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4men-clear-a-farm-for-8-days-and-are-paid-24-How-long-will-6-men-take-to-clear-the-same-farm-if-they-are-paid-360-




Question Number 162951 by Ar Brandon last updated on 02/Jan/22
4men clear a farm for 8 days and are paid 24$  How long will 6 men take to clear the same farm  if they are paid 360$ ?
$$\mathrm{4men}\:\mathrm{clear}\:\mathrm{a}\:\mathrm{farm}\:\mathrm{for}\:\mathrm{8}\:\mathrm{days}\:\mathrm{and}\:\mathrm{are}\:\mathrm{paid}\:\mathrm{24\$} \\ $$$$\mathrm{How}\:\mathrm{long}\:\mathrm{will}\:\mathrm{6}\:\mathrm{men}\:\mathrm{take}\:\mathrm{to}\:\mathrm{clear}\:\mathrm{the}\:\mathrm{same}\:\mathrm{farm} \\ $$$$\mathrm{if}\:\mathrm{they}\:\mathrm{are}\:\mathrm{paid}\:\mathrm{360\$}\:? \\ $$
Commented by mr W last updated on 03/Jan/22
4 men earn in 8 days 24$. ⇒one man in one day: ((24)/(4×8))  6 men earn in x days 360$. ⇒one man in one day: ((360)/(6x))  since it′s a constant how much one man  earns in one day (we must assume,  otherwise not solvable),  ((24)/(4×8))=((360)/(6x))  ⇒x=80 days
$$\mathrm{4}\:{men}\:{earn}\:{in}\:\mathrm{8}\:{days}\:\mathrm{24\$}.\:\Rightarrow{one}\:{man}\:{in}\:{one}\:{day}:\:\frac{\mathrm{24}}{\mathrm{4}×\mathrm{8}} \\ $$$$\mathrm{6}\:{men}\:{earn}\:{in}\:{x}\:{days}\:\mathrm{360\$}.\:\Rightarrow{one}\:{man}\:{in}\:{one}\:{day}:\:\frac{\mathrm{360}}{\mathrm{6}{x}} \\ $$$${since}\:{it}'{s}\:{a}\:{constant}\:{how}\:{much}\:{one}\:{man} \\ $$$${earns}\:{in}\:{one}\:{day}\:\left({we}\:{must}\:{assume},\right. \\ $$$$\left.{otherwise}\:{not}\:{solvable}\right), \\ $$$$\frac{\mathrm{24}}{\mathrm{4}×\mathrm{8}}=\frac{\mathrm{360}}{\mathrm{6}{x}} \\ $$$$\Rightarrow{x}=\mathrm{80}\:{days} \\ $$
Commented by Ar Brandon last updated on 03/Jan/22
Got it now. Thanks Sir.
Answered by Rasheed.Sindhi last updated on 02/Jan/22
 determinant (((men),(days),$),(4,8,(24)),(6,x,(360)))  x:8:: { ((4:6)),((360:24)) :}  x×6×24=8×4×360  x=((8×4×360)/(6×24))=80 days
$$\begin{array}{|c|c|c|}{\mathrm{men}}&\hline{\mathrm{days}}&\hline{\$}\\{\mathrm{4}}&\hline{\mathrm{8}}&\hline{\mathrm{24}}\\{\mathrm{6}}&\hline{{x}}&\hline{\mathrm{360}}\\\hline\end{array} \\ $$$${x}:\mathrm{8}::\begin{cases}{\mathrm{4}:\mathrm{6}}\\{\mathrm{360}:\mathrm{24}}\end{cases} \\ $$$${x}×\mathrm{6}×\mathrm{24}=\mathrm{8}×\mathrm{4}×\mathrm{360} \\ $$$${x}=\frac{\mathrm{8}×\mathrm{4}×\mathrm{360}}{\mathrm{6}×\mathrm{24}}=\mathrm{80}\:\mathrm{days} \\ $$
Commented by Ar Brandon last updated on 02/Jan/22
Really?! OK Sir. Thanks
Commented by Ar Brandon last updated on 02/Jan/22
Is this some sort of cross-multiplication?
