Question

Table 1−7 shows some old measures of liquid volume: To complete the table, what numbers (to three significant figures) should be entered in each column.
The volume of one bag is equal to \[0.1091~{{m}^{3}}\]. If an old story has a witch cooking up some vile liquid in a cauldron of volume \[\text{1}\text{.5}\] chaldrons, what is the volume in cubic meters?

	Wey	chaldron	bag	pottle	gill
1 wey 1 chaldron1 bag 1 pottle 1 gill	\[\text{1}\]	\[\dfrac{\text{10}}{\text{9}}\]	\[\dfrac{\text{40}}{\text{3}}\]	\[\text{640}\]	\[\text{120240 }\]

Answer 1

Hint: In the first row the volume of each liquid in one wey has given. We can equate the given volumes with each other to find the volume of each ingredient. And from the volume of the bag given, we can find the new volume of chaldron.

Complete step by step answer:
Given that,
\[\text{1 wey=}\dfrac{\text{10}}{\text{9}}\text{ chaldron= }\dfrac{\text{40}}{\text{3}}\text{ bag= 640 pottle = 120240 gill}\]
In the first column,
\[\text{1wey = }\dfrac{\text{10}}{\text{9}}\text{chaldron}\Rightarrow \text{1chaldron = 0}\text{.9wey}\]
\[\text{1wey = }\dfrac{\text{40}}{\text{3}}\text{bag}\Rightarrow \text{1bag =}0.075\text{wey}\]
\[\text{1wey = 640 pottle }\Rightarrow \text{1pottle =1}\text{.57}\times \text{1}{{\text{0}}^{-3}}\text{wey}\]
\[\text{1wey = 120240 gill}\Rightarrow \text{1gill =8}\text{.31}\times \text{1}{{\text{0}}^{-6}}wey\]
In the second column,
\[\dfrac{10}{9}\text{chaldron=}\dfrac{40}{3}\text{bag }\Rightarrow \text{1 bag =}\dfrac{\text{30}}{\text{360}}\text{=8}\text{.33}\times \text{1}{{\text{0}}^{-2}}\text{chaldron}\]
\[\dfrac{10}{9}\text{ chaldron}=640\text{ pottle}\Rightarrow \text{1 pottle=}\dfrac{\text{10}}{\text{5760}}\text{=1}\text{.73}\times \text{1}{{\text{0}}^{-3}}\text{haldron}\]
\[\dfrac{\text{10}}{\text{9}}\text{chaldron = 120240 gill}\Rightarrow \text{1 gill=9}\text{.24}\times \text{1}{{\text{0}}^{-6}}\text{chaldron}\]
In the third column,
\[\text{1 bag =}\dfrac{\text{1}}{\text{12}}\text{chaldron}\Rightarrow \text{1 chaldron = 12 bag}\]
\[\dfrac{\text{40}}{\text{3}}\text{bag =640 pottle }\Rightarrow \text{ 1 pottle=}\dfrac{\text{40}}{\text{1920}}\text{=2}\text{.08}\times \text{1}{{\text{0}}^{-2}}\text{bag}\]
\[\dfrac{\text{40}}{\text{3}}\text{bag =120240 gill }\Rightarrow \text{1 gill =}\dfrac{\text{40}}{\text{3 }\!\!\times\!\!\text{ 120240}}\text{=1}\text{.11}\times \text{1}{{\text{0}}^{-4}}\text{bag}\]
In the fourth column,
\[\text{1 pottle=}\dfrac{\text{1}}{576}\text{chaldron}\Rightarrow \text{1 chadron = 576 pottle}\]
\[\text{1 pottle =}\dfrac{\text{1}}{\text{48}}\text{bag}\Rightarrow \text{1 bag = 48 pottle}\]
\[\text{640 pottle = 120240 gill }\Rightarrow \text{1 gill=}\dfrac{\text{640}}{\text{120240}}\text{=5}\text{.32}\times \text{1}{{\text{0}}^{-3}}\text{pottle}\]
In the fifth column,
\[\text{1 gill=}\dfrac{1}{\text{1}\text{.08 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{5}}}}\text{chaldron}\Rightarrow \text{1chaldron =1}\text{.08 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{5}}}\text{gill}\]
\[\text{1 gill =}\dfrac{\text{1}}{\text{9}\text{.01 }\!\!\times\!\!\text{ 1}{{\text{0}}^{3}}}\text{bag}\Rightarrow \text{ 1 bag = 9}\text{.01 }\!\!\times\!\!\text{ 1}{{\text{0}}^{\text{4}}}\text{gill}\]
\[\text{1 gill=}\dfrac{\text{8}}{\text{1503}}\text{pottle}\Rightarrow \text{1 pottle =}188\text{gill}\]
Given that,
Volume of bag\[\text{= 0}\text{.1091 }\!\!~\!\!\text{ }{{\text{m}}^{\text{3}}}\]
Then,
 \[\text{1}\text{.5 chaldron =1}\text{.5}\times \text{12 =18bag}\]
 Since, each bag is \[\text{ 0}\text{.1091 }\!\!~\!\!\text{ }{{\text{m}}^{\text{3}}}\]
 \[\text{1}\text{.5 }\!\!~\!\!\text{ chaldron }\!\!~\!\!\text{ =}\left( \text{18} \right)\left( \text{0}\text{.1091} \right)\text{=1}\text{.96 }\!\!~\!\!\text{ }{{\text{m}}^{\text{3}}}\]

Note:
Important points to remember while determining number of significant figure:
Change of units should not change the number of significant digits.
Use scientific notation to report measurements. Numbers should be expressed in powers of \[\text{10}\] like \[a\times {{10}^{b}}\] where \[\text{b}\] which is called as the order of magnitude.
Dividing or multiplying of exact numbers can have an infinite number of significant digits.