#algebra, #current events - the supposed "meanest problem ever" on the SAT...

I read IFLScience on my Facebook feed daily and today there is a reference to the "meanest test problem ever" on the SAT. So without scrolling down, I decided to decided to tackle it and see what made it so hard. I gotta say I don't see the difficulty...

Question

In a class of p students, the average (arithmetic mean) of the test scores is 70.

In another class of n students, the average of the scores for the same test is 92.

When the scores of the two classes are combined, the average of the test scores is 86.

What is the value of p/n?

Answer

3/8

Analysis

Before we dive into the problem, let's first talk about means (also known as the average). We can find the mean using this general equation:

 

Let's see how this works. With the string of values 1, 2, 3, 4, 5, the sum of the values is 15 and the count is 5, so the mean is 3.



Another way to express this is to see that the sum of the values can be expressed as the product of the mean and the count:



and this is how we solve "the meanest problem ever".

We're told that for the class with p students:



and for the class with n students:



and when the class is put together:



Ok - so let's first work with the individual means and expand them:





and when we put the class together:



we can cross multiply:



and now we'll put p terms on one side and n terms on the other:







~~~~~

Questions and comments always welcome!

#Current Events, #General Science, #Economics, #Household Math - What's a carbon tax?

Question

I've been reading in the news about something called "Carbon Tax". What is it?

Answer

It's a tax on the carbon content of a fossil fuel product at the point of production (or importation).

Analysis

Like most things that are called a "tax", how we look at what it's supposed to accomplish will change the answer. So let me see if I can cover a few viewpoints.

Let me first cover what the tax is first. In essence, it's an amount of money charged by the government on the production of fossil fuels, and the amount of that money charged is dependent on the amount of carbon present in the fuel. For instance, Anthracite (a form of coal) has the highest levels of carbon and natural gas the lowest (according to www.carbontax.org on their webpage on this topic).

One argument for implementing a carbon tax is that the money collected is supposed to be used to defray the costs of the climate change that the burning of the fuel creates. Governments, however, have globally demonstrated an unwillingness, if not complete inability, to spend money targeted to one thing purely on that thing. Instead, the money generally is simply seen as another revenue source that can be spent as desired. This argument, while utopian, ultimately just doesn't pan out.

Another argument for the tax is that it can act as a deterrent on the use of fossil fuels in favour of alternative energy sources or simply reducing energy use. So where would the tax be most noticed by the average person? The gas pump - the carbon tax is passed onto the consumer, who then pays a higher cost for gasoline. Also, the cost of freight also increases since so much of it is fuelled by gasoline and diesel.

A problem here is that alternatives to gasoline fuelled cars are not exactly widely available. Yes, there are things like bicycles and motor scooters that use no and little fuel, respectively, but distance, climate, and use considerations can eliminate them from serious contention (I live in Chiang Mai, Thailand which is a busy metro area. The predominant vehicle here is the motor scooter but it's warm year-round, travel distances tend to be short, and they are used in ways that would never be considered legal in Western countries (multiple riders, stacking goods on the floor of the vehicle, etc). Electric cars are still very expensive and the existent infrastructure often doesn't make them good choices as an all-round car.

Another problem with the "deterrent argument" is that politicians often don't set the carbon tax high enough to act as a deterrent. And so under the guise of a very low carbon tax, politicians feel they can claim the high ground on doing something about climate change, when in fact they are doing very little.

