Double Negatives In Math

Introduction of double negatives in math:

The double negative in math deals with the signed numbers in the math. The double negatives give some rule in which the math rules can be made while the summing of the numbers is made and results to find the solution of the numbers. The double negative can have the values in positive manner. The double negatives come under the arithmetic operations.

Double negatives in math:
The negative numbers are given with (-) signed in front of the numerals. The negative numbers are value lower than the zero. The decreases in the value are made in the negative numbers. The double negative numbers are also named as the addition of the signed numbers. The positive numbers are made to have in the way in which the values are represented that is the values are more than the zero. The signed numbers can be done through the some of the rules which are,

` (-)xx (-) = (+)`

`(-)xx (+) = (-)`

`(+)xx (-) = (-)`

The above are the some of the rules for the negative numbers. The double negative numbers are which the product of the two negative numbers results to the positive of the particular value. Then the summing of the double negatives gives us the summing process with the negative signed value. The subtracting of the double negatives results to the negative signed numbers.


Examples for the double negatives in math:


Example:

What is -6+ (-2)?

Solution:

From above: (- )+(-) becomes a negative sign.

-6+ (-2) = -6 - 2

Answer: -6+ (-2) = -8.

More example problems for adding signed numbers:

(-) 7+ (-2) = -7 - 2 = -9.

(-) 8-(+2) = -8 - 2 = -10.

Example:

What is -5+ (-2)?

Solution:

-5+ (-2) = -5 - 2 = -7.

Answer: -5+ (-2) = -7.

Example: What is -6 - (+3)?

-6 - (+3) = -6 - 3

Answer: -6- (+3) = -9.

Boundary Line Math Definition

Introduction to boundary line math definition:

The boundary line is defined as the line or border around outside of a shape. The Boundary line defines the space or area. The limits of an area can be determined by the boundary line. The boundary line lies instantly inside the boundary. The boundary line indicating an edge of something. There is a boundary line for each and every shape. For each and every shape we can determine the area. The examples of boundary lines in math are given below. They are:

Triangle, Circle, Square, Rectangle, Parallelogram, Trapezoid, Rhombus, Cylinder, Cube and Cone


Examples problems- boundary line math definition:


Example 1: boundary line math definition

Find the area of the right- angled triangle whose height is 10cm and base is 5cm.



Solution:

Step 1: Given that base, b = 5cm and height, h = 10cm

Step 2: The formula for the area of the right-angled triangle is area= `1/2` (`bxx h)`

Step 3: Substitute the values of base and height

Step 4: area=`1/2(5 xx 10)`

Step 5: area=`1/2(50)`

Step 6: area=25cm

Answer: The area of the right- angled triangle is 25cm.

Example 2: boundary line math definition

Find the area of the equilateral triangle whose side is 5cm.



Solution:

Step 1: Given that the side is 5cm.

Step 2: The formula for the area of the equilateral triangle is area= `sqrt(3)/4` ` xx` a2

Step 3: Substitute the values of the side.

Step 4: area= `sqrt(3)/4` ` xx` 5 2

Step 5: area= `sqrt(3)/4 xx` 25

Step 6: area=6.25`xxsqrt(3)`

Step 7: area=6.25 `xx` 1.732 (The value of `sqrt(3) ` is 1.732)

Step 8: area=10.825cm

Answer: The area of the equilateral triangle is 10.825cm.


One more example problem- boundary line math definition:


Example 1: boundary line math definition

Find the area of the circle whose radius is given as 2cm.

      

Solution:

Step 1: Given that the radius is 2cm.

Step 2: The formula for the area of the circle is area= `pi `r2

Step 3: Substitute the values of the radius

Step 4: We know that the value of `pi ` is `22/7` (or) 3.14

Step 5: area= 3.14 `xx` 22

Step 6: area= 3.14 `xx` 4

Step 7: area=12.56cm

Answer: The area of the circle is 12.56cm

Arithmetic math Diagrams

Introduction to arithmetic math diagram:

Arithmetic or arithmetic (from the Greek word ????տ? = number) is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations.

In common usage, it refers to the simpler properties when using the traditional operations of addition, subtraction, multiplication and division with smaller values of numbers.