Commented by Rasheed.Sindhi last updated on 03/Jan/22
The problem is somewhat strange.  It is not dependant on amount of  work!   determinant (((men),(days),$),((4_6    ↓),(8_x   ↑),(24_(360)   ↑)),((men increase_(→days decrease_((inverse proportion)) ) ),,(dollars increase_(→days increase_((direct proportion)) ) )),((4:6),(x:8),(360:24)),((x:8::4:6_(A) ),,(x:8::360:24_(B) )))  A∪B:  x:8:: { ((4:6)),((360:24)) :}  x×6×24=8×4×360  x=((8×4×360)/(6×24))=80 days
$$\mathcal{T}{he}\:{problem}\:{is}\:{somewhat}\:{strange}. \\ $$$${It}\:{is}\:{not}\:{dependant}\:{on}\:{amount}\:{of} \\ $$$${work}! \\ $$$$\begin{array}{|c|c|c|c|c|}{\mathrm{men}}&\hline{\mathrm{days}}&\hline{\$}\\{\underset{\mathrm{6}} {\mathrm{4}}\:\:\:\downarrow}&\hline{\underset{{x}} {\mathrm{8}}\:\:\uparrow}&\hline{\underset{\mathrm{360}} {\mathrm{24}}\:\:\uparrow}\\{\underset{\underset{\left({inverse}\:{proportion}\right)} {\rightarrow{days}\:\boldsymbol{{decrease}}}} {{men}\:\boldsymbol{{increase}}}}&\hline{}&\hline{\underset{\underset{\left({direct}\:{proportion}\right)} {\rightarrow{days}\:\boldsymbol{{increase}}}} {{dollars}\:\boldsymbol{{increase}}}}\\{\mathrm{4}:\mathrm{6}}&\hline{{x}:\mathrm{8}}&\hline{\mathrm{360}:\mathrm{24}}\\{\underset{\mathrm{A}} {\underbrace{{x}:\mathrm{8}::\mathrm{4}:\mathrm{6}}}}&\hline{}&\hline{\underset{{B}} {\underbrace{{x}:\mathrm{8}::\mathrm{360}:\mathrm{24}}}}\\\hline\end{array} \\ $$$$\mathrm{A}\cup\mathrm{B}: \\ $$$${x}:\mathrm{8}::\begin{cases}{\mathrm{4}:\mathrm{6}}\\{\mathrm{360}:\mathrm{24}}\end{cases} \\ $$$${x}×\mathrm{6}×\mathrm{24}=\mathrm{8}×\mathrm{4}×\mathrm{360} \\ $$$${x}=\frac{\mathrm{8}×\mathrm{4}×\mathrm{360}}{\mathrm{6}×\mathrm{24}}=\mathrm{80}\:\mathrm{days} \\ $$
Commented by Rasheed.Sindhi last updated on 03/Jan/22
This is by ′proportion method′ while  sir mr W ′s method (comment to   the question) is ′unitary method′  This is a compound proportion.
$$\mathcal{T}{his}\:{is}\:{by}\:'{proportion}\:{method}'\:{while} \\ $$$${sir}\:{mr}\:{W}\:'{s}\:{method}\:\left({comment}\:{to}\:\right. \\ $$$$\left.{the}\:{question}\right)\:{is}\:'{unitary}\:{method}' \\ $$$$\mathcal{T}{his}\:{is}\:{a}\:{compound}\:{proportion}. \\ $$
Commented by Rasheed.Sindhi last updated on 03/Jan/22
Sir,pl read my above comment again  as I′ve edited it before receiving your  comment.
$$\boldsymbol{{Sir}},{pl}\:{read}\:{my}\:{above}\:{comment}\:{again} \\ $$$${as}\:{I}'{ve}\:{edited}\:{it}\:{before}\:{receiving}\:{your} \\ $$$${comment}. \\ $$
Commented by Rasheed.Sindhi last updated on 03/Jan/22
I have also sampathy for them:)  But sir, the question says:  4 men.....∣6men......clear same farm.  that is do same quantity of work.
