Basic operations involving time

Basic Operation Involving Time

4.3

Basic Operations Involving Time

ADDITION OF TIME IN FRACTIONS AND DECIMALS

HOURS & MINUTES

FRACTION

QUESTION:
\(\frac{1}{2}\text{ hour}+\frac{1}{10}\text{ hour} = \Box\text{ minutes}\)

SOLUTION:
We have to convert it to equivalent fractions.
\(\frac{1}{2}\text{ hour}+\frac{1}{10}\text{ hour}\)

\(=\frac{1\color{red}{\times5}}{2\color{red}{\times5}}\text{ hour}+\frac{1}{10}\text{ hour}\)

\(=\frac{5+1}{10}\text{ hour}\)

\(=\frac{6}{10}\text{ hour}\)

ANSWER:
\(\frac{6}{10}\times60\text{ minutes} = 36\text{ minutes}\)

DECIMALS

QUESTION:
\(0.25\text{ hour} + 3.85\text{ hours} = \Box\text{ hours}\)

SOLUTION:
\(\frac{\begin{array}{lr} &\overset{\color{red}1}{0}.\overset{\color{red}1}{2}5\text{ hour }\\ +&3.85\text{ hours}\\ \end{array}} {\begin{array}{r} \,\,\,&4.10\text{ hours}\\\hline \end{array} }\)

ANSWER:
\(0.25\text{ hour} + 3.85\text{ hours} = 4.1\text{ hours}\)

DAYS & HOURS

FRACTION

QUESTION:
\(1\frac{1}{6}\text{ days}+\frac{1}{8}\text{ day}=\Box\text{ days}\)

SOLUTION:
We can seperate whole numbers from its fractions.

\(1\frac{1}{6}\text{ days}+\frac{1}{8}\text{ day}\)

\(=1\text{ day}+\frac{1\color{red}{\times8}}{6\color{red}{\times8}}\text{ day}+\frac{1\color{red}{\times6}}{8\color{red}{\times6}}\text { day}\)

\(=1\text{ day}+(\frac{8+6}{48})\text{ day}\)

\(=1\text{ day}+(\frac{14}{48})\text{ day}\)

\(= 1\frac{7}{24}\text{ days}\)

ANSWER:
\(1\frac{1}{6}\text{ days}+\frac{1}{8}\text{ day}=1\frac{7}{24}\text{ days}\)

DECIMALS

QUESTION:
\(7.5\text{ days} +19\text{ hours} = \Box\text{ hours}\)

SOLUTION:
Convert to unit hours.
\(7.5\text{ days} +19\text{ hours}\)
\(=(7.5\times24)\text{ hours}+19\text{ hours}\)
\(=180\text { hours} + 19\text{ hours}\)
\(=199\text { hours}\)

ANSWER:
\(7.5\text{ days} +19\text{ hours} = 199\text{ hours}\)

YEARS & MONTHS

FRACTION

QUESTION:
\(1\frac{1}{6}\text{ years} + 4\text{ years }10 \text{ months} = \Box\text{ years}\)

SOLUTION:
We will separate whole numbers and proper fractions.

\(1\frac{1}{6}\text{ years} + 4\text{ years }10 \text{ months}\)

\(=1\frac{1\color{black}{\color{orange}\times2}}{6\color{orange}{\times2}}\text{tahun}+4\frac{10}{12}\text{tahun}\)

\(=1{\frac{2}{12}}\text{ tahun}+4{\frac{10}{12}}\text{ tahun}\)

\(={1+4}+({\frac{2+10}{12}})\text{ tahun}\)

\(={5}\text{ tahun}+{\frac{12}{12}}\text{ tahun} \to{5}\text{ tahun}+{1}\text{ tahun}\)

\(=6 \text{ tahun}\)

ANSWER:
\(1\frac{1}{6}\text{ years} + 4\text{ years }10 \text{ months} = 6\text{ years}\)

DECIMALS

QUESTION:
\(9.25\text{ years}+10.5\text{ years}=\Box\text{ years}\)

