v)$\mathbf{\frac{29}{343}}$

Ans: Given a rational number $\frac{29}{343}$ 

If the denominator of a rational number has prime factors of the form 2n  5m, where, m and n are positive integers. Then the rational number has a terminating decimal  expansion. If the denominator has factors other than 2and 5, then it has non terminating decimal expansion. 

The denominator of the given number is 343. 

Then, factors of 343 are 

⇒343=  7$\times$ 7$\times $4 7 

⇒343 =  73 

Here, the factors of denominator are not in the form 2n 5m. The denominator has factors  other than 2and 5. 

Therefore, $\frac{29}{343}$  has non-terminating repeating decimal expansion. 

(vi) $\mathbf{\frac{23}{2^3 5^2}}$ 

Ans: Given a rational number $\frac{23}{2^3 5^2}$

If the denominator of a rational number has prime factors of the form 2n 5m, where, m and n are positive integers. Then the rational number has a terminating decimal  expansion. If the denominator has factors other than 2 and 5, then it has non terminating decimal expansion. 

The denominator of the given number is 2352 . 

Here, the denominator is of the form 2n 5m . 

Therefore,$\frac{23}{2^3 5^2}$ has terminating decimal expansion. 

(vii)  $\mathbf{\frac{129}{2^2 5^7  7^5}}$

Ans: Given a rational number $\frac{129}{2^2 5^7  7^5}$

If the denominator of a rational number has prime factors of the form  2n  5m, where, m and n are positive integers. Then the rational number has terminating decimal  expansion. If the denominator has factors other than 2 and 5, then it has non terminating decimal expansion. 

The denominator of the given number is  225775 . 

Here, the denominator is of the form 2n 5m but also has factors other than 2 and 5. 

Therefore, $\frac{129}{2^2 5^7  7^5}$has non-terminating repeating decimal expansion. 

(viii)  $\mathbf{\frac{6}{15}}$

Ans: Given a rational number $\frac{6}{15}$

If the denominator of a rational number has prime factors of the form 2n 5m, where, m and n are positive integers. Then the rational number has a terminating decimal  expansion. If the denominator has factors other than 2and 5, then it has non terminating decimal expansion. 

The denominator of the given number is 15. 

Then, factors of 3125 are 

⇒ 15 =  3 $\times$  5 

But we can write the numerator of the given number as 

$\frac{6}{15}$ = $\frac{2 \times 3}{3 \times 5}$ = $\frac{2}{5}$

Here, the factors of denominator are of the form  5m . 

Therefore, $\frac{6}{15}$ has terminating decimal expansion. 

(ix)  $\mathbf{\frac{35}{50}}$

Ans: Given a rational number $\frac{35}{50}$

If the denominator of a rational number has prime factors of the form 2n 5m, where, m and n are positive integers. Then the rational number has a terminating decimal expansion. If the denominator has factors other than 2and 5, then it has non terminating decimal expansion. 

The denominator of the given number is 50. 

Then, factors of 50 are 

⇒ 50  = 10  \$times$ 5 

But we can write the numerator of the given number as 

 $\frac{35}{50}$ =  $\frac{7 \times5}{5 \times 10}$

⇒ 10 =  2 $\times$ 5 

Here, the factors of denominator are of the form 2n  5m  . 

Therefore, $\frac{35}{50}$ has terminating decimal expansion. 

(x)  $\mathbf{\frac{77}{210}}$

Ans: Given a rational number  $\frac{77}{210}$

If the denominator of a rational number has prime factors of the form 2n  5m , where, m and n are positive integers. Then the rational number has a terminating decimal  expansion. If the denominator has factors other than 2 and 5, then it has non terminating decimal expansion. 

The denominator of the given number is 210 . 

Then, factors of 210 are 

⇒ 210 =  2 $\times$ 3 $\times$5 $\times$7 

Here, the denominator has factors other than 2and 5. 

Therefore,$\frac{77}{210}$  has non-terminating repeating decimal expansion. 