$$\left.{I}\:{have}\:{also}\:{sampathy}\:{for}\:{them}:\right) \\ $$$${But}\:\boldsymbol{{sir}},\:{the}\:{question}\:{says}: \\ $$$$\mathrm{4}\:{men}…..\mid\mathrm{6}{men}……{clear}\:\boldsymbol{{same}}\:\boldsymbol{{farm}}. \\ $$$${that}\:{is}\:{do}\:\boldsymbol{{same}}\:\boldsymbol{{quantity}}\:\boldsymbol{{of}}\:\boldsymbol{{work}}. \\ $$
Commented by mr W last updated on 03/Jan/22
“clear the same farm” doesn′t define  the quantity of the work, only the  type of the work, i think.  it could be  that the 4 men cleared the first 120 m^2   of the farm in 8 days for totally 24$,  and the 6 men cleared the remaining  1800 m^2  of the same farm in 80 days  for totally 360$. the same farm pays  the same money for each man and  each day. different farms may pay  less or more. therefore it makes  sinse to state that they cleared the  same farm. this is my opinion.
$$“{clear}\:{the}\:{same}\:{farm}''\:{doesn}'{t}\:{define} \\ $$$${the}\:{quantity}\:{of}\:{the}\:{work},\:{only}\:{the} \\ $$$${type}\:{of}\:{the}\:{work},\:{i}\:{think}.\:\:{it}\:{could}\:{be} \\ $$$${that}\:{the}\:\mathrm{4}\:{men}\:{cleared}\:{the}\:{first}\:\mathrm{120}\:{m}^{\mathrm{2}} \\ $$$${of}\:{the}\:{farm}\:{in}\:\mathrm{8}\:{days}\:{for}\:{totally}\:\mathrm{24\$}, \\ $$$${and}\:{the}\:\mathrm{6}\:{men}\:{cleared}\:{the}\:{remaining} \\ $$$$\mathrm{1800}\:{m}^{\mathrm{2}} \:{of}\:{the}\:{same}\:{farm}\:{in}\:\mathrm{80}\:{days} \\ $$$${for}\:{totally}\:\mathrm{360\$}.\:{the}\:{same}\:{farm}\:{pays} \\ $$$${the}\:{same}\:{money}\:{for}\:{each}\:{man}\:{and} \\ $$$${each}\:{day}.\:{different}\:{farms}\:{may}\:{pay} \\ $$$${less}\:{or}\:{more}.\:{therefore}\:{it}\:{makes} \\ $$$${sinse}\:{to}\:{state}\:{that}\:{they}\:{cleared}\:{the} \\ $$$${same}\:{farm}.\:{this}\:{is}\:{my}\:{opinion}. \\ $$
Commented by mr W last updated on 03/Jan/22
but no matter how it is, 4 men earn  only 24$ in 8 days, that′s definitely  too less, i think :)
$${but}\:{no}\:{matter}\:{how}\:{it}\:{is},\:\mathrm{4}\:{men}\:{earn} \\ $$$${only}\:\mathrm{24\$}\:{in}\:\mathrm{8}\:{days},\:{that}'{s}\:{definitely} \\ $$$$\left.{too}\:{less},\:{i}\:{think}\::\right) \\ $$
Commented by Rasheed.Sindhi last updated on 03/Jan/22
Okay sir, you′re keen obserer,no doubt!  “4 men clear a farm...”means they  clear whole farm.“...6 men take  to clear the same farm”,doesn′t  again it   mean the whole farm?
$$\mathrm{Okay}\:\mathrm{sir},\:{you}'{re}\:{keen}\:{obserer},{no}\:{doubt}! \\ $$$$“\mathrm{4}\:\mathrm{men}\:\mathrm{clear}\:\mathrm{a}\:\mathrm{farm}…''{means}\:{they} \\ $$$${clear}\:\boldsymbol{{whole}}\:{farm}.“…\mathrm{6}\:\mathrm{men}\:\mathrm{take} \\ $$$$\mathrm{to}\:\mathrm{clear}\:\mathrm{the}\:\mathrm{same}\:\mathrm{farm}'',{doesn}'{t}\:\:{again}\:{it}\: \\ $$$${mean}\:{the}\:{whole}\:{farm}? \\ $$
Commented by mr W last updated on 03/Jan/22
actually i thought the same as you.  but then i got a problem with the logic.  so i just changed my opinion in order  to become conform with the solution.
