Caution on Mixed Number

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Fractions are a necessary part of maths.
They come in many forms; improper, proper and mixed number.


Though improper and proper forms are direct in its presentation and interpretation, mixed number form may pose a potential mistake for young learners.

Example:
   
Is this 2 + (3/4) or 2 x (3/4) ?

Caution has to be taken to stress it as 2 + (3/4).

Some students have taken it to mean 2 pieces of (3/4) !
Dangerous isn't it.

But rest assure.
If you understand the language of maths and its "grammar", all will be well and interesting.

:-)

Using Units to Deduce Maths Formula

There are times when we cannot remember some simple formula for a maths application.

Or we have doubts to the some maths working  especially when many parameters got involved.

I ave a simple tip.

Look at the units for the numerical item.

Example:

To calculate distance travelled by a vehicle, given the speed it goes  and time taken,

we look at the speed's units.

Unit:  m / s

What does it tell?

Yes, it gave an indirect answer that speed = distance / time.

Thus if time is given, we are able to know that we just need to multiple speed by time in order to retain only the distance.

(m / s) x s  =  m (only)  ==> Distance

The above allow us to use units to deduce the working (and formula).

Hence, we should not overlook the power of knowing units.

It is simply disappointing to sometimes see people missing out on writing the units for certain parameters. Maths loses its value simply by ignoring this step.

Therefore treasure this little but powerful "units".

:-)
Maths is interesting!

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Simultaneous Equations - Decimal Numbered

In the learning of maths, questions grow challenging as one progress upwards.

One such example is the solving of simultaneous equations.

The easy type:

Solve for x and y using elimination method.
4x + 3y = 10 ---- (1)
3x + 4y = 11 ---- (2)

For the above problem can be solved easily by selecting a coefficient to be commonised.
The post of elimination method is reference here for review.

But moving on (higher) with more challenging maths question ... we may get the below.

0.4x + 0.3y = 1 ---- (A)
3x    +    4y  = 11 ---(B)

What should we do next?

Equation (A) may seems unusual.  It is in decimal form!

But as the blog title claims "Maths Is Interesting!", we should not be worried.

This type of question is actually not new in concept or tricky as it seems.
It is there to test you understanding by being "different".

We have to remove the "catch", which is to change the decimated number to integer.
How we do it here is simply multiplying the coefficients by 10.

This makes equation (A) to be  4x + 3y = 10  (back to the original first set of simultaneous equations at the start of this post.

The above example serves to illustrate the simplicity of changing numbers to suit the condition for easy solving. (Other questions may be multiply by another decimal number, or integer).

Just have a clear mind and a confidence attitude will be enough to allow you to solve most of the maths questions.

Try it and you will believe what I say (or write).

Maths is interesting!

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Decimal Number Simplification

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Algebraic expressions and equations normally come in integer or fraction form.

Examples:
1)  4x - 3 = x
2)  (3/4)x - 3x = 1/(3x)

Simplification of the above examples will not pose much of a problem except maybe in the challenge of bringing the numbers and unknowns over the "equal" sign.

But algebraic equations can come in decimal form too.

Example:
0.3(0.2x - 1) = 0.1x

How do we go about solving the above "decimated" algebraic equation easily?

A simple trick that I can think of (or maybe too simply a technique to call it 'trick").

What I would do is to multiply the expression on both sides by 10.
The idea is to bring the decimal number into the integer range.

BUT do note that the expression on the left side has two decimal numbers.

As such I would have to "x 10" twice.
This means that there is a "x 100" on the left and right side.

The new equation will thus be:

3 (2x - 10) = 10 x
==>  6x - 30 = 10x
==>  -30 = 10x  -  6x  = 4x
==>  x = -30 / 4 = -7.5

Conclusion:
Decimal can be seen to be intimidating when in the decimal form. However, it can be elevated to the familiar integer form through simple multiplication.
However, do take note of how many decimal number has been multiplied.
Left and right sides of the equation has to have the same number of multiplication (or division) to stay equal and valid.

Maths is not that frightening.
It can be interesting, if the method to "attack" it is properly done.

