Answer:
[tex]18.\ Speed\ of\ current=3\ miles/hour\\\\19.\ 6\ hours\\\\20.\ Speed\ of\ Freight\ train=92\ mi/h,\ Speed\ of\ passenger\ train=112\ mi/h\\\\21.\ I\ can\ paint\ the\ dence\ 9\ hours\ and\ my\ sister\ can\ paint\ the\ fence\ in\ 18\ hours.\\\\22.\ Speed\ of\ James=\frac{35}{3}\ mi/h,\ Speed\ of\ Michael=\frac{14}{3}\ mi/h\\\\23.\ Speed\ of\ current=5\ mi/h\\\\24.\ Speed\ of\ dirt\ bike=35\ mi/h,\ Speed\ of\ motorcycle=43\ mi/h[/tex]
Step-by-step explanation:
18.
[tex]Let\ speed\ of\ river's\ current=x\ miles/hour\\\\Speed\ of\ boat=9\ miles/hour[/tex]
Downstream:
[tex]Speed\ in\ the\ downstream=speed\ of\ boat+speed\ of\ current\\\\Speed\ in\ the\ downstream=x+9\\\\distance=24\ miles\\\\time=\frac{distance}{speed}\\\\Time(T_1)=\frac{24}{x+9}[/tex]
Upstream:
[tex]Speed\ in\ the\ upstream=speed\ of\ boat-speed\ of\ current\\\\Speed\ in\ the\ upstream=9-x\\\\distance=24\ miles\\\\time=\frac{distance}{speed}\\\\Time(T_2)=\frac{24}{9-x}[/tex]
[tex]Total\ time=T_1+T_2\\\\Total\ time=6\\\\T_1+T_2=6\\\\\frac{24}{x+9}+\frac{24}{9-x}=6\\\\24(\frac{x+9-x+9}{81-x^2})=6\\\\24(18)=6(81-x^2)\\\\divide\ by\ 6\ both\ sides\\\\4(18)=81-x^2\\\\x^2-81+72=0\\\\x^2=9\\\\x=\pm 3\\\\speed\ can\ not\ be\ negative\\\\x=3\ miles/hour[/tex]
19.
Since they are working together but time taken by each driver to complete the route will be same so minimum time to cover the distance 6 hours.
20.
[tex]Let\ speed\ of\ freight\ train=x\ mi/h\\\\Speed\ of\ passenger\ train=speed\ of\ freight\ train+20\\\\Speed\ of\ passenger\ train=x+20[/tex]
Passenger Train:
[tex]Speed=x+20\\\\distance=392\ mi\\\\Time=\frac{distance}{speed}\\\\Time=\frac{392}{x+20}[/tex]
Freight Train:
[tex]Speed=x\\\\distance=322\ mi\\\\Time=\frac{distance}{speed}\\\\Time=\frac{322}{x}[/tex]
[tex]Time\ taken\ by\ both\ trains\ is\ same.\\\\\frac{392}{20+x}=\frac{322}{x}\\\\Cross\ multiplication\\\\392x=322(20+x)\\\\divide\ by\ 322\\\\\frac{392x}{322}=x+20\\\\\frac{28}{23}x=x+20\\\\\frac{28}{23}x-x=20\\\\\frac{5x}{23}=20\\\\x=\frac{20\times 23}{5}\\\\x=4\times 23\\\\x=92\ mi/h\\\\Speed\ of\ Freight\ train=92\ mi/h\\\\Speed\ of\ passenger\ train=92+20=112\ mi/h[/tex]
21.