And yet another problem is the idea of "cap-and-trade" policies. The idea here is that a business hit with carbon taxes (say a refinery) won't reduce their pollution, but instead will be able to pay for offsets to their carbon tax and do it in such a way that it's cheaper to buy the offsets rather than pay the actual carbon tax. These last two arguments are better fleshed out here.

Critics point to all these problems and claim that a carbon tax essentially adds up to an unfair tax that hampers competitiveness in the global economy. And so they think that the carbon tax system ought to be scrapped.

And now for my opinion on the whole thing.

The problems with carbon taxes outweigh the benefits - a tax that gets collected that doesn't accomplish anything other than giving government a veneer of climate action and protection is merely a distraction. It's smoke and mirrors - a government will claim to be pro-environment by enacting a carbon tax, thus giving it room to pollute (such as building more oil pipelines). And so I don't like carbon taxes.

However, I do believe that actions need to be taken to reduce pollution and the best way to influence consumer choice is financial. And so programs that help support alternate energy production (such as solar - imagine every rooftop being a solar energy collector) and products that use those alternate sources rather than fossil fuels (such as electric cars) should be supported.

#Science - What's an electrolyte? And why should I care?

Question

I'm on a new eating regime and I keep hearing about paying attention to my electrolytes. What are they and why should I care?

Answer

They are minerals and compounds that allow electrical impulses within the body to function. They are important because having an imbalance (whether over or under) can cause a host of symptoms, some of which can be dangerous.

In all things dealing with your health, when in doubt, seek out the advice of a physician! 

Analysis

The human body is an amazing thing. One thing that makes it absolutely amazing is the number of ways information gets processed and transferred throughout the body and one of those ways is by using electrical impulses (this is primarily in the nervous and muscular systems).

In order to achieve electrical impulses, we need things that allow electricity to move - which is to say, when certain materials are dissolved in water (and humans are primarily water), electricity can move (the technical term being conduct).

The number of things that electrolytes are directly involved in are many, but a few are that they regulate blood pH and blood pressure, regulate muscle and nerve function, and heal tissues.

Those materials are called electrolytes and according to www.medicalnewstoday.com in their article on the subject, the following materials are electrolytes in the body:

• sodium 
• potassium 
• calcium
• bicarbonate
• magnesium
• chloride
• phosphate

As a side note, some forms of iron can act as an electrolyte but in the form it resides in the body, it doesn't.

Ok - we've covered what it is. So now to why should you care.

I'll first say that, with you being on a new eating regime, which I'm going to assume is a diet restriction, you're more at risk of electrolyte shortage than of overage (both of which are bad). While the body has the ability to eliminate overages (to a point - the body can be overwhelmed with electrolytes, especially in the face of certain medical conditions - the link has a lot more on that sort of thing), the only way to address an electrolyte shortage is to consume more electrolytes.

This is where things like supplements might come into play or eating certain types of foods to make up the difference.

According to www.healthline.com's article, some of the common symptoms of electrolyte shortage are:

• muscle cramping and weakness
• headaches, confusion, irritability
• fatigue, lethargy
• fast and/or irregular heartbeat
• digestive disorders, such as constipation, diarrhea, and vomiting

Symptoms are different depending on which electrolyte is out of balance. 

And remember - if in doubt, see your doctor!

