Answer Key 9.10
- [latex]\text{Father}=\text{Bill}-2\text{ h}[/latex]
[latex]\therefore \dfrac{1}{B-2\text{ h}}+\dfrac{1}{B}=\dfrac{1}{2\text{ h } 24\text{ min}}[/latex] - [latex]\text{Smaller}=\text{Larger}+4\text{ h}[/latex]
[latex]\therefore \dfrac{1}{L+4\text{ h}}+\dfrac{1}{L}=\dfrac{1}{3\text{ h }45\text{ min}}[/latex] - [latex]\text{Jack}=\text{Bob}-1\text{ h}[/latex]
[latex]\therefore \dfrac{1}{B-1\text{ h}}+\dfrac{1}{B}=\dfrac{1}{1.2\text{ h}}[/latex] - [latex]\dfrac{1}{Y}+\dfrac{1}{B}=\dfrac{1}{T}[/latex]
[latex]\begin{array}{l}Y=6\text{ d}\\ B=4\text{ d}\\ \\ \therefore \dfrac{1}{6}+\dfrac{1}{4}=\dfrac{1}{T} \end{array}[/latex] - [latex]\text{John}=\text{Carlos}+8\text{ h}[/latex]
[latex]\dfrac{1}{C+8\text{ h}}+\dfrac{1}{C}=\dfrac{1}{3\text{ h}}[/latex] - [latex]M=3\text{ d}[/latex]
[latex]N=4\text{ d}[/latex]
[latex]E=5\text{ d}[/latex]
[latex]\begin{array}[t]{l} \dfrac{1}{M}+\dfrac{1}{N}+\dfrac{1}{E}=\dfrac{1}{T} \\ \\ \dfrac{1}{3\text{ d}}+\dfrac{1}{4\text{ d}}+\dfrac{1}{5\text{ d}}=\dfrac{1}{T} \end{array}[/latex] - [latex]\text{Raj}=4 \text{ d}[/latex]
[latex]\begin{array}[t]{l} \text{Rubi}=\dfrac{1}{2}\text{ Raj or }2\text{ d} \\ \\ \therefore \dfrac{1}{4 \text{ d}}+\dfrac{1}{2\text{ d}}=\dfrac{1}{T} \end{array}[/latex] - [latex]\dfrac{1}{20\text{ min}}+\dfrac{1}{30\text{ min}}=\dfrac{1}{T}[/latex]
- [latex]\dfrac{1}{24\text{ d}}+\dfrac{1}{I}=\dfrac{1}{6\text{ d}}[/latex]
[latex]\begin{array}[t]{rrrrl} &&\dfrac{1}{I}&=&\dfrac{1}{6\text{ d}}-\dfrac{1}{24\text{ d}} \\ \\ &&\dfrac{1}{I}&=&\dfrac{4}{24\text{ d}}-\dfrac{1}{24\text{ d}} \\ \\ &&\dfrac{1}{I}&=&\dfrac{1}{8\text{ d}} \\ \\ &&\therefore I&=&8\text{ days} \end{array}[/latex] - [latex]\phantom{a}[/latex]
[latex]\begin{array}[t]{rrrrl} \dfrac{1}{C}&+&\dfrac{1}{A}&=&\dfrac{1}{3.75\text{ d}} \dfrac{1}{5\text{ d}}&+&\dfrac{1}{A}&=&\dfrac{1}{3.75\text{ d}} \\ \\ &&\dfrac{1}{A}&=&\dfrac{1}{3.75\text{ d}}-\dfrac{1}{5\text{ d}} \\ \\ &&\dfrac{1}{A}&=&\dfrac{4}{15\text{ d}}-\dfrac{3}{15\text{ d}} \\ \\ &&\dfrac{1}{A}&=&\dfrac{1}{15\text{ d}} \\ \\ &&\therefore A&=&15\text{ days} \end{array}[/latex] - [latex]\phantom{a}[/latex]