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

(i) $\mathbf{\frac{13}{3125}}$

Ans: To find the decimal expansion of $\frac{13}{3125}$, we will divide the numerator of the  

number by the denominator using a long division method. We get

$\;\;\;\;\;\;\;\;\;\;0.00416\\3125\;\;)\overline{13.00000}\\\;\;\;\;\;\;\;\;\;\; 0 \\\;\;\;\;\;\;\;\;\;\;\overline{13000}\\\;\;\;\;\;\;\;\;\;\;12500\\\;\;\;\;\;\;\;\;\;\;\overline{5000}\\\;\;\;\;\;\;\;\;\;\;3125\\\;\;\;\;\;\;\;\;\;\;\overline{18750}\\\;\;\;\;\;\;\;\;\;\;18750\\\;\;\;\;\;\;\;\;\;\;\overline{0}$

Therefore, the decimal expansion of  $\frac{13}{3125}$ is 0.00416.  

(ii) $\mathbf{\frac{17}{8}}$

Ans: To find the decimal expansion of $\frac{17}{8}$,  we will divide the numerator of the number  by denominator using long division method. We get 

$\;\;\;\;\;\;\;\;\;\;2.125\\8\;\;)\overline{17}\\\;\;\;\;\;\;16\\\;\;\;\;\;\;\;\overline{10}\\\;\;\;\;\;\;\;8\\\;\;\;\;\;\;\;\overline{20}\\\;\;\;\;\;\;\;16\\\;\;\;\;\;\;\;\overline{40}\\\;\;\;\;\;\;\;40\\\;\;\;\;\;\;\;\overline{0}$

Therefore, the decimal expansion of  $\frac{17}{8}$  is 2.125.  

(iii) $\mathbf{\frac{15}{1600}}$

Ans: To find the decimal expansion of $\frac{15}{1600}$, we will divide the numerator of the  

number by the denominator using a long division method. We get 

$\;\;\;\;\;\;\;\;\;\;0.005375\\1600\;\;)\overline{15.000000}\\\;\;\;\;\;\;\;\;\;0\\\;\;\;\;\;\;\;\;\;\;\overline{150}\\\;\;\;\;\;\;\;\;\;\;0\\\;\;\;\;\;\;\;\;\;\;\overline{15000}\\\;\;\;\;\;\;\;\;\;\;14400\\\;\;\;\;\;\;\;\;\;\;\overline{6000}\\\;\;\;\;\;\;\;\;\;\;4800\\\;\;\;\;\;\;\;\;\;\;\overline{12000}\\\;\;\;\;\;\;\;\;\;\;11200\\\;\;\;\;\;\;\;\;\;\;\overline{8000}\\\;\;\;\;\;\;\;\;\;\;8000\;\;\;\;\;\;\;\;\;\;\\ \overline{0}$

Therefore, the decimal expansion of $\frac{15}{1600}$  is 0.009375.  

(iv) $\mathbf{\frac{23}{2^3 5^2}}$

Ans: To find the decimal expansion of    $\frac{23}{2^3 5^2}$

we will divide the numerator of the number  

by the denominator using a long division method. We get

$\frac{23}{2^3 5^2}$ = $\frac{23}{200}$

$\;\;00.115\\\;\;\;\;200)  \overline{23.000}\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; 0\\\;\;\;\;\;\;\;\;\;\; \overline{23}\;\;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\\ \;\;\;\;\;\;\;\;\;\;\;\;\;0 \\\;\;\;\;\;\;\;\;\;\;\; \overline {230}  \\\;\;\;\;\;\;\;\;\;\;\; 200\\\;\;\;\;\;\;\;\;\;\;\; \overline{1000}\\ \;\;\;\;\;\;\;\;\;\;\; 1000 \\\;\;\;\;\;\;\;\;\;\;\; \overline{0}$

Therefore, the decimal expansion of  $\frac{23}{200}$ is 00.115.