~~~~~

As always, questions and comments welcome!

#PreAlgebra - How to multiply fractions...

Question

What's 

Answer

Analysis

The basic equation for multiplication is:



Ok - so why does this work?

Let's look at the question and see why.

We can start with the . If we think of a pizza, we're going to cut it in two and hold onto one piece. Now - if we multiply by a natural number, say like 5, what we're saying is that we're going to add  to itself 5 times, so that looks like:



So what happens when we multiply by a fraction? We split the figure into more bits (the denominator says the number, so in our case, each of our 2 pieces is split into 3 more) and then we'll hold onto more bits as well (the numerator says the number, so in our case, for every 1 we're holding, we're going to hold 2).

Let's talk this out. In the denominator, we had 2 pieces. We multiply by 3, which now gives 6 pieces of pizza. In the numerator, we had 1 piece. We multiply by 2 and so now are holding 2 pieces. Holding 2 pieces of a pizza that is cut into 6 is the same as holding 1 piece when it's cut into 3 pieces:



~~~~~

Questions and comments always welcome!

#PreAlgebra - How to add fractions (and why do I need a common denominator...)

Question

What's ?

Answer

Analysis

First, some fraction stuff! The top number in a fraction is the numerator and the bottom number is the denominator:



Think of a pizza. The denominator tells us the number of pieces there is in 1 whole pizza. The numerator tells us the number of pieces of that pizza that we're concerned with. Notice that if the numerator and the denominator are the same number, we'll end up with the value 1 (or, in other words, the entire pizza):



What our question is asking us to do is to add one slice from the pizza with 4 pieces and 1 slice from the pizza with 6 slices:



Now - if we were simply adding things up, like doing 1+1, all we'd be saying is that I have one slice of pizza here and one there and together they add up to 2:

1+1=2

But with the fractions, we have a bit more information. We can't simply add the numerators together to see that we have 2 pieces of pizza - we need to put more information into our answer. We need to compare the two pizzas on equal terms. To do that, we want the pizzas to both have the same number of pieces in each one - and that is what the Common Denominator is - making the pizzas have the same number of pieces.

Ideally, we'd like to make as few cuts as possible (that pizza is getting cold and I know we're all hungry for some), so we'd like to find the Lowest Common Denominator (LCD). So how do we do that?

I do it by breaking down the two denominators into its prime factors (those numbers that are prime that multiplied together equal the denominator):

4 = 2 X 2
6 = 2 X 3

So now what do we do? The Lowest Common Denominator is the same as the Lowest Common Multiple - we need to include all the primes from each of the biggest groupings. For instance, there are two 2's in the 4, so we need both of those. And there's a 3 in the 6, so we need that too. So we have:

LCD = 2 X 2 X 3 = 12

And now we can scale up our fractions, using clever forms of the number 1, to find the sum.

First remember that anything times 1 equals itself (so we can multiply by 1 and not change the value of the fraction):



And any fraction that has it's numerator and denominator the same is equal to 1. We want to pick values of "1" that when multiplied with our denominators, gives the LCD:





Let's stop here for a second. Notice that with the pizza that had 4 slices, if I cut the pizza into 12 slices, my having 3 of those slices is the same amount of pizza. The fractions are equal. Same with the pizza that had 6 slices - if I cut those slices in half so that there are 12 slices of pizza, and I have 2 of them, it's the same amount of pizza.

Ok, let's finish up:



~~~~~

Questions and comments always welcome!

#Science - Socratic Sunday - Does the Oort cloud exist? If so, why can't we see it?

Hey there,

I'm here again with another Socratic Sunday (I think the name keeps changing slightly. Maybe one of these days I'll actually find one that'll stick...), this time with a question concerning the much theorized Oort cloud:

Question

Does the Oort cloud exist? If so, why can't we see it?

Answer

The Oort Cloud is theorized to exist but due to many factors we can't see it.

Analysis

First let's talk about what the Oort Cloud is (and what it isn't).

The first thing to know is that it is theorized that the cloud exists. While definitive proof will be difficult to come by, the theory helps explain many questions in astronomy (where do long-period comets come from and how to describe their orbits, etc)

From http://space-facts.com/oort-cloud/, it's a region of space that surrounds the solar system where ice, rock, and the occasional larger body (sometimes called Dwarf Planets) exist. The region is a spherical shell and starts about 2,000 AU (Astronomical Units) from the Sun (and to put that into perspective, 1 AU is the average distance from Earth to the Sun. Pluto is 40 AU from the Sun.) and extends well out towards the closest star (perhaps as much as 100,000AU away from the Sun or roughly 1/4 of the distance between the Sun and the closest star).

What it isn't is a cloud in the sense we think about clouds on Earth - big puffy things made up of water vapour that are easy to spot in the sky. Clouds in the astronomical sense are far less dense but are much more vast so that we see them when the light of stars and galaxies is interfered with.

So what would make the cloud hard to see?

Things that we observe in space are observable only if we can see them.

So what does it take to be able to see something? Light.

And where does the light come from? It either is produced by the object (ex. the Sun, a lamp, etc) or the light is reflected off the object (ex. the Moon).

When light is reflected, all the light that strikes the object doesn't bounce back - it gets scattered, absorbed, and the like, so only a tiny fraction of what strikes the object is reflected back (which is why the Moon, even though it can be quite bright at night, isn't anywhere near as bright as the Sun. The Moon absorbs some light and what is reflected is scattered off in all sorts of directions - only a fraction reaches Earth).

Clouds in the Earth's sky are visible because there is a strong light source (the Sun) that allows the clouds to scatter the light and yet still have plenty left for us to look up and see them. There is no light source like that for the Oort Cloud, so what light it gets, when scattered, doesn't come back towards Earth but instead flies off in random directions.

How much light does the Oort Cloud receive? Light diminishes exponentially with distance. A strong light source at distance 1 will only be $\frac{1}{4}$ as strong at distance 2, $\frac{1}{9}$ at distance 3, etc. And this is for both the journey of the light to the object and the reflection back! What that means is that for an object at the closest edge of the Cloud, it receives ${\left(\frac{1}{2000}\right)}^2=\frac{1}{4,000,000}$ the amount of light we get (remember that Earth is at distance 1 AU and the edge of the Cloud is at distance 2000AU)! And then light that is reflected back towards Earth and so we see $\frac{1}{4,000,000}$ of that!

