Math Challenge 24

Math does not purely involve writing mathematical expression .
Sometime what you need is some logically deduction base on, of course, some mathematical principles.

Below is one good example of "deduction" type of math solving.

Let start the challenge, and have some fun!

Above you will find 3 squares.

Do note that the 2 yellows are of the same area and 1 blue of area bigger than the yellow ones.

If the total area of the 3 squares are 57 sq cm, determine the area of the bigger blue square.

I believe you will enjoy this math question.

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Area Displacement Theory

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Maths is not all about calculation.

There are always more to it than meet the eyes.

This is especially true when you are doing geometrical questions where you are involved with area, perimeter and so on.

Displacement theory or its equivalent is always done without the knowledge of many people.

What is this theory about?

Let's look at one example below.

In the diagram above, you will see a path (white coloured) going across a blue platform.
If you are asked to find the area of this path, what can you do to obtain this area?

If no data of dimension is given, it is definitely not possible.

Now if the width of the path and the vertical length of the blue platform is given, can you compute the answer?

Again , this need a bit of thinking.

Displacement theory kicks in here. Look at the diagram on the right.

It is the displaced or closed up portion of the blue platform that does the trick.
Here you will notice the dashed line forming a white rectangluar area on the right-most side of the white blue platform.

Are you able to find the area of this white rectangular piece?
The width of this rectangle piece is ACTUAL the width of the white path!

You should now be able to calculate the area of this rectangular piece since the path width and length of the rectangular block is known or deduced now.

How this is possibe is through the "hidden" clue or step of closing up the path revealing the simpler rectangular area that any decent maths student can calculate.

Hence, maths is wonderful in that it tests you not only about applcations of maths tools, but your other "intelligence".

Having known displacement theory here, I believe you are really for the Math Challenge 23.
Go there and answer the question, and be quick before others grap the position one ... 

:-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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Hidden Clues in Maths Questions

There are different levels in any educational system.
This goes with the learning of mathematics too.

At various level of learning, you will be presented with different level of complexity.

At the elementary stage, you will be shown maths questions that are real straight forward type.
At intermediate, a bit of mind twisting has to be done to resolve any challenge.
At the highest level, the questions come embedded with hidden clues to be discovered by learners and used to continue with the solving process.

But hidden clues are now becoming the norm among intermediate level due to its benefits to prevent pure memorising of mathematical technique.

A example of this interesting "hidden clue" can be seen in my Math Challenge 23.

There anyone taking up the challenge needs another step in order to "see" through the simple trick of solving the issue.
(Note:  The challenge requires only one step to calculate the area of the path).

Multi-discipline is thus needed for merit of helping get the answer.
Knowledge in utilising maths tools and technique are not sufficient these days.

Maths students have to know some basic theory of motional replacement to understand Math Challenge 23.


Hence, to master mathematics, it will be good to read more, especially, topics outside maths.
This enlarge your understanding of real-life cases roped into maths questions.
 
Maths is interesting in this manner since it involves not only one learning discipline but encompasses more.
 
Enjoy maths. It widens your perspective of the world. 
 
:-)

Math Challenge 23

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Albert needed to create a path through a garden of his.

The garden has a size of a rectangle with length of 20m and width of 15m.

He intend to have a path of 3m wide.

His design is shown below.

But he has a problem.

He wanted to know what is the area of this path he is going to lay across the garden.

Can anyone help him calculate that area?

Basic geometry knowledge may helps.

Math Challenge 22

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Given the diagram of boxes below, determine, in the fastest possible way, the area of the dark blue region.

Assume the individual boxes to be 1 unit square in area.

Give your answer in the comment space, please.

Maths does not involve plain counting.

It involves some form of intelligence to get things going.

:-)

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Math Challenge 19

Math does not involve variables that we can see only.

There are the logical deduction type whereby you have to visualise and come out with an answer.

Below is one good example that I would like it to be a challenge.
(Don't be frightened by this, it is just for fun.....)