Arithmetic math diagram and problems:


Example problems for arithmetic math diagram:

Addition problems:

Question 1:

Solve: From the given problem, perform addition operation for 4 red color balls and 6 blue color balls.

Solution:

The given problem is

4 red balls and 6 blue color balls



Total number of balls = red color balls + blue color balls.

= 4 + 6

= 10

Total number of balls = 10 Balls.

Question 2: Solve 41 + 6 + 7 + 3

Solution: The given problem is
= 41 + 6 + 7 + 3
= 54 + 3
= 57

Question 3: Solve 5x + 2x + 4x

Solution: The given problem is

= 7x + 4x
= 11x

Subtraction problems:

Question 1:

Solve: From the given problem, perform subtraction operation for 4 black color balls and 2 green color balls

Solution:

The given problem is

4 black balls and 2 green color balls



Total number of balls = black  color balls - green color balls = red color balls.

= 4 -2

= 2

Total number of balls =2 red balls.

Question 2: Solve 40 - 4 - 2 -3

Solution: The given problem is
= 40 - 4 - 2 -3
= 36 -2 - 3
= 34 -1

= 33

Question 3: Solve 8x - 6y -3x -4y

Solution: The given problem is

= 8x - 6y -3x - 4y

= 8x - 3x - 6y - 4y

= 5x - 2y

Question 4: Solve 14x - 7x - 2x

Solution: The given problem is
=14x -7x -2x

= 7x - 2x

= 5x

Multiplication problems:

Question 1:

Solve: From the given problem, perform Multiplication operation for 2 red color balls and 2 blue color balls.

Solution:

The given problem is

2 red balls and 2 blue color balls



Total number of balls = red color balls * blue color balls = Yellow color balls.

= 2 *2

= 4

Total number of balls = 4 Yellow color balls.

Question 2:

Multiply 381 by 102

Solution:

381 נ102 means 102 times 381

Or 100 times 381 + two times 380

Or 38100 + 760

Or 38860

Therefore 381 נ102 = 38860

Division problems:

Question 1:

Solve: From the given problem, perform Division operation for 4 brown color balls and 2 pink color balls.

Solution:

The given problem is

4 brown balls and 2 pink color balls




Total number of balls = brown  color balls / pink color balls = Green color balls.

= `4 / 2 `

= 2

Total number of balls = 2 Green color balls.

Question 2:  solve (6/10)-(5/10)

Solution:
= (`6/10` )-(`5/10` )
= (6-5) / 10
=  `1 / 10`


Practice problems for arithmetic math diagram:


Practice problems are given below for test preparation,

1) Answer the math expression

32 + (3 * 3) -205

Solution: - 187

2) Answer the math expression

(ֳ)2 + 202 -10 -1

Solution: 200

3) Answer the math expression

3x + 2y -20 + 4y -2x

Solution: x + 6y -2

Problems Involving Coins : 2

Introduction to problems involving coins

The purpose of the article is to help readers understand:

• Represent the value of a given combination of coins.
• Represent a word problem in to an equation.
• How to solve an equation.
• Various denominations (as per US Standards)
• Solve problems consisting probability of coins.
• Representation of worth of amount in various denominations.

Denominations of coins used in solving problems

S.No	COIN	Worth (in Cents)
1	Quarter	25 cents
2	Dime	10 cents
3	Nickel	5 cents
4	Cent	1 cent

We will use the values in the above chart to solve problems involving coins


Examples of word problems involving coins


Ex 1: If a pencil costs 4 quarters and cost of an eraser is 5 dimes then how much will it cost (in dollars) to purchase 2 pencils and 1 eraser?

Sol:  Cost of 1 pencil = 4 quarters = 4 * 25 cents= 100 cents

Cost of an eraser = 5 dimes = 5*10 cents = 50 cents

Total cost of 2 pencils and 1 eraser (in cents) = 2*100 + 50 = 250 cents

Total cost of 2 pencils and 1 eraser (in dollars) = 250/100 = $2.5 ANSWER.

Ex 2: In a fundraiser program a total of $534 were collected in dimes and quarters. If the total number coins were 3000, then how many quarters were in the collection?

Sol: Suppose the number of dimes be equal to‘d’ and number of quarters be equal to ‘q’ in final collection.