SOLUTION:
\(\frac { \begin{array}{lr} &9.25\text{ tahun}\\ +&10.50\text{ tahun} \end{array} } { \begin{array}{lr} &&19.75\text{ tahun}\\\hline\end{array} }\)

ANSWER:
\(9.25\text{ tahun}+10.5\text{ tahun}=19.75\text{ tahun}\)

DECADES & YEARS

FRACTION

QUESTION:
\(\frac{7}{10}\text{ decades}+\frac{2}{5}\text{ decades} = \Box\text{ decades}\)

SOLUTION:
\(\frac{7}{10}\text{ decades}+\frac{2}{5}\text{ decades}\)

\(=\frac{7}{10}\text{ decades}+\frac{2\color{orange}{\times2}}{5\color{orange}{\times2}}\text{ decades}\)

\(=\frac{7+{4}}{10}\text{ decades}\)

\(=\frac{11}{10}\text{ decades}\)

Improper fractions need to be converted to mixed numbers.
\({11\over10}\text{ decades}\to 1\frac{1}{10}\text{ decades}\)

ANSWER:
\(\frac{7}{10}\text{ decades}+\frac{2}{5}\text{ decades} = 1\frac{1}{10}\text{ decades}\)

DECIMALS

QUESTION:
\(4\text{ decades }1\text{ year}+2.3\text{ decades}=\Box\text{ years}\)

SOLUTION:
Seperates decades and years to add the decades.
\(4\text{ decades }1\text{ year}+2.3\text{ decades}\)
\(=4\text{ decades}+1\text{ year}+2.3\text{ decades}\)
\(=4\text{ decades}+2.3\text{ decades}+1\text{ year}\)
\(=6.3\text{ decades}+1\text{ year}\)

Convert decades to years. Then, add both of the years.
\(6.3\text{ decades}+1\text{ year}\)
\(=(6.3\times10)\text{ years}+1\text{ year}\)
\(=63\text{ years}+1\text{ year}\)
\(=64\text{ years}\)

ANSWER:
\(4\text{ decades }1\text{ year}+2.3\text{ decades}=64\text{ years}\)

CENTURIES & DECADES

FRACTION

QUESTION:
\(5\frac{1}{5}\text{ centuries} +2\frac{2}{5}\text{ centuries} =\Box\text{ centuries }\Box\text{ decades}\)

SOLUTION:
\(5\frac{1}{5}\text{ centuries} +2\frac{2}{5}\text{ centuries}\)

\(=(5+2) \text{ centuries}+(\frac{1}{5}+\frac{2}{5})\text{ century}\)

\(= 7\text{ centuries} + \frac{3}{5}\text{ century}\)

Convert the fractions to decades.
\(7\text{ centuries}+(\frac{3}{5}\times10)\text{ decades}\)
\(= 7\text{ centuries } \, 6\text{ decades}\)

ANSWER:
\(5\frac{1}{5}\text{ centuries} +2\frac{2}{5}\text{ centuries} =7\text{ centuries }6\text{ decades}\)

DECIMALS

QUESTION:
\(8.4\text{ centuries}+52.9\text{ centuries}=\Box\text{ centuries }\Box\text{ decades}\)

SOLUTION:
We will add both century values as usual.
\( \frac { \begin{array}{lr} &8.4\text{ abad}\\ +&52.9\text{ abad} \end{array} } { \begin{array}{lr} &&61.3\text{ abad} \\\hline\end{array} } \)

\(61.3 \text{ abad} = 61 \text{ abad}+ 0.3\text{ abad}\)

Convert 0.3 centuries to decades.
\(0.3\text{ century} \to 0.3\times10\text{ decades} \to 3\text{ decades}\)
\(61.3 \text{ centuries} = 61 \text{ centuries } 3\text{ decades}\)

ANSWER:
\(8.4\text{ centuries}+52.9\text{ centuries}=61\text{ centuries }3 \text{ decades}\)