Above you will find a stack of cubic boxes. There are the blue and the yellow cubes.

Question:
What is the least quantity of blue boxes must we use in order to hold the yellow boxes in the same place?

You may present your answer and how you arrive at the answer in the comment section.

Thanks for trying. And I am waiting for that interesting mathematical deduction...

Maths Is Interesting!

Cheers :-D

Tricky Angles | Be Aware!

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Geometry in maths can means dealing with angles from a square or a rectangle.

Normally the question is to determine an unknown angle given some shape and angles.

However, mistakes can happen when basic knowledge of relationship between angles and  shapes are not proper understood.

Here, I will stress on the square and rectangular matters. This is basic but can pose a tricky problem to the unwarys. Poor thing.....

Let's look at the diagram below.

Here, if sides M and N are the same, that is, if the box is a square,  angle A will be 45 degree.

This is so since the corner where angle A lies is 90 degree divided EQUALLY by half due to the diagonal lines reaching to the opposite side. (symmetrical sides).

However, if the side M and N are not equal in length, then angle A WILL NOT be 45 degree. It will depends on the ratio of side M and N.

Note this message and unnecessary mistake can be avoided.

Sometime it is to test the logical thinkng through maths, by not telling you angle A is 45 degree but stating that the box is a square.

This type of maths problem will require you to calculate another angle but using angle A which is not given.

It is tricky but good to have. Your brain will be stretched to make it "flexible" for future use.

Maths is good in this sense as it twists our mind and makes our life interesting!

Work hard as well as smart.

For more examples on avoiding unnecessary mistakes, visit this time calculation post.

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Solving Maths Visually

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Maths is interesting!
There are many exciting concepts and techniques one can apply.
Read on ...

There are many ways to solve a maths question.
Normally it involves taking many steps with related sequence.

However, there are also simple ways to handle a maths problem.

One such solving method is simply through visual steps.
This has no working at all.

What do I mean?

Let's look at an example.

Example:
Determine the angle B from the diagram below. Angle A is 40 degree.

Solution:

Angle B is same as angle A = 40 degree.

There is no working at all. Just the visual determination.

Concept of this trigonometrical question in geometry:
When 2 straight lines cross and meet at an angle to each other, the angle opposite to any one is the same.

As such, angle C is also equal to angle D.

This is visual maths.

Interesting?

:-D

Geometrical Errors Can Be Exciting

The learning of geometry is to allow anyone to have an idea of correct perspective to objects.

Dimensions are important in this area. Angle of view is equally important.

Drawings of object in the fore-ground and background differs because of geometry.

When the concept of geometry is violated in drawing objects, you will get interesting outcome.

This outcome, however, is apparent only in the virtual sense and cannot be physically produced.

Example of links are quoted below:



http://anythinggorgeous.blogspot.com/2009/01/staircase-of-character.html



http://anythinggorgeous.blogspot.com/2009/01/cunning-overlapping-of-objects.html



http://anythinggorgeous.blogspot.com/2008/12/twisted-rods-or-fishy-rods.html



Geometrically deceptive objects are not easily identified, and have to be closely stared at to reveal their "wonders".

Only through learning maths and its relevant topics, you can then appreciate the importance of having done it. This is very obvious in the above few examples.

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Volume and Surface Area | Geometrical Relationship ?

Using geometry, we can determine the volume and surface area of any object.

However, have you wondered what is the relationship between them?

If we want to maintain the volume of an box but reduce its surface area, is it possible?

Or does the volume ALWAYS increases with increase in its surface area?

How about reduction in surface area? Will the volume also reduce?

Let us throw some numbers into an example to figure out the answers.

For simplicity, let us use a simple box. (Diagram 1)



<== Diagram 1 Box of diagram 1 has a : Volume = (2 x 8 ) x 4 = 64 cubic units

Surface area = 2 (2 x 8 ) + 2(4 x 2) + 2(8 x 4) = 112 square units


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Let modify the box (Diagram 2) to another dimension and see its geometric parameters.



