Chapter 20, Problem 18A

To determine

(a)

The ratios $\frac{E}{F},\frac{E}{G},\frac{E}{H},\frac{F}{G},\frac{F}{H},\frac{G}{H},\frac{G}{E}\text{and}\frac{H}{F}$ in the lowest form for first condition.

Answer to Problem 18A

The ratios $\frac{E}{F},\frac{E}{G},\frac{E}{H},\frac{F}{G},\frac{F}{H},\frac{G}{H},\frac{G}{E}\text{and}\frac{H}{F}$ in the lowest form are $\frac{4}{3},\frac{2}{1},\frac{8}{3},\frac{3}{2},\frac{2}{1},\frac{4}{3}\text{,}\frac{1}{2}\text{and}\frac{1}{2}$ respectively.

Explanation of Solution

Given:

Diameters are given in the table below:

  

Calculation:

Divide $E$ by $F$ to get the ratio $\frac{E}{F}$ as follows:

  $\begin{array}{l}\frac{E}{F}=8÷6\\ \frac{E}{F}=\frac{8}{6}\\ \frac{E}{F}=\frac{4}{3}\end{array}$

Divide $E$ by $G$ to get the ratio $\frac{E}{G}$ as follows:

  $\begin{array}{l}\frac{E}{G}=8÷4\\ \frac{E}{G}=\frac{8}{4}\\ \frac{E}{G}=\frac{2}{1}\end{array}$

Divide $E$ by $H$ to get the ratio $\frac{E}{H}$ as follows:

  $\begin{array}{l}\frac{E}{H}=8÷3\\ \frac{E}{H}=\frac{8}{3}\end{array}$

Divide $F$ by $G$ to get the ratio $\frac{F}{G}$ as follows:

  $\begin{array}{l}\frac{F}{G}=6÷4\\ \frac{F}{G}=\frac{6}{4}\\ \frac{F}{G}=\frac{3}{2}\end{array}$

Divide $F$ by $H$ to get the ratio $\frac{F}{H}$ as follows:

  $\begin{array}{l}\frac{F}{H}=6÷3\\ \frac{F}{H}=\frac{6}{3}\\ \frac{F}{H}=\frac{2}{1}\end{array}$

Divide $G$ by $H$ to get the ratio $\frac{G}{H}$ as follows:

  $\begin{array}{l}\frac{G}{H}=4÷3\\ \frac{G}{H}=\frac{4}{3}\end{array}$

Divide $G$ by $E$ to get the ratio $\frac{G}{E}$ as follows:

  $\begin{array}{l}\frac{G}{E}=4÷8\\ \frac{G}{E}=\frac{4}{8}\\ \frac{G}{E}=\frac{1}{2}\end{array}$

Divide $H$ by $F$ to get the ratio $\frac{H}{F}$ as follows:

  $\begin{array}{l}\frac{H}{F}=3÷6\\ \frac{H}{F}=\frac{3}{6}\\ \frac{H}{F}=\frac{1}{2}\end{array}$

Thus, the ratios $\frac{E}{F},\frac{E}{G},\frac{E}{H},\frac{F}{G},\frac{F}{H},\frac{G}{H},\frac{G}{E}\text{and}\frac{H}{F}$ in the lowest form are $\frac{4}{3},\frac{2}{1},\frac{8}{3},\frac{3}{2},\frac{2}{1},\frac{4}{3}\text{,}\frac{1}{2}\text{and}\frac{1}{2}$ respectively.

Conclusion:

The ratios $\frac{E}{F},\frac{E}{G},\frac{E}{H},\frac{F}{G},\frac{F}{H},\frac{G}{H},\frac{G}{E}\text{and}\frac{H}{F}$ in the lowest form are $\frac{4}{3},\frac{2}{1},\frac{8}{3},\frac{3}{2},\frac{2}{1},\frac{4}{3}\text{,}\frac{1}{2}\text{and}\frac{1}{2}$ respectively.

