Chapter 5.7, Problem 27WE

a.

To determine

To Evaluate to three decimal places.

Answer to Problem 27WE

1.5

Explanation of Solution

Given:

Expression: $1+\frac{1}{2}$

Concept used:

To evaluate the given expression the following steps are followed:

1. Find the last fraction in the denominator.
2. Simplify the last fraction and repeat the steps.

Calculation:

Consider the given expression $1+\frac{1}{2},$

On simplifying the expression,

  $\begin{array}{l}\Rightarrow 1+\frac{1}{2}\\ \Rightarrow \frac{1}{1}+\frac{1}{2}\\ \Rightarrow \frac{1\times 2}{1\times 2}+\frac{1\times 1}{2\times 1}\\ \Rightarrow \frac{2}{2}+\frac{1}{2}\\ \Rightarrow \frac{2+1}{2}\\ \Rightarrow \frac{3}{2}\end{array}$

Using a calculator, find the value of $\frac{3}{2}.$

Therefore, $\frac{3}{2}=\mathrm{1.5.}$

Conclusion:

Therefore, the value of $1+\frac{1}{2}=1.5$

b.

To determine

To Evaluate to three decimal places.

Answer to Problem 27WE

1.4

Explanation of Solution

Given:

Expression: $1+\frac{1}{2+\frac{1}{2}}$

Concept used:

To evaluate the given expression the following steps are followed:

1. Find the last fraction in the denominator.
2. Simplify the last fraction and repeat the steps.

Calculation:

Consider the given expression $1+\frac{1}{2+\frac{1}{2}},$

Consider $2+\frac{1}{2},$

On simplifying the expression,

  $\begin{array}{l}\Rightarrow 2+\frac{1}{2}\\ \Rightarrow \frac{2}{1}+\frac{1}{2}\\ \Rightarrow \frac{2\times 2}{1\times 2}+\frac{1\times 1}{2\times 1}\\ \Rightarrow \frac{4}{2}+\frac{1}{2}\\ \Rightarrow \frac{4+1}{2}\\ \Rightarrow \frac{5}{2}\end{array}$

Substituting in the given expression,

  $\begin{array}{l}\Rightarrow 1+\frac{1}{\frac{5}{2}}\\ \Rightarrow 1+\frac{1\times 2}{5}\\ \Rightarrow 1+\frac{2}{5}\\ \Rightarrow \frac{1}{1}+\frac{2}{5}\\ \Rightarrow \frac{1\times 5}{1\times 5}+\frac{2\times 1}{5\times 1}\\ \Rightarrow \frac{5}{5}+\frac{2}{5}\\ \Rightarrow \frac{5+2}{5}\\ \Rightarrow \frac{7}{5}\end{array}$

Using a calculator, find the value of $\frac{7}{5}.$

Therefore, $\frac{7}{5}=\mathrm{1.4.}$

Conclusion:

Therefore, the value of the expression $1+\frac{1}{2+\frac{1}{2}}$ is 1.4.

c.

To determine

To Evaluate to three decimal places.

Answer to Problem 27WE

1.416

Explanation of Solution

Given:

Expression: $1+\frac{1}{2+\frac{1}{2+\frac{1}{2}}}$

Concept used:

To evaluate the given expression the following steps are followed:

1. Find the last fraction in the denominator.
2. Simplify the last fraction and repeat the steps.

Calculation:

Consider the given expression $1+\frac{1}{2+\frac{1}{2+\frac{1}{2}}},$

Consider $2+\frac{1}{2},$

On simplifying the expression,

  $\begin{array}{l}\Rightarrow 2+\frac{1}{2}\\ \Rightarrow \frac{2}{1}+\frac{1}{2}\\ \Rightarrow \frac{2\times 2}{1\times 2}+\frac{1\times 1}{2\times 1}\\ \Rightarrow \frac{4}{2}+\frac{1}{2}\\ \Rightarrow \frac{4+1}{2}\\ \Rightarrow \frac{5}{2}\end{array}$

