Ratio and Proportion:

Ratio: 

A comparison of two quantities by division is called a ratio.

A ratio can be written in different forms like a : b or a/b and is commonly read as, a is to b.

Proportion:

Equality of two ratios is called proportion.

A proportion is written as a : b : : c : d, and is read as a is to b as c is to d. Here, a/b = c/d where b & d are not equal to 0.

There are two types of proportions.

• Direct Proportion
• Inverse Proportion

Direct Proportion:

Direct proportion describes the direct relationship between two quantities. If one quantity increases, the other quantity also increases and vice-versa.

Inverse Proportion:

Inverse proportion describes the relationship between two quantities in which if one quantity increases, the other quantity decreases and vice-versa.

Practice Problems:

1. Write the below ratios in its simplest form.

     a)    3:9            b)    5 : 125            c)  2 : 16

2. The scale on the map is 1: 10000.What actual distance does a length

    of 5 cm on the map represent ?

3. In a school there are 850 pupils and 40 teachers. Write the ratio of 

    teachers to pupils.

4. If 15 liters of petrol cost BD 2.1. Find the cost of 50 liters of petrol.

5. If  3 copies of a book cost $ 27. Calculate the cost of 10 books.

6. Ali and Zahra are given BD 50. They decided to divide the money in

    the ratio 2:3. How much dose each receive ?

7. A train normally complete a 270 km journey in 4.5 hours. How much

   faster would it have to travel to complete the journey in 4 hours ?

 Problems on percentages:

Practice problems on percentages.

1. What is 81 percent of 70 ?

2. What is 95 percent of 50 ?

3. What percent of 72 is  35 ?

4. 92 is 70 percent of what ?

5. $22 is 12 percent of what number ?

6. If the price of a mobile phone changes from 110 to 

   145. Calculate the percent change ?

7. A test has 20 questions. If peter gets 60% correct, 

   how many questions did peter miss?

8. In a school, 20% of the teachers teach basic math. If 

    there are 25 basic math teachers, how many teachers

   are there in the school?

9. 22 students in a class took an algebra test. If 15 

    students passed the test, what percent do not pass?

10. In a class 60% of the students are girls. If the total 

   number of students is 30, what is the number of boys?

***

 Evaluation of numerical expressions using calculator:

1. Practice Problems:

2. Practice Problems involving fractions.

 Simultaneous Linear Equations:

Two or more linear equations that all contain the same unknown variables are called a system of simultaneous linear equations. Solving such a system means finding values for the unknown variables which satisfy all the equations at the same time.

Example:               2x - 3y = 5

                               3x -2y = 0

Also, We can visualize the solution of the linear system of equations by drawing 2 linear graphs and finding out their intersection point.

How To Solve Simultaneous Linear Equations?

The following methods can be used to find the solution of linear system of equations, 

1. Substitution Method.

2. Elimination Method.

3. Graphical Method.

4. Matrix Method.

5.Using Calculator.(See the video below)

 Quadratic Equations:

Definition:

A quadratic equation is an algebraic equation of the second degree in x. The quadratic equation in its standard form is ax2 + bx + c = 0, where a and b are the coefficients, x is the variable, and c is the constant term.

Note: The graph of a quadratic equations is parabola.

Roots of a Quadratic Equation:

The roots of a quadratic equation are the two values of x, which are obtained by solving the quadratic equation. These roots of the quadratic equation are also called the zeros of the equation. For example, the roots of the equation x² - 4x - 5 = 0 are x = -1 and x = 5 because each of them satisfies the equation.

Methods to Solve Quadratic Equations

A quadratic equation can be solved to obtain two values of x or the two roots of the equation. There are five different methods to find the roots of the quadratic equation. The four methods of solving the quadratic equations are as follows.

• Factorizing of Quadratic Equation
• Using quadratic formula
• Method of Completing the Square
• Graphing Method to Find the Roots
• Using Calculator (See the video below)

 Exponential Equations:

An exponential equation is an equation with exponents where the exponent (or) a part of the exponent is a variable. 

For example,   

5x = 125,     3x - 1  = 9, 

In the following video, you will learn how to solve exponential equation using calculator.

 Graphical Representation of Data:

Graphical representation of data is an attractive method of showcasing numerical data that help in analyzing and representing quantitative data visually. A graph is a kind of a chart where data are plotted as variables across the coordinate. It became easy to analyze the extent of change of one variable based on the change of other variables. Graphical representation of data is done through different mediums such as lines, plots, diagrams, etc. 

  Let us discuss how to represent data using following methods:

   1. Pie Chart:

       A Pie Chart is a circular chart which is divided into sectors in which the area of each sector represents the size of the data.

  

   

2. Bar Graph:

        Representation of data in the form of bar or column is known as bar chart.

     

     

 Types of Data

1. Discrete Data: 

    Data collected as integers(whole numbers). 

    For example: Number of employees in construction companies, Number of cars per      

    family, etc.

2. Continuous Data:

    Data than can have any value within certain limits.

    For example: Height of people, daily temperatures, weight of people, cost of material,  

    labor costs, etc.

3. Raw Data:

    In a survey involving the weight(in kg) of first year college students, the following  

   data was obtained:

       60.1,  65.5, 55.0,  58.8, 61.5, 65.9, 56.6, 55.2 and 56.4.

    This is an example of raw data, as it is shown in the manner that it was collected.

4. Grouped Data:

    Data that are arranged in groups is called grouped data. The number of groups will 

   depend on the amount of the data.

    The following is an example of grouped data:

    54.0-56.4,      56.5-58.9,   59.0-61.4,   61.5-63.9,    64.0-66.4

   A group is also called a class and each class is specified by two limits, the lower class  

   limit and the upper class limit. 

   For the first class:  54.0-56.4

               54.0 is the lower limit

               56.4 is the upper limit.

 Measures of Central Tendencies

A measure of central tendency is a single value that attempts to describe a set of data by identifying the central position within that set of data.

The mean, median and the mode are the three measures of central tendencies.

    Mean: 

The mean or average for the ungrouped data is found by adding together the values of  the members of a set and dividing by the number of members in the set.   

      

 Solving Simultaneous Linear Equations:

In this video, you can learn how to solve simultaneous linear equations in 3 minutes using calculator.

 Mean of grouped data:

In this video, you can learn how to find the mean of grouped data in 4 minutes using Microsoft Excel.

 Mean of ungrouped data using Microsoft Excel.

In this video, students can learn how to find the mean of grouped data using the Microsoft Excel:

Representation of Data using Bar Graph and Pie-Chart:

In this video, one can learn how to represent data using the Bar Graph and Pie-Chart.