Converting Repeating Decimal - Fraction

Category: Arithmetic

"Published in Newark, California, USA"

Express 0.454545454545454545454545454545....... into its equivalent fraction.

Solution:

The above decimal is a repeating decimal because the digit, 45 is repeating endlessly. It is usually happened if a fraction cannot be converted into basic fraction. Basic fraction is a fraction whose denominator is a multiple of 10. If the denominator is not a multiple of 10 but the multiples of its factors like 2 and 5 are given, then you can convert it into the multiples of 10. For example, if the denominator is 4, then you need to multiply both the numerator and the denominator by 25 so that the denominator will be 100. 

How about if the denominator is not a multiple of 10 and the factors are not the factors of 10? Well, there's no way to convert it into the multiples of 10. For example if the denominator is 35, then the factors are 5 and 7. Since 7 is not a factor of 10, then there's no way to convert the denominator into the multiples of 10. If you will do the manual division by dividing the numerator by denominator, you will notice that there's no end in division because the remainder is always the same and the digits of the quotient are repeating endlessly that's why the given decimal is called a repeating decimal.

Let's go back to the problem. How do you convert the repeating decimal into a fraction? Well, there's a way to convert the repeating decimal into a fraction. Here's the procedure and solution

Let     x = 0.454545454545454545454545454545.......... 

   100 x = 45.45454545454545454545454545454545......

Subtract x from 100 x, we have

   100 x = 45.4545454545454545454545454545454545......
        - x = -0.454545454545454545454545454545454......
   -------------------------------------------------------------------------------
     99 x = 45

Divide both sides of the equation by 99

As a rule in Mathematics that all fractions must be simplified into a lowest term. Factor 45 and 99 as follows

Their Greatest Common Factor (GCF) is 9. Cancel 9 on both sides of the fraction. Therefore, the answer is

Solving Trigonometric Equations

Category: Trigonometry, Algebra

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Find the values of x in the range from 0º to 360º for

                   Sin x + Sin 2x + Sin 3x = 0

Solution:

Consider the given equation

                    Sin x + Sin 2x + Sin 3x = 0

Apply the Sum and Difference of Two Angles Formula and Double Angle Formula for the above equation

                  Sin x + 2 Sin x Cos x + Sin (x + 2x) = 0 

         Sin x + 2 Sin x Cos x + Sin x Cos 2x + Cos x Sin 2x = 0

 Sin x + 2 Sin x Cos x + Sin x (Cos² x - Sin² x) + Cos x (2 Sin x Cos x) = 0

       Sin x + 2 Sin x Cos x + Sin x Cos² x - Sin³ x + 2 Sin x Cos² x = 0

                Sin x + 2 Sin x Cos x + 3 Sin x Cos² x - Sin³ x = 0

Take out their common factor which is Sin x, we have

               Sin x (1 + 2 Cos x + 3 Cos² x - Sin² x) = 0

               Sin x [1 + 2 Cos x + 3 Cos² x - (1 - Cos² x)] = 0

               Sin x (1 + 2 Cos x + 3 Cos² x - 1 + Cos² x) = 0

               Sin x (2 Cos x + 4 Cos² x) = 0

               (Sin x )(2 Cos x)(1 + 2 Cos x) = 0

Equate each factor in zero

For          Sin x = 0

               x = Sin^-1 0
     
               x = 0º, 180º, 360º

For          2 Cos x = 0

               Cos x = 0

               x = Cos^-1 0

               x = 90º, 270º

For         1 + 2 Cos x = 0

               2 Cos x = -1

               Cos x = - ½

               x = Cos^-1 -½

               x = 120º, 240º

Therefore, 

               x = 0º, 90º, 120º, 180º, 240º, 270º, and 360º

      

Frustum - Right Circular Cone

Category: Solid Geometry, Plane Geometry

"Published in Newark, California, USA"

The frustum of a right circular cone has a slant height of 9 ft. , and the radii of the bases are 5 ft. and 7 ft. Find the lateral area and the total area. What is the altitude of this frustum? Find the altitude of the cone that was remove to leave this frustum. Find the volume of the frustum. 

Solution:

To illustrate the problem, you can draw the figure as follows

Photo by Math Principles in Everyday Life

Since the slant height and the radii of the bases are given, we can get the lateral area, area of the bases, and the total area of the frustum. 

Let's get the circumference of the top base of the frustum as follows

For the bottom base of the frustum

Therefore, the lateral area of the frustum is

Let's get the area of the top base of the frustum as follows

For the bottom base of the frustum

Therefore, the total area of the frustum is

Since the altitude of the frustum is not given in the problem, we have to find the altitude of the frustum with the use of the  vertical section of the frustum. The vertical section of the frustum is a trapezoid.

Photo by Math Principles in Everyday Life

From the figure, we can get the altitude of a trapezoid which is also the altitude of a frustum. As you notice that the end portion of a trapezoid is a right triangle. We can use the Pythagorean Theorem to solve for the altitude of the frustum as follows 

In order to get the altitude of the cone that was removed to leave this frustum, we have use the formula for similar triangles as follows

Finally, we can get the volume of the frustum as follows