So now we can write the following 2 equations:

1)      Equation for matching the total number of coins for both the denominations

d+q=3000 (Equation A)

2)      Equation for matching the total worth of coins and total sum collected (in cents)----10d+25q=534*100 (Equation B)

Representing in table form the problem can be summarized as:

Kind of coin	Number of coins	Value of each coin (in cents)	Total value  (in cents)
Dimes	d	10 cents	10*d
Quarters	q	25 cents	25*q
Total	3000		53400

Solving the two equations:

Multiplying equation A by 2 and dividing equation B by 5 gives

2d+2q = 6000 (Equation C) and 2d+5q= 10680 (Equation D)

Now subtracting Equation C from Equation D gives 3q=4680 and hence q=1560

Putting value of q in Equation A fives value of d= 1440

Hence,

                Number of dimes = 1440 Answer

Number of quarters = 1560 Answer

Ex:3 In a charity box, there is a collection of nickels, dimes, and quarters which amount to $5.90. There are 3 times as many quarters as nickels, and 5 more dimes than nickels. How many coins of each kind are there?

Sol: Let n = the number of nickels.

Then 3n = the number of quarters.

(n + 5) = the number of dimes.

Kind of coin	Number of coins	Value of each coin (in cents)	Total value  (in cents)
Nickels	n	5	5*n
Quarter	3n	25	25*3n
Dimes	n+5	10	10*(n+5)

The total value of all the coins is 590 cents.

5n+25*3n+10*(n+5)=590

5n+75n+10n+50=590

90n=540

n = 6 hence ,

Number of Nickels= 6 Answer

Number of Quarter = 3*6 = 18 Answer   and

Number of Dimes = n+5 = 6+5 = 11 Answer


Problems Based on Probability involving Coins


Concept :  Probabilities are represented as numbers between 0 and 1 (both including) to reflect the chances of occurrence of an event. A probability of 1 represents that the event is certain and a probability of 0 represents that the event is impossible.

When a coin is tossed probability of occurrence of Head or Tail is equal and since sum of probabilities of an event is 1 (at max), we say that probability of occurrence of Head will be equal to probability of occurrence of Tail and both will b equal to half(0.5).

Possible outcomes useful when solving problems involving coins

1) Toss of one coin: T or H

2) Toss of 2 coins: TT, TH, HT, TT

3) Toss of 3 coins: TTT, TTH, THT, THH, HTT, HTH, HHT, HHH

And similarly total outcomes of toss of n coins will be equal to 2n.

Ex 4: A coin is tossed. What is the probability of getting a Head?

Solution : As also discussed in concept above, probability of occurrence of Head in a single toss in a fair coin will be equal to 0.5 OR 50% .

Ex 5: What is the probability of getting three tails, in each flip, if a coin is tossed three times?

Solution : Since the probability of getting tails is 0.5, the probability of three tails would be 0.5*0.5*0.5=0.125 or 12.5 %

Place Value Chart Math

Introduction to place value math chart:

                           In math, the place value of numbers is established by its placed positions. The numbers are located in the particular number of a digit inside a numerical numbers. The place value specifies the position of a numerical system base value. The values are illustrates that numerical value in the standard forms in the identification. The values are established by place of the numerals accessibility.




Numbers on the place value chart in math:


Numbers- words

      The following chart graph is an example of place value chart in math.



In a given diagram, we have illustrate the place values of ones to hundred.

Decimal value of math chart:
 0.5 - 10 th

 0.05 -100 th

 0.005 -1000 th

 0.0005- 10000 th



Math example for decimal number place value:


Math chart Example 1:

       Place value is the sources of our whole number system. A place value in math system is one in which the position of a digit in a number establishes its value.

Look at number 584, the 5 in the hundreds place equal 500, The 8 in tens place equal 80, The 4 in the ones place equal 4.

Math chart Example 2:

 

The 6 is in the ten thousands place. It tells you there is 6 sets of ten thousand in the thousand place.