The Oort Cloud does not make its own light, so we need reflected light to see it. The Cloud is made up of small rocks and pieces of ice with huge distances between them, so most of the light that does manage to make it there passes right through and never reflects. Space rocks and ice are notoriously "dirty" - or covered in a dark coloured "space dust", so most of the light is absorbed by the rocks in the Cloud. And the distances involved are massive, so much so that to see Pluto and a few of the other large objects in the Cloud, it takes very powerful telescopes and a lot of luck in finding them.

The only other way of seeing the Cloud is with the use of a deep space probe. However, because of the vast distances, it's difficult to build a spacecraft that can make the journey. According to https://en.wikipedia.org/wiki/Oort_cloud, Voyager 1, the farthest and fastest moving probe we have and launched in 1977, will be another 300 years before it reaches the Cloud and by then will have no power to explore or send back data. Other probes also making their way out of the solar system will also run out of power long before making contact with the Oort Cloud.

~~~~~

As always, questions and comments are always welcome!

#Budgeting, #Household Math - Tracking spending...

Question

My budget has been written up and everything looks good. So I'm done, right?

Answer

Ummm... no. Now it's time to execute the plan that you've created.

Analysis

It's absolutely fantastic that you've gotten your budget squared away and things look good on paper. Think of the budget as a financial plan - it lays out what you intend to spend over the given month. Now it's time to act on that plan.

There are many ways to accomplish this - some people prefer a more detailed level of bookkeeping while others prefer way less detail. Whatever system works for you, that's the one you should use.

For the more detailed people, receipts are going to be your friend. When you buy something, make sure to get a receipt. If you don't get a receipt, perhaps carry a piece of paper or small notebook to notate what was spent. When you get home, make sure to record those expenses onto a spreadsheet or perhaps a bookkeeping program such as Quickbooks.

For the less detailed people (and I fall into this category), figure out how much you can spend each day and put that amount of money into your wallet. That is what you can spend and when it runs out, your spending for the day is done.

As you track your actual daily expenditures, you may have to tweak your budget to have it match your spending. Alternatively, you may have to tweak your spending to stay within budget.

In the process of recording your expenses, you can record expenses that will have an impact on your tax reporting. This will significantly reduce the stress of doing your taxes at tax time.

This post is part of a series on budgeting - Budgeting 101

~~~~~

Questions and comments always welcome!

#Budgeting, #Household Math - Credit Cards...

Question

What is your opinion of credit cards?

Answer

Analysis

I've tried writing the first sentence to this analysis a few times but eloquence is abandoning me. Elegant writing will simply not produce what it is that I'm trying to say, so I'll go with the simply statement of:

Credit cards are bad.

Actually... let me rephrase that. If you are a purchaser, they are bad. If you are a retailer, they're great. Why? Because studies have shown time and again that people who use credit cards are willing to spend more for products and services than people who use cash.

For whatever reason (this article proposes a few ideas), people spend more when using credit cards than when not. In fact, people are so programmed into this that even when paying by cash, seeing a sign that says that credit cards are accepted will help increase the spending of the average shopper.

One thing that the article above cites is that those who buy with credit cards will tend to focus on the benefits of the purchase. Those who buy with cash tend to focus on the costs of the purchase. It would seem that when the "burden" of paying for something is greatly reduced, now to the point of tapping the credit card on a reader or pulling up a QR code on a phone app for many purchases, it takes away the thoughts on the cost of the purchase. Purchasing in this manner, focusing on the benefits with little regard to costs, is Impulse Buying, and if it isn't the number one reason why people can't stay on a budget, it has to be in the top three.

Staying on a budget requires something that used to be called "sober consideration" - and maybe it still is. It requires a plan and then, far more importantly, execution of that plan. Credit cards act to circumvent that plan, making it far too easy to buy things and to add on to purchases already being made (ex. what's the cost of another drink when dinner is going to be expensive? What's the cost of buying that appliance insurance plan when the cost of the appliance is going to be so much?)

And so, bottom line - credit cards are an occasional necessity but as an integral part of daily life, credit cards are a financial disaster either waiting to happen (or already happening).

