Finding the Percentage of one number and another

In mathematics, a word problem for a set S with respect to a system of finite encoding of its elements is the algorithmic problem of deciding whether two given representatives represent the same element of the set. Percentage is a way of expressing a number as a fraction of 100. It is often denoted using the percent sign, "%". In this article we shall discuss finding the percentage of one number and another number example problem. (Source: Wikipedia)

Finding the percentage of one number and another example problem

To finding the percentage of one number and another following formula used

Percentage formula = Changes in value   x 100%
                                    Original value

Example 1:

Find how much percentage increase from the number 80 to 95.5?

Solution:

Increased value = new value – old value

Increase value = 95.5 – 80 = 15.5

Percentage increase = `15.5/80` x 100%

Increased percentage = 19.3%

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Example 2:

Find how much percentage decrease from the number 105 to 95.5?

Solution:

Decrease value = new value – old value

Decrease value = 95.5 – 105 = 9.5

Percentage decrease = `9.5/105` x100%

Decreased percentage = 9%

Example 3:

Jasmine last month earned salary `$` 1250. He earned this month salary `$` 1600. Find how many percentage increases in her salary amount.

Solution:

Jasmine increased salary = new salary – original salary

`$` 1600 –`$` 1250 = 350

Salary percentage increase = `350/1250` x100% = 28%

Jasmine increased salary percentage is 28%

Example 4:

Kishore last month earned salary `$` 1800. He earned this month salary `$` 1500. Find how much percentage decreased in his salary amount.

Solution:

Kishore decreased salary = new salary – original salary

`$` 1500 –`$` 1800 = 300

Salary percentage increase = `300/1800` x100%= 16.6%

Kishore decreases salary percentage is 16.6%

Finding the percentage of one number and another practice problem

Problem 1:

Joes last month earned salary `$` 1550. He earned this month salary` $` 1800. Find Percentage Increase in her salary amount.

Answer:

Joes increased salary percentage is 13.8%

Problem 2:

Ram last month earned salary `$` 1300. He earned this month salary `$` 600. Find how much percentage decreased in his salary amount.

Answer:

Ram decreases salary percentage is 53.8%

Examples of trig functions

Trig functions generally define the function of angles. To relate the angles of triangle to its length of sides is the common use of trig functions. Trig functions are generally applied in modeling periodic phenomena and study of triangles. Sine, cosine, tangent, cosecant, secant, and cotangent are the most memorable trig functions. Trig functions refer to tabulating the trig functions and trig identities in order. In  this article we shall discuss about examples involved in trig functions.

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Examples of Trig Functions:

Let us see some of the examples problems for trig function.

Examples 1: Prove that, sin^4 x − 2sin^2 x cos^2 x + cos^4 x =  cos^2 (2x).

Proof: L.H.S. = sin^4 x − 2sin^2 x cos^2 x + cos^4 x,

=> (sin^2 x)^2 − 2sin^2 x cos^2 x + (cos^2 x)2,

We know that, (a + b)^2 = a^2 − 2ab + b^2, similarly,

=> (sin^2 x − cos^2 x)^2,

=> (sin^2 x −( 1 − sin^2 x))^2,

=> (sin^2 x − 1 + sin^2 x)^2,

=> (2sin^2 x − 1)^2,

=> (−1 (1 − 2sin^2 x))^2,

=> (1 − 2sin^2 x)^2,

=> (cos(2x))^2,

=> cos^2(2x).

=> R.H.S.

Hence Proved that, sin^4 x − 2sin^2 x cos^2 x + cos^4 x = cos^2(2x).


Examples 2: Prove that, tan^2 x + 1 = sec^2 x.

Proof: We know that, sin^2 x + cos^2 x = 1

Dividing the above equation by cos^2 x, we get,

=> `(sin^2x + cos^2x) / (cos^2x) = 1 / (cos^2x)` ,

=> `(sin^2x) / (cos^2x) + (cos^2x) / (cos^2x) = 1 / (cos^2x)` ,

By using the trigonometric functions; `sinx/cosx` = tan x, and `1/cosx` = secx, we get,

=> tan^2 x + 1 = sec^2 x

Hence proved that, tan^2 x + 1 = sec^2 x.