CENTURIES & YEARS

FRACTION

QUESTION:
\(5\frac{1}{5}\text{ centuries} +2\frac{2}{5}\text{ centuries} =\Box\text{ years}\)

SOLUTION:
\(5\frac{1}{5}\text{ centuries} +2\frac{2}{5}\text{ centuries}\)

\(=(5+2) \text{ centuries}+(\frac{1}{5}+\frac{2}{5})\text{ century}\)

\(= 7\text{ centuries} + \frac{3}{5}\text{ century}\)

Convert to needed unit.
\(7\text{ centuries} + \frac{3}{5}\text{ century}\)
\(= (7\times100)\text{ years} + (\frac{3}{5}\times100)\text{ years}\)
\(=700 \text{ years} + 60\text{ years}\)
\(=760\text { years}\)

ANSWER:
\(5\frac{1}{5}\text{ centuries} +2\frac{2}{5}\text{ centuries} =760\text{ years}\)

DECIMALS

QUESTION:
\(8.47\text{ centuries}+52.9\text{ centuries}=\Box\text{ years }\)

SOLUTION:
\(\frac { \begin{array}{lr} &8.47\text{ centuries}\\ +&52.90\text{ centuries} \end{array} } { \begin{array}{lr} &&61.37\text{ centuries} \\\hline\end{array} }\)

Convert 61.37 centuries to years.
\(61.37 \text{ centuries} \to61.37\times100\text{ years}\to6137\text{ years}\)

ANSWER:
\(8.47\text{ centuries}+52.9\text{ centuries}=6137\text{ years }\)

SUBTRACTION OF TIME IN FRACTIONS AND DECIMALS

HOURS & MINUTES

FRACTION

QUESTION:
\(\frac{3}{4}\text{ hour}-\frac{1}{2}\text{ hour}=\Box\text{ hour}\)

SOLUTION:
We need to equalize the denominator first before subtracting.
\(\frac{3}{4}\text{ hour}-\frac{1}{2}\text{ hour}\)

\(=\frac{3}{4}\text{ jam}-\frac{1\color{orange}{\times2}}{2\color{orange}{\times2}}\text{ jam}\)

\(=\frac{3}{4}\text{ jam}-\frac{2}{4}\text{ jam}\)

\(=\frac{1}{4}\text{ jam}\)

ANSWER:
\(\frac{3}{4}\text{ hour}-\frac{1}{2}\text{ hour}=\frac{1}{4}\text{ hour}\)

DECIMALS

QUESTION:
\(1\text{ hour }40\text{ minutes}-0.95\text{ hour}=\Box\text{ minutes}\)

SOLUTION:
We can convert everything to minute units for ease of operation.
\(1\text{ hour }40\text{ minutes}-0.95\text{ hour}\)
\(=(1\times60)\text{ minutes} + 40\text{ minutes}-(0.95\times60)\text{ minutes}\)
\(=60\text{ minutes} + 40\text{ minutes}-57\text{ minutes}\)
\(=43\text{ minit}\)

ANSWER:
\(1\text{ hour }40\text{ minutes}-0.95\text{ hour}=43\text{ minutes}\)

DAYS & HOURS

FRACTION

QUESTION:
\(\frac{5}{8}\text{ day}-\frac{1}{3}\text{ day}=\Box\text{ hours}\)

SOLUTION:
Equalize the denominators.
\(\frac{5}{8}\text{ day}-\frac{1}{3}\text{ day}\)

\(=\frac{5\color{red}{\times3}}{8\color{red}{\times3}}\text{ day}-\frac{1\color{red}{\times8}}{3\color{red}{\times8}}\text{ day}\)

\(=\frac{15-8}{24}\text{ day}\)

\(=\frac{7}{24}\text{ day}\to\frac{7}{24}\times24\text{ hours}\)

\(=7\text{ hours}\)

ANSWER:
\(\frac{5}{8}\text{ day}-\frac{1}{3}\text{ day}=7\text{ hours}\)