10,000+10,000+10,000+10,000+ 10,000+ 10,000 = 60,000   

The 5 is in the thousands place.  It tells you that there are 5 sets of thousand in the thousands place.1000+1000+1000+1000+1000 = 5000

The 2 is in the hundreds place. It tells you there are 2 sets of hundreds in the hundreds place.100+100=200

The 3 is in the tens place. It tells you there are 3 sets of tens in the tens place 10+10+10=30

The 4 is in the ones place. It tells you there is 4 ones in the ones place 1+1+1+1= 4

60,000 + 5000 + 220 + 30 + 4 = 65234

Fraction Subtraction Rules

Introduction For fraction subtraction rules:
A fraction is a part of a whole group or its region. A fraction written in a form of number with bottom part (denominator) denotes how many parts the whole divided into, and a top part (numerator). Fraction is in few types which can be written as many types which depend upon numerator and denominator by its value.

Fractions are subdivided into
Proper fractions
Improper fractions
Mixed fractions


Classification of Fractions:


Proper Fractions:
A proper fraction, fraction which a denominator shows in number of parts into whole divided and a numerator shows the number of parts which  we taken out. proper fraction also defined as numerator less than denominator.

Example:
`1/3, 2/5`

Improper Fractions:
Improper fractions is a fraction, whose numerator which is greater than the denominator are called improper fractions.

Example:
`3/2, 9/2`

Mixed Fractions:
A  mixed fraction is a fraction defined as which  has combination of a whole and its part.

Example:
2` 1/4,` 2` 2/9` ,


Rules for Fraction Subtraction:


In subtraction, rules for fraction numbers with same denominator, denominator remains same number and we subtract only numerator.
We can't do subtraction in fraction with different denominator rules for that we have to take LCM for all denominator and change different denominator into like denominator by taking LCM and adding fractions

Example Problems in Fraction subtraction rules:
Example 1:
subtract fractions  `1/3` from  `5/3`

Solution :
given fraction is a proper fraction, we have same denominator
`5/3 -1/3` =` (5-1)/3`

=`4/3`

Example 2:
subtract fractions  `2/5` from  `4/5`

Solution :
given fraction is a proper fraction, we have same denominator
`4/5 -2/5` =` (4-2)/5`

=`2/5`

Example 3:
subtract  fractions  `1/5` from `1/3`

Solution:
In this improper fraction we have different denominator  so,we take LCM
The LCM of 3 and 5 is 15.

Therefore, `1/3-1/5` =`(1xx5)/(3xx5)-(1xx3)/(5xx3)`

=`5/15-3/15`

=`2/15`
Example 4:
Subtract fractions  `1 2/5` from `3 3/6`

Solution:
`3 3/6 - 1 2/5` =  `(18+3)/6-(5+2)/5`

Now `21/6-7/5` =`(21xx5)/(6xx5)-(7xx6)/(5xx6) `    since LCM of 5,6 =30

=`105/30-42/30`

=`63/30`

Elementary Math Terms

Introduction to elementary math terms

     Elementary math terms are the basic form of algebra having little or no proper information of mathematics beyond arithmetic. In arithmetic numbers and their arithmetical operations (such as +, −, ×, ÷) occur, in algebra one also uses symbols (such as x and y, or a and b) to denote numbers. Elementary Algebra can be distinguished from abstract algebra, a more advanced field of study.And now we see about the elementary math terms below in simple problems.(Source in wikipedia )

                              

Elementary math terms problems in solving the equations:

• Solve the given problem for  x: 1 - 3(x - 4) = 2(3x + 1) - 7

Solution:-

Now we allocate the  above equation:

1 - 3x +12 = 6x + 2 - 7

Collect like terms:

 -3x +13 = 6x - 5

Add 3x to both sides of the equal sign:

-3x +13 + 3x = 6x - 5 + 3x

13 = 9x - 5

Add 5 to both sides:

18 = 9x

Divide both sides by 9 in above equation :

2 = x

The final result of the given problem fro  x is 2.

• Solvethe given problem for y:   ax + by = c

solution:-

Step 1: Subtract ax from both sides:

Step 2:by = c - ax

Step 3:Divide both sides by b:

           C - ax
y =   -----------  
             b

The final result of the given problem are            C - ax
                                                                       y = -----------                                            
                                                                                b

• Solve the problem :X = 5*(20+5)

Solution:

Step 1:  5*(20+5) = 5*20+5*5

Step 2:  =100+25

               X  =125

The final result the given problem is X= 125

• Solve the problem :X= 9*(6 + 5)

Solution

Step 1: 9*(6 + 5) = 9*6 + 9 *5

Step2: 36 + 45 

Step 3: X =  81     

The final result of the given problem  is X = 81       

• Solve the problem : Y = 7+(4+6)

Solution:

Step 1:7+(4+6)=7+10

                  Y    =17

The final result of the given problem is Y =  17

The above discussed are the  element math terms like addition , subtraction ,Multiplication and division .

