Number of Horses and Time of Work
Name: Karl J.
How can I explain to my 13 year old the solution to
Five horses finish the work in 6 hours yet 3 horses will finish in how
Start with the idea of a "horse hour" -- the amount of work one horse will
do in an hour. This problem assumes that all horses do the same amount of
work in an hour, and that they would do, say, twice as much work in two
hours. In many word problems there are implied assumptions like this, and
one thing people learn from doing word problems is to analyze what is said,
and figure out what is being implied.
The first phrase: "Five horses finish the work in 6 hours" is simply a
statement of how much work is to be done, and you can tell that this is
what the phrase is intended to convey by noticing which words have numbers
attached. We know how many horses; we know how many hours; but we are not
told how much work. That is a clue that the problem "wants" us to figure
out what sort of number, with what sort of units, can be attached to the
Each horse works for 6 hours, and there are 5 horses. We have to find the
total number of horse hours worked. Clearly, we are being asked to do some
sort of arithmetic. I think there is no substitute for letting a kid just
roll this around in their head until they get a mental picture of it that
tells them what arithmetic to do. I suggest you do not even provide any
hints. If you do not get an answer after several minutes, provide lots of
pennies for counting, and just wait however long it takes.
Now that we know how much work our three horses must do, it is not so hard
to figure out how long it will take them to do it, but again a mental
picture is needed, and a kid must understand that it is his/her job to make
one. Again I suggest no hints, an ample supply of pennies, and as much
time as it takes.
I think the best way to talk about this problem is in terms of units. If
five horses finish work in six hours, then the total amount of "work" done
is 30 horse-hours (5 horses x 6 hours). The new question becomes, for the
same amount of "work" (30 horse-hours), how long will it take 3 horses to
complete the work? 3 horses multiplied by (some time) = 30 horse-hours. I
think it is clear that the "some time" must equal 10 hours.
This type of reasoning is used in production scheduling. If a job requires
40 man-hours, you can get it done in 40 hours with one person working on the
job, 20 hours with 2 people working on the job, or in one hour with 40
people working on the job. Of course, there are assumptions built into this
type of reasoning -- like the idea that 40 men (or 5 horses) will all do an
equal share of the work and that adding more men (or horses) will not
interfere with getting the job done.
Todd Clark, Office of Science
U.S. Department of Energy
The book "How to Solve Word Problems" by Mildred & Tim Johnson
(ISBN:0-07-134307-5), available from any local book store or on the Internet
at amazon.com I have found to be an excellent resource on all these "jump
through the hoops" exercises. There are others but I have found this one to
be the most "parent-friendly". I have no financial interest in this
particular book. Now to your "Horse problem":
1. The first step is to find out what fraction EACH horse contributes to
the completion of the job, call that number of hours 'x' (hours/horse). Then
the FRACTION each horse contributes is (1/x) (horses/hour).
2. We are given that 5 horses can do the job in 6 hours, so for the 5
horses we have:
1/x + 1/x + 1/x + 1/x + 1/x = 1/6 --or-- 5/x = 1/6 and solving, we find
that each horse contributes: 1/x = 1/(6*5) = 1/30 th to finishing the job.
3. But now two horses got tired so we can only use 3 horses. Still each
contributes its 1/30th toward completing the job, but it is clear that the
job is going to take longer. How much longer is the question? We have: 1/30
+ 1/30 + 1/30 = 1/J where 'J' is the number of hours it will take 3 horses
to complete the job. Solving: 3/30 = 1/J = 1/10 or:
J = 10 hours.
In the absence of a book like the one above, another route for help is
to do a web search on topics like: "solving work problems", "solving age
problems", "solving coin problems", etc. In all cases I have tried you get
"hits" that will guide you through the particular problem-solving process.
An editorial comment. Word problems are important, that is being able to
translate a verbal statement into a mathematical statement. However, for the
most part it is poorly taught and poorly presented in texts.
Each horse does six hours of work. Five horses working together for six
hours gives you 5x6=30 hours of work. Three horses can do 30 hours of work
in 10 hours.
Math, Science, Engineering
Illinois Central College
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Update: June 2012