To determine

(b)

The ratios $\frac{E}{F},\frac{E}{G},\frac{E}{H},\frac{F}{G},\frac{F}{H},\frac{G}{H},\frac{G}{E}\text{and}\frac{H}{F}$ in the lowest form for second condition.

Answer to Problem 18A

The ratios $\frac{E}{F},\frac{E}{G},\frac{E}{H},\frac{F}{G},\frac{F}{H},\frac{G}{H},\frac{G}{E}\text{and}\frac{H}{F}$ in the lowest form are $\frac{5}{4},\frac{2}{1},\frac{5}{2},\frac{8}{5},\frac{2}{1},\frac{5}{4}\text{,}\frac{1}{2}\text{and}\frac{1}{2}$ respectively.

Explanation of Solution

Given:

Diameters are given in the table below:

  

Calculation:

Divide $E$ by $F$ to get the ratio $\frac{E}{F}$ as follows:

  $\begin{array}{l}\frac{E}{F}=10÷8\\ \frac{E}{F}=\frac{10}{8}\\ \frac{E}{F}=\frac{5}{4}\end{array}$

Divide $E$ by $G$ to get the ratio $\frac{E}{G}$ as follows:

  $\begin{array}{l}\frac{E}{G}=10÷5\\ \frac{E}{G}=\frac{10}{5}\\ \frac{E}{G}=\frac{2}{1}\end{array}$

Divide $E$ by $H$ to get the ratio $\frac{E}{H}$ as follows:

  $\begin{array}{l}\frac{E}{H}=10÷4\\ \frac{E}{H}=\frac{10}{4}\\ \frac{E}{H}=\frac{5}{2}\end{array}$

Divide $F$ by $G$ to get the ratio $\frac{F}{G}$ as follows:

  $\begin{array}{l}\frac{F}{G}=8÷5\\ \frac{F}{G}=\frac{8}{5}\end{array}$

Divide $F$ by $H$ to get the ratio $\frac{F}{H}$ as follows:

  $\begin{array}{l}\frac{F}{H}=8÷4\\ \frac{F}{H}=\frac{8}{4}\\ \frac{F}{H}=\frac{2}{1}\end{array}$

Divide $G$ by $H$ to get the ratio $\frac{G}{H}$ as follows:

  $\begin{array}{l}\frac{G}{H}=5÷4\\ \frac{G}{H}=\frac{5}{4}\end{array}$

Divide $G$ by $E$ to get the ratio $\frac{G}{E}$ as follows:

  $\begin{array}{l}\frac{G}{E}=5÷10\\ \frac{G}{E}=\frac{5}{10}\\ \frac{G}{E}=\frac{1}{2}\end{array}$

Divide $H$ by $F$ to get the ratio $\frac{H}{F}$ as follows:

  $\begin{array}{l}\frac{H}{F}=4÷8\\ \frac{H}{F}=\frac{4}{8}\\ \frac{H}{F}=\frac{1}{2}\end{array}$

Thus, the ratios $\frac{E}{F},\frac{E}{G},\frac{E}{H},\frac{F}{G},\frac{F}{H},\frac{G}{H},\frac{G}{E}\text{and}\frac{H}{F}$ in the lowest form are $\frac{5}{4},\frac{2}{1},\frac{5}{2},\frac{8}{5},\frac{2}{1},\frac{5}{4}\text{,}\frac{1}{2}\text{and}\frac{1}{2}$ respectively.

Conclusion:

The ratios $\frac{E}{F},\frac{E}{G},\frac{E}{H},\frac{F}{G},\frac{F}{H},\frac{G}{H},\frac{G}{E}\text{and}\frac{H}{F}$ in the lowest form are $\frac{5}{4},\frac{2}{1},\frac{5}{2},\frac{8}{5},\frac{2}{1},\frac{5}{4}\text{,}\frac{1}{2}\text{and}\frac{1}{2}$ respectively.

