Cunning Conversion

Cunning. Simple, yet complex. Not difficult to do. Some of us can do it right on the spot. It's not a bird, not a plane, not a monster, not Superman. It's conversion. Conversion. Let us look at the technical definition first. According to dictionary.com, (of course we are not looking at the religious meaning, but the mathematical meaning), it is a change in the form or units of an expression. Basically, here is an example: 1km= 1ooom. Here is another: 1cm= 10mm. The key to doing quick, yet effective conversion questions is to remember what a particular type of unit is in another type of units, something like 1m is always equals equal to 100cm. In questions like: " Tom has a ribbon. Dick gives him another 100cm of ribbon, but Harry steals another 57cm from Tom. However, due to his remorse, he returned another 45cm of ribbon back to Tom. He now has 338cm of ribbon left. What is the length of ribbon Tom has at first? Give your answers in metres."

Well, some of us who are "well versed" in conversion will easily work out this problem. However, I shall work out this problem to show you all my mathematical prowess:

The trick to the main part of this question is to work backwards.
338cm - 45cm = 293cm
293cm + 57cm = 350cm
350cm - 100cm = 250cm
And 250 cm is the answer NOT. Remember to convert to metres.
1cm = 1/1oom
250cm divided by 100 = 2.5m
So the final answer is 2.5m.

The solution is explained this way: "338cm - 45cm = 293cm" - from Tom's final length of ribbon, work backwards, and take away the ribbon Harry returns to him.

"293cm + 57cm = 350cm" - add back the amount of ribbon Harry stole from Tom.

"350cm - 100cm = 250cm" - subtract the amount of ribbon Dick gives Tom, and with this step, you will get the original length of ribbon Tom has, before Dick gave Tom anything.

And of course, you convert to the unit of measurement to the one required. However, another way is to convert the units right from the start, and work accordingly, for example:

338cm = 3.38m, 45cm = 0.45m. 3.38m - 0.45m = 2.93m.

And continue accordingly.

And so, that is about the more simple conversion for units of measurement.

Bye!

National Education in math... yes it is possible

Ha ha. National education. Something that all teachers and parents want children to learn. In hope that they will be become ministers in the future? Probably. Anyway, let me move on to my main points, and some general coverings.

How is math relevant in today's society? We see that math is the base of virtually everything. Important ministries such as the ministry of finance will need math to calculate certain figures, and some of them will be slightly more complex than the regular math problems, these formulas or solutions we have provided will then come into place. Imagine if math is non existent in the world of today. Even if we were in a economic crisis, we would have assumed that we were not, if math were not existent. And, there wo9uld be no statistics for leaders to study, and improve the country, and Singapore would not be the prospering country it is now. Simply because there many good mathematicians in Singapore, our economy is stable, our country keeps improving, and, in a school like ours, which aims to nurture future leaders, this is especially important to us. Without math, we would see that we would not even know what NE is, because we cannot think at all! Of course, I may be incorrect to say that all the problems we have raised here will be used directly in parliament, but then, in one way or another, they will be used indirectly, especially in the finance components, to see what economic status we are in now, and to improve our lives.

Thank you. I hope we have just delivered a teacher's dream.

I love mathematics

I love Mathematics. I love the challenge in dealing with every question. Lately our class is learning FACTORIZATION which I think I am going to be ‘indulged’ in!

Basically there are 3 fundamental BINOMIAL FORMULAS to bear in mind when solving them.

1 (a-b)(a+b)=a² - b²

2 (a+b)(a+b)=a²+2ab+b²

3 (a-b)(a-b)=a²-2ab+b²

I also found out some complicated formulas in BINOMIAL which I think we won’t be learning this year.

The cube of a binomial.

There are similar formulas to factor some special cubic polynomials:

Sums and differences of two cubes.