Practice problems in Elementary math terms :

• Solve algebraic equation       6(-2x - 3) - (x - 2) = -5(2x + 3) + 19  

Solution:    x = 10

• Subtract x³ – 3x² – 1 from 3x³ + 6x² – 4x – 8.

Solution:   2x³ + 9x² – 4x – 7

Intercept Theorem

Introduction:

                  Intercept theorem plays a vital role in elementary geometry. It deals with the ratios of various line segments, which are created if two intersecting lines are intercepted by a pair of parallels. This theorem is equivalent to ratios of similar triangles. Traditionally it is attributed. The intercept theorem is  also called as Thale's Theorem. If a transversal line makes equal intercepts on three or more parallel lines, then any other line cutting them will also make equal intercepts.

Concept of Intercept Theorem:

                  If a transversal line makes equal intercepts on three or more parallel lines, then any other line cutting them will also make equal intercepts. AP || BQ || CR.

Types of intercept Theorem

                  If a transversal line makes equal intercepts on three or more parallel lines, then any other line cutting them will also make equal intercepts. AP || BQ || CR.

What Does Transversal Mean in Math

Introduction to Transversal in Math:

Definition:

A line that cuts (passes through) across two or more (usually parallel) lines then it is said to be a transversal.
It can also be defined as a line that intersects two or more co-planar lines each at a different point is also termed as a Transversal.

Transversal Postulates:
If two lines are parallel to each other and intersected by a transversal then the corresponding angles are congruent.

Some important points:

• Supplementary angles are the angles whose sum is equal to 180⁰.
• Vertical angles are the angles that are not adjacent angles and are formed by two intersecting lines and are always equal to each other.

Properties of transversal of parallel lines:

• If two parallel lines are cut by a straight line (transversal), then the corresponding angles around each intersection are equal in measure or we can say mathematically as these angles are congruent.
• If two parallel lines are cut by a straight line (transversal) then the alternate interior angles are congruent.
• If two parallel lines are cut by a straight line (transversal) then the interior angles on the same side of the transversal are supplementary.

Proofs for transversal by solved examples

Solved Problems to prove the above given points:
Ex 1: In the given figure, If the angles 2 and 3 are congruent then prove that r and s are parallel.

parallel lines 

Sol: Step 1: Given in the problem that angle 2 = angle 3
       Step 2: From the data given above angle 1 and angle 2 are vertically opposite angles and are congruent.
                               therefore, angle 1 = angle 2
       Step 3:The transitive property says that if a=b and b=c then a=c
 Using this property we have,
          angle 2 = angle 3    (given in the problem)
          angle 1 = angle 2    (vertically opposite angles)
so ,    angle 1 = angle 3   
As the angles made by the transversal with the two lines are equal the two lines are parallel.
Ex 2: Show that the transversal shown in the figure cuts the parallel lines.

Sol: In the above figure the angles 4 and 3 are known as alternate interior angles.Accoding to above discusssed points the alternate interior angles are congruent.
           `=>`    angle 4 = angle 3
So, the two lines are parallel lines.
Ex 3: Prove that the lines a and b shown in the figure below are parallel.

Sol: Clearly it is shown that the angles 1 and 2 are equal to` 90@` . That means that they are supplementary angles based upon the above discussion.
As the  angles made by the transversal with the two straigth are supplementary(equal) the two lines are parallel.Hence, proved.

Practice Problem on traversal line

Pro: Find the measures of the unknown angles in the following figure.Given r is parallel to s and angle 1 = `60@.`

Ans:  angle 2 =`120@.`
angle 3 = `60@` .
angle 4 = `120@`
angle 5 = angle 1 = 60⁰
angle 6 = angle 2 = 120⁰
angle 7 = angle 3 = 60⁰
angle 8 = angle 4 = 120⁰