Question

Without actual division, determine which of the following rational number have a terminating decimal representation.

I. 21/128
ii. 27/125
iii. 39/24
iv. -17/90

Answer 1

$\boxed{\boxed{\bf SOLUTION :}}$

 

To find out we have terminating decimals in fractions, we observe prime factors of the denominators.

So,

if we have only 2 and 5 (if only 2 or even only 5 also) as the prime factors of the denominator of a rational number in the lowest form, the given fractional number will have terminating decimal representation.

Here, If we see all numbers,

so

we may note that all the above given numbers (except $\bf \frac{39}{24}$) are in the lowest form.

 

(i) The denominator of   $\bf \frac{21}{128}$   is 128.

Now,

128 = 2 × 2 × 2 × 2 × 2 × 2 × 2

The prime factor of 128 is 2 seven times.

Therefore, $\bf \frac{21}{128}$ has a terminating decimal representation.

________________________

 

(ii) The denominator of  $\bf \frac{27}{125}$   is 125.

Now,

125 = 5 × 5 × 5

The prime factor of 125 is 5 three times.

Therefore, $\bf \frac{27}{125}$ has a terminating decimal representation.

________________________

   

(iii) The denominator of   $\bf \frac{39}{24}$  is 24.

Now,

24 = 2 × 2 × 2 × 3

The prime factor of 24 are 2 and 3. One of the factors is other than 2 and 5. But the rational number is not in the lowest form.

In fact $\bf \frac{39}{24}=\frac{13\times \cancel{3}}{8\times \cancel{3}}=\frac{13}{8}$ , whose denominator 8 = 2 × 2 × 2.

Therefore, $\bf \frac{39}{24}$ has a terminating decimal representation.

________________________

   

(iv) The denominator of   $\bf \frac{17}{90}$   is 90.

Now,

90 = 2 × 3 × 3 × 5

The prime factor of 90 are 2, 3, 5.

One of the factors is other than 2 and 5.

Therefore, $\bf \frac{17}{90}$ will not have a terminating decimal representation.

Answer 2

SOLUTION:

To find out we have terminating decimals in fractions, we observe prime factors of the denominators.

So,

if we have only 2 and 5 (if only 2 or even only 5 also) as the prime factors of the denominator of a rational number in the lowest form, the given fractional number will have terminating decimal representation.

Here, If we see all numbers,

so

we may note that all the above given numbers (except $\bf \frac{39}{24}$

) are in the lowest form.

(i) The denominator of $\bf \frac{21}{128}$

is 128

Now,

128 = 2 × 2 × 2 × 2 × 2 × 2 × 2

The prime factor of 128 is 2 seven times.

Therefore, $\bf \frac{21}{128}$

has a terminating decimal representation.

________________________

(ii) The denominator of $\bf \frac{27}{125}$

is 125.

Now,

125 = 5 × 5 × 5

The prime factor of 125 is 5 three times.

Therefore, $\bf \frac{27}{125}$

has a terminating decimal representation.

________________________

(iii) The denominator of $\bf \frac{39}{24}$

is 24.

Now,

24 = 2 × 2 × 2 × 3

The prime factor of 24 are 2 and 3. One of the factors is other than 2 and 5. But the rational number is not in the lowest form.

In fact $\bf \frac{39}{24}=\frac{13\times \cancel{3}}{8\times \cancel{3}}=\frac{13}{8}$

whose denominator 8 = 2 × 2 × 2.

Therefore, $\bf \frac{39}{24}$

has a terminating decimal representation.

________________________

(iv) The denominator of $\bf \frac{17}{90}$

is 90.

Now,

90 = 2 × 3 × 3 × 5

The prime factor of 90 are 2, 3, 5.

One of the factors is other than 2 and 5.

Therefore, $\bf \frac{17}{90}$

will not have a terminating decimal representation.