<== Diagram 2 Volume = ( 4 x 4 ) x 4 = 64 cubic units

Surface area = (4 x 4) x 6 sides = 96 square units

Note, the volume remained but....
the surface area has reduced!

This saves material, right?

How about another dimension (Diagram 3)?



<== Diagram 3 ( A tall box!) Volume = (2 x 2 ) x 16 = 64 cubic units

Surface area = 2(2x2) + 4(2x 16) = 136 square units

What now?

The volume again remained,
but the surface area has increased!

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What is the conclusion from here?

It is observed that althought the volume of the box has not changed, their surface areas has changed. The direction of change( increase or decrease), however, is on a case by case basis.

This concluded that there is no relationship between volume and surface area of any object.

Therefore don't be tricked into reckoning that surface area increase will cause a definite increase in volume. Likewise for reduction also.

Hope you gain much from here.

Geometry is exciting and mysterious at the same time, right?

For relationship comparsion between another set of geometrical parameters, perimeter and area, click here.

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Who Says The Compass Triangle Is Right-Angled?

A simple way to obtain a right-angled triangle is to draw a triangle with its corners touching the edge of a circle with its diameter as one side of the triangle.

Look at this post for details on what I am talking (or rather writing) about.

However, the question is how do we know it is truly a right-angled triangle?

The answer is "Go back to basic and prove it is!".... (or it is not)

This ability to prove will enhance your understanding of its principles.

Let the spirit of the mathematician starts flourishing ...

A diagram to simplify explanation is shown herein.

Our target: To prove that Angle C is 90 degree.



Two key items exist here. They are:

1) Length DA = DC & Length DC = DB,
since they are the radius of the circle

2) Angle C1 = Angle A & Angle C2 = Angle B,
due to the facts in item 1) above.

These 2 key points will ensure that angle C is 90 degree (or right-angle).

Moving on ...

Angle d2 = 180⁰ - Angle C2 - Angle B = 180⁰ - 2 x (Angle C2)

Angle d1 = 180⁰ - Angle d2

Which makes Angle d1 = 180⁰ - [ 180⁰ - 2 x (Angle C2) ]

This produces an important observation: Angle d1 = 2 x (Angle C2).

Also Angle A + Angle C1 + Angle d1= 180⁰, which also means,
===> [ 2 x Angle C1 ] + Angle d1= 180⁰.

Re-writing the above gives [ 2 x Angle C1 ] + 2 x (Angle C2) = 180⁰

Factorising the above equation yields:
2 x (Angle C1 + Angle C2) = 180⁰ ---(1)

Hang on, we are coming to the end ...

(Our target) Angle C = Angle C1 + Angle C2

From equation (1),
we can show that Angle C = Angle C1 + Angle C2 = 90⁰!

Therefore, it is true that
Compass Triangle has Angle C as right-angle.

Hurray!! Eureka!!

Fall back to maths basic if in doubt. The skill picked up will serve you good in the application of maths theory.

:P

How To Draw Right-Angled Triangle Using Geometric Compass

We can draw a right-angled triangle using various ways.

One way is to make use of a protractor.
Another way is to use a try square.

But what if you have only a geometric compass, how do you do it?
(Not the compass that give you directional bearings of north, south, east and west!
It is the one that let us draw circle with.)

Don't worry. The trick is simple!

The steps are :

1. Draw a (light) circle to serve as boundary


2. Draw a line across the diameter of this circle to serve as one side of the triangle


3. From the 2 ends of the line drawn in step 2, produce 2 more lines, joining them at any point along the circle boundary.


4. Smile! You have created a wonderful right-angled triangle true to target.

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See the diagram 1 beside for an overview of the steps used.

Diagram 1 ===>

Try out this method and see for yourself!

Proof using a protractor or try square at angle C to verify the 90⁰.