This post is part of a series on budgeting - Budgeting 101

~~~~~

Questions and comments always welcome!

#Admin - Math Fact-orials 2.0!

Hey all,

The editing is done, the formatting finished, the transformation complete!

New content will be arriving today and I'm looking forward to posting it up using my new formatting system.

Something you'll notice is the hashtag-ed subjects at the top of the page - the subject line of each post will have at least one tag so that you can (more) easily navigate to content you want to see and avoid anything that will give you math-mares.

I've found a great online graphing calculator at https://www.desmos.com/calculator and so where I can graph, I will (with this and the math formatter, algebra will be much much easier to do).

Which brings me to the links at the top of the page - there's the one for going to my Facebook page (the blog's embassy at Facebook, if you will...) and then links to the tools that I reach for in making the blog - an online set of calculators at www.calculatorsoup.com and also the equation editor I'm using at www.codecogs.com. All of these enable me to bring better quality content to the blog.

And as always, your questions and comments drive the blog forward. It's like a symbiotic relationship - you help me by asking questions, I help you by providing answers. Win Win!

Parz

#Admin, #Math gif - Math gif's...

Hello all,

I'm still working my way through the blog editing process - it's moving along well although I do wish it were done already!!! - and so my answering questions is again going to be on halt today.

But I ran into this on a site I follow and just had to share.

It's from a site called... let's just use IFLScience... and it's a series of gif mini-movies they've collected and put into a single post. It's brilliant, fun, and really the closest thing to math TV out there!

I hope you enjoy it as much as I do...

https://www.iflscience.com/brain/math-gifs-will-help-you-understand-these-concepts-better-your-teacher-ever-did/all/3

#Admin - Another admin-y post...

Hello again everyone!

So it turns out the blog needs a bit more TLC before I'll be adding new content to it - things like changing the labeling system so that it's easier to find content a reader might be interested in, editing hyperlinks, cleaning up equations... there's a lot of work to be done! But in the end I hope it makes the blog more useful to anyone who swings in to read about... well... whatever it is you're looking to read about - the numbers of ways of doing things, or probability, or algebra (with an equation editor available, I can now do algebra!), or science, or an opinion piece, or household finance questions.

And so stay tuned! I don't know how long it will take to get it up and running again, but when it is going, it'll be better than ever!

#Budgeting, #Economics - Why is it that governments can run huge deficits for years? Why can't I?

Question

So my budget has a deficit - I apparently spend more than I earn. But governments do it all the time. Why can't I?

Answer

There is a world of difference between using a currency (people/companies) and issuing a currency (governments).

Analysis

There is a world of difference between the way money flows affect a government/country and the way they affect a person/family.

With a person/family, and even with a company, financial health is dictated directly by the ability to have more income than expenses. When expenses are bigger than income, and particularly when we're talking about a long period of time or a large deficit income, then assets will decline (bank balances, investments, etc, will drop) or liabilities will increase (bigger and bigger credit card balances, bigger loans from the bank, etc). And unfortunately there's really no way around it - if expenses are higher than income, there's a problem that is either front and centre or is waiting in the wings.

A government is a different animal. While people/families/companies use money, governments issue it. From that situation comes how governments can operate at a deficit for so long.

Think of government's operations this way - when governments take in money (taxes are one frequent way), they are taking money out of the nation's monetary system. When governments spend money (whether on the military, a social safety net, or any other way), money is put into the nation's monetary system. One of the goals of government is to maintain a healthy balance between taking money in putting money back into the system.

So what happens in the case of a government that is constantly spending more than it's taking in? Let's walk this one through to see what happens:

• Government spends more than it takes in, which puts more money into the economy 
• More money in the economy means that businesses and people have more money (on average) to spend
• With more money to spend, businesses and people desire to buy more things
• Demand for things to buy (more and more buyers) goes up, also risk tolerance increases (which basically means that more and more people will be willing to put more and more money into stocks and investments that are riskier and riskier)
• With demand increasing and supply not catching up, prices rise (which is called "inflation")

Inflation is a topic unto itself, but suffice it to say for now that low inflation is a mixed bag of good and bad and high inflation is (pretty much) all bad.

And so when governments overspend on a routine basis, there are ripples throughout the nation's economy, some of which are good and some which are bad. When people and companies do it, they simply end up in a bad situation.

~~~~~

As always, questions and comments welcome!

