Question

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:
1) 6/250
2)128/455
3)231/1260
4)45/1600
5)39/343

Answer 1

Answer:

Let x=  p/q    be a rational number, such that the prime factorisation of q is of the form 2ⁿ5ˣ where n, x are non-negative integers. Then, x has a decimal expansion which terminates.

i) 250 = 2¹×5³

∴6/250 will be have a terminating decimal expansion.

ii) 455 = 5×7×13

∴128/445 will be have a non-terminating repeating decimal expansion.

iii) 1260 = 2² x 3² x 5¹ x 7¹

∴231/1260 will be have a non-terminating repeating decimal expansion.

iv) 1600 = 2⁶x 5²

∴45/1600 will be have a terminating decimal expansion.

v) 343 = 7³

∴39/343 will be have a non-terminating repeating decimal expansion.

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Answer 2

$\frac{6}{250} \text { and } \frac{45}{1600}$  have a terminating decimal expansion.

$\frac{128}{455}, \frac{231}{1260}, \text { and } \frac{39}{343}$ have a non-terminating repeating decimal expansion.

Concept used:

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

Given :

Rational numbers:

$\text { 1) } \frac{6}{250}\\\\\\\text { 2) } \frac{128}{455}\\\\\\\text { 3) } \frac{231}{1260}\\\\\\\text { 4) } \frac{45}{1600}\\\\\\\text { 5) } \frac{39}{343}$

To find:

Whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion without actually performing the long division

SOLUTION:

(1) Given: $\frac{6}{250}$

Step 1: $\frac{6}{250}$ is not in the simplest form. We have to make it in the simplest form by taking the HCF of both the numbers and dividing both the numbers by their HCF.  HCF (6, 250) = 2

$\frac{6 \div 2}{250 \div 2}=\frac{3}{125}$

Step 2: Write the factors of the denominator:

$\frac{3}{125}=\frac{3}{\left(2^{0} \times 5^{3}\right)}$

Step 3: Check whether the denominator is of the form $2^n\times 5^m$ or not:

Hence, the factors of the denominator 125 are $2^0 \times 5^3$ which is in the form$2^n\times 5^m$.

Hence, $\frac{6}{250}$ has a terminating decimal expansion.

(2) Given: $\frac{128}{455}$

Step 1:  Write the factors of the denominator:

$\frac{128}{455}$ = $\frac{128}{(5 \times 7 \times 13)}$

Step 2: Check whether the denominator is of the form $2^n\times 5^m$ or not:

Here, the factors of the denominator 455 are  5 × 7 × 13 which is not in the form $2^n\times 5^m$.

Hence, $\frac{128}{455}$ has a non-terminating repeating decimal expansion.

(3) Given: $\frac{231}{1260}$

Step 1:  Write the factors of the denominator:

$\frac{231}{1260}$ = $\frac{231}{( 2^2 \times 3^2 \times 5^1 \times 7^1)}$

Step 2: Check whether the denominator is of the form $2^n\times 5^m$ or not:

Here, the factors of the denominator 1260  are  $2^2 \times 3^2 \times 5^1 \times 7^1$ which is not in the form $2^n\times 5^m$.

Hence, $\frac{231}{1260}$ has a non-terminating repeating decimal expansion.

(4) Given: $\frac{45}{1600}$

Step 1: It is not in the simplest form. We have to make it in the simplest form by taking the HCF of both the numbers and dividing both the numbers by their HCF.  HCF(45,1600) = 5

$\frac{\frac{45}{5}}{\frac{1600}{5} } = \frac{9}{320}$

Step 2:  Write the factors of the denominator:

$\frac{9}{320} = \frac{9}{( 2^6 \times 5^1)}$

Step 3: Check whether the denominator is of the form $2^n\times 5^m$ or not:

Here, the factors of the denominator 1600 are $2^6 \times 5^1$ which is in the form $2^n\times 5^m$.

Hence, $\frac{45}{1600}$ has a terminating decimal expansion.

(5) Given: $\frac{39}{343}$

Step 1:  Write the factors of the denominator:

$\frac{39}{343}$ = $\frac{39}{7^3}$

Step 2: Check whether the denominator is of the form $2^n\times 5^m$ or not:

Here, the factors of the denominator 343 are 7³ which is not in the form $2^n\times 5^m$.

Hence, $\frac{39}{343}$ has a non-terminating repeating decimal expansion.

Learn more on Brainly:

Without actually performing the long division, state whether state whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion.

(i)

(ii)

(iii)

(iv)

(v)

(vi)

brainly.in/question/6751559

Write down the decimal expansions of the following rational numbers by writing their denominators in the form , where,m,n are non-negative integers.

(i)

(ii)

(iii)

(iv)

(v)  [NCERT]

brainly.in/question/6760419

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