:-)

Basic Decimal Conversion

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Every thing falls back to basic.

If the fundamentals are weak, any maths learners will have a hard time moving forward in their maths learning journey.

Let me quote an example.

How do we change 4.75 to fraction.

We can use 475 / 100 and reduce it through long division.  This will give 4 and 3/4.

However, if we know that 4.75 is actually consisting of 4 add to 0.75, the conversion will be simpler.

4 and 0.75 means 4 + (3/4) which leads directly to 4 whole and 3/4.  Same as answer of above.
(NOTE:  0.75 is a quarter which equates to 3/4).

Simple isn't it?

Maths is interesting.

:-)

Maths Symbol in Our Applications

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There are many symbols in maths.
To learn and understand maths, we need to know the meaning of the symbols.

This is very much like talking to a foreigner. Without understanding each other's language, no communication can be carried out (other than the international body language!)

Hence, knowing the usage of the symbols in a mathematical expression helps.

But is it really so?

Partially.

Why do I say that?

Yes, you may know the symbol while doing maths, but if the same symbol is used elsewhere, do you still understand?

One example is:

y = x + 1
This means x is added by one and their total is represented by the variable "y".
This is for the maths operator "+".

But what about the expression x++ ?
This looks odd, isn't it?

To the maths learner, this may be a typo error, or something is missing.

"x++" is actually commonly used in C programming.

What it means is   x = x + 1.
It is a short-cut way of writing the addition of x and replacing it by the same variable "x".

Thus this example showed the use of "+" in another application.

It is still maths in some sense, but written in another form.

Maths is therefore always around us. It is a matter of us applying them and understanding them.

Only by learning their "language", can we communicate with them.

Interesting?  I bet you agree!

Other applications can be " += ", " :-) " and " x>>4 ". 
Can you find their meaning?

:D

Tips on Avoiding Mistakes (Unit writing)

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Maths involves many traps.

Any one of this traps will make the solution looks odd or even to the extent of wrong answer.

What are this traps ?

Mathematical operators, symbols, units, transferring of numbers, size of the written symbols, decimal points are some of the examples of traps contributing to the error.

Here I would like to mention about "unit".

In maths, calculation of items are aplenty. One of them is the study of speed.

In the topic of speed, students are dealing with three basic elements.
They are the distance, time and their ratio (speed).

All these three elements have different units all to themselve.

Distance ==  metre
Time ==  second
Speed == metre / sec

There are variations of the above.
km, mintues, hours, km / h, m / min, etc

Do you now see the danger?

If you are dealing with so many units in one maths question, what are the chance of making mistakes?
If you are careful, the chance is low, but it does not mean zero.

You still have to be careful.

How to avoid having mistakes due to this undesired slip?

One tip is to write down the units in the working steps.
Do not leave the numerical answer (in the working) without any unit indicated.

Make clear the item of interest, whether it is distance or time by reflecting the unit besides the number.

Example:  5 km,  40 sec.

A complete maths example will push the message across, thus ....

Example :
Alan travelled at a speed of 60 km / h for 2 h. After that, he slowed down by 20 km / h and travelled the last quarter of the journey at this new speed. How long did he take to travel?

Working:
60  x20 = 120
120 / 3 = 40
60 - 20 = 40
40 / 40 = 1
2 + 1 = 3
Answer: 3 hrs.

What is your comment on the working?

I personally feel uncomfortable.  What about you?

The danger in that sort of working is the lack of showing the actual item in the calculation.
It does not allow a good way for checking after completing the worksheet (if many maths problems are within).

Clearly writing the units will, at least, make checking later an easier task.

It also allows the marker (teacher) a clearer picture instead of guessing what you intend to show.

Along the way, during the working, you will also have a lesser chance of getting confuse as the items are listed with the proper message (through the units).

So are you convince proper unit presentation is worth the while?

A pointer for your thoughts.....

Cheers  :-D

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Percentage Increase in Area

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Is there any formula or maths expression showing the ncrease in area when its length and its breadth are increase by m% ?

If you cannot find one, it does not matter. You can easily derive one!

Let us work on this and show the others how simple maths can help us solve daily issue.

