9.3 Simplification of denominate numbers

Simplify 19 in.

Since $\text{12}\text{ in}\text{.}=\text{1 ft}$ , and $\text{19}=\text{12}+7$ ,

$\begin{array}{ccc}\hfill \text{19 in.}& =& \text{12 in.}+\text{7 in.}\hfill \\ & =& \text{1 ft}+\text{7 in.}\hfill \\ & =& \text{1 ft 7 in.}\hfill \end{array}$

Simplify 4 gal 5 qt.

Since $4\text{ qt}=\text{1 gal}$ , and $5=4+1$ ,

$\begin{array}{ccc}\hfill \text{4 gal 5 qt}& =& \text{4 gal}+\text{4 qt}+\text{1qt}\hfill \\ & =& \text{4 gal}+\text{1 gal}+\text{1 qt}\hfill \\ & =& \text{5 gal}+\text{1 qt}\hfill \\ & =& \text{5 gal 1 qt}\hfill \end{array}$

Simplify 2 hr 75 min.

Since $\text{60 min}=\text{1 hr}$ , and $\text{75}=\text{60}+\text{15}$ ,

$\begin{array}{ccc}\hfill \text{2 hr 75 min}& =& \text{2 hr}+\text{60 min}+\text{15 min}\hfill \\ & =& \text{2 hr}+\text{1 hr}+\text{15 min}\hfill \\ & =& \text{3 hr}+\text{15 min}\hfill \\ & =& \text{3 hr 15 min}\hfill \end{array}$

Simplify 43 fl oz.

Since $\text{8 fl oz}=\text{1 c}$ (1 cup), and $\text{43}÷8=5\text{R3}$ ,

$\begin{array}{ccc}\hfill \text{43 fl oz}& =& \text{40 fl oz}+\text{3 fl oz}\hfill \\ & =& 5\cdot \text{8 fl oz}+\text{3 fl oz}\hfill \\ & =& 5\cdot \text{1 c}+\text{3 fl oz}\hfill \\ & =& \text{5 c}+\text{3 fl oz}\hfill \end{array}$

But, $\text{2 c}=\text{1 pt}$ and $5÷2=2\text{R1}$ . So,

$\begin{array}{ccc}\hfill \text{5 c}+\text{3 fl oz}& =& \text{2}\cdot \text{2 c}+\text{1 c}+\text{3 fl oz}\hfill \\ & =& 2\cdot \text{1 pt}+\text{1 c}+\text{3 fl oz}\hfill \\ & =& \text{2 pt}+\text{1 c}+\text{3 fl oz}\hfill \end{array}$

But, $\text{2 pt}=\text{1 qt}$ , so

$\text{2 pt}+\text{1 c}+\text{3 fl oz}=\text{1 qt 1 c 3 fl oz}$

Add 6 ft 8 in. to 2 ft 9 in.

$\begin{array}{cc}\hfill \text{6 ft 8 in}\text{.}& \\ \hfill \underline{+\text{2 ft 9 in}\text{.}}& \\ \hfill \text{8 ft 17 in}\text{.}& \text{Simplify this denominate number.}\hfill \end{array}$

Since $\text{12 in}\text{.}=\text{1 ft}$ ,

$\begin{array}{ccc}\hfill \text{8 ft}+\text{12 in.}+\text{5 in.}& =& \text{8 ft}+\text{1 ft}+\text{5 in.}\hfill \\ & =& \text{9 ft}+\text{5 in.}\hfill \\ & =& \text{9 ft 5 in.}\hfill \end{array}$

Subtract 5 da 3 hr from 8 da 11 hr.

$\begin{array}{c}\hfill \text{8 da 11 hr}\\ \hfill \underline{-\text{5 da 3 hr}}\\ \hfill \text{3 da 8 hr}\end{array}$

Subtract 3 lb 14 oz from 5 lb 3 oz.

$\begin{array}{c}\hfill \text{5 lb 3 oz}\\ \hfill \underline{-\text{3 lb 14 oz}}\end{array}$

We cannot directly subtract 14 oz from 3 oz, so we must borrow 16 oz from the pounds.

$\begin{array}{cccc}\hfill \text{5 lb 3 oz}& =& \text{5 lb}+\text{3 oz}\hfill & \\ & =& \text{4 lb}+\text{1 lb}+\text{3 oz}\hfill & \\ & =& \text{4 lb}+\text{16 oz}+\text{3 oz}\hfill & (\text{Since 1 lb}=\text{16 oz.})\hfill \\ & =& \text{4 lb}+\text{19 oz}\hfill & \\ & =& \text{4 lb 19 oz}\hfill & \end{array}$

$\begin{array}{c}\hfill \text{4 lb 19 oz}\\ \hfill \underline{-\text{3 lb 14 oz}}\\ \hfill \text{1 lb 5 oz}\end{array}$

Subtract 4 da 9 hr 21 min from 7 da 10 min.

$\begin{array}{c}\hfill \text{7 da 0 hr 10 min}\\ \hfill \underline{-\text{4 da 9 hr 21 min}}\end{array}$ Borrow 1 da from the 7 da.

$\begin{array}{c}\hfill \text{6 da 24 hr 10 min}\\ \hfill \underline{-\text{4 da 9 hr 21 min}}\end{array}$ Borrow 1 hr from the 24 hr.

$\begin{array}{c}\hfill \text{6 da 23 hr 70 min}\\ \hfill \underline{-\text{4 da 9 hr 21 min}}\\ \hfill \text{2 da 14 hr 49 min}\end{array}$