Algebra Probability Help

Mathematical numbers are studied by algebra. Algebra is also used to learn the polynomials and the equations etc. Probability is a method of state knowledge or principle that an occurrence will happen. In mathematics the idea has been given a correct sense in probability theory, that is used widely in such areas of learn as mathematics, finance, statistics etc. Here we will see the examples and solved with the help of algebra probability.

Example Problems for Algebra

1) What is the multiplication of following two numbers with the help of algebra? 115*118

Solution

115*118 = (100+15)*(100+18)

= (100)2+(100*18)+(15*100)+(15*18)

=10000+1800+1500+270

=13570.

2) 18x+12y+12x+8a. Simplify the given equation in algebra.

Solution

The given equation is 18x+12y+12x+8a

There are two related groups are available. So connect the groups.

The new equation is,

(18x+12x)+12y+8a

Add the numbers inside the bracket. We get 30x+12y+8a.

Assemble the numbers and we get the correct format.

=8a+30x+12y.

Example for Probability

There are 45 things are available in a shop. In those things, 16 are the books, 12 are the bags and 17 are the caps. What is the probability for the following outcomes?

i) Select the books

ii) Select the bags

iii) Select the caps.

Solution:

Total number of things n(S) =45

Number of books n (A) =16

Number of bags n (B) =12

Number of caps n(C) =17

i) Assume P(A) is the probability for select the books.

P(A)=`(n(A))/(n(S))`

=`(16)/(45)` .

ii) Assume P(B) is the probability for select the bags.

P(B) =`(n(B))/(n(S))`

= `(12)/(45)`

=`(4)/(15)` .

iii) Assume P(C) is the probability for select the caps.

P(C)=`(n(C))/(n(S))`

=`(17)/(45)` .

Practice Problems

Practice problem 1

What is the product of the following numbers in algebra ? 111*128

Answer:

101*108=14208.

Practice problem 2

8x+14y+12a+12y. Simplify the terms with the help of algebra.

Answer:

12a+8x+26y.

Practice problem 3

Jenifer has the 6 papers, Stephen has 7 papers. What is the probability for select the Jenifer’s papers.

Answer:

Probability for select the Jenifer’s papers=6/13.

These algebra and probability problems are helping to study these concepts

Trigonometry Sine Function

In trigonometry one of the ratios is the sine function. It is defined by sinx = (Opposite side/Hypotenuse) = (Perpendicular/Hypotenuse)

This trigonometry sine function has applications on solving some practical problems in finding the height of the wall, length of a ladder leaning on the wall and the distance between the wall and the foot of the ladder.

Let us learn some sine values for standard angles.





The above table will help us in solving problems involving sine functions.  Now let us solve few problems on the topic trigonometry sine function.

Example Problems on Trigonometry Sine Function

Ex 1: From the below diagram, find the value of x using sine function.

Sol: Sin30 = `x/12`

This is implies, x = 12 `xx` sin30

                            = 12 `xx` `(1/2)` [table value for sin30 = `1/2` ]

                            = 6 cm.

Therefore, the value of x = 6cm.

Ex 2: From the below diagram, find the value of x using sine function.

Sol: Sin30 = `3/x`

               x = `3/sin30`

                  = `3/(1/2)`

                  = 3 `xx` 2 = 6cm.

Therefore, the value of x = 6cm.

Ex 3: If 2 sinA = 1, what is the value of A?

Sol: Given: 2sinA = 1

                     SinA = `1/2`

This implies that the value of A = `30^0` .     [Table value Sin30 = `1/2` ]

Ex 4: Simplify: sin60 + sin30.

Sol: sin60 + sin30 = `sqrt(3)/2` + `(1/2)`

                              = `(((sqrt(3)) + 1)/2)` .            [Table value Sin60 = `sqrt(3)/2` ]

Ex 5: In the triangle, find the value of x^0.

Sol: We know that Sin`x^0` = `sqrt (3)/2`

This implies `x^0` = `60^0` [Table value, sin60 = `sqrt (3)/2` ]

Ex 6: If 4sin²x – 3 = 0, x is an acute angle, find (i) sinx (ii) x.

Sol: Given: 4sin²x – 3 = 0

                     Sin²x = `(3/4)`

                     Sinx = `+-` `sqrt(3)/2`

(ii) Since, sinx = `+-` `sqrt(3)/2` ,

As per the table value, x =  `60^0` .

Therefore, x = `+-` `60^0` .

Ex 7: Solve for x:

Sin(x + 10) = ½

Sol: Since Sin(x + 10) = `(1/2)`

                        X + 10 = `30^0`

Therefore, x = 30 – 10

                     = `20^0` .

Therefore, the value of x = `20^0` .