Here are two more formulas to handle special cases of cubic polynomials:

(a+b)³ = a³ + 3a²b + 3ab² + b²

(a-b)³ = a³-3a²b + 3ab² – b³

Say, we like to factor x³ + 8. By formula (6), we can write

X³ + 8 =x³ + 2³ = (x + 2) (x² – 2x + 4)

In this case the factorization is complete, since the polynomial x² – 2x + 4 is an irreducible quadratic polynomial.

These are some questions to ponder:

Exercise 1.

Factorize the polynomial x² + 6x + 9 completely.

Exercise 2.

Factorize the polynomial x² + 6x - 9 completely.

Exercise 3.

Factorize the polynomial x⁴ - 2 completely.

Exercise 4.

Factorize the polynomial 125 – 8x³ completely.

Have fun!!!

Algebra Math Quotes

"As long as algebra is taught in school, there will be prayer in school. " -- Cokie Roberts

"It is hard to convince a high-school student that he will encounter a lot of problems more difficult than those of algebra and geometry." -- Edward W. Howe

Arithmetic - Interest

What is interest?
Interest is a fee paid on borrowed assets. It is the price paid for the use of borrowed money or, money earned by deposited funds. Assets that are sometimes lent with interest include money, shares, consumer goods through hire purchase, major assets such as aircraft, and even entire factories in finance lease arrangements. The interest is calculated upon the value of the assets in the same manner as upon money. Interest can be thought of as "rent of money". For example, if you want to borrow money from the bank, there is a certain rate you have to pay according to how much you want loaned to you.
Interest is compensation to the lender for foregoing other useful investments that could have been made with the loaned asset. These foregone investments are known as the opportunity cost. Instead of the lender using the assets directly, they are advanced to the borrower. The borrower then enjoys the benefit of using the assets ahead of the effort required to obtain them, while the lender enjoys the benefit of the fee paid by the borrower for the privilege. The amount lent, or the value of the assets lent, is called the principal. This principal value is held by the borrower on credit. Interest is therefore the price of credit, not the price of money as it is commonly believed to be. The percentage of the principal that is paid as a fee (the interest), over a certain period of time, is called the interest rate.

Simple interest-
Simple interest is calculated only on the principal amount, or on that portion of the principal amount which remains unpaid.

The formula is as follows:

I = Prt

I = interest
P = principal
r = interest rate (per year)
t = time (in years or fraction of a year)

A real-life example:
My Dad deposited $400 earning simple interest of 4% per year. Calculate the simple interest at the end of one year and at the end of five months.

Solution: I = $400 x 0.04 x 1= $16.00 (substitute the known values)


What is Compound Interest?
When you borrow money from a bank, you pay interest. Interest is really a fee charged for borrowing the money, it is a percentage charged on the principle amount for a period of a year - usually.If you want to know how much interest you will earn on your investment or if you want to know how much you will pay above the cost of the principal amount on a loan or mortgage, you will need to understand how compound interest works.

The formula:

P is the principal (the initial amount you borrow or deposit)

r is the annual rate of interest (percentage)

n is the number of years the amount is deposited or borrowed for.

A is the amount of money accumulated after n years, including interest.

When the interest is compounded once a year:
A = P(1 + r)n

Annually = P × (1 + r) = (annual compounding)
Quarterly = P (1 + r/4)4 = (quarterly compounding)
Monthly = P (1 + r/12)12 = (monthly compounding)

An example for compound interest:

My Dad takes a loan of $30,000 from DBS for 2 months to start a business with his 2 other friends.The interest per year is 2%.He would pay once every month.So the total amount is $30,000 X 2/100 X 2/12 =$100 (2 months).By the nd of the 2 months, he needs to pay the principle+the interest=$(30000+100)=$30100.

Exciting Arithmetic - Percentages

In mathematics, a percentage is a way of expressing a number as a fraction of 100 (per cent meaning "per hundred"). It is often denoted using the percent sign, "%". For example, 45% is equal to 45 / 100, or 0.45.

Percentages are used to express how large one quantity is, relative to another quantity. The first quantity usually represents a part of, or a change in, the second quantity which should be greater than zero.

