Math: Simple Interest and Compound Interest

Financial math gets more coverage in high school now.  As a math tutor, you need to explain the difference between simple and compound interest.

To discuss either type of interest, we need to define some variables:

A=the end amount:  the total value at time t

t=the elapsed time in years

r=the interest rate as a decimal (not a percent)

P=the principal amount (the amount of money deposited at the beginning)

I=the value of the interest earned

The simple interest on an investment is calculated as follows:

I=Prt

Of course, the total value includes the principal as well as the interest:

A=P + Prt

You can factor out P and get the other form:

A=P(1+rt)

Example 1:  calculate the value of a $5000 investment kept in the bank for 6 years at 3.2% simple interest.

Solution:  First, we note the value of each variable given:

A=what we have to find

t=6 years

r=0.032 (to get the decimal, divide the percent by 100).

P=$5000, which is the amount invested.

Plugging into the formula gives us

A=5000(1+0.032(6))

We simplify to arrive at

A=5960

So, if we put $5000 in an account that pays simple interest of 3.2% and leave it in there for six years, the balance will be $5960.

To explain compound interest, we need to define compounding.  In financial math, compounding means taking the interest earned and adding it to the principal.  Once that interest is added to the principal, it can earn interest as well.

Hence the difference between simple interest and compound interest:  with simple interest, only the original deposit can earn interest.  With compound interest, the interest itself can earn interest.

With compound interest, people usually find the total value at the end, A, rather than the interest itself.  Of course,

I=A-P

To calculate the end amount, A, using compound interest, you need to know how many times per year the interest is compounded.  For today’s post, we’ll start with the easiest case:  annual compounding.  Then our formula for the end amount, A, after time t is

A=P(1+r)t

Example 2:  Find the value of a $5000 investment kept in the bank for 6 years at 3.2% compounded annually.

Solution:

A=5000(1+0.032)6

From the calculator, we get

A=6040.16

So, if we leave $5000 in an account paying 3.2% compounded annually for six years, the balance at the end will be $6040.16.

Comparing Example 2 with Example 1, you see that with all else equal, the account paying compound interest grows faster than the one paying simple interest.  As time goes on, the difference gets more pronounced.  We’ll have more to say about that in a future post.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Math: Trigonometry: The Tangent Ratio

As a math tutor, you explain the tangent ratio a few times a year.

Trigonometry involves finding unknown sides and angles of triangles.  At first, it only involves “right” triangles – that is, ones that contain a 90º angle.

At beginner’s level, there are three trigonometric functions: sin, cos, and tan.  (Of course, tan is short for tangent.)  Note their presence on any scientific calculator.  By the way:  in most cases, if a calculator has sin, cos, and tan keys, it’s probably got all you need for high school.

Understanding sin, cos, and tan means understanding how the sides of a triangle are named.

The hypotenuse is always the longest side.

The remaining two sides are called the legs.  The leg touching the angle of interest is called the adjacent side; the other leg is the opposite.

Note that the following diagram, like most diagrams in trig, is not to scale.

The capital letters refer to angles A, B, and C.  If A is the angle of interest, then the adjacent side is 11, and the opposite is 13.  If, on the other hand, B is the angle of interest, then the adjacent side is 13, while the opposite is 11.

The definition of tan is as follows:

(1)   \begin{equation*}tan=\frac{opposite}{adjacent}\end{equation*}

Therefore, in the diagram above,

(2)   \begin{equation*}tanA=\frac{13}{11}\end{equation*}

Here’s where we get practical:  if you know the angle of interest, then your calculator knows its tan ratio.  For instance, tan32º=0.625, rounded to three decimal places.  (Make sure your calculator is set to degrees.)

Let’s use the tangent ratio (known affectionately as tan) to solve a height question:

Problem:

When the sun is at 40º elevation, a tree casts a shadow 13m long.  How high is the tree?

Solution:

First, we draw a diagram:

Note that the box in the corner means 90º.

Looking at the diagram, we see that relative to the 40º angle, the height, h, is the opposite side.  13m is the adjacent side.  Remembering that

(3)   \begin{equation*}tan=\frac{opposite}{adjacent}\end{equation*}

it follows that, in our case,

Of course,

So then

Using the method of cross-multiplication described previously in this post, we proceed:

so that we have

h(1)=13(tan40º)

h=10.9m

Apparently the tree is 10.9m high.

Hope this gets you on the way to calculating those heights that seemed out of reach until now:)

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Biology: Arteries vs Veins

When you tutor Biology 12, you cover the circulatory system.

