Working Together Math Formula
Hedvig Horning
Work word problems usually involve two or more entities working together to complete a task. Work problems have direct real-life applications. We often need to determine how many people are needed to complete a task within a given time. Alternatively, given a limited number of workers, we may need to determine how long it will take to finish a project. Here, we will deal with the basic math concepts of how to calculate work word problems.
Example 2:
It takes Maria 10 hours to pick forty bushels of apples. Kayla can pick the same amount in 12 hours.How long will it take if they work together? Round your answer to the nearest hundredths.
Example 1:
Jane, Paul and Peter can finish painting the fence in 2 hours. If Jane does the job alone she can finishit in 5 hours. If Paul does the job alone he can finish it in 6 hours. How long will it take for Peterto finish the job alone?
Example 1:
A tank can be filled by pipe A in 3 hours and by pipe B in 5 hours. When the tank is full, it can bedrained by pipe C in 4 hours. if the tank is initially empty and all three pipes are open, how manyhours will it take to fill up the tank?
It is possible to solve word problems when two people are doing a work job together by solving systemsof equations. To solve a work word problem, multiply the hourly rate of the two people working togetherby the time spent working to get the total amount of time spent on the job. Knowledge of solving systemsof equations is necessary to solve these types of problems.
Example:
Latisha and Ricky work for a computer software company. Together they can write a particular computerprogram in 19 hours. Latisha can write the program by herself in 32 hours. How long will it take Rickyto write the program alone?
Example:
A swimming pool is being drained through the drain at the bottom of the pool, and filled by the hoseat the top. If the hose can fill the pool in 21 hours and the drain can empty the pool in 24 hours,how many hours will it take to fill the pool if the drain is left open? Express the answer in hoursand round the answer to the nearest hour if needed.
Try the free Mathway calculator andproblem solver below to practice various math topics. Try the given examples, or type in your ownproblem and check your answer with the step-by-step explanations.
Doing math together over the phone worked well because students had to articulate problem sets verbally, which improved general language proficiency as well as their ease with the particular language of math. The downside was not being able to see the material they were discussing. We experimented with Skype and FaceTime to incorporate that missing visual element, but, like the phone, neither platform offered a way to save and refer to information later.
So nothing was quite hitting the spot, until a teacher friend added me to a Marco Polo group that she was using to organize a birthday party. Suddenly, I had my own light bulb moment: Why not use Marco Polo with my students?
#1: Accommodates different learning styles and needs. With Marco Polo you can exchange information verbally, visually, and in written form. Students are able to review specific saved Polos when preparing for exams, and their academic vocabulary increases immensely.
If it takes Felicia 4 hours to paint a room and her daughter Katy 12 hours to paint the same room, then working together, they could paint the room in 3 hours. The equation used to solve problems of this type is one of reciprocals. It is derived as follows:
In many cases, you will work with other people or have employees. You will always have a feeling that your colleagues are not performing as you or some of your employees are lazing around. Because of that, the shared work word problems always exist.
Two painters can paint a house in 12 hours when working together. If one painter can take 18 hours painting the house alone, how long would it take the other painter to paint the same house when working alone?
Tap A takes 5 hours to fill an overhead water tank and tap B takes 8 hours to empty the same tank. If both taps are accidentally opened at the same time, find the time it would take the two taps to fill the tank?
This problem is not as direct as the previous two. Here, we have two taps working together. Tap A is contributing positively while tap B is contributing negatively. From this statement, the formula becomes:
Regardless of the nature of a shared work problem, you can always find the solution by applying the right form of the formula. It also does not matter how many people or machines are sharing the work. Just extend the formula to accommodate every contributor.
This lesson was prepared by Robert O. He holds a Bachelor of Engineering (B.Eng.) degree in Electrical and electronics engineering. He is a career teacher and headed the department of languages and assumed various leadership roles. He writes for Full Potential Learning Academy.
This snippet should update the preview everytime the keypress event is fired. Instead on page onload the tex is rendered fine but as soon as I start typing the $...$code is printed in the preview box.
There are several problems with your current setup. First, the example you have borrowed from is an example of updating a single equation, not paragraphs that includes multiple equations. For that, you would need to consider the second dynamic example (from the MathJax examples page). You should be getting an error message in your browser console that will have to do with a null value (math will be null unless you start out with some math in the editor to begin with).
But there is a second issue, which is that the wmd editor will be updating the wmd-preview area, and you should coordinate with it to do the MathJax updating, as otherwise it might change the content of the div while MathJax is working on it. Wmd is also smarter about when it does updates than just on every keypress (e.g., arrow keys don't cause updates), so it will be more efficient to coordinate as well. The SE version of wmd has hooks to allow you to do that, and I suspect the one you are using does as well.
Finally, you are going to have to do more work to protect the mathematics from the Markdown engine so that things like underscores and backslashes don't get processed by Markdown when they appear in mathematics. That is a bit tricky, but without that, you will run into lots of problems with your TeX code not getting processed properly.
1) Karen and Betty have been hired to pain a house. Working together, they can paint the house in $\dfrac 23$ the time it would take Karen working alone. Betty could paint the house by herself in $6$ hours. How long would it take Karen to paint the house working alone?
The way I advised my brother to do these is to write the "$d=rt$" equation for every case: each person working alone, and both people working together; introduce as many variables as you need. From then, I just told him to try to manipulate the equations to solve for the unknown variable. However, he found this approach difficult. He said that somebody really smart could understand what's going on better and not have to rely on algebraic manipulations. Is there a better way to do these?
Problem 130. A swimming pool is divided into two equal sections. Each section has its own water supply pipe. To fill one section (using its pipe) you need $a$ hours. To fill the other section you need $b$ hours. How many hours would you need if you turn on both pipes and remove the wall dividing the pool into sections?
This involves the so-called harmonic mean, which is not a particularly well-known or used mean among high schoolers. If we had the same intuition for that as the regular (additive/arithmetic) mean, maybe this kind of problem would be obvious.
However, I bet that with time and thought one could come up with a nice algorithm that avoided algebra to solve them, and likely such problems are of great enough antiquity that people did come up with them. The problem then is that you have to memorize some non-algebraic algorithm; the algebraic advantage is that you are directly translating everything and so one can recover the problem from the algebra.
I don't know much about math education, but I can give my thoughts on how to solve this problem. I think the main issue is whether to express the rate of work as a amount of work per unit time or its reciprocal, the amount of time per unit work. When you have two people working together, it is the amount of work per unit time that combines additively, so that is the natural thing to work with. (In another situation where you have one person painting two houses of different sizes, and you were asked how long it would take, you would want to work with the amount of time per unit work.)
If they can paint the house in 2/3 the time it takes Karen to paint the house, then they can paint 3/2 houses in the time it takes Karen to paint one house. Therefore, when Karen has painted one house, Betty has only painted a half a house. This means Betty is half as fast as Karen. So if it takes Betty 6 hours to paint a house, it takes Karen only 3 hours. (Here taking the reciprocal was the key step that allowed us to compare how many houses each was able to do in a fixed amount of time, and thereby compare how quickly they work.)
Stan and Hilda mowed a lawn, with Hilda doing twice as much as Stan. This means Stan only mowed 1/3 lawn in forty minutes, so it takes him 3*40 minutes to mow a whole lawn. (Again we decomposed the rate 1 lawn / 40 minutes into the sum of two contributions: 1/3 lawn in 40 minutes and 2/3 lawn in 40 minutes. Notice these are both amounts of work per unit time.)
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