Saturday, March 17, 2012

Economics: Interest

Interest: amount of money earned by given principal money.
Borrower's Viewpoint - interest is the amount of money to be paid for the use of a borrowed capital.
Lender's Viewpoint - interest is the income generated by the capital that was lent.

  • Simple Interest - varies directly with time since it is computed after or at the end of the invested period
    • Ordinary Simple Interest - base on one banker's year
    • Exact Simple Interest - based on exact number of days of a year, which considers ordinary year of 365 days and leap years that have 366 days
Formula: Future Worth = Present Worth + Interest Earned 
(F = P + i)

  • Compound Interest - computed every end of each interest period which is called compounding interest and the interest is added to the principal (which is now interest plus principal) that will produce a bigger interest 

Formula: Future Worth = Present Worth (1 + Interest Earned) ^total number of compounding period 
(F = P [1 + i]^n) 

Friday, March 9, 2012

Engineering Mechanics: Introduction

Engineering Mechanics - is a field of science that deals with forces and its effect on rigid bodies. Engineering Mechanics are subdivided into two parts:

  •  Statics - rigid bodies remains at rest while absorbing the effects and distributions of forces
    • Force System - any arrangement where in two or more forces act on a body or on group of related bodies. 
      • Three Major Divisions of Force System
        • Concurrent - all forces pass through a common point
        • Non-Concurrent - all of the lines of action of the forces in this system do not meet at one point
        • Parallel - forces whose line of action are parallel, in same or opposite direction
    • Applications - trusses, centroids, friction
  • Dynamics - rigid bodies motion caused by the force applied to it, it deals with bodies in motion
    • Kinematics - is the geometry of motion, the motion of a particle without considering the forces causing the motion
      • Motion of Particles
        • Translation - motion of rigid bodies where in a straight line pass through any two of its particles always remain to be parallel on its original position
        • Rotation - motion of rigid bodies where in the particles move in circular paths with centers or axis of rotation on a fixed straight line
        • Plane Motion - motion of rigid bodies where in all particles in the body remain at a constant distance form a fixed reference plane
    • Kinetics - relates the force action on the body to its mass and acceleration
      • Newton's Law of Motion
        • a body at rest will remain to be at rest or in motion will remain in motion along a straight path unless acted upon by unbalanced force
        • a particle acted upon by an unbalanced force system has an acceleration in line with and directly proportional to the resultant of the force system and inversely proportional to its mass
        • in every action there is always an equal and opposite reaction
      • D'Alembert's Principle
        • the resultant of the external forces applied to a body rigid or non-rigid composed of a system of particles is equivalent to the vector summation of the effective forces acting on all particles


Wednesday, March 7, 2012

Solid Geometry: Polyhedrons

Polyhedrons - solids whose faces are plane polygons.

Regular Polyhedrons - are polyhedrons that have identical faces. There are only five known kinds of polyhedrons.

Let:
a = length of the edge
n = number of vertices
f = number of faces
m = number of polygons meeting at a vertex

  • Tetrahedron - four faces - a pyramid
    • faces: 4
    • edges: 6
    • vertices: 4
    • number of polygons meeting at a vertex: 3
    • surface area: a square * square root of 3
    • volume: a cube / 6 * square root of 2 
  • Hexahedron - six faces - a cube
    • faces: 6
    • edges: 12
    • vertices: 8
    • number of polygons meeting at a vertex: 3
    • surface area: 6 * a square
    • volume: a cube
  • Octahedron - eight faces
    • faces: 6
    • edges: 12
    • vertices: 8
    • number of polygons meeting at a vertex: 4
    • surface area: 6 * a square
    • volume: a cube
  • Dodecahedron - twelve faces
    • faces: 12
    • edges: 30
    • vertices: 20
    • number of polygons meeting at a vertex: 3
  • Icosahedron - twenty faces
    • faces: 6
    • edges: 12
    • vertices: 8
    • number of polygons meeting at a vertex: 5
For any Polyhedron:
Number of Edges: nf/2
Number of Vertices: nf/m

Tuesday, March 6, 2012

Tips and Tricks: Multiplication


The multiplication table of 1 is a no-brainer, BUT, let's face it! even adults sometimes are having a hard time doing numbers in their heads!

