Find three decimals between each of the two following pairs of decimals. a. 9.6 and $$ 9 . \overline{6} $$ b. $$ 167 . \overline{74} $$ and $$ 167 . \overline{7} $$ a. Choose the correct answer below. A. $$ 9.665,9.664,9.663 $$ B. $$ 9.663,9.665,9.667 $$ C. $$ 9.663,9.664,9.668 $$ D. $$ 9.665,9.667,9.668 $$ b. Choose the correct answer below. A. $$ 167.7472,167.7473,167.7476 $$ B. $$ 167.7472,167.7473,167.7474 $$ C. $$ 167.7474,167.7475,167.7477 $$ D. $$ 167.7475,167.7476,167.7477 $$

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We first rewrite each repeating decimal as an infinite decimal. --- **(a) Between $$9.6$$ and $$9.\overline{6}$$:** • The number $$9.6$$ is the same as $$9.600$$. • The number $$9.\overline{6}$$ means $$ 9.\overline{6}=9.6666\ldots $$ Any decimal number with three digits after the decimal must lie strictly between $$9.600$$ and $$9.6666\ldots$$. Examine the choices: - **Choice A:** The three numbers are $$ 9.665,\quad 9.664,\quad 9.663. $$ All of these satisfy $$ 9.600 < 9.663 < 9.664 < 9.665 < 9.6666\ldots $$ - **Choice B:** Contains $$9.667$$ which is greater than $$9.6666\ldots$$. - **Choice C:** Contains $$9.668$$ which is too high. - **Choice D:** Also contains numbers that exceed the upper limit. Thus the correct answer for (a) is **Choice A**. --- **(b) Between $$167.\overline{74}$$ and $$167.\overline{7}$$:** • The number $$167.\overline{74}$$ means $$ 167.\overline{74} = 167.74747474\ldots $$ • The number $$167.\overline{7}$$ means $$ 167.\overline{7}=167.77777777\ldots $$ We now want three decimals (given in the answers with four digits after the decimal) that lie strictly between these two numbers. Examine the choices: - **Choice A:** The first number is $$167.7472$$. Notice that $$ 167.7472 = 167.747200\ldots $$ but $$ 167.747474\ldots>167.747200. $$ Thus $$167.7472$$ does not lie above $$167.\overline{74}$$. - **Choice B:** Begins with $$167.7472$$ as well, so it too is not acceptable. - **Choice C:** The first number is $$167.7474$$. Writing it as $$167.747400\ldots$$, we compare with $$167.747474\ldots$$ and see that $$ 167.747400 < 167.747474\ldots, $$ so it is not strictly greater than $$167.\overline{74}$$. - **Choice D:** The numbers are $$ 167.7475,\quad 167.7476,\quad 167.7477. $$ Compare the first one: $$ 167.7475 = 167.747500, $$ and since $$500 > 474$$, we have $$ 167.7475 > 167.747474\ldots. $$ The other two are similarly between $$167.747474\ldots$$ and $$167.777777\ldots$$. Thus the correct answer for (b) is **Choice D**. --- **Final Answers:** a. Choice A b. Choice D a. Choice A b. Choice D

Find three decimals between each of the two following pairs of decimals.
a. 9.6 and
b. and
a. Choose the correct answer below.
A.
B.
C.
D.
b. Choose the correct answer below.
A.
B.
C.
D.

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Find three decimals between each of the two following pairs of decimals. a. 9.6 and b. and a. Choose the correct answer below. A. B. C. D. b. Choose the correct answer below. A. B. C. D.