The circle

In geometry, the circle is the locus of points at the same distance from a given point.

An easier way to define such geometric figure is to say that it is a closed curved line with all points equally distant from the center.

The following shows an example:



Real life examples bicycle wheels, coins, such as dimes and pennies, CDs, and MP3 players.

A line segment joining two points on the figure is a chord.The following are examples of two chords



When a chord passes through the center,we call it a diameter. A diameter usually divides such figure into two equal halves. Each half is called a semi-circle



Half a diameter is called a radius.

In other words, 2 radii= diameter



Buy a comprehensive geometric formulas ebook. All geometric formulas are explained with well selected word problems

Common geometry formulas

Here, we provide you with common geometry formulas for some basic shapes

---

Rectangle:



Perimeter = l + l + w + w = 2 × l + 2 × w

Area = l × w

---

Square:



Perimeter = s + s + s + s = 4 × s

Area = s²

---

Parallelogram:



Perimeter = a + a + b + b = 2 × a + 2 × b

Area = b × h

---

Rhombus:



Perimeter = b + b + b + b = 4 × b

Area = b × h

---

Triangle:



Perimeter = a + b + c

Area = (b × h)/2

---

Trapezoid:



Perimeter = a + b + c + d

---

Circle:



Perimeter = 2 × pi × r or Perimeter = pi × d

Area = pi × r² or Area = (pi × d²)/4

Area of shapes

By definition, the area of shapes is the amount of space inside those shapes.

To get the amount of space inside a figure, we use a square to represent 1 unit and we say that the area is measured in square units

Take a look at the following rectangle. To get the area, we are going to draw squares of equal sizes inside of it.



1 square represents 1 square unit. The rectangle has 8 squares, so the area for this rectangle is 8 square units.

We can also write 8 units² and it will mean the same

Notice,it is very important, that you can get the same answer if you multiply 2 square units by 4 square units because 2 × 4 = 8

2 square units represent the measure of the width and 4 square units represent the measure for the length.

Thus, in general, to get the area for a rectangle, just use the following formula:
Area of rectangle = length × width



In practice, when looking for the area of shapes, you will be using real life units such, inches, yards, feet, and so forth

The following examples demonstrate how to do this



Notice here the unit we are using is inch. That means we are going to use squares, which have a side of 1 inch to get the area for the rectangle.

Area = length × width = 5 × 2 = 10 square inches or 10 inches²

This also means that we can fit 10 squares with a side of 1 inch inside this rectangle.

Find the area of the following rectangle



Area = length × width = 10 × 2 = 20 square inches or 20 inches²

Now,that you understand how to get the area for a rectangle, it is going to be easy to get the area of shapes such as squares, triangles, and trapezoid.

Finding the perimeter

Finding the perimeter of a shape means that you are looking for the distance around the outside of that shape.

Example #1

Find the distance around for the following irregular polygon



Distance around = 5 inches + 4 inches + 2 inches + 3 inches + 6 inches = 20 inches

Example #2

Find the distance around for the following trapezoid



Distance around = 5 inches + 8 inches + 4 inches + 3 inches = 20 inches

Example #3

Find the distance around the following triangle



Distance around = 5 inches + 4 inches + 2 inches = 11 inches

Example #4

Find the distance around the following rectangle



Distance around = 3 inches + 3 inches + 6 inches + 6 inches = 18 inches

Example #5

Find the distance around the following square



distance around = 5 inches + 5 inches + 5 inches + 5 inches = 20 inches
You can get the same answer by doing 4 × 5 = 20

Free math problem solver

The free math problem solver below is a sophisticated tool that will solve any math problems you enter quickly and then show you the answer.

I recommend that you use it only to check your own work because occasionally, it might generate strange results.







After you enter your math problems, click on "Math Format" to make sure you have effectively entered the math problem you really want it to solve

You can also enter word problems, but don't be too fancy. Use plenty of math operators and keep it as simple as possible

If you want step by step solution after the final answer is entered, click on "View Steps"

This will take you to the developer's site where you sign in. A modest fee may be required to view all steps

Enjoy this online math problem solver!

Basic geometry

Basic geometry is the study of points, lines, angles, surfaces, and solids.The study of this topic starts with an understanding of these. Let's define them.




