Lesson Using quadratic equations to solve word problems on joint work

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Using quadratic equations to solve word problems on joint work

Some joint work problems require to use quadratic equations for their solution.
In this lesson you will learn how to solve such problems using quadratic equations.
The problems like "Painting a wall", "Painting a house", "Covering a roof", "Filling a reservoir", "Emptying a reservoir" etc. may fall to this category.

Problem 1. Working together and separately to complete a job

Andrew and Bill, working together, can cover the roof of a house in 6 days.
Andrew, working alone, can complete this job in 5 days faster than Bill.
How long will it take Bill to make this job?

Solution
Let

be the number of days for Bill to cover the roof, working himself.
If Andrew works alone, he can complete this job in x-5

days.
Thus, in one single day Andrew covers 1%2F%28x-5%29

part of the roof area, while Bill covers 1%2Fx

part of the roof area.
Working together, Andrew and Bill make 1%2F%28x-5%29%2B1%2Fx

of the whole work in each single day.
Since they can cover the entire roof in 6 days working together, the equation for the unknown value

is as follows:

6%2F%28x-5%29+%2B+6%2Fx+=+1

.

To simplify this equation, multiply both sides by %28x-5%29%2Ax

, then transfer all terms from the right side to the left
with the opposite signs, then collect like terms and adjust the signs. In this way you get
6x+%2B+6%28x-5%29+=+x%28x-5%29

.
You get the quadratic equation. Apply the quadratic formula (see the lesson Introduction into Quadratic Equations) to solve this equation. You get

.
The equation has two roots: x%5B1%5D=%2817%2B13%29%2F2=15

and

.
The second root x%5B2%5D=2

does not fit the given conditions (if Bill covers the roof in two days, then Andrew has 2-5=-3 days, what makes no sense).

So, the potentially correct solution is x%5B1%5D=15

: Bill covers the roof in 15 days.
Let us check it. If Bill gets the job done in 15 days, then Andrew makes it in 10 days, working separately.
Since 6%2F10%2B6%2F15=1

, this solution is correct.

Answer. Bill covers the roof in 15 days.

Problem 2. Filling the liquid reservoir

Two tubes, working together, can fill the reservoir with the liquid in 12 hours.
The larger tube, if works separately, can fill the reservoir in 18 hours less than the smaller tube.
How long will it take to fill the reservoir using the smaller tube only?

Solution
Let

be the number of hours to fill the reservoir using the smaller tube only.
Then x-18

is the time in hours to fill the reservoir using the larger tube only.
Thus, in one single hour the smaller tube will fill 1%2Fx

part of the reservoir volume, while the larger tube will fill 1%2F%28x-18%29

part of the reservoir volume in each single hour.
Working together, these two tubes will fill 1%2Fx+%2B+1%2F%28x-18%29

part of the reservoir volume in each single hour.
Since two tubes working together fill the tank in 12 hours, this gives the equation

12%2Fx+%2B+12%2F%28x-18%29+=+1

.

To simplify this equation, multiply both sides by

-x%5E2+%2B+12x+%2B+12x+%2B+18x+-+218+=+0

.
You get the quadratic equation. Apply the quadratic formula (see the lesson Introduction into Quadratic Equations) to solve this equation. You get

.
The equation has two roots: x%5B1%5D=%2842%2B30%29%2F2=36

and

.
The second root x%5B2%5D=6

does not fit the given conditions (if smaller tube fills the reservoir in 6 hours, then larger one should make it in 6-18=-12 hours, which makes no sense).

So, the potentially correct solution is x%5B1%5D=36

: smaller tube fills the reservoir in 36 hours.
Let us check it. If smaller tube fills the reservoir in 36 hours, then the larger one makes it in 36-18=18 hours, working separately.
Since 12%2F36%2B12%2F18=1

, this solution is correct.

Answer. Smaller tube fills the reservoir in 36 hours.

Problem 3

Alvin and Lito washed the family car in 18 minutes. When each washed the car alone, Lito took
15 minutes longer to do the job than Alvin. How long did it take Alvin to washed the car?

Solution

Let x be the time in minutes for Alvin to wash the car alone.

Then the time for Lito to wash the car alone is (x+15) minutes, according to the condition.


So, in one minute Alvin makes    of the job, working alone;

                  Lito  makes    of the job, working alone.


Thus, working together, they make   +   of the job,

and this combined rate of work is equal to  ,  according to the condition.


So, we have this equation to find x

     +  = .     (*)


Equation (*) says that the combined rate of work is equal to the given value.


At this point, the setup is just completed.  Equation (*) is your governing equation to find x.



To solve equation (*), multiply both siodes by 18*x*(x+15}}}.  You will get

    18*(x+15) + 18x = x*(x+15).


Simplify it step by step.


    18x + 270 + 18x = x^2 + 15x

    36x + 270 = x^2 + 15x

    x^2 + 15x -36x - 270 = 0

    x^2 - 21x - 270 = 0.


Fortunately, the left side admits factoring

    (x-30)*(x+9) = 0.


So, this equation has two different roots, x= 30 and x= -9.

Of them, only the root x= 30 is meaningful.


So, Alwin can complete the job in 30 minutes, working alone;  Lito can do it in 30+15 = 45 minutes.    ANSWER


CHECK.   +  =  +  =  = .     ! Correct !

Problem 4

Joe can paint a room in 6 hours less time than Jay.
If they can paint the room in 4 hours working together, how long would it take each to paint the room working alone ?

Solution

Let x be the time for Joe to complete the job alone, in hours;

then the time for Jay is (x+6) hours.


In one hour, Joe makes    part of the entire job, working alone;

             Jay makes    part of the entire job, working alone.


Working together, they make    of the job in one hour.


It gives an equation


     +  = .      (1)


It is your basic equation.  As soon as you get it, the setup is done.


To solve the equation, multiply both sides by  4x*(x+6)  and simplfy.   You will get


    4(x+6) + 4x = x^2 + 6x

    x^2 - 2x - 24 = 0.


Factor left side


    (x-6)*(x+4) = 0.


Of two roots,  x= 6  and  x= 4, only positive x= 6 is meaningful.


It gives the ANSWER to the problem:


    Joe con make the entire work in 6 hours, working alone;  Jay can do it in 6+6 = 12 hours.


CHECK.    +  =  =  =  = .   ! Correct.  Equation  (1)  is held !

My other lessons on joint work problems in this site are
    - Rate of work problems
    - Using Fractions to solve word problems on joint work
    - Solving more complicated word problems on joint work
    - Solving rate of work problem by reducing to a system of linear equations
    - Solving joint work problems by reasoning
    - Selected joint-work word problems from the archive
    - Joint-work problems for 3 participants
    - HOW TO algebreze and solve these joint work problems ?
    - Had there were more workers, the job would be completed sooner
    - One unusual joint work problem
    - Special joint work problems that admit and require an alternative solution method
    - Snow removal problem
    - Entertainment problems on joint work
    - Joint work word problems for the day of April, 1
    - OVERVIEW of lessons on rate-of-work problems

Use this file/link ALGEBRA-I - YOUR ONLINE TEXTBOOK to navigate over all topics and lessons of the online textbook ALGEBRA-I.

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