Pretty much everyone knows arteries and veins are different.  However, we’ll focus on their similarities first:

1)  They both contain a reinforcing layer of smooth muscle.

2)  For both veins and arteries, the central opening which conducts the blood is called the lumen.

Now the differences between them:

1)  Arteries carry blood away from the heart, whereas veins conduct blood back to the heart.

2)  Arteries carry blood under pressure, whereas the blood pressure in veins is minimal.

3)  Arteries have thicker walls than veins.  The reason:  since the blood in arteries is under pressure, the reinforcing layer of muscle in an artery is much thicker than in a vein.

4) Veins have valves, whereas arteries don’t.  Veins need the valves to prevent backflow; remember, the blood in them is under minimal pressure.

5)  Arteries are, for the most part, buried deep in the body, while veins are commonly visible through the skin.  (The wrist, where you take your pulse, is an exception: at that location an artery is close to the surface.)

Although some of the facts above are familiar, there might be a few surprises:)

Sources:

Biology 12, Module 3:  Human Biology I.  2007:  Open School BC.

Inquiry Into Life, Eleventh Edition, Sylvia S. Mader.  2006:  McGraw-Hill.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Biology: Your friend bile

Tutoring Biology 12, you cover human digestion – which mentions bile.

Bile is an oft-referred-to secretion:  in literature, it’s related to anger.  Specifically, if someone’s “bile is up”, they’re in a bad mood.  I don’t understand the association, but there it is.

Bile is a greenish fluid made by the liver but stored in the gall bladder.  It’s made, partly, from worn-out red blood cells.

In the digestive system, bile emulsifies fats – which means it separates large fat droplets into many more small ones.  Once the fat is spread out in tiny droplets, it can be digested by the enzyme lipase.

Via the bile duct, bile is released in the duodenum – the lead section of the small intestine – so as to mix with the food passing through.

Hope this helps:)

Sources:

Biology 12, Module 3:  Human Biology I.  2007:  Open School BC.

Inquiry Into Life, Eleventh Edition, by Sylvia S. Mader.  2006:  McGraw-Hill.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Biology: Carbohydrates

When you tutor Biology 12, which is needed for nursing, you need to define sugars and carbohydrates.

Put simply, a carbohydrate is a compound consisting only of carbon, hydrogen, and oxygen.  The ratio between the three is roughly 1:2:1.  “Hydrate” suggests water (H2O): note that in carbohydrates, the same 2:1 ratio exists between hydrogen and oxygen. Sugar, starch, and glycogen are all carbohydrates.

In biology 12, sugars are either monosaccharides or disaccharides.  A monosaccharide is a simple sugar. Technically, it can have three to seven carbon atoms.  However, in Bi-12, we mainly think of glucose (6 carbons), fructose (6 C), galactose (found in milk, 6 C as well), or ribose (5 C).  All are single-ring structures.

A disaccharide is two monosaccharides fused together; hence, it’s a two-ring structure. Sucrose is an example:  it comes from the union of glucose and fructose.  Bond two glucoses together and you get maltose. Lactose is glucose plus galactose.

If you bond many monosaccharides together, you get a polysaccharide.  Three instances of polysaccharides are starch, glycogen, and cellulose.  All are polymers of glucose molecules – meaning that they consist of large numbers of glucose molecules strung together.  (Glucose is the monomer, whereas starch, for example, is the polymer.)  Starch is the molecule that plants use to store glucose; glycogen is what animals use.  In cellulose, the glucose molecules are joined so as to be indigestible; cellulose gives plants their erect, rigid structure.

There’s the “skinny” on carbohydrates:)

Sources:

Inquiry into Life, Eleventh Edition, by Sylvia S. Mader.  McGraw-Hill: 2006.

Biology 12, Module I: Cell Biology I.  Open School BC: 2007.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC

Punctuation: Everyday use of the Colon

When you tutor English, punctuation is a constant concern.  Appropriate use of the colon can add a nice touch to a writing assignment.

My wife has been questioning my use of colons for months now:  “Don’t you need to follow a colon with a capital letter?” she asks.

Well, according to the McGraw-Hill Handbook of English (1986), you don’t.  A sentence like the following is perfectly acceptable:

My one goal this year is simple:  to fit into my size-32 jeans.

The first writer I ever noticed using colons in the middle of sentences was Charles Dickens.  I don’t remember his following them with capital letters.

The colon’s purpose is to call the writer’s attention to what follows.  You needn’t ever use a colon in the middle of a sentence, of course.    Some writers don’t write in a style that suggests such use of colons.  However, it helps to “change things up” in a longer piece of writing.

Good luck, if you decide to experiment:)

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.