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Examples 3: Prove that, `(tan A + cot B)/(cotA + tanB) = tan A /tanB` .

Proof: L.H.S. = `(tan A + cot B)/(cotA + tanB)` ,

We know that, cot x = `1/tanx` , we get,

=> `((tan A + 1/tanB))/((1/tanA + tanB))` ,

By taking L.C.M. We get,

=> `(((tanAxxtanB + 1)/tanB))/(((1 + tanAxxTanB)/tanA))` ,

By taking reciprocal, we get,

=>  `((tanAxxtanB + 1)/tanB)xx(tanA/(1 + tanAxxTanB))` ,

By simplification, we get,

=> `tan A /tanB` ,

=> R.H.S.

Hence proved that, `(tan A + cot B)/(cotA + tanB) = tan A /tanB` .

Practice Problems of Trigonometric Functions:

Problem 1: Prove that, `(1 + tan x)/(1+cotx) = (sin x + tanx)/(1 + cosx)`.

Problem 2: Prove that, `cosx /(1+sinx)` = sec x − tan x.

Problem 3: Prove that, `(1+cosx)/(1-cosx)` = (csc x + cot x)2.

Calculating Mean Median mode

In statistics, the mode is the value that occurs the most frequently in a data set or a probability distribution.

In statistics, mean has two related meanings:

the arithmetic mean.

the expected value of a random variable, which is also called the population mean.

Median is described as the numeric value separating the higher half of a sample

-         Source Wikipedia

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Examples for calculating mean median mode:

Example 1:

Calculating the mode for given number set.17, 13, 18, 14, 19, 12, and 18.

Solution:

First, we have to arrange the numbers in ascending order. Therefore, we get,

12, 13, 14, 17, 18, 18, 19

The mode of the given number set is 18.

Example 2:

Calculating the mean of the given number set. 17, 13, 18, 14, 19, 12, and 18.

Solution:

First, we have to arrange the numbers in ascending order. Therefore, we get,

12, 13, 14, 17, 18, 18, 19

Mean = (12 + 13 + 14 + 17 + 18 + 18 + 19) / 7

Mean = 111/7

The mean of the given number set is 15.9.

Example 3:

Calculating the Median of the given number set. 17, 13, 18, 14, 19, 12, and 20.

Solution:

First, we have to arrange the numbers in ascending order. Therefore, we get,

12, 13, 14, 17, 18, 19, 20

Here, the given number set is odd. So we have to pick out the middle number.

The median of the given number set is 17.

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Example 4:

Calculating the Median of the given number set. 21, 17, 13, 18, 14, 19, 12, and 20.

Solution:

First, we have to arrange the numbers in ascending order. Therefore, we get,

12, 13, 14, 17, 18, 19, 20, 21

Here, the given number set is even. So we have to pick out the middle of two numbers and we have to find average or mean of those numbers.

The median of the given number set is (17 + 18) / 2 = 17.5.

Practice problems for calculating mean median mode:

Problem 1:

Calculating the mode for given number set.27, 23, 28, 24, 29, 22, and 28.

Solution:

The mode of the given number set is 28.

Problem 2:

Calculating the mean of the given number set. 27, 23, 28, 24, 29, 22, and 28.

Solution:

The mean of the given number set is 25.9.

Problem 3:

Calculting the Median of the given number set. 27, 23, 28, 24, 29, 22, and 30.

Solution:

The median of the given number set is 27.

Problem 4:

Calculating the Median of the given number set. 21, 27, 23, 28, 24, 29, 22, and 30.

Solution:

The median of the given number set is (17 + 18) / 2 = 25.5.

Approximation Calculus

In approximation calculus, we use both the differentiation and integration process. Approximation calculus gives the approximate values. It does not give the complete solution of the problem. The resultant value of the function is not a exact solution of the problem. Approximation calculus mainly uses the differentiation process. Linear function variables are also used in approximation calculus.

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Example problems for approximation calculus

Approximation calculus problem 1:

Find the approximate value of (3.9)4. The given function f(x) = x4. The equation of the tangent line to f(x) at x = 4 can be given as y = mx + b.

