Without actual division, show that each of the following rational numbers is a terminating decimal. Express each in decimal form:
(ii) $\frac{24}{125}$

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Solution

Solution:
Step 1: Showing that the given rational number is a terminating decimal:
A rational number having the factors in the denominator as a power of $2$ or $5$ or a product of powers of $2$ and $5$ is a terminating decimal.
Given rational number is $\frac{24}{125}$ .
In the given rational number, the denominator has a factor as the power of $5$ , that is $125 = 5^{3}$ . So, it is a terminating decimal.
Step 2: Expressing in decimal form:
$\frac{24}{125} = \frac{24}{5^{3}} \Rightarrow \frac{24}{125} = \frac{1000}{125} \times \frac{24}{1000} [Dividing and mutiplying by 1000] \Rightarrow \frac{24}{125} = 8 \times 0.024 \Rightarrow \frac{24}{125} = 0.192$
Final answer : Hence, the rational number $\frac{24}{125}$ is a terminating decimal and its decimal form is $0.192$

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Q. Without actual division, show that each of the following rational numbers is a terminating decimal. Express each in decimal form:

(i)

\frac{23}{(2^{3} \times 5^{2})}

(ii)

\frac{24}{125}

(iii)

\frac{171}{800}

(iv)

\frac{15}{1600}

(v)

\frac{17}{320}

(vi)

\frac{19}{3125}

Without actual division, show that each of the following rational numbers is a terminating decimal. Express each in decimal form.

$(i) \frac{23}{(2^{3} \times 5^{2})} (i i) \frac{24}{125} (i i i) \frac{171}{800} (i v) \frac{15}{1600} (v) \frac{17}{320} (v i) \frac{19}{3125}$

Without actual division, show that each of the following rational numbers is a nonterminating repeating decimal.

$(i) \frac{11}{(2^{3} \times 3)} (i i) \frac{73}{(2^{2} \times 3^{3} \times 5)} (i i i) \frac{129}{(2^{2} \times 5^{3} \times 7^{2})} (i v) \frac{9}{35} (v) \frac{77}{210} (v i) \frac{32}{147} (v i i) \frac{29}{343} (v i i i) \frac{64}{455}$

Without actual division, show that each of the following rational numbers is a terminating decimal. Express each in decimal form:

(iv) $\frac{171}{800}$