To determine

(c)

The ratios $\frac{E}{F},\frac{E}{G},\frac{E}{H},\frac{F}{G},\frac{F}{H},\frac{G}{H},\frac{G}{E}\text{and}\frac{H}{F}$ in the lowest form for third condition.

Answer to Problem 18A

The ratios $\frac{E}{F},\frac{E}{G},\frac{E}{H},\frac{F}{G},\frac{F}{H},\frac{G}{H},\frac{G}{E}\text{and}\frac{H}{F}$ in the lowest form are $\frac{4}{3},\frac{2}{1},\frac{4}{1},\frac{3}{2},\frac{3}{1},\frac{2}{1}\text{,}\frac{1}{2}\text{and}\frac{1}{3}$ respectively.

Explanation of Solution

Given:

Diameters are given in the table below:

  

Calculation:

Divide $E$ by $F$ to get the ratio $\frac{E}{F}$ as follows:

  $\begin{array}{l}\frac{E}{F}=12÷9\\ \frac{E}{F}=\frac{12}{9}\\ \frac{E}{F}=\frac{4}{3}\end{array}$

Divide $E$ by $G$ to get the ratio $\frac{E}{G}$ as follows:

  $\begin{array}{l}\frac{E}{G}=12÷6\\ \frac{E}{G}=\frac{12}{6}\\ \frac{E}{G}=\frac{2}{1}\end{array}$

Divide $E$ by $H$ to get the ratio $\frac{E}{H}$ as follows:

  $\begin{array}{l}\frac{E}{H}=12÷3\\ \frac{E}{H}=\frac{12}{3}\\ \frac{E}{H}=\frac{4}{1}\end{array}$

Divide $F$ by $G$ to get the ratio $\frac{F}{G}$ as follows:

  $\begin{array}{l}\frac{F}{G}=9÷6\\ \frac{F}{G}=\frac{9}{6}\\ \frac{F}{G}=\frac{3}{2}\end{array}$

Divide $F$ by $H$ to get the ratio $\frac{F}{H}$ as follows:

  $\begin{array}{l}\frac{F}{H}=9÷3\\ \frac{F}{H}=\frac{9}{3}\\ \frac{F}{H}=\frac{3}{1}\end{array}$

Divide $G$ by $H$ to get the ratio $\frac{G}{H}$ as follows:

  $\begin{array}{l}\frac{G}{H}=6÷3\\ \frac{G}{H}=\frac{6}{3}\\ \frac{G}{H}=\frac{2}{1}\end{array}$

Divide $G$ by $E$ to get the ratio $\frac{G}{E}$ as follows:

  $\begin{array}{l}\frac{G}{E}=6÷12\\ \frac{G}{E}=\frac{6}{12}\\ \frac{G}{E}=\frac{1}{2}\end{array}$

Divide $H$ by $F$ to get the ratio $\frac{H}{F}$ as follows:

  $\begin{array}{l}\frac{H}{F}=3÷9\\ \frac{H}{F}=\frac{3}{9}\\ \frac{H}{F}=\frac{1}{3}\end{array}$

Thus, the ratios $\frac{E}{F},\frac{E}{G},\frac{E}{H},\frac{F}{G},\frac{F}{H},\frac{G}{H},\frac{G}{E}\text{and}\frac{H}{F}$ in the lowest form are $\frac{4}{3},\frac{2}{1},\frac{4}{1},\frac{3}{2},\frac{3}{1},\frac{2}{1}\text{,}\frac{1}{2}\text{and}\frac{1}{3}$ respectively.

Conclusion:

The ratios $\frac{E}{F},\frac{E}{G},\frac{E}{H},\frac{F}{G},\frac{F}{H},\frac{G}{H},\frac{G}{E}\text{and}\frac{H}{F}$ in the lowest form are $\frac{4}{3},\frac{2}{1},\frac{4}{1},\frac{3}{2},\frac{3}{1},\frac{2}{1}\text{,}\frac{1}{2}\text{and}\frac{1}{3}$ respectively.