Point: A point is a location in space. It is represented by a dot. Point are usually named with a upper case letter. For example, we refer to the following as "point A"

Line: A line is a collection of points that extend forever. The following is a line. The two arrows are used to show that it extends forever.

We put two points in order to name the line as line AF. However, there are an infinite amount of points. You can also name it line FA

Line segment: A line segment is part of a line. The following is a segment. A segment has two endpoints. The endpoints in the following segments are A and F. Notice also that the line above has no endpoints.

Ray: A ray is a collection of points that begin at one point (an endpoint) and extend forever on one direction. The following is a ray.

Angle: Two rays with the same endpoint is an angle. The following is an angle.

Plane: A plane is a flat surface like a piece of paper. It extends in all directions. We can use arrows to show that it extends in all directions forever. The following is a plane

Parallel lines When two lines never meet in space or on a plane no matter how long we extend them, we say that they are parallel lines The following lines are parallel.

Intersecting lines: When lines meet in space or on a plane, we say that they are intersecting lines The following are intersecting lines.



Vertex: The point where two rays meet is called a vertex. In the angle above, point A is a vertex.

Number trick with 1089

The number trick with 1089 has been around for centuries. To impress someone with this trick, he or she will need paper and pencils:


Here is how it goes:

Step #1:

Have the person write down any three digits number with decreasing digits (432 or 875).

Step #2:

Reverse the number you wrote in step #1.

Step #3:

Subtract the number obtained in step #2 from the number you wrote in step #1

Step #4:

Reverse the number obtained in step #3

Step #5:

Add the numbers found in step #3 and step #4


Example #1:


Step #1:

Have the person write down any three digits number with decreasing digits: 752

Step #2:

Reverse the number you wrote in step #1: 257

Step #3:

Subtract the number obtained in step #2 from the number you wrote in step #1: 752 - 257 = 495

Step #4:

Reverse the number obtained in step #3: 594

Step #5:

Add the numbers found in step #3 and step #4: 495 + 594 = 1089


Example #2:


Step #1:

Have the person write down any three digits number with decreasing digits: 983

Step #2:

Reverse the number you wrote in step #1: 389

Step #3:

Subtract the number obtained in step #2 from the number you wrote in step #1: 983 - 389 = 594

Step #4:

Reverse the number obtained in step #3: 495

Step #5:

Add the numbers found in step #3 and step #4: 594 + 495 = 1089


Example #3:


Step #1:

Have the person write down any three digits number with decreasing digits: 210

Step #2:

Reverse the number you wrote in step #1: 012

Step #3:

Subtract the number obtained in step #2 from the number you wrote in step #1: 210 - 012 = 198

Step #4:

Reverse the number obtained in step #3: 891

Step #5:

Add the numbers found in step #3 and step #4: 198 + 891 = 1089

This number trick with 1089 works with any 3 digits number as long as you choose a number with decreasing digits in step #1

So, it does not matter what number the person choose in step #1, have the person do the math and pretend you are not looking. When you call the answer as 1089, the person will be shocked! Cool!

Finger multiplication

The use of finger multiplication has been widespread through the years. It is not the traditional way of doing multiplication in school today.

However, if your kid has trouble remembering the whole multiplication table, multiplication with fingers is a good alternative

To multiply with fingers, you are only required to remember the multiplication table up to 5 × 5.

After that, all multiplication can be performed with your fingers

Here is the technique:

The two numbers to be multiplied are each represented on a different hand

Each hand may have some raised fingers and some closed fingers at the same time

Number of fingers to raise = Factor − 5. Remember that a factor is a number in a multiplication problem

The sum of the raised fingers is the number of tens

The product of the closed fingers is the number of ones

Example #1:

7 × 8

For 7, use your left hand and raise 2 fingers (Factor − 5 = 7 − 5 = 2) . This means there are 3 closed fingers

For 8, use your right hand and raise 3 fingers. This means that there are 2 closed fingers

Sum of raised fingers = 2 + 3 = 5. This means we have 5 tens or 50

Product of closed fingers = 3 × 2 = 6. This means that we have 6 ones

50 + 6 = 56

Example #2:

9 × 6

For 9, use your right hand and raise 4 fingers. This means there is 1 closed finger

For 6, use your left hand and raise 1 finger. This means that there are 4 closed fingers