[tex]Let\ I\ can\ paint\ the\ fence\ in\ x\ hours\\\\My\ sister\ can\ paint\ the\ fence\ in\ 2x\ hours\ (as\ I'm\ two\ times\ faster\ than\ her)\\\\[/tex]
In 1 hour:
[tex]Per\ hour\ painiting=\frac{1}{hours\ taken\ to\ paint\ whole\ fance}\\\\I\ can\ paint\ per\ hour=\frac{1}{x}\\\\My\ sister\ can\ paint\ per\ hour=\frac{1}{2x}\\\\Together\ we\ can\ paint\ per\ hour=\frac{1}{x}+\frac{1}{2x}\\\\it\ takes\ 6\ hours\ to\ both\ of\ us\ to\ paint\ the\ fence \\\\Per\ hour\ painting\ =\frac{1}{6}\\\\\frac{1}{x}+\frac{1}{2x}=\frac{1}{6}\\\\\frac{3}{2x}=\frac{1}{6}\\\\2x=18\\\\x=9\\\\I\ can\ paint\ the\ fence\ in\ 9\ hours[/tex]
[tex]My\ sister\ can\ paint\ the\ fence\ in\ 9\times2=18\ hours[/tex]
22.
[tex]Let\ speed\ of\ Micheal\ = x\ mi/hr\\\\Speed\ of\ James\ =speed\ of\ James\ +7\\\\Speed of James =x+7\\\\Time\ of\ Michael =\frac{distance}{speed}=\frac{8}{x}\\\\Time\ of\ james\ =\frac{distance}{Speed}=\frac{20}{x+7}\\\\Both\ time\ are\ same\\\\\frac{8}{x}=\frac{20}{x+7}\\\\8x+56=20x\\\\20x-8x=56\\\\12x=56\\\\x=\frac{56}{12}\\\\x=\frac{14}{3}\ mi/h\\\\x+7=\frac{14}{3}+7=\frac{35}{3}\ mi/h\\\\Speed\ of\ Michael\ =\frac{14}{3}\ mi/h\\\\ Speed\ of\ James\ =\frac{35}{3}\ mi/h[/tex]
23.
[tex]Let\ speed\ of\ current\ =x\ mi/hr\\\\ Upstream\ speed\ =speed\ of\ boot-speed\ of\ river\\\\Upstream\ speed\ =15-x\\\\Downstream\ speed=sped\ of\ boat\ +speed\ of\ river=15+x\\\\Upstream\ time=\frac{distance}{speed}=\frac{10}{15-x}\\\\Downstream=\frac{distance}{speed}=\frac{10}{15+x}\\\\Downstream\ =1.5\ hours\\\\\frac{10}{15-x}+\frac{10}{15+x}=1.5\\\\10\left(\frac{15+x+15-x}{225-x^2}\right)=\frac{3}{2}\\\\\frac{10\times30}{225-x^2}=\frac{3}{2}\\\\225-x^2=10\times30\times\frac{2}{3}[/tex]
[tex]225-x^2=200\\\\x^2=225-200\\\\x^2=25\\\\x=\pm5\\\\Speed\ can\ not\ be\ negative\\\\x=5\\\\Hence\ speed\ of\ current\ =5\ mi/hr[/tex]
24.
[tex]Let\ speed\ of\ dirt\ bike=x\ mi/h\\\\Speed\ of\ motorcycle\ =speed\ of\ dirt\ bike\ +8\\\\Speed\ of\ motorcycle\ =x+8\\\\Time\ taken\ on\ dirt\ bike=\frac{distance}{speed}=\frac{52.5}{x}\\\\Time\ taken\ on\ motorcycle=\frac{distance}{speed}=\frac{64.5}{x+8}\\\\Both\ the\ times\ are\ equal\\\\\frac{52.5}{x}=\frac{64.5}{x+8}\\\\Cross\ multiplication\\\\52.5(x+8)=64.5x\\\\\frac{64.5}{52.5}x=x+8\\\\\frac{43}{35}x=x+8\\\\\frac{43}{35}x-x=8\\\\\frac{8x}{35}=8\\\\x=35[/tex]
[tex]Speed\ of\ dirt\ bike=35\ mi/hr\\\\Speed\ of\ motorcycle=35+8=43\ mi/hr[/tex]
