Equilateral and Equiangular Polygons

A polygon is a 2 dimensional geometric figure bound with straight sides. A polygon is called Equilateral if all of its sides are congruent. Common examples of equilateral polygons are a rhombus and regular polygons such as equilateral triangles and squares.  Now, a polygon is equiangular if all of its internal angles are congruent.  Some important facts to consider The only equiangular triangle is the equilateral triangle If P is an equilateral polygon that has more than three sides, it does not have to be equiangular. A rhombus with no right angle is an example of an equilateral but non-equiangular polygon.  Rectangles, including squares, are the only equiangular quadrilaterals Equiangular polygon theorem. Each angle of an equiangular n-gon is  $$\Bigg(\frac{n-2}{n}\Bigg)180^{\circ} = 180^{\circ} -   \frac{360^{\circ}}{n} $$ Viviani's theorem   Vincenzo Viviani (1622 – 1703) was a famous Italian mathematician. With his exceptional intelligence in math...
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Time and Work Problems

Key Facts

  • If a person can finish a job in n days, then the work done by the person in 1 day is $\frac{1}{n}$th of the total job
  • If a person completes $\frac{1}{n}$th of the total job in 1 day, then the time taken by the person to finish the complete job is n days. 
  • Another version of work is pool/tank problems where there is an inlet of water and an outlet as well.  
    • If an inlet fills a tank in n hours, then it fills $\frac{1}{n}$th part of the tank in 1 hour. i.e. work done by it in 1 hour is $\frac{1}{n}$
    • If an outlet empties a full tank in m hours, then it will empty $\frac{1}{m}$th part of the tank in 1 hour. i.e. work done by it is $-\frac{1}{m}$

The concept is not hard to understand, however the application can be extremely tricky. So it would only help to solve as many problems as you can on this subject. Let's look at some examples.

Ex 1. A copy machine can copy a paper in $36$ minutes. If a second copy machine were to be used at the same time, copying the paper would take a total of $24$ minutes. How long, in minutes, would it take the second copy machine to complete the copying if it were working alone?

Sol. 
Let the time it takes the second copy machine to complete the copying be $x$ minutes. We know that the sum of the rates of each copy machine equals the rate of both of them working together. In other words,
$$\frac{1}{36}+\frac{1}{x} = \frac{1}{24} \Rightarrow \frac{1}{x} = \frac{1}{24} - \frac{1}{36} =\frac{3-2}{72}= \frac{1}{72}$$
$$\Rightarrow \frac{1}{x} = \frac{1}{72} \Rightarrow x=\boxed{72 \text{ minutes}}$$

Ex 2.  $A$ and $B$ can finish a job in $72$ days; $B$ and $C$ can do it in $120$ days; $A$ and $C$ take $90$ days. How long will it take $A$ to do the job by himself.

Sol.  
$(A+B)$'s $1$ day of work = $\frac{1}{72}$ of the job
$(B+C)$'s $1$ day of work = $\frac{1}{120}$ of the job
$(A+C)$'s $1$ day of work = $\frac{1}{90}$ of the job.

Then, $[(A+B) + (B+C) + (A+C)]$'s one day of work is $\frac{1}{72}+ \frac{1}{120} + \frac{1}{90} = \frac{5+3+4}{360} = \frac{12}{360}=\frac{1}{30}$ of the job.

$\Rightarrow 2(A+B+C)$'s $1$ day of work $= \frac{1}{30} $
$\Rightarrow (A+B+C)$'s $1$ day of work is $\frac{1}{60}$.

In other words, all three of them do $\frac{1}{60}$ of the job in one day, and if only $B$ and $C$ work, they do $\frac{1}{120}$ of the work. Thus, to find how much work $A$ does in one day, we subtract these two rates and get
$$A = (A+B+C)-(B+C) = \frac{1}{60} - \frac{1}{120} = \frac{1}{120} $$.
This means that $A$ takes $\boxed{120}$ days to complete the job if he was working alone.


Ex 3. Alvin can do a certain job in $12$ days. Bob is $60%$ more efficient that Alvin.. What is the number of days it takes Bob to do the same job?

