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Question:

Two women, Ganga and Saraswati, working separately can mow a field in 8 and 12 hrs respectively. If they work in stretches of one hour alternately, Ganga beginning at 9 a.m., when will the mowing be finished?

Doubt:

I tried taking the number of days as a common multiple to get rid of fractions. That way let us consider 24 units of work. So 4 pair of hours will be done or performed. Now I am getting stuck up. I know that the process was right but what do I think of next. after the 4 pair of hours please assist me with the rest of the steps.

Knowledge Seeker
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2 Answers2

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You're on the right track. Ganga and Saraswati mow $\frac{1}{8}^{\text{th}}$ and $\frac{1}{12}^{\text{th}}$ of a field respectively.

They work alternatively with Ganga beginning at $9$ a.m., so every two hours Ganga and Saraswati mow $\left(\frac{1}{8}+ \frac{1}{12}\right)^{\text{th}}$ of the field. So by $5$ p.m. i.e. $8$ hours later, they mow $4 \times \left( \frac{1}{8}+ \frac{1}{12} \right) = \frac{5}{6}^{\text{th}}$ of the field.

Now for the remaining part of the field, which is $1- \frac{5}{6}=\frac{1}{6}^{\text{th}}$, Ganga begins the mowing as it is the $9^{\text{th}}$ hour and hence by $6$ a.m., she completes mowing $\frac{5}{6}+ \frac{1}{8}= \frac{23}{24}^{\text{th}}$ of the field. There's only $1- \frac{23}{24}= \frac{1}{24}^{\text{th}}$ of the field left to mow.

Next up is Saraswati, who can mow $\frac{1}{12}^{\text{th}}$ of the field in an hour, so in half an hour, she can mow $\frac{1}{24}^{\text{th}}$ of the field.

So the whole field is mowed by $6:30$ p.m. $\blacksquare$

Ibrahim
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You could calculate the speed combined of them (G + S) over 2 hours, $$G = \dfrac 18 h^{-1}$$ $$S = \dfrac 1{12} h^{-1}$$ $$G+S = (\dfrac 18 + \dfrac 1{12}) {(2h)}^{-1}=\dfrac 5{24}{(2h)}^{-1}$$ To reach $1$, without exceeding, you take $4$ of that, meaning, in $8$ hours you get $$4 \times \dfrac5{24}=\dfrac{20}{24}$$ of the "mowing" done. Then you had $1$ hour of G, $$\dfrac{20}{24}+\dfrac 18=\dfrac{23}{24}$$ then you need a bit of S to complete the task, how much? $\dfrac1{24}$ or half an hour, thus the total should be $$4\times2+1+\dfrac12=9.50$$or 9 hours and 30 minutes, the work should end at $18:30$ (or 6:30pm)

Déjà vu
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  • Why is 1/24 equivalent to half an hour and 3/24 equivalent to an hour?How is 20/24 equivalent to 4 X 2? – Knowledge Seeker Jun 10 '21 at 12:09
  • $\dfrac1{12}$ is the part of the mowing that S (Saraswati) does in 1 hour, thus she does $\dfrac1{24}$ (half of 1/12, what we need to complete the $\dfrac1{24}$) in half an hour. They mow, G then S (in 2 hours), $\dfrac5{24}$, thus 4 times this ($\dfrac{20}{24}$) is what they mow in 4 x 2 hours. – Déjà vu Jun 10 '21 at 12:17
  • I have one request let us assume 24 as the total units of work now can you list only the numerators as units and explain this to me. – Knowledge Seeker Jun 10 '21 at 12:22
  • Ok, G does $\dfrac18 = \dfrac3{24}$ , so G does $3$ according to your comment, and S does $\dfrac1{12}=\dfrac2{24}$, ie S does $2$ in 1 hour. So when G works for 1 hour and S works for 1 hour, they have done $3+2=5$. If they do that 4 times, $4\times5=20$, in $8$ hours. We then add 1 hour of G to get $23$, and the last $1$ ($\dfrac1{24}$) is done by S in half an hour. – Déjà vu Jun 10 '21 at 12:27
  • 1 unit is 3hr or 3 unit is 1 hour. Please explain this to me. – Knowledge Seeker Jun 10 '21 at 12:45