Enjoy yourself! Geometry is fun! :)

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Exciting Area and Perimeter in Geometry

In geometry studies, you would come across, definitely, the terms "area" and "perimeter".

I suppose, you are also good at calculating their numerical value.

However, in the midst of doing the many calculations and heavily involved with different funny shapes, the fundamentals of geometry have to be maintained. The principles are always the main issue while studying mathematics. Take a thinker below.

Is perimeter related to area in any way?

Does the perimeter expands with increase in area?

Does the area shrink with decrease in perimeter?

These questions serve to address understanding of these 2 key geometrical parameters.

If you are able to answer them correctly, give yourself a pat.
Otherwise, give yourself a slap!

Let's look further into them, if you have any queries.



The 2 diagrams to the left has the SAME area, but look at their perimeter. One is longer than the other even when the areas are the same.

Amazing right?



Now, look at the 2 diagrams on the left. These 2 shapes has the SAME perimeter, but how about their areas?

The areas are highly different!

Exciting area and perimeter relation, right?

Do note, however, that the above 2 cases are just to illustrate concepts of area and perimeter in geometry. There are also normal cases where increase in area results in increase in perimeter, and likewise for decreasing condition.

Stick to principles and understanding is assured. Cheers!

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Why Is Net Important In Geometry?

Geometry does involve objects in the 3-D perspective.

The height factor of the geometrical outlook is presented in the 3-dimensional view.

But does it really present the object with a complete understanding of its shape?

There may be assumptions made if the angle of view is not appropriate.
The assumptions may be wrong!

Let me cite one example below.



From looking at the shape presented on the left, can you tell whether the shape is that of a tetrahedron or pyramid?

Rather difficult isn't it?

It is almost impossible to tell!

Therefore, NET comes in useful in this aspect.

So, what is this net about?

Net is a different way of presenting shape. It utilises only 2-dimensional view. It is actually a 3-D object "flattened" into 2-D perspective.

The usefulness of net can be seen from the answer it presents below using the above object as reference.



Here, if the net of the object is shown, as compared to its 3-D version, the message is clearly presented.

If the object is a tetrahedron, then the net is that of the left.
If the object is a pyramid, then the net is that of the right.

Thus, net in geometry serves its purpose in showing a clearer idea of shape when the 3-D perspective is not sufficient to reveal all details.

One common application of net is in ORIGAMI, an interesting paper folding activity liked by many.

:)

Pyramid - An Ancient Shape

Geometry has lasted a long time ago.

The Egyptian Pyramids of Giza in Egypt is a famous example.

What is so special about the shape of this pyramid?

It differs from other shape in that, besides the base, all its other surfaces end up in a vertex, or pointed tip at one end.

Actually this pyramid is closely related to the prism. In fact, pyramid is part of the prism, and therefore is mathematically linked.

The volume of a prism is given as
"Area of Base x Height".

The volume of the pyramid is thus given as one-third that of the prism.

In other words, volume of pyramid = (1/3) x (Area of Base) x Height.

For more information about this volume relationship and why is it one-third, you may read Volume of Common Solids.

To better understand pyramid, test yourself with the question below.

Look at the diagram below and tell which volume is the biggest.
Note that they have the same base area and height, but has different slanting angles.



Any ideas? Read on for the answer...

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The answer is that they are all equal in volume.

This is because the angle of the slanted edge does not play a part in the computation the pyramid volume. The formula tells the "story".

After understanding the concept of pyramid, do you have a better admiration for the "Great Pyramids" of Egypt?

I sure do! :)

Why Learn About Surface Area?

One key topic in geometry is the calculation of surface area for a certain shape.

Why the need for this knowledge? Where is this parameter of Surface Area applied?

Look at the pictures on the right.

Do you see them often? I believe so!

Here, any maker of these items has to know the surface area in order to purchase material to produce them. Geometry calculation in this aspect becomes important for him.