Sum of raised fingers = 4 + 1 = 5. This means we have 5 tens or 50

Product of closed fingers = 1 × 4 = 4. This means that we have 4 ones

50 + 4 = 54

Example #3:

8 × 5

For 8, use your left hand and raise 3 fingers. This means there are 2 closed fingers

For 5, use your left hand and raise no finger. This means that there are 5 closed fingers

Sum of raised fingers = 3 + 0 = 3. This means we have 3 tens or 30

Product of closed fingers = 2 × 5 = 10. This means that we have 10 ones or 1 ten or 10

30 + 10 = 40

Squaring any two digit number ending in five

In this lesson, we will show you how straightforward squaring any two digit number ending in five can be


First look at the following multiplication

Did you make the following important observation?

The answer always ends in 25. This will be the case when you square any two digit number ending with 5

Now, how did we get the number(s) before 25? Easy!

Look for the number in the tens place.For the multiplications above, these are 2, 4, and 1

Multiply each number by its next higher digit

So,

2 × 3 = 6

4 × 5 = 20

1 × 2 = 2

Put these numbers on the left of 25

Other examples

1) 35 × 35

The digit in the tens place is 3

Multiply 3 by its next higher digit, which is 4

3 × 4 = 12

Write down 12 and put 25 next to it

The answer is 1225

2) 55 × 55

The digit in the tens place is 5

Multiply 5 by its next higher digit, which is 6

5 × 6 = 30

Write down 30 and put 25 next to it

The answer is 3025

3) 95 × 95

The digit in the tens place is 9

Multiply 9 by its next higher digit, which is 10

9 × 10 = 90

Write down 90 and put 25 next to it

The answer is 9025

Important note:

Before you can really apply these tricks, you must know your multiplication table by heart

Have you learned your muliplication table yet?

If having problems to learn the multiplication table by heart, have you tried finger muliplication ?

Multiplication by 11

To understand the multiplication by 11 trick, take a look at the following multiplication below:

Did you make the following two important observations?

First, notice that the digits of the number that is multiplied by 11 appear again in the answer

Second, the number in the middle of the answer is always found by adding the digits of the number that is multiplied by 11

This is the basic facts that you have to remember to quickly multiply numbers by 11

Therefore, if you want to use this trick, all you have to do is add the digits of the number and put the answer between the number

Other examples

1) 35 × 11

Just add 3 and 5 to get 8

Put 8 between 3 and 5

The answer is 385

2) 18 × 11

Just add 1 and 8 to get 9

Put 9 between 1 and 8

The answer is 198

3) 43 × 11

Just add 4 and 3 to get 7

Put 7 between 4 and 3

The answer is 473


Now, look at this example 48 × 11
Add 4 and 8 to get 12

Just put 12 between 4 and 8???

No, it does not work like this!

You can still put 2 in between. However, since the number is bigger than 9, you have to carry the 10 represented with a 1 over the 4

1 + 4 = 5. Thus the answer is 528

4) 85 × 11

Just add 8 and 5 to get 13

Add 1 to 8 to get 9

Put 3 between 9 and 5

The answer is 935


5) 78 × 11

Just add 7 and 8 to get 15

Add 1 to 7 to get 8

Put 5 between 8 and 8

The answer is 858

Multiplying by powers of ten

Follow the following shortcut when multiplying by powers of ten

Whole numbers multiplied by powers of 10

When multiplying a whole number by a power of ten, just count how many zero you have and attached that to the whole number

Examples:

1) 56 × 10

There is only one zero, so 56 × 10 = 560

2) 45 × 10,000

There are 4 zeros, so 45 × 10000 = 450000

3) 18 × 10,000,000

There are 7 zeros, so 18 × 10,000,000 = 180,000,000

Decimals multiplied by powers of 10

When multipying a decimal by a positive power of ten (positive exponent), move the decimal point one place to the right for each zero you see after the 1

Examples:

1) 0.56 × 10

There is only one zero, so move the decimal point one place to the right.

0.56 × 10 = 5.6

2) 0.56 × 100

There are 2 zeros, so move the decimal point two places to the right

0.56 × 100 = 56

3) 0.056 × 1000

There are three zeros, so move the decimal point 3 places to the right.

