Question

42/52,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:

42/52 = 2 × 3 × 7 / 2× 2× 13 = 21/26

Here, the factors of the denominator 26 are are 13¹ × 2¹ which is not in the form 2ⁿ 5^m .It has one factor 13 other than 2 & 5.

Hence , 42/52 has non terminating decimal expansion .

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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 $\frac{p}{q}$, 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. $\frac{42}{52}$

⇒ Given number is $\frac{42}{52}$ and HCF (42, 52) = 2

∴ $\frac{42}{52} = \frac{42 ÷ 2}{65 ÷ 2} = \frac{21}{26}$

⇒ Now, 26 = (2 × 13) and none of 2, 13 is a factor of 21.

∴ $\frac{21}{26}$ is in its simplest form.

⇒ Also, 26 = (2 × 13) ≠ $(2^{m} × 5^{n})$

[Denominator has 13 in denominator so denominator is not in form $2^{m} × 5^{n}$]

∴ $\frac{21}{26}$ and hence $\frac{42}{52}$ is a non-terminating decimal.

⇒ $\frac{42}{52}$ does not have terminating decimal expansion beacuse it has one more prime factor 13 other than 2 and 5.