(v) $\mathbf{\frac{6}{15}}$

Ans: To find the decimal expansion of $\frac{6}{15}$, we will divide the numerator of the number  by the denominator using a long division method. We get 

$\frac{6}{15}$ = $\frac{2 \times 3}{3 \times 5}$= $\frac{2}{5}$

$\;\;\;\;\;\;\;\;0.4\\\;\;\;\;\;5)\overline{2.0}\\ \;\;\;\;\;\; \;\;\;0 \\ \;\;\;\;\;\;\;\;\overline{20}\\\;\;\;\;\;\;\;\;20\\ \;\;\;\;\;\;\;\; \overline{0}$ 

Therefore, the decimal expansion of $\frac{6}{15}$ is 0.4. 

(vi) $\mathbf{\frac{35}{50}}$

Ans: To find the decimal expansion of $\frac{35}{50}$,  we will divide the numerator of the number  by the denominator using a long division method. We get 

$\;\;\;\;\;\;\;\;0.7\\\;\;\;\;\;50)\overline{35.0}\\ \;\;\;\;\;\; \;\;\;0 \\ \;\;\;\;\;\;\;\;\overline{350}\\\;\;\;\;\;\;\;\;350\\ \;\;\;\;\;\;\;\; \overline{0}$

Therefore, the decimal expansion of $\frac{6}{15}$  is 0.7. 

3. The following real numbers have decimal expansions as given below. In each  case, decide whether they are rational or not. If they are rational, and of the form $\mathbf{\frac{p}{q}}$ what can you say about the prime factors of q? 

(i) 43.123456789 

Ans: Given a decimal expansion 43.123456789. 

The given number has terminating expansion, we can write the number as $\frac{43124356789}{1000000000}$, , which is of the form pq. 

Therefore, the number 43.123456789is a rational number. 

Since the number has terminating decimal expansion, the factors of q must be of the  form 2n 5m . 

(ii) 0.120120012000120000...... 

Ans: Given a decimal expansion 0.120120012000120000.......

When we observe the given expansion we can say that the number has non-terminating  and non-repeating decimal expansion. Hence we cannot express it in the form of $\frac{p}{q}$ Therefore, the number is irrational. 

(iii) 43.123456789 

Ans: Given the decimal expansion 43.123456789. 

The given number has non-terminating but repeating decimal expansion. So the  number will be of the form$\frac{p}{q}$ . 

Therefore, the number 43.123456789is a rational number. 

But the factors of denominator are not of the form 2n 5m. Denominator also has factors  other than 2and 5.

NCERT Solutions for Class 10 Maths Chapter 1 Real Numbers Exercise 1.4

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Decimal Expansions of Rational Numbers

Rational Numbers: Rational numbers are numbers that are written in the form p/q, (p in the numerator and q in the denominator) where p and q are integers and q≠0. We can say all fractions are rational numbers.

Examples of rational numbers: 1/3, 3/10, -7/10,−3

Terminating and nonterminating decimals

1. Terminating decimals: The decimal values whose value is definite and terminate at a certain digit are said to be Terminating decimals.

Example: 0.4, 5.8, and so on.

2. Non-terminating: The decimal values in which the digits after the decimal point do not terminate and the decimal value is not exactly known are called Non-terminating decimals.

Example: 0.333333….., 0.13135235343…

Non-terminating decimals can be classified again as:

• Non-terminating recurring Decimals: In this type of decimal number, a part of the decimal indefinitely repeats itself. Eg: 0.142869142869142869….(where 142869 is repeating after the decimal point).

• Non-terminating non-recurring Decimals: In this type of decimal number, no part of the decimal repeats indefinitely. Example: π=3.1415926535…

How to check whether a given rational number is terminating or not?

If a rational number a/b is given, then its decimal expansion of the rational number will terminate if it satisfies the below two conditions:

a) The H.C.F of a (numerator) and b (denominator) of the given rational number should be 1.

b) The denominator of the given rational number(b) can be expressed as a prime factorization of 2 and 5 i.e b=2m×5n where m or n, or both can be 0. If the prime factorization of b contains any number other than 2 or 5, then the decimal expansion of that number will be a non-terminating decimal.

NCERT Solutions for Class 10 Maths Chapter 1 Exercises