Revisiting Rational Numbers and Their Decimal Expansions

The following real number have decimal expansions as given below. Decide whether they are rational or not. If they are rational and of the form $\frac{\mathrm{p}}{\mathrm{q}}$, what can you say about the prime factors of $\mathrm{}\mathrm{q}$?
$0.120120012000120000...$
 

The following real number has a decimal expansion as given below. Decide whether it is rational or not. If it is rational, and of the form $\frac{\mathrm{p}}{\mathrm{q}}$, what can you say about the prime factors of $\mathrm{}\mathrm{q}$?

$43.\overline{123456789}$

Without actually performing the long division, state whether the following rational number will have a terminating decimal expansion or a non-terminating repeating decimal expansion:

$\frac{77}{210}$

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:
$\frac{35}{50}$
 

Without actually performing the long division, state whether the following rational number will have a terminating decimal expansion or a non-terminating repeating decimal expansion:
$\frac{6}{15}$
 

Without actually performing the long division, state whether the following rational number will have a terminating decimal expansion or a non-terminating repeating decimal expansion:
$\frac{129}{2^2{5}^7{7}^5}$

Without actually performing the long division, state whether the following rational number will have a terminating decimal expansion or a non-terminating repeating decimal expansion.
$\frac{23}{2^3{5}^2}$

Without actually performing the long division, state whether the following rational number will have a terminating decimal expansion or a non-terminating repeating decimal expansion:
$\frac{29}{343}$

Without actually performing the long division, state whether the following rational number will have a terminating decimal expansion or a non-terminating repeating decimal expansion.
$\frac{15}{1600}$

Without actually performing the long division, state whether the following rational number will have a terminating decimal expansion or a non-terminating repeating decimal expansion.
$\frac{64}{455}$

Without actually performing the long division, state whether the following rational number will have a terminating decimal expansion or a non-terminating repeating decimal expansion.
$\frac{17}{8}$

The following real numbers have decimal expansions as given below. In each case, decide whether they are rational or not. If they are rational, and of the form $\frac{\mathrm{p}}{\mathrm{q}}$, what can you say about the prime factors of $\mathrm{}\mathrm{q}$?
$43.123456789$
 

Write down the decimal expansions of the rational numbers given below which have terminating decimal expansions.

(i) $\frac{13}{3125}$

(ii) $\frac{17}{8}$

(iii) $\frac{15}{1600}$

(iv) $\frac{23}{2^3{5}^2}$

(v) $\frac{6}{15}$

(vi) $\frac{35}{50}$

Without actually performing the long division state whether the following rational number will have a terminating decimal expansion or a non-terminating repeating decimal expansion.
$\frac{13}{3125}$