[latex]\begin{array}[t]{rrrrl} \dfrac{1}{S}&+&\dfrac{1}{F}&=&\dfrac{1}{\text{job}} \dfrac{1}{3\text{ d}}&+&\dfrac{1}{6\text{ d}}&=&\dfrac{1}{\text{job}} \\ \\ \dfrac{2}{6\text{ d}}&+&\dfrac{1}{6\text{ d}}&=&\dfrac{1}{\text{job}} \\ \\ &&\dfrac{3}{6\text{ d}}&=&\dfrac{1}{\text{job}} \\ \\ &&\dfrac{1}{2\text{ d}}&=&\dfrac{1}{\text{job}} \\ \\ &&\therefore \text{job}&=&2 \text{ days} \end{array}[/latex] - [latex]\phantom{a}[/latex]
[latex]\begin{array}[t]{rrrrl} \dfrac{1}{T}&+&\dfrac{1}{J}&=&\dfrac{1}{\text{job}} \\ \\ \dfrac{1}{10\text{ h}}&+&\dfrac{1}{8\text{ h}}&=&\dfrac{1}{\text{job}} \\ \\ \dfrac{4}{40\text{ h}}&+&\dfrac{5}{40\text{ h}}&=&\dfrac{1}{\text{job}} \\ \\ &&\dfrac{9}{40\text{ h}}&=&\dfrac{1}{\text{job}} \\ \\ &&\therefore \text{job}&=&\dfrac{40\text{ h}}{9} \\ \\ &&\text{job}&=&4\dfrac{4}{9}\text{ h}= 4.\bar{4}\text{ h} \end{array}[/latex] - [latex]\phantom{a}[/latex]
[latex]\begin{array}[t]{rrrrl} &&\text{slow}&=&2\times \text{fast} \\ \\ &&\dfrac{1}{2}\text{slow}&=& \text{fast}\\ \\ \dfrac{1}{F}&+&\dfrac{1}{S}&=&\dfrac{1}{6\text{ h}} \\ \\ \dfrac{1}{\dfrac{1}{2}S}&+&\dfrac{1}{S}&=&\dfrac{1}{6\text{ h}} \\ \\ \left(\dfrac{2}{2}\right)\dfrac{1}{\dfrac{1}{2}S}&+&\dfrac{1}{S}&=&\dfrac{1}{6\text{ h}} \\ \\ \dfrac{2}{S}&+&\dfrac{1}{S}&=&\dfrac{1}{6\text{ h}} \\ \\ &&\dfrac{3}{S}&=&\dfrac{1}{6\text{ h}} \\ \\ &&\dfrac{S}{3}&=&6\text{ h} \\ \\ &&S&=&6(3) \\ \\ &&S&=&18\text{ h} \end{array}[/latex] - [latex]\phantom{a}[/latex]
[latex]\begin{array}[t]{rrcrl} &&\text{slower}&=&3\times \text{faster} \\ \\ \dfrac{1}{F}&+&\dfrac{1}{S}&=&\dfrac{1}{3\text{ h}} \\ \\ \dfrac{1}{F}&+&\dfrac{1}{3F}&=&\dfrac{1}{3\text{ h}} \\ \\ \dfrac{3}{3F}&+&\dfrac{1}{3F}&=&\dfrac{1}{3\text{ h}} \\ \\ &&\dfrac{4}{3F}&=&\dfrac{1}{3\text{ h}} \\ \\ &&\therefore \dfrac{4}{3}&=&\dfrac{F}{3\text{ h}} \\ \\ &&F&=&4\text{ h} \end{array}[/latex] - [latex]\phantom{a}[/latex]
[latex]\begin{array}[t]{rrcrl} &&\text{full}&=&8\text{ h} \\ \\ &&\text{empty}&=&2\times\text{full or }16\text{ h} \\ \\ \dfrac{1}{F}&-&\dfrac{1}{E}&=&\dfrac{1}{T} \\ \\ \dfrac{1}{8\text{ h}}&-&\dfrac{1}{16\text{ h}}&=&\dfrac{1}{T} \\ \\ \dfrac{2}{16\text{ h}}&-&\dfrac{1}{16\text{ h}}&=&\dfrac{1}{T} \\ \\ &&\therefore \dfrac{1}{16\text{ h}}&=&\dfrac{1}{T} \\ \\ &&T&=&16\text{ h} \end{array}[/latex] - [latex]\phantom{a}[/latex]
