Question

after how many decimal places will be the rational number 1251/ 1250 terminate

Answer 1

Hey there!

We are asked to find after how many decimal places the given rational number
$\frac{1251}{1250}$

Now, First let us confirm if it really terminates or it has a non terminating decimal expansion.

We know that,
A decimal has a terminating expansion, if the denominator of the fraction equivalent to decimal has its prime factorization in terms of 2 alone , 5 alone or both 2 & 5 .

So, Here the denominator = 1250 .

Let's find out prime factorization of 1250

= 1250

= 2 * 625

$= 2 \times {5}^{4}$

So , We observe that 1250 can be expressed in terms of 2 & 5 , So it has a terminating decimal expansion

Now, Perform the actual division,

$1250 \: ) \: 1251 \: ( \: 1 .0008\\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: 1250 \\ \: \: \: \: = = = = = = = = = = \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: 10000 \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: 10000 \\ \: \: \: = = = = = = = = = = = = \\ \: \: \: \: \: \: \: \: \: \: \: \: 0$

Therefore, We found our that 1251/1250 = 1.0008 .

So it terminated after 4 decimal places.

We can do this without performing actual division too,

1251/1250 * 8/8 = 10008/10000 = 1.0008

$\therefore$ Decimal expansion of 1251/1250 terminates after 4 decimal places

Answer 2

hii dear !!


》1251/1250
Multiplying the both numerator and denominator by 8 to make the denominator 10000, we get.

= (1251*8)/(1250*8)
= 10008/10000
= 1.0008

After 4 decimal places, the rational number 1251/1250 will terminate.

hope it helps!!