#Science - Socratic.org Sunday throwback - Why are arteries thicker than veins?

As I've mentioned before, I used to be quite active on Socratic.org, a free-to-use question and answer website. However, the powers that be at that site decided to turn it into read-only (it actually went read-only this past week), and so I highlight past questions and answers that seemed to catch people's attention. Today's question is:

Question

Why are arteries thicker than veins?

Answer

To withstand the high pressure of blood coming from the heart.

Analysis

The "heart" of the circulatory system is the heart. It pumps blood through arteries (starting with the biggest one of them all, the aorta). The blood runs through smaller and smaller arteries until it gets to a capillary, where the blood drops off oxygen and nutrients to cells and receives carbon dioxide from the cells. 

Also at the level of the capillaries, a bit of plasma "leaks" into the space between cells, becoming interstitial fluid (which is recaptured in a main vein just above the heart through the lymphatic system).

On the return trip, blood flows through veins and ends up back at the heart (there is activity with the lungs but we'll ignore that for now).

The whole system is under pressure - a high pressure on the outflow from the heart and a lower pressure on the inflow. Because of that high pressure on the outflow from the heart, the walls of the arteries need to be thicker.  Veins, which are under far less pressure, have thinner walls. In order to help the blood flow back to the heart, veins have "check valves" - blood can flow through towards the heart but can't flow backwards away from it.

~~~~~

As always, questions and comments welcome!

#Budgeting, #Household Math - Budgeting - what happens if the numbers add up to bad news?

Question

I've done the steps of listing out my expenses, putting in the numbers, and then identifying my income and it turns out my expenses are higher than my income. What now?

Answer

First verify the numbers are correct. Then begin to address areas that can be improved. Lastly, be open to change - if there's a large deficit, you may have to be open to things such as government assistance and the like.

Analysis

Before we get into what might you do to adjust your living situation, let's first verify that the budget is accurate. Ironically, the bigger the difference between income and expenses, the easier it'll be to verify the numbers.

If the difference is quite big and it has been happening for some period of time (and the budget is accurate), then the money that you've been spending has to have been coming from somewhere. Are your credit card balances increasing? or is there a running balance on your cards? Is your bank balance or investments balance decreasing? Are you borrowing larger and larger sums of money and having difficulties paying it back?

Again, the bigger the deficit and the longer the period of time this has been the case, the easier it'll be to see where the money is coming from. (If it turns out your bank, investment, and debt balances are not changing, then it may be the case that something has been left out of the budget or there's a math mistake).

Let's say you've verified the numbers and they are correct - expenses are higher than income. What can be done?

First off, stop and breathe. Be proud of yourself for bringing the situation to light. Coming to grips with a difficult to accept situation is praise-worthy, so take a minute and give thanks that you now have the knowledge that your financial health isn't what you thought it was.

Ok - now we can address the situation. What is the magnitude of the deficit?  If it's small then it may be that eating out a little less or other types of luxuries can be cut back and that will solve the problem.

With deficits that are larger, it'll require more work. For some people, it may require obtaining outside help, such as public assistance, to be able to keep some version of your financial life intact. For others, it may require a severe downsizing or moving from a high-cost location to a lower-cost location.

Again, the most important thing is to work with what is known. The worst thing to do when faced with a deficit in the budget is to ignore it and pretend it doesn't exist. That is the road to disaster.

So what if income is higher than expenses? Great news! Now - did you verify the budget is correct by checking your bank/investment/debt balances? Are they moving in the correct directions? If so - good! If not - you've missed something!

With the budget numbers in front of you, do you see anything that looks like you could do better? Is the dining out budget too high or is it right on? Are there opportunities to save even more money?

With all of this, remember that the budget is a living document in that it should be changed when there is a life change.

This post is part of a series on budgeting - Budgeting 101

~~~~~

As always, questions and comments always welcome!

#Admin - A bit of admin...

Math Fact-orials has been up and running for roughly 3 weeks and already been hitting milestones I didn't think would be possible so soon, including having nearly 1,000 page views!

I owe much of this success and exposure to two audiences: former users of Socratic.org (which is now a read-only site) and the art community of Sketchbook Skool.

