let us consider following problem:

The amount of time taken to paint a wall is inversely proportional to the number of painters working on the job. If it takes 3 painters 5 days to complete such a job, how many days longer will it take if there are only 2 painters working? so let us recall terminology of inversely proportional,if one number is inversely proportional to another number it means that

$k=c*1/x$ where $k$ is number which is inversely proportional of number $c$ and $1/x$ is inversely proportional coefficient,so in our case

$5=3*1/x$ so $x=3/5$ ,it means that one painter paints wall in $3/5=0.6$ days right? and $2$ painter will paint $2/0.6=3.3$ days right?so longer day would be $5-3.3=1.7$,but here


answer is $2.5$,pleas help me


1 Answer 1


Let $d(n)$ be the number of days required when there are $n$ painters. We’re told that $d(n)$ is inversely proportional to $n$, so there is a constant $c$ such that $$d(n)=\frac{c}n\;.$$ We’re also told that $d(3)=5$, so $$5=d(3)=\frac{c}3\;,$$ and therefore $c=3\cdot5=15$. Therefore $$d(2)=\frac{15}2=7.5\;.$$ Thus, two painters will require $7.5-5=2.5$ more days than $3$ painters.

Another way to look at inverse proportionality is this: if $d$ and $n$ are inversely proportional, then multiplying $n$ by some factor $a$ causes $d$ to be multiplied by $\frac1a$. When we reduce the number of painters from $3$ to $2$, we’re multiplying $n$ by $\frac23$; $\frac1{2/3}=\frac32$, so the effect is to multiply $d$ by $\frac32$. And $\frac32\cdot5=\frac{15}2=7.5$, so once again we see that the increase in time required is $2.5$ days.

  • $\begingroup$ i was thinking that because it is inversely proportional,then many painter will take more time $\endgroup$ Jul 30, 2013 at 7:05
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    $\begingroup$ @dato: No, the inversely part of inversely proportional ensures that the two quantities move in opposite directions: if the number of painters goes up, the number of days goes down, and vice versa. $\endgroup$ Jul 30, 2013 at 7:09
  • $\begingroup$ @dato: You’re very welcome. (It must be difficult reading mathematics in a language as different as English is from Kartuli, which I’m guessing is your first language.) $\endgroup$ Jul 30, 2013 at 7:14
  • $\begingroup$ yes yes it is,i am preparing for GRE,it is in 15 august,in this center americancouncils.ge/ge $\endgroup$ Jul 30, 2013 at 7:16
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    $\begingroup$ @dato: I think that you’re doing very well, and I wish you the best of luck on the GRE. $\endgroup$ Jul 30, 2013 at 7:19

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