Practice Problems on Trigonometry Sine Function

1. Solve for x: Sin² x + sin²30 = 1.

[Answer: x = `+-` `60^0` ]

2. Find the acute angle A and B, if Sin (A+B) = 1 and Sin(A – B) = `(1/2)`

[Answer: A = `60^0` , B =` 30^0` ].

Areas of Combinations of Plane Figures

Areas of Combinations of plane figure:

Areas of Combinations of plane figure is the process of calculating the areas of different combinations of figures. these types of figures  We come across in our daily life and also in the form of various interesting designs.

Flower beds, drain covers, window designs, We come across, designs   on th etable covers, are some of such examples.We illustrate the  process of calculating  areas of these  figures through some examples.

The following   examples are combined with some plane figures.

Areas of Combinations of Plane Figure Problems:

Example:

Two circular flower beds have been shown on two sides of  a square lawn ABCD  of side  50m.If the center of each circular flower bed is the point of intersection O of the diagonals of the square lawn, find the sum of the areas of the lawn and the flower beds.





Solution:

Area of  the square lawn ABCD = 50 x 50 m2  --------------------- (1)

Let   OA = OB = x metres

So            x2 + x2 = 502

Or             2x2 = 50 x  50

X2  = 25 x 50                                        -----------------------(2)

Now ,

Area of sector OAB  = `(90)/(360)` *`Pi` *x2

=`(1)/(4)` * `Pi` i*x2

= `(1)/(4)`  x `(22)/(7)` x 25 x 50m2     [from (2)] ------(3)

Also, area of  `Delta` OAD = `(1)/(4)` * 50 * 50 m2  (<AOB=90)------(4)

So, area of flower bed                 AB = (`(1)/(4)`* `(22)/(7)`*25*50 – `(1)/(4)`*50*50)m2   [from (3) and (4)]


= `(1)/(4)`*25*50(`(22)/(7)` -2)m2

=`(1)/(4)`*25*50* `(8)/(7)` m2   ----------------(5)

Similarly area of the other flower bed

= `(1)/(4)` * 25 * 50 * `(8)/(7)` m2    -------------------------(6)

Therefore,

Total area  =(50*50 + `(1)/(4)` *25*50* `(8)/(7)` +`(1)/(4)`*25*50*`(8)/(7)`)m2    [from (1),(5) and (6)]

=25*50(2+`(2)/(7)`+`(2)/(7)`)m2

=25*50* `(18)/(7)` m2

=  3214. 29 m2

Areas of Combinations of Plane Figure Example 2:

Example 2:

Find the area of the shaded region in the following figure , where ABCD is a square of a side  10 cm.





Solution:

Area of square ABCD

= 10 * 10 cm2

= 100 cm2

Diameter of each circle    = `(10)/(2)`cm=5cm

So, radius of each circle      =(5)/(2) cm

<br>

=`(22)/(7)` *`(5)/(2)` * `(5)/(2)`cm2

=`(550)/(28)` cm2

Therefore area of the four circles = 4* `(550)/(28)`cm2 =  78.57Cm2

Hence area of the shaded region = (100-78.57)cm2= 21.43Cm2

Polynomials Chart

In mathematics, a polynomial is an expression of finite length constructed from variables (also known as indeterminate) and constants, using only the operations of addition, subtraction, multiplication, and non-negative, whole-number exponents. For example, x2 − 4x + 7 is a polynomial, but x2 − 4/x + 7x3/2 is not, because its second term involves division by the variable x and because its third term contains an exponent that is not a whole number. (Source Wikipedia)

In this article polynomial chart we see about basic concepts of polynomial, its types of polynomial ,some example problems

Polynomial Types:

Basic concepts of polynomials:
Polynomial is nothing but algebraic expression and also concept of algebras. More types of polynomials are available in the algebra depends on the number of terms. Based on the number of terms polynomial was classified four types

Different types of polynomial:

Polynomial chart



Monomial:

If the expression having one term mean it was called as monomial

Example: 7x ,8x2

Binomial:

If the expression having two terms mean it was called as binomial

Example: 6x+4x

Trinomial

If the expression having three terms mean it was called as trinomial

Example: 3x+8x2+9

Polynomial:

If the expression having more than three terms mean it was called as polynomial

Example: 5x2+12x3+9x+10

Polynomial operations are addition of polynomial, subtraction of polynomial, multiplication of polynomial, division of polynomial.

Example Problems in Polynomial:

Example problems in Polynomial degree chart:

Polynomial addition chart:

Example 1:

Add the polynomial 3x2+5x+2 and 5x+6

Given polynomials: 3x2+5x+2,5x+6

Now we have to arrange the  terms for addition

After than add the terms one by one.

This is a polynomial addition chart



Example 2:

Polynomial multiplication chart:

(2x+5)(3x+1)

Now we have to multiply the one terms with another terms

And then add the terms



Example 3:

Degree of polynomial:

(9z9 +8 z4 − 6z5 + 8) Find the degree of polynomial for each term?