For example, an increase of $ 0.15 on a price of $ 2.50 is an increase by a fraction of 0.15 / 2.50 = 0.06. Expressed as a percentage, this is therefore a 6% increase.

Percentages are correctly used to express fractions of the total. For example, 25% means 25 / 100, or one quarter, of some total.

Percentages larger than 100 can be meant literally (such as "a family must earn at least 125% over the poverty line to sponsor a spouse visa").

Here are other examples:

1. What is 200% of 30?
   

Answer: 200% × 30 = (200 / 100) × 30 = 60.

1. What is 13% of 98?
   

Answer: 13% × 98 = (13 / 100) × 98 = 12.74.

1. 60% of all university students are male. There are 2400 male students. How many students are in the university?
   

Answer:400 = 60% × X, therefore X = (2400 / (60 / 100)) = 4000.

1. What is 13 of 20?

Answer:13/20=65/100=65%

1. A stall selling clothes of with a discount of 50% had no people buying its clothes,so it increased its discount to 99%. The original price of a piece of clothing is $200.

(a)What is its first discounted price?

(b)What is its final discounted price?

Answer= (a) 1/2 x $200=$100

(b) 1/100x $200 = $2

Acknowledgements:

http://en.wikipedia.org/wiki/Percentage

Actually Arithmetic- Speed

Speed is the rate of motion, or equivalently the rate of change of distance.Speed is a scalar quantity with dimensions length/time; the equivalent vector quantity to speed is velocity. Speed is measured in the same physical units of measurement as velocity, but does not contain the element of direction that velocity has. Speed is thus the magnitude component of velocity.



Types of Speed:

1. meters per second, (symbol ms-1; m/s)
   


2. kilometers per hour, (symbol km/h)
   


3. miles per hour, (symbol mph)
   


4. knots (nautical miles per hour, symbol kt)
   


5. Mach number, speed divided by the speed of sound
   


6. speed of light in vacuum (symbol c) is one of the natural units

What Is Average speed?

It is simply put into a formula:

Let r = average speed

Let D = total distance

Let t = total time

Here's your formula:

r = D/t

An example:

John wants to travel from City A to City B which is 900 Kilometres away from each other. He drives at a constant speed of 100 kilometres per hour. How long does he need to travel from City A to City B if he does stops and rest for 2 hours??

Answer= 11 hours

Solution= 900 km/hr divided by 100km/hr

= 9 hours

9 hrs + 2 hrs

=11hrs

Acknowledgements:

http://en.wikipedia.org/wiki/Speed

http://www.askmehelpdesk.com/middle-school/formula-average-speed-134278.html

Arithmetic - Proportion

In mathematics, two quantities are called proportional if they vary in such a way that one of the quantities is a constant multiple of the other, or equivalently if they have a constant ratio.

Inverse Proportion-
Two proportional variables are sometimes said to be directly proportional. This is done so as to contrast proportionality with inverse proportionality.

Two variables are inversely proportional (or varying inversely, or in inverse variation, or in inverse proportion or reciprocal proportion) if one of the variables is directly proportional with the multiplicative inverse (reciprocal)of the other, or equivalently if their product is a constant.

Direct Proportion-
Sometimes a change in one quantity causes a change, or is linked to a change, in another quantity. If these changes are related through equal factors, then the quantities are said to be in direct proportion.
For example, suppose that cans of soup at the store cost 50 cents, or $0.50, each.

Case #1:

Suppose that you buy 4 cans.You would pay $2.00.

Case#2:

You buy 8 cans. You would pay $4.00.

So, changing the number of cans that you buy will change the amount of money that you pay.Notice that the number of cans changed by a factor of 2, since4cans times2is8 cans.Also, notice that the amount of money that you must pay also changed by a factor of 2, since $2.00 times 2 is $4.00.Both the number of cans and the cost changed by the same factor, 2.When quantities are related this way we say that they are in direct proportion. That is, when two quantities both change by the same factor, they are in direct proportion.