Let the length be x and breadth be y.

If x and y increase by m%,
length becomes x + x(m/100), and breadth becomes y + y(m/100).

Area is length x breadth.

Thus new area becomes  [ x + x(m/100)] [ y + y(m/100)]

This gives us an area of  xy + (m/100)xy + (m/100)xy + (m/100)(m/100)xy.

From the above maths expression, we can deduce that increase in area is:
2 x m%  +  (m%  x  m%)/100

Example with numbers will convince readers better, therefore ......

Example

If the increase in perimeter is 10%, what is the increase in area?

Answer is 2 (10%)  + (10%  x  10%)  /100  = 20%  +  1%  = 21%

Easy isn't it? 

For other post related to this concept in percentage increase, see the post Percentage Increase in Perimeter.

Maths is interesting. 

;-D

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Percentage Increase in Perimeter

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Percentage is a nice and mystery word in maths.

Why do I say that?

Look at the example below:

If the perimeter has increased by 30%, does the length also increases by the same amount?

The answer is obviously YES.

Next,

If the perimeter is increased by 30%, does the area covered by it also increases by the same amount?

???  The answer needs some pondering, right?

Answer to this:
If the perimeter is increased by 30%, the length and width will both increase by 30%.
This makes the area increase by 2(30%) + (30% x 30%) = ?

(I will explain this maths calculation in a later  post.)
For now, let's concentrate on the maths operation.

What do you get from 30% x 30%?
30% = 0.3
Thus 30% x 30% = 0.3 x 0.3 = 0.09 = 9%

This is a potential mathematical  mistake.
Error:  30% x 30% = 900% !

So, increase in area becomes 60% + 9% = 69%

Interesting how the mind works.

If the mind is not clear when doing maths, common mistakes do occur.
With more practice, however, this form of mistakes will be lesser.

Hence, be careful when dealing with parameter such as perimeter, length and AREA.
Know their relation and be aware of the "catch" when this type of maths question is being asked.

Do not fall for the maths trick.

:-)
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Math Challenge 21

Some math questions are simple and can be answered easily.

See this math challenge 20 and its answer in the comment.

But a twist of the questioning will and can make it more challenging without changing the math expression.

Below is one:

Base on math challenge 20, if all unknowns CANNOT be repeated, what are they?
Again they are integers and below 10.

Enjoy the math thrill answering this.

Merry Christmas!
^.^

Math Challenge 20

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Can anyone come out the answers for A, B, C and D in the below math expression?

A² + B² + 2C² = D²

The rule is that the unknowns are all integers and below 10.

Happy trying, and
don't forget that maths is interesting!

:-)

Percentage | Common Mistake

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Percentage problems can be tricky at times when you are careless.

Let's us look at one maths word problem related to it.

Question:
There are 3 persons, John, Mary and Jane.
John is richer than Mary by 10%, and Mary is richer than Jane by 20%.
Is John richer than Jane by (10 + 20)% = 30% ?

Most students, upon quick thinking, will acknowledge that 30% is the correct answer.

Is it so?

To verify the answer, let us assume that Jane has $1000.
As such, Mary will have (100+20)% of $1000 = 1.2 x $1000 = $1200.
John is then 1.1 x $1200 = $1320 richer than Jane ==> By 32%.

If 30% is correct, we should get 1.3 X $1000 = $1300.

The latter number (dollar) is not the same as the first worked out solution.
Why?

Mistake in understanding what is percentage:
To assume that John is 10% + 20% richer than Jane is incorrect.
This is due to the fact that percentage has to take a common reference for this to be correct.

In the word problem, the percentages of comparison are not to a common reference.
The first one is to Mary, while the next is to Jane.

These made the denominator of the ratio different.
Thus adding the percentage up is a mistake, and an easy one too!

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Square Root | An Exciting Outcome

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Everyone knows what a square root is for and what it does to a number.

But have you tried multiple square rooting ?

What do I mean?

Let's take an example.

Start with a number, say, 3.

i)  Square root this 3.
ii) You will get a number after step one.
iii) Square root this new number again.
iv) Continue the steps above and look closely at the number.

