Question

26/65,State whether given rational number have terminating decimal expansion or not and if it has terminating decimal expansion, find it.

Answer 1

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

SOLUTION:26/65 = 2 × 13 / 5 × 13 = 2/5

Here, the factors of the denominator 65 are are 5¹ × 2^0 which is in the form 2ⁿ 5^m .

26/65 has terminating decimal expansion which can be expressed as : 26/65 = ⅖ = 2¹× 2¹ / 5¹× 2¹ = 4 /(5×2)¹= 4/10= 0.4

Hence, the terminating decimal expansion of 26/65 is 0.4.

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Answer 2

Hey there!

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→ To check whether a given rational number is a terminating or repeating decimal :

Let x be a rational number whose simplest form is , where p and q are integers and q ≠ 0. Then,

(i) x is a terminating decimal only when q is of the form $(2^{m} × 5^{n})$ for some non-negative integers m and n.

(ii) x is a non-terminating repeating decimal, if $(2^{m} × 5^{n})$

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Q. $\frac{26}{65}$

⇒ Given number is $\frac{26}{65}$ and HCF (26,65) = 13

∴ $\frac{26}{65} = \frac{26 ÷ 13}{65 ÷ 13} = \frac{2}{5}$

⇒ Now, 5 = (1 × 5) and 5 is not a factor of 2.

∴ $\frac{2}{5}$ is in its simplest form.

⇒ Also, 5 = $(5^{1} × 2^{0})$ = $(2^{m} × 5^{n})$

∴ $\frac{2}{5}$ and hence $\frac{26}{65}$ is a terminating decimal.

⇒ Now, $\frac{26}{65}$ has a terminating decimal expansion which can be expressed as

∴ $\frac{26}{65} = \frac{2}{5} = \frac{2 × 2}{5 × 2} = \frac{4}{10} = 0.4$