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How does he determine the surface area for the cone?





Given the dimensional specification of the cone, he can spread it out as shown in the diagram on the right.

The spread-out surface area of the desired cone consists of 2 pieces:-
- A base that is a complete circle
- A sector of a larger circle that forms the side of the same cone.

Why spread out the 3-dimensional object into its flat surface (2-dimensional view)?

Advantages are that the ease of computation become apparent and all surfaces to be calculated will be exposed, reducing the chance of missing any surfaces of the object.

For more information on the computation of the surface area of a sector, click here.

Geometry is a very useful mathematical tools in our daily life. Its usage is aplenty and knowing it becomes a benefit to mankind.

Maths is genuinely wonderful! :P
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Surface Area and Volume | Their Difference

Geometry involves many interesting analysis. One of the key items is the computation of Surface Area. Another is the Volume of various 3-dimensional shapes.

What is the main difference between these 2 parameters of geometrical study?

Surface Area: The study of total areas on the outer surface. It is what user can feel.

Volume: The study of how much the shape (or object) can contain within its boundaries.

Let me explain more using a simple cube.


Let me start with surface area of the cube (diagram above).

It has 6 equal surfaces (being a cube for this example only)

If the length of one edge is, say, A units, then the surface area of ONE side is A² unit².

Total surface area = 6 x (A²) unit² = 6A² unit².
This is the total area of all surfaces that any person can feel (physically). OK? Clear?

If Surface Area is understood, then let's move on to Volume.

Volume is the amount of stuffs we can fill the object in question with (totally without gaps).

Volume for the cube (above) = width x length x height

(This formula is specific to cubic structure).

Interpretation of Volume, from the formula, is actually,
Area x Height!

Answer of Volume for the cube = A x A x A unit³ = A ³ unit³ .

Note: For the same object, the numerical value for surface area may not be smaller than its volume.

Common mistakes made:
- Missing out on some surfaces (or inability to visualise ==> need more practices)
- Wrong unit dimensioning (Area: unit² and Volume: unit³)

:P

Sector Area In Circle | Calculation Concepts

In geometry, there are occassions where you have to compute the area of a sector belonging to a circle.

What is this sector about?

Sector is an area of a part of the complete circle. It is not a length!

To understand more, let us do a maths question on finding the sector area.

Question:



Find the area of the green sector given the ratio of the length of the green outline to the whole circle outline is r /L.

("L" being the radius of the circle)

Before I present the solution and steps in achieveing the answer, I would like you to understand the principles and concept of each steps. This way of studying will resort less on memory but more on long term retention of knowledge.

Solution:
Area of a sector is given by the formula S = (1/2) x (radius)² x q,

where q, is angle in radian (unit).

Click here for more information on sector area.

Since length of the sector outline is given, we need to understand the property of this parameter. This parameter is also term "Arc" length.

Arc length is given by the formula Length of Arc = r q.


Here, you see that the arc length is proportional to the angle compassing the arc or sector width. (Property of Arc Length)

In the question, the ratio of the outlines is given as r / L.

This also meant that r / L = q / 2 p. === (A)

Why?

Because "r" is proportional to the angle q, and "L" is the angle of the whole circle.
(Refer to the Property of Arc Length).

From the relation (A) above, you can re-write the expression as q = (r/L) x 2 p.


Replace this new q expression into the sector area formula.
Sector Area S = (1/2) x (radius)² x q = (1/2) x L² x (r/L) x 2 p

= (1/2) x L x (2pr)

In another words, you can say that the Sector Area is determined by :

half the radius of the large circle multiplied by the circumference of a circle with radius "r", "r" being a hidden parameter that equals the sector's arc length.

How's the going? Tough?

Hope not. But if it is, do not despair.

Review the concept a few times and it will be clearer.

Remember, knowing and mastering the concepts and principles has a long term benefit than that of pure memory of the Sector Area formula. Cheers!

:)