Degree of polynomial for first term=9

Degree of polynomial for second term =4

Degree of polynomial for third term=5

Degree of polynomial for fourth term=0

Highest degree of polynomial is 9

Ratio to Fraction Converter

Ratio :

In mathematics, The ratio can be used to relate two quantities by using the symbol : Also it can be expressed as follows,

• x is to y
• the ratio of x to y
• x : y

Fraction :

In mathematics , Part of the whole can be expressed as fraction. There are three kinds of fraction

• Proper fraction
• Improper fraction
• Mixed fraction

In this article we are going to see about how to simplify the ration as fraction by using the ratio to fraction converter.

Ratio to Fraction Converter :

Converter:

The electronic or software device that can perform the operations Quickly. The ratio to fraction converter can be used to convert fraction for the given ratio.



Fig(i) Ratio to fraction converter

Let us see some problems on ratio to fraction convertor.

Problems on Ratio to Fraction Converter :

Problem 1:

Convert the ratio 45 : 180 into simplified fraction

Solution:

Given,The ratio 45 : 180

We need to convert the given ratio into fraction .

we know that 45 : 180 = ` 45/ 180`

Divided by  45 on both numerator and denominator,

`45/180` = `( 45 / 45 ) / ( 180 / 180 )`

= `1 / 4`

Answer: The simplified fraction of the given fraction is  `1/4` .


Problem 2:

In a bag, there is Blue  and Green balls, the ratio of Blue balls to Green balls is 5:6. If the bag contains 180 Blue balls, how many green balls are there?

Solution:

Given The ratio of the Blue and green balls = 5 : 6

Number of blue balls = 180

Let us take x = green balls

To find the green balls we need to convert the given ratio into fraction,

Write the items in the ratio as a fraction.

`(blue) / (green)` = `5/6` = `x / 180`

`5/6` = `x / 180`

Multiply by 6 on both sides,

5 = `x / 180 `

5 = `x / 30`

Now multiply by 30 on both sides,

5 * 30 = x

150 = x

x = 150

Total number of black balls = 150

Answer: Green balls = 150

Square Root of Complex Number

A square root of an x is a numeral r such to r2 = x, or a numeral r whose square is x. each non-negative real number x have a exclusive non-negative square root, recognized the principal square root, signify by a essential symbol as . For optimistic x, the main square root knows how to as well be written in exponent information, as x1/2.  (Source: Wikipedia)

Square Root of Complex Number:

A complex number is a number regarding of a real also imaginary part. It know how to be written in the structure a + bi, where a and b are real numbers, also i is the ordinary imaginary unit among the property i 2 = −1. The complex numbers include the ordinary factual numbers, however expand them by adding in extra numbers and equally increasing the considerate of addition also multiplication.

Every positive numbers x contain two square roots. One of them is` sqrt(x)` , to be constructive, with the other `-sqrt(x)` that is negative. Collectively, these two roots are signifying `+-sqrt(x)` . Square roots of unconstructive numbers know how to be converse in the structure of complex numbers. More commonly, square roots know how to be thinks in any context in that a notion of square of several arithmetical objects is definite.

Example for Square Root of Complex Number:

Example 1:

Solve square root of compel numbers `sqrt(3) - isqrt(6)`

Solution:

Step 1: the given square root complex number is `sqrt(3) - isqrt(6)`

Step 2: `x = sqrt(3) and y = -sqrt(6)`



Step 3:    `r =sqrt( x^2+y^2)`

Step 4: substitute x and y values

`r =sqrt( (sqrt(3)^2)+(-sqrt(6)^2))`

Step 5:   r = 3

Example 2:

Solve square root of comple numbers 4+13i

Solution:

Step 1: the given square root complex number is   4+13i

Step 2: the given problem of the form is a+bi

Step 3: `x = 4 and y = 13`

Step 4:    `r =sqrt( x^2+y^2)`

Step 5: substitute x and y values

`r =sqrt( (4^2)+(13^2))`

Step 6:    `sqrt(377)`

Step 7:   r = 19

Step 8: finding b value

` b =sqrt((r-x)/2)) `

Step 9:    ` b =sqrt(((19)-4)/2) `

Step 10:    ` b =sqrt((15)/2) `

Step 11:  b = 2

substitute y and b in a

Step 12:  a = `y/(2b)`

a = `13/(2(2))`

a = 3

Step 13: find square root of  r1 and r2

r1 =a + bi = 3 + 2i

r2 = -a-bi = -3-2i

so the square root of complex numbers are 3+2i and -3-2i

Simplifying Radicals Activity

Simplifying radicals activity involves the process of solving radicals equation with step by step solution. Activity is the process of solving equations with radicals symbol. The square root symbol is also represented as radicals. Simplifying radicals is easily carried out by performing squaring operations on the given equation is known as simplifying radicals activity. The following are the example problems which explain the radicals activity.

