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Question Number 157753 by Engr_Jidda last updated on 27/Oct/21
A farmer has numbers of animals  in his farm made up of hens and goats.  One morning he counted 20 heads and 66 legs  of the animals. How many hens and  goat were counted by the farmer.
$${A}\:{farmer}\:{has}\:{numbers}\:{of}\:{animals} \\ $$$${in}\:{his}\:{farm}\:{made}\:{up}\:{of}\:{hens}\:{and}\:{goats}. \\ $$$${One}\:{morning}\:{he}\:{counted}\:\mathrm{20}\:{heads}\:{and}\:\mathrm{66}\:{legs} \\ $$$${of}\:{the}\:{animals}.\:{How}\:{many}\:{hens}\:{and} \\ $$$${goat}\:{were}\:{counted}\:{by}\:{the}\:{farmer}. \\ $$
Answered by Kunal12588 last updated on 27/Oct/21
let number of hens be x  and number of goats be y  Total number of heads = 20  ⇒x+y=20 ....[1]  Total number of legs = 66  ⇒2x+4y=66  ⇒x+2y=33 ...[2]  subtracting [1] from [2]  y=13  putting value if y in equation [1]  x=7  Therefore there were 7 hens and 13 goats
$$\mathrm{let}\:\mathrm{number}\:\mathrm{of}\:\mathrm{hens}\:\mathrm{be}\:{x} \\ $$$$\mathrm{and}\:\mathrm{number}\:\mathrm{of}\:\mathrm{goats}\:\mathrm{be}\:{y} \\ $$$$\mathrm{Total}\:\mathrm{number}\:\mathrm{of}\:\mathrm{heads}\:=\:\mathrm{20} \\ $$$$\Rightarrow{x}+{y}=\mathrm{20}\:….\left[\mathrm{1}\right] \\ $$$$\mathrm{Total}\:\mathrm{number}\:\mathrm{of}\:\mathrm{legs}\:=\:\mathrm{66} \\ $$$$\Rightarrow\mathrm{2}{x}+\mathrm{4}{y}=\mathrm{66} \\ $$$$\Rightarrow{x}+\mathrm{2}{y}=\mathrm{33}\:…\left[\mathrm{2}\right] \\ $$$$\mathrm{subtracting}\:\left[\mathrm{1}\right]\:\mathrm{from}\:\left[\mathrm{2}\right] \\ $$$${y}=\mathrm{13} \\ $$$$\mathrm{putting}\:\mathrm{value}\:\mathrm{if}\:{y}\:\mathrm{in}\:\mathrm{equation}\:\left[\mathrm{1}\right] \\ $$$${x}=\mathrm{7} \\ $$$$\mathrm{Therefore}\:\mathrm{there}\:\mathrm{were}\:\mathrm{7}\:\mathrm{hens}\:\mathrm{and}\:\mathrm{13}\:\mathrm{goats} \\ $$
Answered by som(math1967) last updated on 27/Oct/21
no ofhens=x no of goats=y   x+y=20....i)  2x+4y=66⇒x+2y=33.....ii)  ii) −i) y=13  x+y=20  x=20−13=7  7 hens ,13goats
$${no}\:{ofhens}={x}\:{no}\:{of}\:{goats}={y} \\ $$$$\left.\:\boldsymbol{{x}}+\boldsymbol{{y}}=\mathrm{20}….\boldsymbol{{i}}\right) \\ $$$$\left.\mathrm{2}\boldsymbol{{x}}+\mathrm{4}\boldsymbol{{y}}=\mathrm{66}\Rightarrow\boldsymbol{{x}}+\mathrm{2}\boldsymbol{{y}}=\mathrm{33}…..\boldsymbol{{ii}}\right) \\ $$$$\left.\boldsymbol{{i}}\left.\boldsymbol{{i}}\right)\:−\boldsymbol{{i}}\right)\:\boldsymbol{{y}}=\mathrm{13} \\ $$$$\boldsymbol{{x}}+\boldsymbol{{y}}=\mathrm{20}\:\:\boldsymbol{{x}}=\mathrm{20}−\mathrm{13}=\mathrm{7} \\ $$$$\mathrm{7}\:\boldsymbol{{hens}}\:,\mathrm{13}\boldsymbol{{goats}} \\ $$
Commented by Engr_Jidda last updated on 27/Oct/21
thanks
$${thanks} \\ $$
Answered by mr W last updated on 27/Oct/21
we have 20 animals and 66 legs.  now each animal should hide two legs,  then 66−20×2=26 legs remain. these  legs are the remaining legs from   the goats, because all legs from the  hens are hidden. since each goat has   2 legs remaining, so the number of  goats is 26/2=13. then the number   of hens is 20−13=7.
$${we}\:{have}\:\mathrm{20}\:{animals}\:{and}\:\mathrm{66}\:{legs}. \\ $$$${now}\:{each}\:{animal}\:{should}\:{hide}\:{two}\:{legs}, \\ $$$${then}\:\mathrm{66}−\mathrm{20}×\mathrm{2}=\mathrm{26}\:{legs}\:{remain}.\:{these} \\ $$$${legs}\:{are}\:{the}\:{remaining}\:{legs}\:{from}\: \\ $$$${the}\:{goats},\:{because}\:{all}\:{legs}\:{from}\:{the} \\ $$$${hens}\:{are}\:{hidden}.\:{since}\:{each}\:{goat}\:{has}\: \\ $$$$\mathrm{2}\:{legs}\:{remaining},\:{so}\:{the}\:{number}\:{of} \\ $$$${goats}\:{is}\:\mathrm{26}/\mathrm{2}=\mathrm{13}.\:{then}\:{the}\:{number}\: \\ $$$${of}\:{hens}\:{is}\:\mathrm{20}−\mathrm{13}=\mathrm{7}. \\ $$
Commented by Engr_Jidda last updated on 27/Oct/21
thanks
$${thanks} \\ $$

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