Socratic.org is where I cut my teeth on answering questions online, where I found my love of all things factorial, and saw in action how topics such as budgets, household math, finance, accounting, and so many other topics that are important to people are ignored by academic math sites. One of my main goals is to bring household math into a place where people can use math as a tool and not regard it as a nightmarish exercise in frustration (I feel like I can hear people, whether consciously thinking or unconsciously feeling something like: "Please... whatever we do... whether we choose to buy the new car or lease it... don't make me figure out which one is actually better! Make the numbers stop dancing in my head!").

With that ideal in mind, I've been posting here and there about budgeting and will be adding a few more posts to that conversation. There are already a few posts about household finances and more will be forthcoming (both from me and from you - my readers!)

Sketchbook Skool's response to the post about the numbers of trade items and also the number of unique trades has been nothing short of phenomenal. A thousand and one thank you's to Aleesha, my artistic wife, the source of many of the questions on the blog, and the inspiration for the blog post about Sketchkon and who put the post onto Sketchbook Skool's Facebook page.

Looking ahead, I've found a way to satisfactorily put equations that look like equations onto the blog (thanks to latex.codecogs.com) and so that is what I'll be doing over the course of the next few days. I'm really looking forward to seeing math rendered the way it should look! 

I'll also keep on with Socratic.org Sunday throwbacks, where I'll grab questions and answers from a host of different topics that seemed to grip people's attention. This week's question will be Why Are Arteries So Much Thicker Than Veins?

I'm extremely grateful to my current, past, and future readership and I hope that as the readership grows and develops, the blog can do so alongside so that it is always meaningful and helpful. And the best way to help make that happen is to send me emails, leave comments, and ask questions! 

#PreAlgebra - The Order of Operations

Question

What's the Order of Operations? And why does it matter?

Answer

It's akin to grammar for math - it helps to make sure that what is being written by one person is understood the same way by someone else reading it.

Analysis

When we write a sentence, we use grammar to help make sense of what's being written. One of my favourite examples of this is on my son's T-shirt

It's time to eat grandma

It's time to eat, grandma

Commas save lives

That little bit of grammar tells us that we don't intend to eat grandma but rather that we're letting her know it's time to eat.

The Order of Operations is like grammar for math. It tells us in what order we're supposed to do operations. For instance, with the following:

5 + 3 X 6

is the answer 23 (= 5 + 18)? is it 48 (= 8 X 6)? The Order of Operations helps the person reading the math and the person writing it to both see the same meaning (and answer).

The Order is sometimes called PEMDAS (or PEDMAS or BEMDAS or some other acronym like that. It refers to:

P or B = Parentheses (or Brackets)
E = Exponentials
M = Multiplicaton
D = Division
A = Addition
S = Subtraction

M and D have the same weight so we do those in order from left to right. The same is true for A and S.

In 5 + 3 X 6, using PEMDAS, we can see one addition and one multiplication. We do the multiplication first, so that gives

5 + 18 = 23

Let's do something a little harder:

(6 + 3^2) / 3 + 2

We do the bracket first. Within the bracket there is an addition and an exponential - we do the exponential first:

(6 + 9) / 3 + 2

Now we have division and addition. Division comes first:

15 / 3 + 2

5 + 2 = 7

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Questions and comments always welcome!

#Combinametrics, #Probability - The chances of having the same birthday...

Question

How many people need to be in the same room for there to be a 50% chance that two of those people will have the same birthday?

Answer

23 people

Analysis

When working out this kind of problem, it's all in the approach. For this problem, let's work up to a method of doing it.

One of to do it would be to find the number directly - as in calculating the probability "from the ground up". The other way to do it is to see what the probability is of not having the situation and subtracting that from 1 (1 being the sum of all probabilities possible). It's this "subtracting" approach we'll use. To see how, let's follow smaller groups of people and work up to bigger ones.

Let's say we have 1 person in a room (person A). A can have a birthday that can be any one of 365 days (we'll ignore leap days). We can express this as



If we put a second person into the room (person B), B too can have any one of 365 days as their birthday. If it's not the same day as A, then it can be any one of 364 days:



probability that it's different.

Which also means that there is a

 

probability it's the same.

If we put a third person into the room (person C), C can also have any of the 365 days as their birthday. For A, B, and C to all have different birthdays, we can say:



probability that they are all different, which means there's a 0.0083 probability (1 - 0.9917) one of them is the same.