$${actually}\:{i}\:{thought}\:{the}\:{same}\:{as}\:{you}. \\ $$$${but}\:{then}\:{i}\:{got}\:{a}\:{problem}\:{with}\:{the}\:{logic}. \\ $$$${so}\:{i}\:{just}\:{changed}\:{my}\:{opinion}\:{in}\:{order} \\ $$$${to}\:{become}\:{conform}\:{with}\:{the}\:{solution}. \\ $$
Commented by mr W last updated on 03/Jan/22
this is the logic problem i mentioned.  “clear a farm” and “clear the same  farm” may not mean the same work  done, because one can not pay for the  same work done in one time only 24$   and in an other time 360$. certainly   in reality it′s possible, but in   mathematics it′s not possible.  because there is no mathematics if  there is no ligic. therefore “clear a   farm” and “clear the same farm”  can logically only mean the type of  the work, i.e. “clear in a farm” as   well as “clear in the same farm”.  i think the guy who originally wrote   this question is no lawyer, even no  native speaker of english,  like you  and me. therefore we should look more  at the logic behind it instead of the  wording.
$${this}\:{is}\:{the}\:{logic}\:{problem}\:{i}\:{mentioned}. \\ $$$$“{clear}\:{a}\:{farm}''\:{and}\:“{clear}\:{the}\:{same} \\ $$$${farm}''\:{may}\:{not}\:{mean}\:{the}\:{same}\:{work} \\ $$$${done},\:{because}\:{one}\:{can}\:{not}\:{pay}\:{for}\:{the} \\ $$$${same}\:{work}\:{done}\:{in}\:{one}\:{time}\:{only}\:\mathrm{24\$}\: \\ $$$${and}\:{in}\:{an}\:{other}\:{time}\:\mathrm{360\$}.\:{certainly}\: \\ $$$${in}\:{reality}\:{it}'{s}\:{possible},\:{but}\:{in}\: \\ $$$${mathematics}\:{it}'{s}\:{not}\:{possible}. \\ $$$${because}\:{there}\:{is}\:{no}\:{mathematics}\:{if} \\ $$$${there}\:{is}\:{no}\:{ligic}.\:{therefore}\:“{clear}\:{a}\: \\ $$$${farm}''\:{and}\:“{clear}\:{the}\:{same}\:{farm}'' \\ $$$${can}\:{logically}\:{only}\:{mean}\:{the}\:{type}\:{of} \\ $$$${the}\:{work},\:{i}.{e}.\:“{clear}\:{in}\:{a}\:{farm}''\:{as}\: \\ $$$${well}\:{as}\:“{clear}\:{in}\:{the}\:{same}\:{farm}''. \\ $$$${i}\:{think}\:{the}\:{guy}\:{who}\:{originally}\:{wrote}\: \\ $$$${this}\:{question}\:{is}\:{no}\:{lawyer},\:{even}\:{no} \\ $$$${native}\:{speaker}\:{of}\:{english},\:\:{like}\:{you} \\ $$$${and}\:{me}.\:{therefore}\:{we}\:{should}\:{look}\:{more} \\ $$$${at}\:{the}\:{logic}\:{behind}\:{it}\:{instead}\:{of}\:{the} \\ $$$${wording}. \\ $$
Commented by Rasheed.Sindhi last updated on 03/Jan/22
T_(H_(AN) K) S S_I_(R!)    👍👍👍👍👍
$$\mathbb{T}_{\mathbb{H}_{\mathbb{AN}} \mathbb{K}} \mathbb{S}\:\mathbb{S}_{\mathbb{I}_{\mathbb{R}!} } \\ $$👍👍👍👍👍

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