Algebra: How to Translate verbal Statements into symbols

An equation is an important tool in problem solving but another tool which is just as necessary is representing related quantities by means of symbols. It must be recalled thatb letter like x can be used to represent the numerical value of a certain quantity. quantities related to the latter may also be expressed in terms of x . For example , if x represents the first of two consecutive integers, then the next consecutive integer( which is greater by 1) will be represented by x+1.

 There are certain rules which must be followed in representing quantities. generally, it is sufficient to use x or any chosen symbol to represent the smaller of two quantities or the smallest of three or more quantities. another way is by representing the independent quantity by x. suppose , for example, that Pedro is twice as old as Juan. this condition shows that the age of Pedro depends upon the age of Juan and hence it is better to let x represent the age in years of Juan. Pedro's age in years will re represented by 2x

Algebra: What is mean Proportion and How to solve it?

A proportion such as 1:2 =2:4 or a:b = b:c  is called a mean proportion because the second and the third terms are equal. In the latter b is the mean proportional between the first and the last terms a and c, respectively and c is the third proportional to a and b.

Solving for the mean proportional
 Given
 in proportion, the product of the extremes is equal to the product of the means.
  extracting the square roots of booth members of the equation

Note; like roots and like powers of equal numbers are equal

It follows that:
The mean proportional between two quantities is equal to the square root of their product.

example 1 :
Solve for x in

example 2 :
Solve for x in

Algebra: How to solve the Equations containing decimals

Equations containing decimals are equations with fractions. hence the solution of decimal equations is similar to that of fractional equations with this difference-in decimal equation the multiplier of both members is that power
of ten that will clear the equations of its decimal.

Example1.
Solve for x:                                 .3x + 5 = .2x
Multiplying both members          
of the equation by 10,                 3x + 50 = 2x
                                                  x = -50

Check                                     .3x(-50) + 5 = .2(-50)
                                              -15+5  = -10
                                               -10 = -10

Example no 2.                                      Solve for y in the equation
                                                         .5y/3 - .18/4 = .1y/6
Clear the equation of decimals by multiplying both members by 100
50y/3 - 18/4 = 10y/6
Clear the equation of fractions by multiplying both members by 12, the LCD
200y - 54 = 20y
180y = 54
y = 54/180 = .3

check:
.5y/3 - .18/4 = .1y/6
(.5(.3))/3 -.18/4  = (.1(.3))/6
.05-.045 = 0.005
.005 = .005



                                            

Algebra : How to solve a Literal equation

Equations where the known quantities are expressed as letters are called literal equations. Formulas are literal equations. Generally, the last letter of the alphabet are used to represent the unknown quantities, and the first letters, the known quantities.

Examples:

Solve for the unknown and check

1. 3x + 2a = 2x+5a     transposing
    3x - 2x = 5a - 2a      combining like terms
           x = 3a  


check
3.3a + 2a = 2.3a + 5a
9a + 2a = 6a + 5a
11a = 11a


2. 8z + 6m = 5z + 9m
    8z - 5z = 9m - 6m        transposing
    3z = 3m                   combining like terms
    z = m                       axiom of division

check
   8.m + 6m = 5.m + 9.m
   8m + 6m= 5m + 9m
   14m = 14m  

Power Factor for a more efficient circuit and loading

Power factor is an important aspect to consider in AC circuit, because any factor less than 1 means that the circuit wiring has to carry more current than what would be necessary with zero reactance in the circuit to deliver the same true power to the resistive load.
    When expressed as fraction, the ratio between true power and  apparent power is called the power factor of the circuit.
                                                     Kw
The power factor is given cosƟ = --------
                                                    KVA
in the case of single phase supply

                     VI                    KVAx1000
KVA=     ----------  or I=    -----------  
                     1000                    V

In case of three-phase supply
            sqrt(3)VL IL                 KVAx 1000
KVA= --------------     or IL=-------------
                1000                           sqrt(3)VL
In the case of three phase supply
   All current will cause losses in the supply and distribution system. A load with a power of 1.0 result in the most efficient loading of the supply and a load with a power of 0.5 will result in much higher losses in the supply such as an induction motor, power transformer, lighting ballast,welder or variable speed drive, switched mode power supply, discharge lighting or other electronic load.
  The heating and lighting loads supplied from three-phase supply have power factors ranging from 0.95 to unity. But motor loads have usually low lagging power factor ranging from 0.5 to 0.9. Single phase motors may have a power factor of as low as 0.4 and electric welding units have even lower power factor of 0.2 or 0.3
 A poor power factor due to an inductive load can be improved by the addition of power factor correction,but a poor power factor due to a distorted current waveform requires a change in equipment design or expensive harmonic filter s to gain an appreciable improvement. many inverters are quoted as having a power factor of better than 0.95 when in reality, true power factor is between 0.5 and 0.75. the Figure of 0.95 is based on the cosine of the angle between the voltage and current but does not take into account that the current waveform is discontinues and therefore contributes to increased losses on the supply
   in each  equation above, the KVA is directly proportional to the current. The chief disadvantage of a low power factor is that the current requires for a given power is very high. this fact leads to the following undesirable results.

1. Large KVA for a given amount of power.
    All electric machines such as alternators, transformers, cables are limited in their current-carrying capacity by the permissible temperature rise which is proportional to I^2 .hence, they may all be fully loaded with respect to their rated KVA without delivering their full power.

2. Poor voltage regulation . When loading a low lagging power factor is switched on,there is a large voltage drop in the supply lines and transformers. this drop in voltage adversely affects the starting torque of motors and needs expensive voltage stabilizing equipment for keeping the consumers voltage fluctuations within the statutory limits