I can mentally calculate tables 2 to 5, and table of 10 is as no-braner as the multiplication of 1. :D
Then comes, 6, 7, 8 and 9.

You might already knew this, and forgotten about is, so this will help you refresh a Math skill!
  • The Multiplication by Hands - applicable for 6, 7, 8, 9 and 10 as multiplier and multiplicand
    • Step 1 - look at your hands, your thumbs represents number 10, the index finger as 9 and so on, hence, your pinky is number 6.
    • Step 2 - let's set an example. So lets say its 7 x 8. (No calculators pls! that's cheating!) What you do is connect the fingers representing 7 and 8.
    • From the connecting fingers, count each fingers downwards by ten. So thats 10, 20, 30, 40, 50.
    • Step 3 - Count the remaining left fingers and add it to the sum of the remaining right fingers. so that's 3 x 2 = 6
      • Step 4 - Add the sum of step 5 and 6, so that will give you 50 + 6 = 56. Awesom I    know! :D
Take a look:
Photo Courtesy of helpingwithmath.com
I first observed that the digits of the answers if added will always be 9.
And aside from tables 1 and 10, 9 is a no-brainer too! you just have to know the trick!

Example: 9x7. Instead of counting 9 + 9 + 9 + 9 + 9 + 9 + 9... All you need to do is add zero to the digit you are multiplying to 9, in this case 7 will become 70. Then we have to minus that digit to the new set of number, 70 - 7 = 63.

In short: 8 x 9 = 80 - 8 = 72.

Get it, got it!

Do you have more math tricks to share? Comment down below, its time to share yours! Click like for more math tricks! :P

Monday, March 5, 2012

Algebra: Laws and Theorems of Equality and Inequality

Equality - two equations or quantities are of equal and the same value.
Inequality - two equations or quantities are lesser or greater than each other.


Basic Laws of Equality
  • Reflexive Property: x = x
  • Symmetric Property: if x = y, then y = x
  • Transitive Property: 
    • if x = y, and y = z, then, x = z. (x = y = z)
    • if w = x, and y = z, then, w + y = x + z.
    • if w = x, and y = z, then, wy = xz.
  • First Conclusion: things equal to the same thing are equal to each other.
  • Second Conclusion: equals added to equals are equals
  • Third Conclusion: equals multiplied to equals are equals

Basic Laws of Inequality
  • x > y;        x is greater than y
  • x < y;        x is less than y
  • x </= y;     x is less than or equal to y
  • x>/=y;       x is greater than or equal to y

Basic Theorems on Inequalities
  • x > y if and only if -x < -y
  • if x > 0, then -x < 0 
  • if -x > 0, then x < 0
  • if x > y, z < 0, then ac < bc
  • if x > y, w > z, then (x + w) > (y + z)
  • if x > y, w> z, and w, x, y, z > 0, then xw > yz
  • if x > 0, y > 0, x > y, then 1/x < 1/y

Saturday, March 3, 2012

Algebra: Law of Natural Numbers


Natural numbers refer to members of the set of positive integers or set of non-negative integers.

In Law of Natural Numbers, let x, y and z be any number, hence,

Law 1 - closure for addition: 
x + y

Law 2 - commutative law for addition: 
x + y = y + x

Law 3 - associative law for addition:
(x + y) + z = x + (y + z) = x + y + z

Law 4 - closure for multiplication:
x * y


Law 5 -  commutative law for multiplication:
x * y = y * x


Law 6 - associative law for multiplication:
x(y*z) = (x*y)z

Law 7 - distributive property of addition and multiplication:
x(y +z) = x*y + x*z