0.056 × 1000 = 56

4) 0.056 × 100,000

0.056 × 100,000 = 0.056 × 1000 × 100 = 56 × 100 = 5600

When multipying a decimal by a negative power of ten (negative exponent), move the decimal point one place to the left for each zero you see before the 1

Note that 0.1 = 10^-1, 0.01 = 10^-2, 0.001 = 10^-3, and so forth....

We call 10^-1, 10^-2, and 10^-3 negative powers of 10 because the exponents are negative

Examples:

1) 56 × 0.1

There is only one zero, so move the decimal point one place to the left.

56 × 0.1 = 5.6

2) 560 × 0.01

There are 2 zeros, so move the decimal point two places to the left

560 × 0.01 = 5.6

2) 560 × 0.001

There are 3 zeros, so move the decimal point two places to the left

560 × 0.001 = 0.560

3) 0.56 × 0.1

There is only one zero, so move the decimal point one place to the left.

0.56 × 0.1 = 0.056

4) 0.56 × 0.01

There are 2 zeros, so move the decimal point two places to the left

0.56 × 0.01 = 0.0056

Any questions about multiplying by powers of ten? Let me know...

Properties of exponents

We will show 8 properties of exponents. Let x and y be a number not equal to zero and let n and m be any integers

Property #1

x⁰ = 1

Example: 4⁰ = 1 and (2500000000000000000000)⁰ = 1

Property #2

xⁿ × x^m = x^{n + m}

Example: 4⁶ × 4⁵ = 4^{6 + 5} = 4¹¹

Property #3

xⁿ ÷ x^m = x^{n − m}

Example: 4⁶ ÷ 4⁵ = 4^{6 − 5} = 4¹

Property #4

(xⁿ)^m = x^{n × m}

(5²)⁴ = 5^{2 × 4} = 5⁸

Property #5

(x × y)ⁿ = xⁿ × yⁿ

(6 × 7)⁵ = 6⁵ × 7⁵

Property #6

x^-n = 1 ÷(xⁿ) = 1/(xⁿ)

8^-4 = 1 ÷ (8⁴) = 1 / (8⁴)

Property #7

(x/y)ⁿ = xⁿ / yⁿ

(8/5)⁴ = 8⁴ / 5⁴

Property #8



Any questions about properties of exponents? Let me know.

Properties of inequality

We will show 6 properties of inequality. When appropriate, we will illustrate with real life examples of properties of inequality.

Let x, y, and z represent real numbers

Addition property:

If x < y, then x + z < y + z

Example: Suppose Sylvia's weight < Jennifer's weight, then Sylvia's weight + 4 < Jennifer's weight + 4

Or suppose 1 < 4, then 1 + 6 < 4 + 6

If x > y, then x + z > y + z

Example: Suppose Sylvia's weight > Jennifer's weight, then Sylvia's weight + 9 > Jennifer's weight + 9

Or suppose 4 > 2, then 4 + 5 > 2 + 5

Subtraction property:

If x < y, then x − z < y − z

Example: Suppose Sylvia's weight < Jennifer's weight, then Sylvia's weight − 4 < Jennifer's weight − 4

Or suppose 4 < 8, then 4 − 3 < 8 − 3

If x > y, then x − z > y − z

Example: Suppose Sylvia's weight > Jennifer's weight, then Sylvia's weight − 9 > Jennifer's weight − 9

Or suppose 8 > 3, then 8 − 2 > 3 − 2

Multiplication property:

If x < y, and z > 0 then x × z < y × z

Example: Suppose 2 < 5, then 2 × 10 < 5 × 10 ( Notice that z = 10 and 10 > 0)

If x > y, and z > 0 then x × z > y × z

Example: Suppose 20 > 10, then 20 × 2 > 10 × 2

If x < y, and z < 0 then x × z > y × z

Example: Suppose 2 < 5, then 2 × -4 > 5 × -4 ( -8 > -20. z = -4 and -4 < 0 )

If x > y, and z < 0 then x × z < y × z

Example: Suppose 5 > 1, then 5 × -2 < 1 × -2 ( -10 < -2 )

Division property:

