Converting Decimal Number - Fraction

Category: Arithmetic

"Published in Newark, California, USA"

Express 0. 0125 into its equivalent fraction. 

Solution:

The above decimal is known as a common decimal or terminating decimal. Terminating decimal means that the decimal digits are not repeating endlessly and there's an end in division if you divide the numerator by the denominator by manual division. If the denominator is a factor of 2, 5, and 10, then the quotient is a terminating decimal. On the other hand, if the denominator is not a factor of 2, 5, and 10, then the quotient is a repeating decimal. On December 4, 2012, we discussed about the conversion of a repeating decimal into a fraction.

How do you convert a terminating decimal into a fraction? Well, let's consider this procedure and solution

Let                x = 0.0125

         10,000 x = 125

(We multiply both sides of the equation by 10,000 because we want the right side of the equation to move the decimal point to the right. One decimal movement is equivalent to a power of 10. If you move a decimal position to the right, then you multiply a number by 10. If you move two decimal position to the right, then you multiply a number by 100, and so on.)

Divide both sides by 10,000

As a rule in Mathematics that all fractions must be simplified into a lowest term. Factor 125 and 10,000 as follows

Cancel three 5's as their Greatest Common Factor (GCF) in a fraction

Therefore, the answer is 

Derivative - Algebraic Functions, Radicals

Category: Differential Calculus, Algebra

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Find the derivative for

Solution:

Consider the given equation

Express the above equation in terms of an exponential power as follows

First, get the derivative of a function in terms of a power

Next, get the derivative of a function inside the exponential function in terms of a rational function

Simplify the above equation

We can make a negative exponent into a positive exponent by getting the reciprocal of a rational function inside the exponential function as follows

As a rule in Mathematics, we have to rationalize the denominator in order to eliminate the cube root sign at the denominator as follows

Therefore,

Two Coincident Lines

Category: Analytic Geometry, Algebra

"Published in Newark, California, USA"

Find the points of intersection of the following lines:

                                2x - y = 8

                              4x - 2y = 16

Solution:

Since the given equations are all first degree, then they are linear equations. They are straight lines. We can graph the two lines by getting their slope and y-intercept. 

For 2x - y = 8

                                  2x - y = 8

                                        -y = -2x + 8

                                         y = 2x - 8

                                  slope (Δy/Δx), m = 2

                                  y-intercept, b = -8

To trace the graph, plot -8 at the y-axis. This is your first point of the line (0, -8). Next, use the slope to get the second point. From the first point, count 1 unit to the right and then 2 units upward. 

For 4x - 2y = 16

                                  4x - 2y = 16

                                       -2y = -4x + 16

                                          y = 2x - 8

                                 slope (Δy/Δx), m = 2

                                  y-intercept, b = -8


To trace the graph, plot -8 at the y-axis. This is your first point of the line (0, -8). Next, use the slope to get the second point. From the first point, count 1 unit to the right and then 2 units upward.

Photo by Math Principles in Everyday Life

From the graph, the two lines are coincide to each other because their slopes and y-intercepts are the same which are 2 and -8. The two lines will have an infinite number of points of intersection. When you solve for x and y from the two given equations, their x, y, and constant will be equal to zero. From the two given equations,

                                  2x - y = 8

                                4x - 2y = 16

Multiply the first equation by 2 and -1 at the second equation. Add the two equations and let's see what will happen to x, y, and constant.

        2 (2x - y = 8)                             4x - 2y = 16
                                         →
     -1 (4x - 2y = 16)                         -4x + 2y = -16
                                                   _______________

                                                                  0 = 0

Since everything in the equation are all equal to zero, then there's no way that we can solve for x and y. Therefore, the two lines are coincide to each other.