Introduction to ratio and proportion summary

Introduction to ratio and proportion summary:

              The ratio a: b is equivalent to the quotient `a/b` . A is relation of give the equality of two ratios, in the form `a/b` = `c/d` . This relation has called as proportion. Proportion is a states in which numbers are called as proportionality relation giving of two ratios, in the form `a/b` = `c/d` . Here a & d are known as extreme and b & c are known as means.                                                    

Problems on Proportion Summary:

     The ratio a: b is equivalent to the quotient `a/b` . The ratio of two given numbers a and b is the fraction, usually expressed in reduced form.

Ratio summary - example 1:  

                  A classroom has 60 men and 20 women. What is the ratio of men and women?        

Solution:

                   Men to women = 60 to 20 or 60: 20,

                   which reduces to = 6 to 2 or 6:2

                   The ratio of men to women is 6 to 2, or `6/2` , or 6 : 2.

Ratio summary - example 2:  

              A triangle has angle measures of 30°, 60°, and 90°. In simple form, what is the ratio of the given angles to each other?

Solution:

            30: 60: 90 = 3: 6: 9 (10 is a common divisor).That is,

                     (1) Ratio of the first to the second is 3 to 6.

                     (2) Ratio of the first to the third is 3 to 9.

                     (3) Ratio of the second to the third is 6 to 9.

Some Problems for Proportion Summary:

    A is relation give the equality of two ratios, in the form `a/b ` = `c/d` . This relation has called as proportion. Proportion is a states in which numbers are called as proportionality relation giving of two ratios, in the form `a/b` = `c/d` . Here a & d are known as extreme and b & c are known as means.                                                   

Proportion summary - example 1:

           If `x/5` = `y/2` , find the ratio of  `x/y` .(proportion)

Solution:

                    `x/5` =  `y/2`                                                          

                    `x/y` =  `5/2`

       Therefore the ratio of x and y is 5:2

Proportion summary - example 2:

           If  `x/3` = `2/6` , then find x?

Solution:

           Using cross multiplication (Property 1)

                    2x = 3 `xx` 6

                    2x = 18

                     x = `18/2`

                     x = 9

            So the value of x = 9

Introduction to Solving Straight Line Equation

Introduction to Solving Straight Line Equation:

Straight line is a curve which has the same slope along its length. Basically  line is a series of points that extends in two opposite directions without end.There are many ways to solve the equations of straight lines. A straight line contains x-intercept, y-intercept and slope (m). An equation is a mathematical statement that two expressions are equal.
The general form of an equation of a straight line is
                                                     y = mx+b.
 Where m is the slope of the straight line and b is the intercept made by the straight line on y-axis.
If a Straight line passing trough two points (x₁,y₁) and (x₂,y₂) then slope of that line is `(y2-y1)/(x2-x1)`

• If two lines are parallel then their slopes are equal.
• If two lines are perpendicular to each other then product of their slopes is equal to -1.

There are different forms of equation of a straight line which are given below :

1. Slope-intercept form 
2. Intercept form 
3. Point-slope form 
4. Two-point form

The descriptions of each method will be explained below with an example. 

Different Forms of Equation of Straight Line:

1. Slope - Intercept form of a Straight Line:

Statement: The equation of the straight line with slope 'm' and Y-intercept 'c' is y = mx+c

Proof: Let L be the straight line whose slope is m and which cuts off an intercept 'c' on the Y-axis 

                   If P(x,y) is a point on the xy-plane, then

               P lies on L   => m = slope of L = (y-c)/(x-0)

                                      => y = mx+c

               Conversely, if P(x,y) satisfies the equation y = mx+c, then x = 0 => y = c.

2. Intercept form of a Straight Line:

Statement: The equation of the straight line which cuts off intercepts a and b on the X-axis and Y-axis respectively is`x/a+y/b=1`

Proof: The straight line L which cuts off intercepts a and b on the X-axis and Y -axis respectively meets these axis at A(a,0) and B(0,b) and therefore slope of the line = -b/a.

Hence, the equation of L , by the slope-intercept form is  `y= (-b/a)x +b` or `x/a+y/b=1` .

Example: Solve the Equation of straight line which makes intercepts whose sum is 5 and product is 6 

Solution : Let a and b are the intercepts made by the line , then 

given in the problem that a + b = 5 and ab = 6 

solving these equations we obtain a = 2 and b = 3 

Then, the required equation of the straight line is `x/2+ y/3=1` i.e 3x+2y-6 = 0

                                                                   or `x/3+ y/2=1` i.e 2x+3y-6 = 0

3. Point - Slope form of a Straight Line:

Statement: The equation of the straight line with slope m and passing through the point (x1,y1) is y - y1 = m(x-x1).

Proof: Equation of any straight line with slope m is of the form y = mx+c.

This line passes through the point (x1,y1) is y1 - mx1 = c

Therefore, the equation of the line with slope m containing the point (x1,y1) is 

         y - mx=c = y1 - m1x1

    i.e y-y1 = m(x-x1)

Example: Solve the equation of straight line which has the slope -1 and passes through the point (-2,3) 

Solution: The slope of the given straight line m = -1 

and the point on the line is ( -2,3) 

hence the equation of the line is 

                           y-3= -1(x+2) 

                           x+y-1=0.

4. Two-Point form of a Straight Line:

Statement: The equation of the straight line passing through the point A(x1,y1) and B(x2,y2) is

(x-x1)(y1-y2)=(y-y1(x1-x2).