It works exactly the same way as multiplication

If x < y, and z > 0 then x ÷ z < y ÷ z

Example: Suppose 2 < 4, then 2 ÷ 2 < 4 ÷ 2

If x > y, and z > 0 then x ÷ z > y ÷ z

Example: Suppose 20 > 10, then 20 ÷ 5 > 10 ÷ 5

If x < y, and z < 0 then x ÷ z > y ÷ z

Example: Suppose 4 < 8, then 4 ÷ -2 > 8 ÷ -2 ( -2 > -4 )

If x > y, and z < 0 then x ÷ z < y ÷ z

Example: Suppose 5 > 1, then 5 ÷ -1 < 1 ÷ -1 ( -5 < -1 )

Transitive property:

If x > y and y > z, then x > z

Example: Suppose 10 > 5 and 5 > 2, then 10 > 2

x < y and y < z, then x < z

5 < 10 and 10 < 20, then 5 < 20

Comparison property:

If x = y + z and z > 0 then x > y

Example: 6 = 4 + 2, then 6 > 4

The properties of inequality are more complicated to understand than the property of equality.

Allow yourself plenty of time as you go over this lesson.Any questions about the properties of inequality, let me know.

Properties of equality

We will show 8 properties of equality. When appropriate, we will illustrate with real life examples of properties of equality.

Let x, y, and z represent real numbers

Reflexive property: x = x

Example: 2 = 2 or I am equal to myself

Symetric property: If x = y, then y = x

Example: Suppose fish = tuna, then tuna = fish

transitive property: If x = y and y = z, then x = z

Example: Suppose John's height = Mary's height and Mary's height = Peter's height, then John's height = Peter's height

Addition property: If x = y, then x + z = y + z

Example: Suppose John's height = Mary's height, then John's height + 2 = Mary's height + 2

Or suppose 5 = 5, then 5 + 3 = 5 + 3

Subtraction property: If x = y, then x − z = y − z

Example: Suppose John's height = Mary's height, then John's height − 5 = Mary's height − 5

Or suppose 8 = 8, then 8 − 3 = 8 − 3

Multiplication property: If x = y, then x × z = y × z

Example: Suppose Jetser's weight = Darline's weight, then Jetser's weight × 4 = Darline's weight × 4

Or suppose 10 = 10, then 10 × 10 = 10 × 10

Division property: If x = y, then x ÷ z = y ÷ z

Example: Suppose Jetser's weight = Darline's weight, then Jetser's weight ÷ 4 = Darline's weight ÷ 4

Or suppose 20 = 20, then 20 ÷ 10 = 20 ÷ 10

Substitution property: If x = y, then y can be substituted for x in any expression

Example: x = 2 and x + 5 = 7, then 2 can be substituted in x + 5 = 7 to obtain 2 + 5 = 7

Any questions about the properties of equality, let me know.

Identity property of multiplication

The identity property of multiplication, also called the multiplication property of one says that a number does not change when that number is multiplied by 1.

Examples

3 × 1 = 3

10 × 1 = 10

6 × 1 = 6

68 × 1 = 68

1 × 4 = 4

1 × -9 = - 9

x × 1 = x

(a + b) × 1 = a + b

Multiplicative inverse property

If you multiply two numbers and the product is 1, we call the two numbers multiplicative inverses or reciprocals of each other

For example, 4 is the multiplicative inverse of 1/4 because 4 × 1/4 = 1

1/4 is also the multiplicative inverse of 4 because 1/4 × 4 = 1

Notice that the multiplicative inverse of 1 is 1. In fact, 1 and -1 are the only two numbers that can be their own multiplicative inverse

Notice also that any number divided by 1 return the same number

We call this the identity property of division

Examples

2 ÷ 1 = 2

50 ÷ 1 = 50

-5 ÷ 1 = -5

Properties of zero

The two properties of zero are the addition property and the multiplication property.

Addition property:

The addition property says that a number does not change when adding or subtracting zero from that number

Examples

2 + 0 = 2

12 + 0 = 12

5 − 0 = 5

48 − 0 = 48

0 + 1 = 1

0 − 9 = - 9

x + 0 = x

(a + b) + 0 = a + b

Additive inverse property

If you add two numbers and the sum is zero, we call the two numbers additive inverses or opposites of each other

For example, 2 is the additive inverse of -2 because 2 + -2 = 0

-2 is also the additive inverse of 2 because -2 + 2 = 0

Multiplication property

The multiplication property says that zero times any number is equal to zero

Examples

2 × 0 = 0

0 × 12 = 0

-5 × 0 = 0

23344555677888882 × 0 = 0

x × 0 = 0

(x + y + z + r )× 0 = 0

Distributive property

We will explain the distributive property with three good examples

Example #1:

Look at the following illustration. How would you get the area?