Findings: 
You will notice that the number after multiple square rootings, will give you closer and closer to or approaching the number "1".


Now let's start with another number. This time, let's choose, 0.4

After doing the same procedures, you will again notice that the answer is getting and approaching the number "1"!

Amazing isn't it?

What other wonders can you find from this Square Root maths operator?
Share with me in the comment section.

Maths is interesting.........

:D

What Does "of" Means In Mathematics?

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Reading maths question is a critical skill to learn.

Without understanding the question, you will not be able to solve the challenge correctly.

A typical maths problem uses the word "of" to express ratio.

What really does this simple word means?

To an adult this is the understanding of English.
But to a kid, this is not English but a maths question!

What then is "of" in maths?

Answer:  It means MULTIPLY.

Examples:
half of the time ==>  0.5 x time

2/3 of the class ==> (2/3) x (class total)

If 40% of the apples are rotten, how many are left? ==> 0.4 x apples are rotten

Learn English well to handle maths.

This simple word "of" may make you happy or sad.
The choice is yours.

But remember, "Maths Is Interesting!".
So despair not, enjoy your maths.

Cheers  :-D

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Time Mathematics | Tips On Avoiding Mistakes

Dealing with time and its calculation can be a bit tricky sometimes.

There are the hours, the minutes and the seconds to handle.

We can add or subtract them.
We do not operate in the hundreds, or ones, like any simple arithmetric.

We are calculating in terms of sixties.
1 hour = 60 minutes
1 minute = 60 seconds

Hence when given a maths question on subtracting two times, how do we go about to avoid mistakes?

Let's take an example to illustrate.

Question:
John start his journey to the market at 09:15am. If he arrived at the market at 10:05am, how long did he travelled?

Approach 1:
We can use the conventional method of carrying back 60 to the minutes, since the ending minute is smaller than the starting minute. And reducing the 10 to 9 (hour).
Next, we can then subtract with the new numbers.
That is 9:65 - 9:15 = 50 minutes.
This answer is fine.

Approach 2:
Change all the times to mintes.
10:05 ==> (10 x 60) + 5 = 605
09:15 ==> ( 9 x 60) + 15 = 555

Subtracting the two new numbers gives 50 minutes directly.
And that is the answer!

Remarks:
Comparing approach 1 with approach 2, you would notice that the latter seems to be simpler.

Why so?
This is because we have simplified the mixture of hours and minutes to only one dimension, that is, the minutes.
Thus, subtracting the newly created numbers involved only the minutes, avoiding the distraction of handling the hours.

To do maths, clear the mind of the unnecessary.

In the example above, we have removed the "hours" factor to focus purely on the "minutes".

Maths is interesting in that it is up to us to "play" with the techniques.
We can work with it or go against it.
The choice is up to us to select.

Hope you pick up some tips to make maths interesting.
Cheers!

:-D

Simple Tricky Maths Pattern

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Maths forms interesting patterns if you care to look around.

One such example is highlighted below:

12  11  10  09  08  07  06   <== row 1  (decreasing by 1)
01  02  03  04  05  06  07   <== row 2  (increasing by 1)

If you add the numbers in row 1 to that in row 2, you will get, surprisingly, the same answer throughout!

13  13  13  13  13  13  13

Why is it so?

It is a simple trick to the unwaries, actually.

Maths is not that difficult, to start off.
The answers to the above additions seem to be accidental in having the same number.
It is actually not accidental if you think abit.

The answers were intended to be 13 to begin with.

13 = 12 + 1
13 = 11 + 2
13 = 10 + 3  ....  and so on.

Though the additions seems magical with one row in ascending mode while the other row in descending mode, it is the visual maths trick that corrupts and confuses the mind.

If your principles of mathematics is good and strong, you will break through this simple trick in no time.

To excite young minds, this is a good one to try on them.

Happy playing with maths.
Maths is interesting!

:D

Misconception of Percentage

Maths is interesting!

 

It catches you nicely when you are not aware of its full implication.

 

Many of its techniques and concepts has wide boundaries.