Proof: Let L be the Straight line containing the points A(x1,y1) and B(x2,y2) 

Case 1: suppose L is  non-vertical. Then x1 ≠x2 and slope of L = `(y1-y2)/(x2-x1)`

Therefore, the equation of L is `y-y1 = (y1-y2)(x-x1)/ (x1-x2)`

 i.e  (x-x1)(y1-y2) = (y-y1)(x1-x2)     ……(1)

Case 2: If L is vertical , then x1=x2 and y1≠y1. Equation of L, in this case, is x = x1 [form (1)]

Example: Solve the equation of straight line which is passing through the point (1,-2) and (-2,3)

Solution: The slope of the line containing (1,-2) and (-2,3) is

               m= (3+2)/(-2-1) = -5/3

and hence equation is y+2=-5/3(x-1)

 i.e      5x+3y+1=0

Exercise:

Problem 1: The following equation are in slope-intercept form. In each case, specify the slope and  y-intercept.

i) y=2x+7   (Ans: slope = 2, y-intercept = 7)

ii)  y=−4x+2    (Ans: slope = -4, y-intercept = 2)    

iii)  2x+5y=15.    (Ans: slope = `-2/5` , y-intercept = 3)         

Problem 2: Find the equation of line passing through two points (2,3) and (5,6).

Ans: y = x+1.

Applying scatter plot

Define scatter plot:
The definition of scatter plot is better understood using examples. Let us consider the following scatter plot problem:
The data of two variables X and Y is given in the table below, where X is the independent variable and Y is the dependent variable. Construct a scatter plot for the given data.

X 1 2 3 4 5 6 7
Y 8 6 7 4 6 8 8


Solution:
All we need to do is to plot the above X and Y values as (x,y) ordered pairs on a graph. So we get:

a graph that looks like above. This is called a scatter plot. In simple words a scatter plot is just plotted set of data as points on the co-ordinate axis.

We usually see scatter plot problems in statistics. The data for annual precipitation v/s the annual crop yield for a particular region, the data for demand of a particular soft drink against the month of the year, the research data of how long a particular pain killing drug takes to relieve pain in patients, etc. all can be depicted using a scatter plot. So we see that a scatter plot has applications in lots of fields, viz., business, finance, national growth and development, health care, aeronautics, astronomy etc.

Scatter plot correlation:
We know about frequency distribution of a single variable. But suppose now we have two variables instead of one. That means, each member of the population will exhibit two values, one for each variable under consideration. A population of this kind is called a bivariate population. To understand how these two variables relate to each other, one of the methods we use is a scatter plot correlation.
If the corresponding values of the two variables are noted for each member, then we can group our data into a table of double entry showing the frequencies of the pairs of values lying within given class intervals.
Each row in such a table gives the frequency distribution of the first variable for the members of the population in which the second variable likes within the limits stated on the left of the row.
Now if these two variables are termed x and y and a scatter plot as explained above is made out of the  data, then we can try finding some correlation between the two variables. The correlation may be linear or parabolic, or inverse or exponential. To find the equation of the line or curve that best fits the scattered points is studied under correlation and regression.

Know more about the statistics help for students, Math Homework Help. This article gives basic information about Applying scatter plot. Next article will cover more statistics concept and its advantages,problems and many more. Please share your comments.

Answer math peoblem

Get help with geometry answers free
Online is a free process to getting help from others .In schools, teachers will teach the lessons for the whole class but in the way of online method .The student can study the lessons with out any disturbances and he can take more notes freely from online process, also get help with geometry problem solver.
Geometry is a important part of mathematics that involves with shape, position of figures and the property of space. In study the geometry math problems are in different shapes presented. In that we have to find the area, perimeter, volume and surface area.

Introduction to basic math formula sheet

• In general, basic formulas are constructed by the symbols and by certain formation rules of the given logical language.
• In mathematics, Formulas are applied to provide a solution and the formulas are the key in solving the equations.
• Preparation of basic math formulas are very essential. In this article of basic math formula sheet, various basic mathematical formula sheets related to algebra, geometry and trigonometry are given.

also get help with trigonometry table

Generally geometry applications are mainly used for the everyday life for measuring area, perimeter and volume of the every object like building, land etc the shape of the buildings are different like the square ,rectangle ,circle etc, so according to the shape the formulas are different here we are going to learn the geometry formula for the grade 9 student .

Learning Graphing Calculator

Introduction to graphing calculator online for free

Graphing is one of the subtopics in the mathematics. Graphing is usually used to represent the given set of data in graphical format. Graphing is used to study the characteristics of the given data. Graphing is mainly used in algebra problems. In algebra, the topics, slope intercept form, linear equation and non linear equation are usually represented in graph format.

In online, learning graphing is very much interactive and fun. In online, free graphing calculator is used to solve the given problem. In online free calculator, when the equation is entered, it automatically generates the graph for the given equation. Free graphing calculators are easy to solve and those free calculators gives step by step explanation. In online, students can learn about various graphing topics. Through online, students have one to one learning.

Arithmetic sequence and polynomial problems

The sequence in which the each term, except the first is obtained by adding a fixed number to the term immediately preceding it is called "Arithmetic Sequence".
This fixed number is the difference of two successive terms and is called as "Common Difference" usually denoted by 'd'.Quantities are said to be in Arithmetic Sequence when they increase or decrease by a common difference.

Polynomial arithmetic is one of the interesting topics in mathematics. It is the process of performing different types of arithmetic operations such as addition, subtraction, multiplication and division in polynomial. It is the sums of a finite number of monomials are called as polynomial. Polynomial has more than one term and it has a constant value for the given each term, for that variable power of integral is raised to more than two.