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A man builds $\dfrac{1}{8}$ th part of wall each day but $20\%$ of the wall built on each day falls down. In how many days will the construction of the wall be finished?

My doubt particularly is why we are choosing the unit of work here. $\dfrac{1}{8}$th is interpreted as whole number. Why is that so? Also here the problem is more difficult than it seems. A detailed theory about this is most welcome.

Sathvik
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Nayanthara Thomas
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    Why do you say that 1/8 is treated as a whole number here? – David Jun 04 '21 at 14:40
  • The instructor took the work as 80 units. – Nayanthara Thomas Jun 04 '21 at 14:55
  • It may help to walk through this building process one day at a time. On day 1, the builder completes $\frac18$ of the wall. At the end of day 1, $20%$ or $\frac2{10}$ of that wall breaks down, which means $\frac18\times\frac2{10}=\frac1{40}=2.5%$ of the wall is undone, leaving the builder with $\frac18-\frac1{40}=\frac5{40}-\frac1{40}=\frac4{40}=\frac1{10}=10%$ of the wall completed. What happens on day 2? – user170231 Jun 04 '21 at 14:55
  • Up to 9 th day it is predictable. What about the 10 th day? – Nayanthara Thomas Jun 04 '21 at 15:21
  • On any given day, $\frac18$ of the wall is built, and $20%$ of what was completed on that day is destroyed. So the builder is completing the same amount each day. – user170231 Jun 04 '21 at 15:39
  • And what about the last day? Is there any exception. If yes please explain that theoretically. – Nayanthara Thomas Jun 04 '21 at 15:52
  • Some problems have a quantity that can be anything without changing the answer. In this case, the wall could be any length and the answer to the question would be the same. So it's a valid way to work the problem by choosing a convenient length. Since $1/8$ and $20%$ of $80$ are easy to compute, the instructor chose $80$ feet (or meters or furlongs.) You could use $40$ or $800$. Or $352.17$ if you're a masochist. – B. Goddard Jun 04 '21 at 15:53
  • Can you explain the rest of the part? – Nayanthara Thomas Jun 04 '21 at 16:06
  • If the part of the wall that falls down is $ 20% $ of the portion that was built on a given day, then the fraction of the wall built each day is $ \frac{1}{8} (1 - 0.2) = \frac{1}{10} $. Thus, it will take 10 days to complete the wall. But if the part that falls down is $ 20% $ of the whole wall that was already built, the construction will never be finished. – ИΛJΛ Jun 04 '21 at 17:37

1 Answers1

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Well, if $\frac{1}{8}$ of the wall is built, but $20$%, or $\frac{1}{5}$, of that falls down, then each day, the man builds $\frac{4}{5} \cdot\frac{1}{8} = \frac{4}{40} = \frac{1}{10}$ of the wall in a day. So the wall will be completed in $10$ days.

I hope this helped!

Mathemagician314
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