In the above example the number of soup cans is in direct proportion to the cost of the soup cans. The number of soup cans is directly proportional to the cost of the soup cans.The formal definition of direct proportion:

Another example:

You had a container holding 6 quarts of a liquid, and that liquid weighed 3 pounds. If only 3 quarts remained, that liquid would now weigh 1.5 pounds. So, the volume of the liquid changed by a factor of 1/2, since it went from 6 to 3 quarts. The weight of the liquid also changed by a factor of 1/2 since it went from 3 to 1/5 pounds. Both the volume and the weight changed by the same factor, 1/2. So, in this example the weight and volume of the liquid are in direct proportion.

Acknowledgement:
http://en.wikipedia.org/wiki/Proportionality_(mathematics)
http://id.mind.net/~zona/mstm/physics/mechanics/forces/directProportion/directProportion.html

Amazing Arithmetic- Rate

In mathematics, a rate is a ratio between two measurements, often with different units. If the unit or quantity in respect of which something is changing is not specified, usually the rate is per unit time. However, a rate of change can be specified per unit time, or per unit of length or mass or an other quantity. The most common type of rate is speed, heart rate and flux. Rates that have a non-time denominator include exchange rates, literacy rates and electric flux.

In describing the units of a rate, the word "per" is used to separate the units of the two measurements used to calculate the rate (for example a heart rate is expressed "beats per minute"). A rate defined using two numbers of the same units (such as tax rates) or counts (such as literacy rate) can be expressed as a percentage or fraction or as a multiple.

Often "rate" is a synonym of rhythm or frequency, a count per second e.g. radio frequencies or heart rate or sample rate.

A question:

Jill lent Bruce 4800 for 5 months. At the end Bruce had to pay Jill an intrest of 85 dollars. What was the rate of simple interest per annum?

Solution:

Rate of interest

=100%($85.00/[$4,800.00{5/12}])

=100%($85.00/$2,000.00)

=100%(17/400)

=4.25%

Answer: 4.25%

Amusing Arithmetic - ratio

What is the definition of a ' ratio ' ??A ratio is an expression which compares quantities relative to each other.A ratio can comprise of two ,three,four,five or even more quantities but the most common examples involve two quantities, but in theory any number of quantities can be compared. Mathematically, they are represented by separating each quantity with a colon, for example the ratio 4:3, which is read as the ratio "four to three" as shown on the right.I n general, a ratio of 4:3 means that the amount of the first quantity is (four thirds) of the amount of the second quantity – this pattern works with ratios with more than two terms.

However, a ratio with more than two terms cannot be completely converted into a single fraction; a single fraction represents only one part of the ratio. If the ratio deals with objects or amounts of objects, this is often expressed as "for every four parts of the first quantity there are three parts of the second quantity".

If these two quantities are the only quantities in a particular situation, a good example would be the length and the breadth of a wall , it is sometimes said that "the whole" contains seven parts, made up of four parts length and three parts breadth.

In this case, or 4/7 of the whole is the length and 3/7 of the whole is the breadth. This comparison of a specific quantity to "the whole" is sometimes called a proportion. Proportions are sometimes expressed as percentages as demonstrated above.
Note that ratios can be reduced like fractions, so that the ratio 8:6 or 12:9 is identical in meaning to the ratio 4:3.

For example:

There are 6 apples and 90 oranges in a basket where there are no other fruits. What is the ratio of the apples to the oranges?

Answer= 1:15

Solution = 6:90

= 3:45

= 1:15

Thank You!!

We the Kings

Heya, new blog! But I am not a newbie now. And I, or rather my team mates will be blogging about math. So basic introduction: Of course there's me, blogking007, and here is my blog. And there's Sean, and here's his blog; there's Jeremy, here's his blog, and there's Yi Shin, and of course here's his blog.

So a few fun loving kids we are, as you can infer. But today we are just going to slack for a while, no math first, just basic intro. Next post, you will be very full. So that's about it.

See you again!