Question

The Res, a Roman measure of length, is approximately equal to 11.65 inches. It is also equivalent to 16 smaller Roman units called digits. Based on these relationships 75 Roman digits is equivalent to how many feet, to the nearest hundredth? \[\left( {{\rm{12 inches}} = {\rm{1 foot}}} \right)\]
(a). 3.55
(b). 4
(c). 4.55
(d). 5.55

Answer 1

Hint: To solve this question, we will develop a relation between ‘digits’ and ‘inches’ with the help of information given in question. After developing a relation between them, we will convert the digits into feet with the help of relation: 1 foot = 12 inches.

Complete step-by-sep answer:

It is given that 1 Res is approximately equal to 11.65 inches. Thus, we have the following relation:
\[{\rm{1 Res}} = {\rm{11}}{\rm{.65 inches}}\]. . . . . . . . . . . . . . . . . . . . (i)
Another relation between the two units given in question is that 1 Res is equal to 16 smaller units called digits. Thus, we have the following relation:
\[{\rm{1 Res}} = {\rm{16 digits}}\]. . . . . . . . . . . . . . . . . . . . . . . . . (ii)
Thus, from the equations (i) and (ii), we will obtain the following relation with digits and inches:
\[{\rm{11}}.{\rm{65 inches}} = {\rm{16 digits}}\]. . . . . . . . . . . . . . . . . . . (iii)
Now, we will divide the equation (iii) with 16. Thus, we get the following equation:
\(\begin{array}{l} \Rightarrow {\rm{ }}\dfrac{{11.65{\rm{ inches}}}}{{16}}{\rm{ = }}\dfrac{{16{\rm{ digits}}}}{{16}}\\ \Rightarrow {\rm{ }}\dfrac{{11.65{\rm{ }}}}{{16}}{\rm{inches = 1 digit}}\\ \Rightarrow {\rm{ 1 digit = }}\dfrac{{11.65{\rm{ }}}}{{16}}{\rm{inches}}\end{array}\)
Now, we have to find the value of 75 digits in terms of feet. Thus, we have the following result:
\(\begin{array}{l}75{\rm{ digits = 75 }} \times {\rm{ }}\left( {\dfrac{{11.65{\rm{ }}}}{{16}}{\rm{inches}}} \right)\\\end{array}\)
\(75{\rm{ digits = 54}}{\rm{.609 inches }}\) . . . . . . . . . . . . . . . . . . (iv)
It is also given in question that:
\[{\rm{1 foot}} = {\rm{12 inches}}\]
\( \Rightarrow {\rm{ }}\dfrac{1}{{12}}foot{\rm{ = 1 inch}}\)
\( \Rightarrow {\rm{ 1 inch = }}\dfrac{1}{{12}}foot\) . . . . . . . . . . . . . . . . . . . . . . .(v)
Now, from equation (iv) and (v), we have:
\(\begin{array}{l} \Rightarrow {\rm{ 75 digits = 54}}{\rm{.609 }} \times {\rm{ }}\dfrac{1}{{12}}foot\\ \Rightarrow {\rm{ 75 digits = }}\dfrac{{54.609}}{{12}}feet\\ \Rightarrow {\rm{ 75 digits = 4}}{\rm{.55 feet}}{\rm{.}}\end{array}\)
Hence, option (c) is correct.

Note: The above answer can also be obtained in a reversed way. From equation (i) and (v), we have:
\(\begin{array}{l}1{\rm{ Res = }}\dfrac{1}{{12}}{\rm{ }} \times {\rm{ 11}}{\rm{.65 foot}}\\{\rm{1 Res = 0}}{\rm{.97 feet}}\end{array}\)
On comparing the above equation with the equation (ii), we will obtain the following result:
\[0.{\rm{97 feet}} = {\rm{16 digits}}\]
Dividing the above equation by 16, we get the following:
\(\begin{array}{l} \Rightarrow {\rm{ }}\dfrac{{0.97{\rm{ feet}}}}{{16}}{\rm{ = }}\dfrac{{16{\rm{ digits}}}}{{16}}\\ \Rightarrow {\rm{ 1 digit = }}\left( {\dfrac{{0.97}}{{16}}} \right)feet\\ \Rightarrow {\rm{ 1 digit = 0}}{\rm{.06 feet}}\end{array}\)
Now, we have to calculate the value of 75 digits. Thus, we have:
\(\begin{array}{l}75{\rm{ }}digits{\rm{ = 75 }} \times {\rm{ }}\left( {0.06{\rm{ }}feet} \right)\\ \Rightarrow {\rm{ 75 digits = 4}}{\rm{.545 feet}}\\ \Rightarrow {\rm{ 75 digits = 4}}{\rm{.55 feet}}\end{array}\)