Substituting in the given expression,

  $\begin{array}{l}\Rightarrow 2+\frac{1}{\frac{5}{2}}\\ \Rightarrow 2+\frac{1\times 2}{5}\\ \Rightarrow 2+\frac{2}{5}\\ \Rightarrow \frac{2}{1}+\frac{2}{5}\\ \Rightarrow \frac{2\times 5}{1\times 5}+\frac{2\times 1}{5\times 1}\\ \Rightarrow \frac{10}{5}+\frac{2}{5}\\ \Rightarrow \frac{10+2}{5}\\ \Rightarrow \frac{12}{5}\end{array}$

Substituting in the given expression,

  $\begin{array}{l}\Rightarrow 1+\frac{1}{\frac{12}{5}}\\ \Rightarrow 1+\frac{1\times 5}{12}\\ \Rightarrow \frac{1}{1}+\frac{5}{12}\\ \Rightarrow \frac{1\times 12}{1\times 12}+\frac{5\times 1}{12\times 1}\\ \Rightarrow \frac{12}{12}+\frac{5}{12}\\ \Rightarrow \frac{12+5}{12}\\ \Rightarrow \frac{17}{12}\end{array}$

Using a calculator, find the value of $\frac{17}{12}.$

Therefore, $\frac{17}{12}=\mathrm{1.416.}$

Conclusion:

Therefore, the value of the expression $1+\frac{1}{2+\frac{1}{2+\frac{1}{2}}}$ is 1.416.

d.

To determine

To Evaluate to three decimal places.

Answer to Problem 27WE

1.413

Explanation of Solution

Given:

Expression: $1+\frac{1}{2+\frac{1}{2+\frac{1}{2+\frac{1}{2}}}}$

Concept used:

To evaluate the given expression the following steps are followed:

1. Find the last fraction in the denominator.
2. Simplify the last fraction and repeat the steps.

Calculation:

Consider the given expression $1+\frac{1}{2+\frac{1}{2+\frac{1}{2}}},$

Consider $2+\frac{1}{2},$

On simplifying the expression,

  $\begin{array}{l}\Rightarrow 2+\frac{1}{2}\\ \Rightarrow \frac{2}{1}+\frac{1}{2}\\ \Rightarrow \frac{2\times 2}{1\times 2}+\frac{1\times 1}{2\times 1}\\ \Rightarrow \frac{4}{2}+\frac{1}{2}\\ \Rightarrow \frac{4+1}{2}\\ \Rightarrow \frac{5}{2}\end{array}$

Substituting in the given expression,

  $\begin{array}{l}\Rightarrow 2+\frac{1}{\frac{5}{2}}\\ \Rightarrow 2+\frac{1\times 2}{5}\\ \Rightarrow 2+\frac{2}{5}\\ \Rightarrow \frac{2}{1}+\frac{2}{5}\\ \Rightarrow \frac{2\times 5}{1\times 5}+\frac{2\times 1}{5\times 1}\\ \Rightarrow \frac{10}{5}+\frac{2}{5}\\ \Rightarrow \frac{10+2}{5}\\ \Rightarrow \frac{12}{5}\end{array}$

Substituting in the given expression,

  $\begin{array}{l}\Rightarrow 2+\frac{1}{\frac{12}{5}}\\ \Rightarrow 2+\frac{1\times 5}{12}\\ \Rightarrow \frac{2}{1}+\frac{5}{12}\\ \Rightarrow \frac{2\times 12}{1\times 12}+\frac{5\times 1}{12\times 1}\\ \Rightarrow \frac{24}{12}+\frac{5}{12}\\ \Rightarrow \frac{24+5}{12}\\ \Rightarrow \frac{29}{12}\end{array}$

Substituting in the given expression,

  $\begin{array}{l}\Rightarrow 1+\frac{1}{\frac{29}{12}}\\ \Rightarrow 1+\frac{1\times 12}{29}\\ \Rightarrow \frac{1}{1}+\frac{12}{29}\\ \Rightarrow \frac{1\times 29}{1\times 29}+\frac{12\times 1}{29\times 1}\\ \Rightarrow \frac{29}{29}+\frac{12}{29}\\ \Rightarrow \frac{29+12}{29}\\ \Rightarrow \frac{41}{29}\end{array}$

Using a calculator, find the value of $\frac{41}{29}.$

Therefore, $\frac{41}{29}=\mathrm{1.413.}$

Conclusion:

Therefore, the value of the expression $1+\frac{1}{2+\frac{1}{2+\frac{1}{2}}}$ is 1.413.