[latex]\begin{array}[t]{rrcrl} \dfrac{1}{E}&-&\dfrac{1}{F}&=&\dfrac{1}{T} \\ \\ \dfrac{1}{3}&-&\dfrac{1}{5}&=&\dfrac{1}{T} \\ \\ \dfrac{5}{15}&-&\dfrac{3}{15}&=&\dfrac{1}{T} \\ \\ &&\dfrac{2}{15\text{ min}}&=&\dfrac{1}{T} \\ \\ &&\therefore T&=&\dfrac{15\text{ min}}{2}=7.5\text{ min} \end{array}[/latex] - [latex]\phantom{a}[/latex]
[latex]\begin{array}[t]{rrcrl} \dfrac{1}{\text{full}}&-&\dfrac{1}{\text{empty}}&=&\dfrac{1}{2 T} \\ \\ \dfrac{1}{10\text{ h}}&-&\dfrac{1}{15\text{ h}}&=&\dfrac{1}{2 T} \\ \\ \dfrac{3}{30\text{ h}}&-&\dfrac{2}{30\text{ h}}&=&\dfrac{1}{2 T} \\ \\ &&\dfrac{1}{30\text{ h}}&=&\dfrac{1}{2 T} \\ \\ &&2T&=&30\text{ h} \\ \\ &&T&=&\dfrac{30\text{ h}}{2} \\ \\ &&T&=&15\text{ h} \end{array}[/latex] - [latex]\phantom{a}[/latex]
[latex]\begin{array}[t]{rrcrl} \dfrac{1}{\text{full}}&-&\dfrac{1}{\text{empty}}&=&\dfrac{3}{4T} \\ \\ \dfrac{1}{6\text{ min}}&-&\dfrac{1}{8\text{ min}}&=&\dfrac{3}{4T} \\ \\ \dfrac{4}{24\text{ min}}&-&\dfrac{3}{24\text{ min}}&=&\dfrac{3}{4T} \\ \\ &&\therefore \dfrac{1}{24\text{ min}}&=&\dfrac{3}{4T} \\ \\ &&T&=&\dfrac{3}{4}(24\text{ min}) \\ \\ &&T&=&18\text{ min} \end{array}[/latex] - [latex]\phantom{a}[/latex]
[latex]\begin{array}[t]{rrcrl} \dfrac{1}{H}&+&\dfrac{1}{C}&=&\dfrac{1}{T} \\ \\ \dfrac{1}{H}&+&\dfrac{1}{3.5\text{ min}}&=&\dfrac{1}{2.1\text{ min}} \\ \\ &&\dfrac{1}{H}&=&\dfrac{1}{2.1\text{ min}}-\dfrac{1}{3.5\text{ min}} \\ \\ &&\dfrac{1}{H}&=&\dfrac{50}{105\text{ min}}-\dfrac{30}{105\text{ min}} \\ \\ &&\dfrac{1}{H}&=&\dfrac{20}{105\text{ min}} \\ \\ &&\dfrac{1}{H}&=&\dfrac{4}{21\text{ min}} \\ \\ &&H&=&\dfrac{21}{4}\text{ min} \\ \\ &&H&=&5.25 \text{ min} \end{array}[/latex] - [latex]\phantom{a}[/latex]
[latex]\begin{array}[t]{rrcrl} \dfrac{1}{A}&+&\dfrac{1}{B}&=&\dfrac{1}{T} \\ \\ \dfrac{1}{4.5\text{ h}}&+&\dfrac{1}{B}&=&\dfrac{1}{2\text{ h}} \\ \\ &&\dfrac{1}{B}&=&\dfrac{1}{2\text{ h}}-\dfrac{1}{4.5\text{ h}} \\ \\ &&\dfrac{1}{B}&=&\dfrac{9}{18\text{ h}}-\dfrac{4}{18\text{ h}} \\ \\ &&\dfrac{1}{B}&=&\dfrac{5}{18\text{ h}} \\ \\ &&B&=&\dfrac{18\text{ h}}{5} \\ \\ &&B&=&3.6\text{ h} \end{array}[/latex]
