Categories
Book Notes Problems Variable Substitutions

Mathematical Quickies No.122

This problem kind of bugs me. Here is the problem and my first step:

Mathematical Quickies No.122

So I like my first step – but then my way of solving {a(a-1)(a-2)(a-3) = 120} is wonky. I want to expand it and use the quadratic equation – which is not slick enough to be what they are after. I am not seeing synthetic division, or any usual suspects – so I think I am missing something obvious.

Rick, someone I work with, looked at the problem and immediately said you could tell that the solution needed to be divisible by 5, which happens to be a valid solution here. However I don’t believe that is always the case though.

My way to get the solution shows a solution of {-2,5}. Work shown here.

Categories
Book Notes Problems Variable Substitutions

Mathematical Quickies No.84

This is another example of a substitution problem. Normally I would have just worked it out long hand, expanding thing and solving. That’s a lot of work and frankly kind of a foolish way to attack the problem looking at it now. Since I started looking for substitutions first – the problem collapses to simple in one substitution. Unfortunately its still fairly un-elegant – so I don’t think this is the solution they book is looking for.

The problem is stated as: “Solve: (6x+28)^1/3 – (6x-28)^1/3 = 2”

My solution

Categories
Algrebra Book Notes Problems

Mathematical Quickies No.205

I kind of hate problems where the solution is to find the most elegant answer possible. This problem is an excellent example of why. My solution (here), solves the problem as stated – but I took a grinder solution just wearing the problem away. My gut says that there is no way that this is the solution they want – however I do solve it with multiple applications of the quadratic equation, which the problem framing hinted at, so I am posting it for now.

Knowing that this is not the solution they wanted, I want to go look at the back of the book to confirm that is the case, and see what they were after. The problem with looking though is that I ruin the problem. So, I pretty much never want to look – hence my dislike of problems I where I can not prove I have the right solution. They all seem to be asking me to prove that someone smarter than I am could not solve the problem in a more elegant manner, but since the person in question is by definition smarter than I am – well – it seems a might ridiculous.

Here we have the problem:

“Solve this equation using nothing higher tan quadratic equations:

X = Sqrt( (X-(1/X)) + Sqrt(1-(1/X)) )”

My Solution

Categories
Math Puzzles Variable Substitutions

Mathematical Quickies No.226

So for some reason I never started looking for substitutions when solving equations. It was certainly something I learned to do when analyzing circuits, but when I see a math problem I never started looking for substitutions that might simplify the problem. Until recently.

Looking at, I think this problem, it “just clicked” – and I started looking for substitutions. Then used them to nock out answers for the next half dozen puzzle problems I tackled. Weird, since I am not doing anything I did not know before, but I just started looking at problems differently.

This one is fairly straight forward:

Solve: (x-a)/b + (x-b)/a = b/(x-a) + a/(x-b)

It becomes way easier to solve after a simple substitution.
My Solution.

Categories
Algrebra Programming

Correction – find missing / extra element in set of values 1..(N-1)

So I already noted this correction in the previous post – but this is what they find a repeated / missing element in a collection of values from 1…N-1 should have looked like.

Correction - find a repeated / missing element in set of values 1..(N-1)
Correction - find a repeated / missing element in set of values 1..(N-1)

Just writing this problem down is making me realize how much I have forgotten. Time to crack open the books. I mean I even forgot my notation for mapping into a set with conditions!

Categories
Algrebra Problems

Up and Down

This is another one from hard to solve brain teasers. It is just another algebra story problem that I have no idea why the included in the book. You just write down what they tell you as equations and end up with two linear equations with two unknowns – so it is fairly straightforward to solve . I do like the way they present the information here as it acts as a good example problem for introducing the idea of graphing or drawing a problem in order help set up equations to solve it.

It would still be a fairly simple problem to solve, but making the problem 3 equations in two unknowns, and using the extra information to nail one of several solutions might make it more fun. So one like this should go in the book.

Good example on graphical representations to solve a problem
Good example on graphically representing a story problem
Categories
Algrebra Problems

Irregular Circuit

So this was problem 5 from hard to solve brain teasers. I am working through the book and this problem is representative of about 60-70% of the problems which are just annoying. This would not be a hard problem in a 6th or 7th grade algebra course. I am fairly sure the book is targeted at adults – so are these problems hard because you are supposed to not remember junior high? Ugh.

The problem:

“Two cars start from point A at the same time and drive around a circuit more than one mile in length. While they are driving laps around the circuit, each car must maintain a steady speed. SInce one car is faster than the other, one car will pass the other at certain points. The first pass occurs 150 yards from point A.

At what distance from A will one car pass the other again?”

Irregular CIrcuit
Irregular CIrcuit

The thing I don’t like about this problem is it really is just a story problem where you just get the answer reading it. How? Because the problem lacks enough information to have any other way to set it up. You either get that starting at 0 and ending at A away from 0 means the second pass will be 2A away from the origin as long as the track is longer than 4A in length.

This could be turned into an interesting problem if the speed around the track was not constant in such a way that for the Nth lap A would be in the lead, for the Mth B would be in the lead. Then knowing who wins the race would require knowing the track length. If that was not given but was instead easily calculable the way to solve the problem would not jump out so much. I will have to try and write up a version like that for the book.

Update: Interesting – the problem directly before this one in the book is the same problem class I was proposing to modify this too. I wonder if the author was thinking that when clustering these problems of if that was just on my mind since I just solved that problem two hours ago or so. Huh.

Categories
Algrebra

“Twin statistics”

Ok, this is another problem that was kind of lame as worded but could be turned into a useful example

Problem as stated: Suppose that 3% of births give rise to twins. What percentage of the population is a twin 3%, less than 3%, or more that 3%.

So the problem is obviously trying to get the reader to realize the birth of twins increases the population by two people as opposed to “normal” births increasing the population by a single person. Then intuit what that means to the percentage of twins in the population.

No reason that the reader should not be able work out the exact percentage given the problem as stated and the assumption that triplets, quintuplets, and all other multiple births are a negligibly small percentage of the population.

Stated this way I can use this problem to introduce readers to the idea of taking two equations dependent on some third and unknown variable that cancels out later.

"Twin statistics"
"Twin statistics"

Categories
Algrebra Math Problems

Grazing problem

This is basic “interview algebra” – it is *not* a differential equations problem. I suspect the whole trick to this problem is making you *think* you are supposed to be setting up and solving a differential equation – so basically if you are getting asked a problem like this in an interview the interviewer is trying to see if you make things hard for yourself under pressure. Useful information to know for both parties.

Grass grows in a field at some rate r, where r is the units of grass grown per day. It is known that if 10 sheep are turned out in the field, the grass will be gone in 20 days. On the other hand, if 15 sheep are turned out in the field, the grass will be gone in 10 days. If 25 sheep are turned out in the field, when will the grass be gone? ( Another Microsoft question from http://vijay.techi.googlepages.com/puzzles).

So I ran across pretty much this exact same problem on pg 36 of Mathematical Recreations from 1907 and they attributed the problem to Isaac Newton.

Grazing rate problem
Grazing rate problem
Categories
Algrebra

Men crossing a desert

An explorer wishes to cross a barren desert that requires 6 days to cross, but one man can only carry enough food for 4 days. What is the fewest number of other men required to help carry enough food for him to cross, the constraint is that each man should survive? (One of the Microsoft questions from http://vijay.techi.googlepages.com/puzzles).