Solution:

Given function f(x) = x4

Differentiate the above function with respect to x, we get

f'(x) = 4x3

Find the slope of tangent line at x = 4 is given as,

f'(4) = 4 * (4)3

= 256

In point slope form, the line passes through the points (4, 44) and has the slope 256 is given as,

y - 256 = 256 * (x - 4)

y = 256x - 768

Therefore,

x4 = 256x - 768 at x = 4

Finally,

(3.9)4 = 256(3.9) - 768 = 230.4

Answer:

The final answer is 230.4

Approximation calculus problem 2:

Find the approximate value of (4.5)3. The given function f(x) = x3. The equation of the tangent line to f(x) at x = 5 can be given as y = mx + b.

Solution:

Given function f(x) = x3

Differentiate the above function with respect to x, we get

f'(x) = 3x2

Find the slope of tangent line at x = 5 is given as,

f'(4) = 3 * (5)2

= 75

In point slope form, the line passes through the points (5, 53) and has the slope 75 is given as,

y - 125 = 75 * (x - 5)

y = 75x - 250

Therefore,

x3 = 75x - 250 at x =5

Finally,

(4.5)3 = 75(4.5) - 250 = 87.5

Answer:

The final answer is 87.5

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Practice problems for approximation calculus

Approximation calculus problem 1:

Find the approximate value of (3.7)4. The given function f(x) = x5. The equation of the tangent line to f(x) at x = 2 can be given as y = mx + b.

Answer:

The final answer is 140.8

Approximation calculus problem 2:

Find the approximate value of (5.9)2. The given function f(x) = x2. The equation of the tangent line to f(x) at x = 6  can be given as y = mx + b.

Answer:

The final answer is 34.8

what is the median in math

In statistics theory, the median is the middle value of the set of data after the arrangement of ascending or descending order. If the total of elements in a set is even then take the two middle elements (n and n+1) after the arrangement of the elements and find the average (n + (n+1))/2 which is the required median.

Median in Math – Example Problems

Example 1: In a class nine students’ marks as follows 28, 37, 45, 98, 76, 71, 65, 49, and 58. What is the median?

Solution:

Arrange the elements of a data set in ascending order {28, 37, 45, 49, 58, 65, 71, 76, 98}

Median:

{28, 37, 45, 49, 58, 65, 71, 76, 98}

Total number of elements in a data set is 9 which is an odd number. So find the middle number which is the median.

Here, 58 is the middle value.

Therefore 58 is the median.

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Example 2: In a class ten students’ weights as follows 36, 42, 54, 68, 72, 67, 49, 57, 62, and 74}. What is the median?

Solution:

Arrange the elements of a data set in ascending order {36, 42, 49, 54, 57, 62, 67, 68, 72, 74}

Median:

{36, 42, 49, 54, 57, 62, 67, 68, 72, 74}

Total number of elements in a data set is 10 which is an even number.

Divide 10 by 2

10 / 2 = 5 = n

So take the 5th (n) and 6th (n+1) elements and find the average.

(57 + 62) / 2 = 59.5

Therefore 59.5 is the median.

Example 3: Ten students’ height in a class as follows 157, 157, 178, 160, 120, 146, 165, 135, 182, and 184. What is the median?

Solution:

Arrange the elements of a data set in ascending order {120, 135, 146, 157, 157, 160, 165, 178, 182, 184}.

Median:

{120, 135, 146, 157, 157, 160, 165, 178, 182, 184}

Total number of elements in a data set is 10 which is an even number.

Divide 10 by 2

10/12 = 5 = n

So take the 5th (n) and 6th (n+1) elements and find the average.

(157 + 160) / 2 = 158.5

Therefore 158.5 is the median.

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Median in Math – Practice Problems

Problem 1: Ten students’ weight in a class as follows 56, 42, 64, 72, 89, 79, 56, 61, 65, and 56. What is the median?

Answer: Median: 62.5

Problem 2: Nine students’ height in a class as follows 165, 146, 170, 140, 162, 137, 193, 179, and 160. What is the median?

Answer: Median: 162