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# Decimals

Coins are often used to express a fraction of a currency. In the US, a quarter is $0.25 and a dime is $0.10. Counting change is just one of many ways you can put decimals to use in everyday life.

If the following infinite series \(S\) is evaluated as a decimal,what is the 37th digit to the right of the decimal place?

\[ \large S=\frac { 1 }{ 9 } +\frac { 1 }{ 99 } +\frac { 1 }{ 999 } +\ldots + \frac { 1 }{ { 10 }^{ n }-1 } + \ldots \]

Which of the following is equal to

\[ 0.123 \, 123 \, 123 \, 123 \, \ldots ? \]

Without actual division find out which of the following rational numbers has a terminating decimal expansion

P)\( \frac { 17 }{ 90 } \)

Q)\(\frac { 33 }{ 50 } \)

R)\(\frac { 121 }{ { 2 }^{ 2 }\times { 5 }^{ 3 } } \)

S)\(\frac { 121 }{ { 2 }^{ 3 }\times { 3 }^{ 2 }\times { 7 }^{ 5 } } \)

**True or False**:

\[ 1 > 0.99999\ldots \]

\[0.20\overline{16}=0.2016161616\ldots =\frac{a}{b}\]

Given that \(a\) and \(b\) are coprime positive integers, find \(a+b\).

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