And so on. Note that we end up with a numerator that is a permutation and a denominator that is 365 taken to the power of the number of people in the room. Therefore, we can generalize the calculation as:



And now we want to find where our expression is 50%:



This is a very gnarly problem to work with algebra - it'd be better to use a graphing technique or a spreadsheet to work through values of n. And so I worked it using a spreadsheet and found that at 23 people, there's roughly a 50% probability of at least 1 shared birthday.

As a side note, at 57 people, there is a better than 99% probability of at least 1 shared birthday.

The spreadsheet is here:
https://docs.google.com/spreadsheets/d/1XPWcgzaXS5zcXOF5M0VzofFMxJGtsFwlrrX_3ybPxtE/edit?usp=sharing

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Questions and comments always welcome!

#Household Math - High school, career paths, and financial outlooks...

Question

For a high school student who's thinking about the future, it seems to me (a parent) that there are three basic choices: working straight out of high school and getting into some sort of un- or limited- skill labour; going to university and (presumably) learning some sort of skill that can translate into a job directly (ex. accounting), and going to trade school and learning some sort of trade. What are your thoughts on this?

Answer

Before talking about potential net earnings, I think it's most important to talk about the desires and nature of the student. That said, and with some very basic (and perhaps widely inaccurate) assumptions, the trade school grad does best over time. However, going to work straight out of school and being able to get a good wage gives that person a leg up over the college graduate for close to 20 years.

Analysis

There's quite a bit to unpack here, so let's take it step by step.

My first thoughts don't go to finances but rather to happiness - where is it that any given person will be his/her happiest. This is an individual decision that will be guided heavily by the interests and desires of that person. For instance, I know that at that age, the thought of doing anything other than going to University was a non-starter. Thankfully, I had a scholarship that covered many of the costs! However, as I've grown older and worked with students, it's become clear that for many students, it's a better choice, due to their desires and temperament, that another choice such as a trade school would be far more desirable. It used to be the case that a college education was a guarantee to a good job but that hasn't been the case for many many years. For example, Lululemon, the clothing retailer, only hires college grads as sales people. How are they able to do that? Because there are so many of them that can't get jobs in their chosen fields that Lululemon can be extremely choosy in who they hire.

Let's say for arguments' sake that the choice really is up in the air - the student is planning a future and that all of the three choices (working, university, trade school) are appealing. We can now look at this in terms of finances.

I'm going to make a lot of assumptions on this question, so if you spot an assumption that needs adjusting, please do point it out!!! For ease, I'll assume a 2000 hour work year.

Working straight out of school

This part of the question needs a heavy dose of assumption. I'm going to assume that a limited-skill job can be obtained - something that brings in the following amounts:

$14/hr for the first two years
$16/hr for the next two years
$18/hr for the next two years
$20/hr for all years thereafter

University

This part of the question also needs a heavy dose of assumptions. I'm going to assume that the cost of University is all borne by the student and is all paid for with loans. I'll set the loan interest at 10% per year for 10 years. The cost of university I'll assume to be $20,000 per year for four years, all in (tuition, lodging, food, books, lab fees, etc) (public schools being a bit less and public schools being potentially considerably more). Coming out of school, I'll also assume a skill has been learned that will pay $40,000 entry with a 5% raise per year afterwards.

Trade School

And time for more assumptions... I'll assume a trade school is one year and $20,000 all in, with the same repayment schedule as for the university (10 years, 10% interest). I'll assume the graduate is able to get a job for $20/hr starting and receives a raise of $5/hr after two years (I'm assuming moving from an apprentice to a journeyman) and then 5% thereafter.

I'll use a spreadsheet to figure out the results over time.

https://docs.google.com/spreadsheets/d/1_SpdTjdH2DJr8grYs8Mfl90crsECIih8mVvo5YozoII/edit?usp=sharing

And so again - these results are the result of the assumptions made and one or more of them may be wildly off, depending on individual circumstances - please take that into account if using this as a basis for decision making! Or let me know your particular circumstances and I can adjust the sheet appropriately.

If we project the net earnings of these three tracks over the course of 30 years, we find the following:

At year 4, the Trade School graduate overtakes the person who started working straight out of school.
At year 19, the University graduate overtakes the person who started working straight out of school.

In terms of overall earnings over the course of 30 years, we have:

• Working straight out of school = 1,152,000
• University = 1,920,538
• Trade School = 2,782,456

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Questions and comments always welcome!