Expert: Josh - 6/16/2006

Question
Hi Josh,

It takes 6 workers 6 weeks to lay 1000 ft of pipe. How long will it take 8 workers to lay 1200ft of pipe?

The work amounts differ, but the rates equal.
W=rt

I worked it out as a proportion.

6workers x 6weeks / (1000ft) = 8workers x (weeks)/ 1200ft

weeks = 5.4 for 8 workers to complete 1200 ft.

Can I do that or what approach would you have taken?  

Answer
Here are my thoughts in relation to some of the comments you've made.

It's a cliche, but practice makes perfect.
What this means is doing representative questions to expand your knowledge and experience. Use your time profitably, do not dwell on a set of questions if they are repetitive or subtle variation of one another. Familiarize yourself with the techniques mentioned in the syllabus, learn common strategies and become confident in applying these strategies to solve each class of problem. Your daily effort counts toward your exam preparation.

You cannot study effectively, if you do not organize ideas and develop a plan. Start writing concise summaries for each topic when you recap on the lessons each week. If you keep this habit, you can develop a clear picture and literally see where you are heading. It's nice to have all the essential formula and concepts summarized succinctly on a piece of A4 paper. It is then very easy to overview/revise each topic, understand the main concepts, relate ideas and assimilate information. This way, you can measure your progress and have absolute clarity of the expected learning outcomes. You can even review a topic while you catch a train home, without flipping through a stack of notes. Print the summary sheet each week on your computer, and put it in your pocket.

It's much easier doing this by hand or using a word processor with floating tables. But just to give you an idea, I've outlined a sample summary to get you started. Keep in mind that the topics are chosen somewhat arbitrarily, of course, you'll make appropriate changes to suit your needs.

=============================
PERSONAL SUMMARY

MATHS SKILLS
1.0 Algebraic Techniques
   1.1 Solving equations with 2 unknowns
   1.2 Physics problems
       1.2.1 projectile motion which involves quadratic equation
       1.2.2 mechanical problems (working out speed, time etc.)
   1.3 Financial mathematics (calculating interest rates etc.)
2.0 Deductive Reasoning
   2.1 Geometry
   2.2 Graphical methods (solving simultaneous equations)
   2.3 Probability and combinatorial problems
3.0 Calculus
   3.1 Differentiation and integration
   3.2 Maximizing/Minimizing the cost function (e.g., dimensions of fencing to maximize area)
4.0 Proofs
   4.1 Mathematical induction
   etc.

TOPICS SUMMARY
Geometry:
* Properties of a circle      [Ref: Week 2]
 - Area: A=pi*r^2; Circumference: C=2*pi*r
 - Theorem: Angles subtended by a common arc at any point along the circumference are the same.
* Properties of a triangle    [Ref: Week 3, morning class]
 - Congruency and similarity
-----------------------------
Word problems:
* Concept of invariance (holding a variable constant; before and after) [Ref: Week4]
* Knowledge of work rate/ manipulation of key equations
   W = Rt [W=amount of work done; R=effective work rate; t=time duration]
   R = Nr [r=individual work rate; N=number of individuals]
 Note: careful when converting between different units
...
 you fill the details, very easy to do on a computer:)
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Questions coverage - A question is worthwhile doing if it reinforces the main concepts, and teaches you something relevant or new. When I study for exams, I tend to concentrate on understanding the techniques (pay special attention to how a solution is determined), rather than working out a numerical answer (the mechanical part).

Always remind yourself before the exam, that time management is crucial. Don't persist on any one problem for too long. Spend as much time as they are worth. Attempt all questions and pace yourself. What you do not attempt will earn you no marks. All revision should be done months in advance, cramping in the last minute is not going to help. Trust me:) And most importantly, get a good night's sleep. The half hour before the test, you should keep you mind off the topic and let your nerves settle. Hopefully, all good work has already been done.

Here are two links that you may find useful.

Maths Problem Solving Recipe:
http://www.mav.vic.edu.au/PSTC/general/strategy.htm

A variety of GCSE Questions on key areas:
http://www.themathsforum.co.uk/
Go to GCSE support, [Mathematics] or [Physics] link

You may also consider getting past papers from friends and relatives who have been there before you. They often contain model solutions. You pick up a lot of skills off other people, simply by noting how others approach problems of the same kind.
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Hi Steph,

I couldn't immediately say whether you are right or not. This comes down to exam technique, which I have been criticized a lot when I was in school.

The approach I would take (though by no means the only acceptable way) is to first emphasize in the answer, that I've understood that the rate is invariant. (Meaning, the rate is fixed, the value before, and after, are the same).

You're right in quoting W=Rt, it's good practice to explain what these symbol means. I won't repeat these here, since we've talked about this last time.

Relating W=Nrt back to the question,

Initial figures:
Length of pipe, W=1000 (ft);
Time taken, t=6 (wks);
Number of workers, N=6;
Individual work rate, r, is unknown.

r=W/(Nt)=1000/(6*6)=1000/36.

New situation:
Length of pipe, W'=1200 (ft);
Time taken, t', is unknown;
Number of workers, N'=8;
Individual work rate, r'=r (no change).

The relevant equation is W'=N'*r'*t.
Making time the subject, t'=W'/(N'*r')

t'=1200/(8(1000/36))
 =1200*36/(8*1000)
 =5.4 (wks)
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So, yes. You are absolutely right.

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Comment: To score easy marks in a test, make sure you communicate effectively to the examiner. Spell out what you are doing. Explain what theory or equation you are going to use, briefly explain what each symbol means and show how each quantity is related to each symbol. I am not trying to criticize you or anything, I'm just trying to give you some useful feedback, for your sake, so you can boost your performance in a test/exam. It doesn't help if we just plug in the numbers in an equation all at once, without some justification as to what you are doing.
For this question, it's fair enough. Understanding how proportion works is probably enough. The thing is this, if you get the answer right, the teacher is probably not going to deduct any mark. However, if you get the answer wrong, it would be extremely frustrating (it takes some effort) for the teacher to follow what you're doing. It doesn't do you any favor. For more complex problems, you don't want to make it harder for teachers to give you marks.
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