Question

Identify which of the following rational numbers have a terminating and which of them have a non terminating decimal expansions?
A)15/1600
B)17/6
C)23/8
D)35/50
Pls answer fast!!!!!!!!!!

Answer 1

Question :-- Identify which of the following rational numbers have a terminating and which of them have a non terminating decimal expansions?

A)15/1600

B)17/6

C)23/8

D)35/50

Solution :-

Let See Easiest Trick For This Question :-

Answer (1) :- 15/1600

Find Prime Factors of Denominator Now.

→ 1600 = 2 * 2 * 2 * 2 * 2 * 2 * 5 * 5

Now,

Remember :- if Prime Factors of Denominator are 2 or 5 or both than the Number is a terminating Number. And if Prime Factors are other than 2 or 5 , the number is non terminating recurring..

So,

we can say That, our First Number (15/1600) is Terminating As Prime Factors of Denominator are 2 and 5 only.

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Answer (2) :- 17/6

Prime Factors of Denominator :-

→ 6 = 2 * 3

Prime Factors are other than 2 or 5. (3).

So, (17/6) is a non terminating recurring.

decimal expansion.

______________________________

Answer (3) :- (23/8)

Prime Factors of Denominator :-

→ 8 = 2 * 2 * 2

Prime Factors of Denominator are 2 only.

So, (23/8) is a terminating decimal expansion.

______________________________

Answer (4) :- (35/50)

Prime Factors of Denominator :-

→ 50 = 2 * 5 * 5

Prime Factors of Denominator are 2 & 5 both.

So, (35/50) is a terminating decimal expansion.

______________________________

Answer 2

${ \boxed{ \mathtt{To \: find}}}$

The given rational numbers have terminating or non terminating decimal expansions .

( Without performing actually division )

${\boxed{\mathtt{Solution}}}$

☆ The main key point to find it out is . If the prime factors of the denominator are in the form 2^m. 5^n then the number have terminating decimal expansion . Or if they are not in this form then they have non terminating repeating decimal expansion .

1st rational number :-

⇝ 15/1600

⇝ Prime factors of 1600 are → 2 × 2 × 2 × 2 × 2 ×2 × 5 × 5

⇝ 2^6 . 5² = 2^m . 5^n

So here m = 6 and n = 2 . Hence factors are in 2^m . 5^n form . So this have terminating decimal expansion.

2nd number :-

⇝ 17/ 6

⇝ Prime factors of 6 = 2× 3 .

So it's not in the form 2^m . 5^n . So this is a non terminating repeating decimal expansion.

3rd number :-

⇝ 23/8

⇝ Prime factors of 8 = 2 × 2 × 2 × 1

Here we have only 2 . But we also have 1 as it's prime factor . So let's do a little bit change .

⇝ 5^0 = 1 . Do replacing 1 by 5^0 we'll got the required form .

⇝ 2^3 . 5^0 = 2^m . 5^n

Where m = 3 and n = 0 .

Hence this have a terminating decimal expansion.

4th number :-

⇝ 35/50 :-

⇝ Prime factors of 50 = 2 × 5× 5

⇝ 2^1 . 5^2 = 2^m . 5^n

Here also we have the required form . So this also have a terminating decimal expansion.