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:

(i) 13/3125 (ii) 17/8 (iii) 64/455 (iv) 15/1600 (v) 29/343 (vi) 23/(2352) (vii) 129/(225775) (viii) 6/15 (ix) 35/50 (x) 77/210

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Answer 1

Solution :-

Note: If the denominator has only factors of 2 and 5 or in the form of 2m ×5n then it has terminating decimal expansion.

If the denominator has factors other than 2 and 5 then it has a non-terminating decimal expansion.

(i) 13/3125

=> Factorizing the denominator, we get,

=> 3125 = 5 × 5 × 5 = 5^{5}

Since, the denominator has only 5 as its factor, 13/3125 has a terminating decimal expansion.

(ii) 17/8

=> Factorizing the denominator, we get,

=> 8 = 2×2×2 = 2^{3}

Since, the denominator has only 2 as its factor, 17/8 has a terminating decimal expansion.

(iii) 64/455

=> Factorizing the denominator, we get,

=> 455 = 5×7×13

Since, the denominator is not in the form of 2m × 5n, thus 64/455 has a non-terminating decimal expansion.

(iv) 15/ 1600

=> Factorizing the denominator, we get,

=> 1600 = 2^{6}5^{2}

Since, the denominator is in the form of 2^{m} × 5^{n}, thus 15/1600 has a terminating decimal expansion.

(v) 29/343

=> Factorizing the denominator, we get,

=> 343 = 7×7×7 = 7^{3}

Since, the denominator is not in the form of 2^{m} × 5^{n} thus 29/343 has a non-terminating decimal expansion.

(vi)23/(2^{3}5^{2} )

=> Clearly, the denominator is in the form of 2^{m } × 5^{n}.

Hence, 23/ (2^{3}5^{2}) has a terminating decimal expansion.

(vii) 129/(2^{2}5^{7}7^{5})

=> As you can see, the denominator is not in the form of 2^{m} × 5^{n}.

Hence, 129/ (2^{2}5^{7}7^{5} ) has a non-terminating decimal expansion.

(viii) 6/15

=> 6/15 = 2/5

Since, the denominator has only 5 as its factor, thus, 6/15 has a terminating decimal expansion.

(ix) 35/50

=> 35/50 = 7/10

=> Factorising the denominator, we get,

=> 10 = 25

Since, the denominator is in the form of 2^{m} × 5^{n} thus, 35/50 has a terminating decimal expansion.

(x) 77/210

=> 77/210 = (7× 11)/ (30 × 7) = 11/30

=> Factorising the denominator, we get,

=> 30 = 2 × 3 × 5

As you can see, the denominator is not in the form of 2^{m}× 5^{n} .Hence, 77/210 has a non-terminating decimal expansion.

Answer 2

Terminating rational numbers:

The rational numbers of the form p/q when solved, if gives a decimal value that has an end digit, then those kind of rational numbers are said to be terminating rational numbers.

Non terminating repeating rational numbers:

The rational numbers of the form p/q when solved, if gives a decimal value that doesn't have an end digit and the block of numbers are repeated continuously, then those kind of rational numbers are known as non terminating repeating rational numbers.

We can tell which rational number will have terminating decimal expansion and which will have non terminating repeating decimal expansion by observing the denominator.

If denominator is of the form

${2}^{m} \times {5}^{n}$

Then the rational number will have terminating decimal expansion.

If the denominator contains other numbers then the rational number will have non terminating repeated decimal expansion.

1.

$13 \div 3125$

Let's observe the denominator. Change the denominator into the multiples of prime numbers.

$3125 = {5}^{5}$

The denominator is in the form

${2}^{0} \times {5}^{5}$

So, it has a terminating decimal expansion

2.

$17 \div 8$

Let's observe the denominator. Change the denominator into the multiples of prime numbers.

$8 = {2}^{3}$

The denominator is in the form

${2}^{3} \times {5}^{0}$

It has a terminating decimal expansion.

3.

$64 \div 455$

Let's observe the denominator. Change the denominator into the multiples of prime numbers.

$455 = 5 \times 91$

It is not in the form

${2}^{m} \times {5}^{n}$

Hence it has non terminating repeating decimal expansion.

4.

$15 \div 1600$

$1600 = {2}^{6} \times {5}^{2}$

It has terminating decimal expansion.

5.

$29 \div 343$

It cannot be changed to the form

${2}^{m} \times {5}^{n}$

It has non terminating repeating decimal expansion

6.

$23 \div 2352$

$2352 = {2}^{4} \times 147$

It has non terminating repeating decimal expansion

7.

$129 \div 225775$

$225775 = {5}^{2} \times 9031$

It has non terminating repeating decimal expansion

8.

$6 \div 15$

$15 = 5 \times 3$

It has non terminating repeating decimal expansion

9.

$35 \div 50$

$50 = {5}^{2} \times {2}^{1}$

It has terminating decimal expansion

10.

$77 \div 210$

$210 = 5 \times 2 \times 21$

It has non terminating repeating decimal expansion.

Hence, 1,2,4,9 has terminating decimal expansion while others have non terminating repeating decimal expansion.

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