DECIMALS

QUESTION:
\(0.25\text{ day}-0.125\text{ day}=\Box\text{ day}\)

SOLUTION:

\(\frac{ \begin{array}{lr} &\color{red}{^{4}}\,\color{red}{^{10}}\,\,\,\,\,\,\,\,\,\,\\&0.\,2\,{\not}5\,{\not}0\text { day}\\ -&0.\,1\,2\,5\text{ day} \end{array} }{ \begin{array}{lr} &\,\,\,\,\,0.\,1\,2\,5\text{ day}\\\hline \end{array} }\)

ANSWER:
\(0.25\text{ day}-0.125\text{ day}=0.125\text{ day}\)

YEARS & MONTHS

FRACTION

QUESTION:
\(4\frac{1}{2}\text{ years} - 1\frac{5}{12}\text{ years}=\Box\text{ years }\Box\text{ months }\)

SOLUTION:
We need to equalize the denominator of the fraction first.

\(4\frac{1\color{orange}\times6}{2\color{orange}\times6}\text{ years} - 1\frac{5}{12}\text{ years}\)

\(=4\frac{6}{12}\text{ years} - 1\frac{5}{12}\text{ years}\)

\(=3\frac{1}{12}\text{ years}\)

We need to convert fractions to units of months.

\(3\frac{1}{12}\text{ years}\)

\(=3\text{ years }+(\frac{1}{12}\times24)\text{ months}\)

\(=3\text{ years }2\text{ months}\)

ANSWER:
\(4\frac{1}{2}\text{ years} - 1\frac{5}{12}\text{ years}=3\text{ years }2\text{ months }\)

DECIMALS

QUESTION:
\(7.75\text{ years}-4.5\text{ years} = \Box\text{ years }\Box\text{ months}\)

SOLUTION:
We can convert decimals to monthly units first.
\(0.75\times12\text{ months}=9\text{ months}\)
\(0.5\times12\text{ months}=6\text{ months}\)

\(7.75\text{ years}-4.5\text{ years}\)
\(=7\text{ years }9\text{ months}-4\text{ years }6\text{ months}\)
\(=3\text{ years }3\text{ months}\)

ANSWER:
\(7.75\text{ years}-4.5\text{ years} = 3\text{ years }3\text{ months}\)

DECADES & YEARS

FRACTION

QUESTION:
\(9\frac{7}{10}\text{ decades}-2\text{ decades }3\text{ years}=\Box\text{ years}\)

SOLUTION:
Convert all to units of years.
\(9\frac{7}{10}\text{ decades}\)
\(=(9\times10)+(\frac{7}{10}\times10)\text{ years}\)
\(=(90 +7) \text{ years}\\ =97\text{ years}\)

\(2\text{ decades }3\text{ years}\)
\(=(2\times10)+3\text{ years}\\\)
\(=(20+3)\text{ years}\\ =23\text{ years}\)

\(97\text{ years}-23\text{ years}=74\text{ years}\)

ANSWER:
\(9\frac{7}{10}\text{ decades}-2\text{ decades }3\text{ years}=74\text{ years}\)

DECIMALS

QUESTION:
\(8.1\text{ decades} - 27\text{ years}=\Box\text{ decades }\Box\text{ years}\)

SOLUTION:
Convert 8.1 decades to decades and years.
\(8.1\text{ decades}\)
\(=8\text{ decades}+(0.1\times10)\text{ years}\\=8\text{ decades }1\text{ year}\)

\(27\text{ years}\)
\(=27\div10\text{ decades}\\=2.7\text{ decades}\\=2\text{ decades}+(0.7\times10)\text{ years}\\=2\text{ decades }7\text{ years}\)

Perform a subtraction operation.
\(8.1\text{ decades} - 27\text{ years}\)
\(=8\text{ decades }1\text{ year}-2\text{ decades } 7\text{ years}\)
\(=7\text{ decades }{11}\text{ years}-2\text{ decades } 7\text{ years}\)
\(=5\text{ decades } 4\text{ years}\)