Area = width × length

Since width = 6 and length = 4 + 10, area = 6 × (4 + 10)

You can do the math two ways

You can add 4 and 10 and multiply what you get by 6.

Otherwise, you can use the distributive property illustrated above by multiplying 6 by 4 and 6 by 10 and adding the results

Example #2:

You go to the supermarket. 1 bag of apples costs 4 dollars. 1 gallon of olive oil costs 10 dollars. You get 6 bags of apples and 6 gallons of olive oil.

How much money do you pay the cashier?

Total cost = # of items you get × (cost for apples + cost for olive oil)

Total cost = 6 × (4 + 10) = 6 × 4 + 6 × 10 = 24 + 60 = 84 dollars.

Same answer you would get for example #1!

Example #3:

Robert has 8 notebooks and his brother has 6. If we double both amount, how many do they now have altogether?

We get 2 × ( 8 + 6) = 2 × 8 + 2 × 6 = 16 + 12 = 28

Notice that we get the same asnwer if we add 8 and 6 and multiply the result by 2

Associative property

The word associate in associative property, may mean to join or to combine

For examples, suppose I go to the supermarket and buy ice cream for 12 dollars, bread for 8 dollars, and milk for 15 dollars.

How much money do I owe the cashier? The situation above is associative

When I do my total in my head, I can combine or add the price of the ice cream and the bread first and add the result to the price of milk.

Otherwise, I can combine or add the price of bread and milk first and add the result to the price of ice cream

Both ways of approaching the problem gives the same answer

Mathematically, you are trying to do the following:

12 + 8 + 15

You can add these three numbers in the order they appear

12 + 8 = 20 ( This is adding price of ice cream and bread first)

20 + 15 = 35

You can use parentheses to show the order in which you are adding

(12 + 8) + 15

Another way to add is to add not according the order in which they appear

You may decide you will add first 8 and 15

8 + 15 = 23 ( This is adding price of bread and milk first)

12 + 23 = 35

Again, using parentheses to show the order in which you are adding, you get:

12 + (8 + 15)

We conclude that (12 + 8) + 15 = 12 + ( 8 + 15)

The above example illustrates the associative property of addition

Terms added in different combinations or grouping yield the same answer

Associative property of multiplication

Again, we know that

(3 × 4) × 5 = 3 × (4 × 5)

(2 × 6) × 7 = 2 × (6 × 7)

(1 × 9) × 8 = 1 × (9 × 8)

All three examples given above will yield the same answer when the left and right side of the equation are multiplied

For example, 3 × 4 = 12 and 12 × 5 = 60

Also, 4 × 5 = 20 and 3 × 20 = 60

Warning! Although mutiplication is associative, division is not associative

Notice that ( 24 ÷ 6) ÷ 2 is not equal to 24 ÷( 6 ÷ 2)

( 24 ÷ 6) ÷ 2 = 4 ÷ 2 = 2

However, 24 ÷( 6 ÷ 2) = 24 ÷ 3 = 8

Therefore, different combination may yield different results.

Notice that it may happen that a different grouping gives the same result.

( 24 ÷ 6) ÷ 1 = 24 ÷( 6 ÷ 1)

( 24 ÷ 6) ÷ 1 = 4 ÷ 1 = 4 and 24 ÷( 6 ÷ 1) = 24 ÷ 6 = 4

However, we shall not make a rule out of this because it is not true for all cases

Finally, note that unlike the commutative property which plays around with two numbers, the associative property combines at least three numbers

Other examples:

( 1 × 5) × 2 = 1 ×( 5 × 2)

( 6 × 9) × 11 = 6 ×( 9 × 11)

( 1 + 5) + 2 = 1 + ( 5 + 2)

( 6 + 9) + 11 = 6 +( 9 + 11)

( x + 5) + 4 = x + ( 5 + 4)

( 6 + z) + 1 = 6 +( z + 1)

( x + y) + z = x + ( y + z)

( x × y) × z = x × ( y × z)

Commutative property

Certainly, in commutative property, we see the word commute which means exchange from the latin word commutare

The word exchange in turn may mean switch. For examples, washing my face and combing my hair is a good example of this property.