Sol. 
$A$'s one day of work $=\frac{1}{12}$.
Thus, $B$'s one day of work $ = 1.6 \cdot \frac{1}{12} = \frac{2}{15}$.
So, the number of days $B$ takes to complete the job is $\frac{15}{2}$ days or $\boxed{7.5 \text{ days}}$


Ex 4. $10$ men working $6$ hours a day can paint a fence in $18$ days. How many hours a day must $15$ men work to paint the sane fence in $12$ days?

Sol. 
To complete the work in $18$ days, $10$ men work $6$ hours a day.
To complete the work in $18$ days, one man will work $(6 \cdot 10)$ hours a day.
To complete the work in one day, $1$ man will work $(6 \cdot 10 \cdot 18)$ hours a day.
To complete the work in one day, $15$ men will work $\frac{6 \cdot 10 \cdot 18}{15}$ hours a day.
Therefore, to complete the work in $12$ days, $15$ men will work $\big(\frac{6 \cdot 10 \cdot 18}{12 \cdot 15}\big) = \boxed{6 \text{ hours a day}}$


Ex 5. Andrew can build a wall in $25$ days and Bill can build it in $20$ days. They work together for $5$ days, and then Andrew leaves. How long does it take for Bill to finish the remaining work (in days)?

Sol. 
Time taken by Andrew to finish the job is $=25$ days.
$\Rightarrow$ Andrew's one day of work $=\frac{1}{25}$.
Time taken by Bill to build the wall $=20$ days.
$\Rightarrow$ Bill's one day of work $=\frac{1}{20}$.

Together, the build $\frac{1}{25} + \frac{1}{20} = \frac{9}{100}$ of the wall in one day.

We know that they work together for $5$ days, so they do $5 \cdot \frac{9}{100} = \frac{9}{20}$.
Remaining work after Andrew leaves $= 1 - \frac{9}{20} = \frac{11}{20}$

We know Bil does $\frac{1}{20}$ of the wall per day, so he will need
$$\frac{\frac{11}{20}}{\frac{1}{20}} = \boxed{11 \text{ days}}$$


Ex 6. An inlet pipe can fill a pool in $12$ hours and the outlet pipe can empty the pool in $15$ hours. If both pipes are left open, how long will it take to fill the pool?

Sol.
The rate of the inlet pipe is $\frac{1}{12}$.
The rate of the outlet pipe is $-\frac{1}{15}$. (Negative because it is emptying water from the pool).
The net rate at which the pool is being filled $ = \frac{1}{12} - \frac{1}{15} = \frac{1}{60}$.
Thus, it takes $\boxed{60 \text{ hours}}$ for the pool to be filled with both inlet and outlet pipes open.

These examples should cover the majority of the different types of work and time problems you will encounter. Please reach out to me if you need additional explanation.



Comments

  1. AnonymousOctober 3, 2019 at 6:27 PM

    Hi
    can you explain the problem of the copying machine.

    It says 36 minutes and 24 minutes but why does it say 1/36 and 1/24

    Thanks,
    Nivaen

    ReplyDelete
    Replies
    1. akaliaOctober 3, 2019 at 7:39 PM

      Hi Nivaen,
      When you have to solve work problems, think about Unit Rate.

      If a person can finish a job in $4$ days, then he does $\frac{1}{4}$th of the job in $1$ day. This $\frac{1}{4}$ becomes your unit rate.

      Going back to the copy machine problem, if the machine takes $36$ minutes to copy one paper, it will copy $\frac{1}{36}$th of the paper in $1$ minute. The two machines together would complete the job in $24$ minutes, which means they would do $\frac{1}{24}$th of the copy job per minute.

      We are asked to find the time taken by the second machine for the copy job if it were working alone. If the time taken by the second machine was $x$ minutes, then it would accomplish $\frac{1}{x}$th of the job per minute.

      For the final step, the rate of the two machines together has to be equal to the sum of the rates of the two machines working independently which is how we get to $$\frac{1}{36} + \frac{1}{x} = \frac{1}{24}$$.

      Hopefully this helps.

      Delete
    2. AnonymousOctober 5, 2019 at 5:57 AM

      This is great! It really helps me understand time and work problems.

      Delete

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