ANSWER:
\(8.1\text{ decades} - 27\text{ years}=5\text{ decades }4\text{ years}\)

CENTURIES & DECADES

FRACTION

QUESTION:
\(2\text{ centuries } 4\text{ decades}-1\frac{1}{5}\text{ centuries}=\Box\text{ decades}\)

SOLUTION:
Convert to decade units.
\(2\text{ centuries } 4\text{ decades}\)
\(=(2\times10)\text{ decades}+4 \text{ decades}\\\)
\(=20\text{ decades}+4\text{ decades}\\=24\text{ decades}\)

\(1\frac{1}{5}\text{ centuries}\)
\(=\frac{6}{5}\times10\text{ decades}\\\)
\(=12\text{ decades}\)

\(2\text{ centuries } 4\text{ decades}-1\frac{1}{5}\text{ centuries}\\\)
\(=24\text{ decades}-12\text{ decades}\\ =12\text{ decades}\)

ANSWER:
\(2\text{ centuries } 4\text{ decades}-1\frac{1}{5}\text{ centuries}=12\text{ decades}\)

DECIMALS

QUESTION:
\(8.1\text{ centuries}-6.7\text{ centuries}=\Box\text{ centuries }\Box\text{ decades}\)

SOLUTION:

\(\frac{ \begin{array}{lr}&\color{red}{^{7}}\color{red}{^{11}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\\&{\not}8.\,{\not}1\,\text { centuries}\\ -&6.\,7\,\text{ centuries} \end{array} }{ \begin{array}{lr} &\,\,\,\,\,1.\,4\,\text{ centuries}\\\hline \end{array} }\)

\(1.4\text{ centuries}\\\)
\(= 1 \text{ century} +(0.4\times10)\text{ decades}\\ = 1 \text{ century } 4\text{ decades}\)

ANSWER:
\(8.1\text{ centuries}-6.7\text{ centuries}=1\text{ centuries }4\text{ decades}\)

CENTURIES & YEARS

FRACTION

QUESTION:
\(2\frac{1}{2}\text{ centuries} -\frac{3}{4}\text{ century}=\Box\text{ years}\)

SOLUTION:
\(2\frac{1}{2}\text{ centuries}\\\)
\(=\frac{5}{2}\times100\\=250\text{ tahun}\)

\(\frac{3}{4}\text{ century}\\\)
\(=\frac{3}{4}\times100\\=75\text{ years}\)

\(\frac{ \begin{array}{lr} &\color{red}{^{4}}\,\color{red}{^{10}}\,\,\,\,\,\,\,\,\,\,\,\\&2\,{\not}5\,{\not}0\text { years}\\ -&\,7\,5\text{ years} \end{array} }{ \begin{array}{lr} &\,\,\,\,\,1\,7\,5\text{ years}\\\hline \end{array} }\)

ANSWER:
\(2\frac{1}{2}\text{ centuries} -\frac{3}{4}\text{ century}=175\text{ years}\)

DECIMALS

QUESTION:
\(9.1\text{ decades}-6.3\text{ decades}=\Box\text{ decades }\Box\text{ years }\)

SOLUTION:

\(\frac{ \begin{array}{lr} &\color{red}{^{8}}\,\color{red}{^{11}}\,\,\,\,\,\,\,\,\,\,\,\\&{\not}9.\,{\not}1\text { decades}\\ -&6.\,3\text{ decades} \end{array} }{ \begin{array}{lr} &\,\,\,\,\,2.\,8\text{ decades}\\\hline \end{array} }\)

\(2.8\text{ decades}\\\)
\(=2\text{ decades}+(0.8\times10)\text{ years}\\ =2\text{ decades } 8\text{ years}\)

ANSWER:
\(9.1\text{ decades}-6.3\text{ decades}=2\text{ decades }8\text{ years }\)

Basic Operation Involving Time

Chapter : Time

Topic : Basic operations involving time

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