To determine

(d)

The ratios $\frac{E}{F},\frac{E}{G},\frac{E}{H},\frac{F}{G},\frac{F}{H},\frac{G}{H},\frac{G}{E}\text{and}\frac{H}{F}$ in the lowest form for fourth condition.

Answer to Problem 18A

The ratios $\frac{E}{F},\frac{E}{G},\frac{E}{H},\frac{F}{G},\frac{F}{H},\frac{G}{H},\frac{G}{E}\text{and}\frac{H}{F}$ in the lowest form are $\frac{5}{4},\frac{3}{2},\frac{5}{2},\frac{6}{5},\frac{2}{1},\frac{5}{3}\text{,}\frac{2}{3}\text{and}\frac{1}{2}$ respectively.

Explanation of Solution

Given:

Diameters are given in the table below:

  

Calculation:

Divide $E$ by $F$ to get the ratio $\frac{E}{F}$ as follows:

  $\begin{array}{l}\frac{E}{F}=15÷12\\ \frac{E}{F}=\frac{15}{12}\\ \frac{E}{F}=\frac{5}{4}\end{array}$

Divide $E$ by $G$ to get the ratio $\frac{E}{G}$ as follows:

  $\begin{array}{l}\frac{E}{G}=15÷10\\ \frac{E}{G}=\frac{15}{10}\\ \frac{E}{G}=\frac{3}{2}\end{array}$

Divide $E$ by $H$ to get the ratio $\frac{E}{H}$ as follows:

  $\begin{array}{l}\frac{E}{H}=15÷6\\ \frac{E}{H}=\frac{15}{6}\\ \frac{E}{H}=\frac{5}{2}\end{array}$

Divide $F$ by $G$ to get the ratio $\frac{F}{G}$ as follows:

  $\begin{array}{l}\frac{F}{G}=12÷10\\ \frac{F}{G}=\frac{12}{10}\\ \frac{F}{G}=\frac{6}{5}\end{array}$

Divide $F$ by $H$ to get the ratio $\frac{F}{H}$ as follows:

  $\begin{array}{l}\frac{F}{H}=12÷6\\ \frac{F}{H}=\frac{12}{6}\\ \frac{F}{H}=\frac{2}{1}\end{array}$

Divide $G$ by $H$ to get the ratio $\frac{G}{H}$ as follows:

  $\begin{array}{l}\frac{G}{H}=10÷6\\ \frac{G}{H}=\frac{10}{6}\\ \frac{G}{H}=\frac{5}{3}\end{array}$

Divide $G$ by $E$ to get the ratio $\frac{G}{E}$ as follows:

  $\begin{array}{l}\frac{G}{E}=10÷15\\ \frac{G}{E}=\frac{10}{15}\\ \frac{G}{E}=\frac{2}{3}\end{array}$

Divide $H$ by $F$ to get the ratio $\frac{H}{F}$ as follows:

  $\begin{array}{l}\frac{H}{F}=6÷12\\ \frac{H}{F}=\frac{6}{12}\\ \frac{H}{F}=\frac{1}{2}\end{array}$

Thus, the ratios $\frac{E}{F},\frac{E}{G},\frac{E}{H},\frac{F}{G},\frac{F}{H},\frac{G}{H},\frac{G}{E}\text{and}\frac{H}{F}$ in the lowest form are $\frac{5}{4},\frac{3}{2},\frac{5}{2},\frac{6}{5},\frac{2}{1},\frac{5}{3}\text{,}\frac{2}{3}\text{and}\frac{1}{2}$ respectively.

Conclusion:

The ratios $\frac{E}{F},\frac{E}{G},\frac{E}{H},\frac{F}{G},\frac{F}{H},\frac{G}{H},\frac{G}{E}\text{and}\frac{H}{F}$ in the lowest form are $\frac{5}{4},\frac{3}{2},\frac{5}{2},\frac{6}{5},\frac{2}{1},\frac{5}{3}\text{,}\frac{2}{3}\text{and}\frac{1}{2}$ respectively.