In logarithm, the radical pictogram is represented by √. The symbol n√x , n, x and √ are represented by index , radical and and radical symbol. Steps to multiplying radicals is the similar way of ordinary multiplications, additionally it has the following conditions. When multiplying two or more radicals, we must multiply the numbers exterior radicals and then multiply the numbers in the interior radicals.

Steps to multiplying radicals:

The following steps to multiplying radicals, when the radicals have the indistinguishable key, n:

Step 1:  Utilize the multiplying radicals rule for nth roots to find the product of the radicals.

Step 2:  Simplify the product by factoring and taking the nth root of the factors that are ideal nth powers.

Rules on multiplying radicals:

The following rules for steps to multiplying radicals helps the easy way of understanding and simplifying the radical expression.

Rule 1:

n√x . n√y  =  n√(x . y)

Rule 2:

n√( x + y) . n√(x – y) = n√(x2 – y2)

Rule 3:

(x. n √y)n  =  xn . y

Simplifying Radicals Activity Example Problems:

Ex:1 Solve the radicals.

`sqrt(u^2-5u+7) = 1`

Sol:

Given equation is
`sqrt(u^2-5u+7) = 1`

To solve the radical symbol perform squaring operation on both sides
`[sqrt(u^2-5u+7)]^2 = (1)^2`

And simplify.
u 2 – 5u+7= 1

Make the above equation in factor form.
u 2 - 5 u + 6 = 0

The above equation is in quadratic equation form with two solutions
u = 3 and u = 2 is the answer.

Ex:2 Solve the radicals.

`sqrt(3u-5) = u - 1`

Sol:

Given equation is
`sqrt(3u-5) = u - 1`

To solve the radical symbol perform squaring operation on both sides

`[sqrt( 3u-5)]^2 = (u-1)^2`

Simplify the above equation
3 u - 5 = u 2 - 2 u + 1

Change the above equation in factor form.
u 2 - 5 u + 6 = 0

The above equation is in quadratic equation form with two solutions
u = 2 and u = 3 is the answer.

Ex:3 Solve the radicals.

`sqrt(u^2-16u+37) = 3`

Sol:

Given equation is
`sqrt(u^2-16u+37) = 3`

To solve the radical symbol perform squaring operation on both sides
`[sqrt(u^2-16u+37)]^2 = (3)^2`

And simplify.
u 2 – 16u+37= 9

Make the above equation in factor form.
u 2 - 16 u + 28 = 0

The above equation is in quadratic equation form with two solutions
u = 2 and u = 14 is the answer.

Practice Problems on Simplifying Radicals:

Q:1 Solve the radicals.

` sqrt(u^2-12u+29) = 3`

Answer: u = 2 and u = 10.

Q:2 Solve the radicals.

`sqrt(4u-3) = u -2`

Answer: u = 7 and u = 1

Multiplying Radicals:

Ex:1 (a)     Steps to multiplying radicals √2 . √36.

Sol:

1.   Multiply the i nside radical numbers  :          = √(2 .36)   = √72

2.   Simplify radicals if possible             :          =  √(2 . 62 )

3.   Answer                                           :          = 6 √2

(b) Steps to multiplying radicals  √5 . √7

Sol:

1.    Multiply the inside radical numbers = √(5 . 7)

2.    Simplify radicals if possible     =  √35

(c)   Steps to multiplying radicals                                      4√7 . 7√5

Sol:

1.   Multiply the outside of radical numbers first      (4 . 7) = 28

2.   Multiply the inside radical  numbers              √(7 . 5)  = √35

3. . Put steps 1 and 2 together                                   28√35

4.  Answer                                                                 28√35

Ex:2 (a) Steps to multiplying radicals √(4 + 6) . √(4 - 6)

Sol:

√(4 + 6) . √(4 - 6)       (given)

= √(42 – 62)

= √(16 – 36)

= √-20                  (Simplify radicals if possible )

= 2√-5                           (Answer)

(b) Steps to multiplying radicals √(3 + 2) . √(3 - 2)

Sol:

√(3 + 2) . √(3 - 2)           (given)

= √(32 – 22)

= √(9 -4)                  (Simplify radicals if possible )

= √5                                  (Answer)

Examples on rule:

(a) Steps to multiplying radicals  (7√3)2

Sol:

(7√3)2          (given)

= 72 . 3

= 49 . 3

= 147                  (Answer)

(b) Steps to multiplying radicals  (3√2)2

Sol:

(3√2)2            (given)

=  32 . 2

=  9 . 2

=  18               (Answer)