 

Any maths learners has to understand its fundamentals and basic concepts in order to "escape" being caught with wrong usage.

 

Percentage is one area I wish to mention in this post.

 

You may find that percentage or percent is a simple term in maths.

 

Besides being a ratio, it is a comparative element.

 

It tells how big the target is to the original, or how small it is.

 

Misconception of Percentage:

 

The numerator is always smaller than the denominator (which is the total).

 

Is it true?

 

A definite NO!

 

Reason:

If a ratio has its numerator less than the denominator ==>  The numerator is relatively smaller in size than the total.

If the numerator is larger than the denominator ==> The numerator is bigger in size than the total.

An example can illustrate the concept.

Example:
If a costume is now priced at 90% of its original, $100, it means that the price is not only $90.
It is lesser than the original, since the ratio is 90 / 100 or 0.9.

If the costume is newly priced at 120% of its original, $100, it means that the price is now at $120!
A price value more than the original.
It is a practical real-life number.

• Percentage can be more than 100%.
• The numerator can be more than the denominator.

Understanding this concept will serve anyone good. Things in life goes up as well as down. Percentage reflects this sense through its number.

Interesting? You bet.

:)  Happy maths learning.
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Usage of Percentage

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In maths learning, every learners will come across the terms "percent" or "percentage".

Percentage is a ratio.
It is a ratio between two numbers.
The denominator is normally the total of an item.

The unit used is %.

The above may be common knowledge for anyone doing percent maths problems.

However, what is the presentation of the solution that is appropriate?

Let's us do an example.

Example:
Class A has 30 students. If 40% of them are girls, how many are boys?

Solution:
Number of boys in the class = 30 - 40%

Now, is this "30 - 40%" correct?

The idea may be there, but someting is amiss.

What is this "40%"?
Can we just write 40% as it is?

The answer is NO!

Percentage is a ratio of the total (the class size of 30).
We cannot simply write 40% if we want to know the actual number.

The correct way is to present the step or working as:  (40/100) x 30 = 12 students

We cannot simply write :   30 - 40%.
The correct way has to be  30 - (40/100) x 30 = 30 - 12 = 28 students.

2 mistakes, in concept, were made:

1) Number cannot subtract a percentage.  Their units are different.
    
2) Percentage is a ratio, an indication of the proportion of a piece to the total. It is a relative term, not an absolute number. Thus, an absolute number cannot operate with a relative term.

Understanding this concept of "percent" and "percentage" will be handy and avoid the unnecessary trouble and anxiety of doing maths.

Tips: 
Thinks of a slice of cake when doing "Percent". It has meaning only when compared to the whole cake.

Remember: Maths Is Interesting!  And it WILL be get more interesting. 


:D

Average Explained

When you encounter the word "average", it means that all the items within the consideration has the same number.

Example: 
When 3 tests has an average score of 80 marks. All the tests are 80, 80 and 80 marks each.

But nte that the three 80 marks each may not be true.
It is assumed to be to make the average correct.

The actual marks may differ from the 80 marks.

They may be 70, 80 and 90 actually, but their average is 80marks.

You may see the post on averaging if you need more information.

:D

Meaning of "Average"

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I managed to see a word problem shown below:

" Mary obtained an average of 80 marks for 2 tests. What marks has she to get for an average of 80 marks for 3 tests?"

If a maths student understand the meaning of "average", she can do this without even working out the mathematical steps.

The first statement in the question stated that Mary scored on both tests an average of 80 marks. This implied that for one test, she obtained 80 marks. Same as for the second test.

For the next test (the third one), to get an average of again 80 marks, it meant that all the test she has to get 80 marks each. This is the power of "average". That is to say, all the test can be concluded as the same score.

The actual differences between the individual marks can be offset from each other to achieve a final "average" of same marks (here, 80 marks).

The maths problem is to test the understanding of the word "average" in maths, and its concept.

Thus, knowing concepts do help in solving maths questions.
It does not mean working out the mechanical computation steps to get answers.

With strong maths concept, anyone can solve simple problems without doing working.
But that is maths also (in the brain, the step-less way).

Interesting. Maths is so. Fear not.

:-)