Another good example is doing my math homework and then finishing my science reading.

The important thing to notice in the two examples above is that the order we do things can be switched, so does not matter or will never cause any problems or conflicts.

However, reading a math lesson and then answering the review questions is not commutative.

Here the order does matter because I have to read the lesson before knowing how to answer the review questions

In mathematics, we know that

2 + 5 = 5 + 2

12 + 4 = 4 + 12

-1 + 8 = 8 + -1

All the above illustrates the commutative property of addition. This means that when adding two numbers, the order in which the two numbers are added does not change the sum

All three examples given above will yield the same answer when the left and right side of the equation are added

For example, 2 + 5 = 7 and 5 + 2 is also equal to 7

The property is still valid if we are doing multiplication

Again, we know that

3 × 4 = 4 × 3

12 × 0 = 0 × 12

9 × 6 = 6 × 9

Again, 3 × 4 = 12 and 4 × 3 = 12

More examples: Take a close look at them and study them carefuly

(3 + 2) × 4 = 4 × (3 + 2)

x + y = y + x

x × y = y × x

2 × x = x × 2

(x + z) × (m + n) = (m + n) × (x + z)

4 + y = y + 4

Warning! Although addition is commutative, subtraction is not commutative

Notice that 3 − 2 is not equal to 2 − 3

3 − 2 = 1 , but 2 − 3 = -1

Therefore, switching the order yield different results

Metric measurement

The metric measurement is a system of measuring using the liter, the gram, and the meter as basic units.

The gram is used to measure mass or weight.

For instance, a person weighing 170 pounds in customary measurement, weighs 77110 grams or 77.11 kilograms in the metric system.

the liter is used to measure capacity.

for instance, 1 liter of coke measures about 33.81 ounces in customary measurement.

The meter is used to measure length.

For instance, a person whose height is 1 meter is equivalent 3.2808 feet in customary measurement.

In the metric system, other units are named by using prefixes such as kilo, hecto, deka, deci, centi, and milli etc...

1 kilo = one thousand and the symbol we use is k

For example, 1 kilometer = 1000 meters

1 hecto = one hundred and the symbol we use is h

1 hectogram = 100 grams

1 deka = ten and the symbol we use is da

1 dekameter = 10 meters

1 deci = one-tenth and the unit is d

1 deciliter = 0.1 liter

1 centi = one-hundredth and the unit is c

1 centimeter = 0.01 meter

1 milli = one-thousandth and the unit is m

1 millimeter = 0.001 meter

Finger multiplication

The use of finger multiplication has been widespread through the years. It is not the traditional way of doing multiplication in school today.

However, if your kid has trouble remembering the whole multiplication table, multiplication with fingers is a good alternative

To multiply with fingers, you are only required to remember the multiplication table up to 5 × 5.

After that, all multiplication can be performed with your fingers

Here is the technique:

The two numbers to be multiplied are each represented on a different hand

Each hand may have some raised fingers and some closed fingers at the same time

Number of fingers to raise = Factor − 5. Remember that a factor is a number in a multiplication problem

The sum of the raised fingers is the number of tens

The product of the closed fingers is the number of ones

Example #1:

7 × 8

For 7, use your left hand and raise 2 fingers (Factor − 5 = 7 − 5 = 2) . This means there are 3 closed fingers

For 8, use your right hand and raise 3 fingers. This means that there are 2 closed fingers

Sum of raised fingers = 2 + 3 = 5. This means we have 5 tens or 50

Product of closed fingers = 3 × 2 = 6. This means that we have 6 ones

50 + 6 = 56

Example #2:

9 × 6

For 9, use your right hand and raise 4 fingers. This means there is 1 closed finger

For 6, use your left hand and raise 1 finger. This means that there are 4 closed fingers

Sum of raised fingers = 4 + 1 = 5. This means we have 5 tens or 50

Product of closed fingers = 1 × 4 = 4. This means that we have 4 ones

50 + 4 = 54

Example #3:

8 × 5

For 8, use your left hand and raise 3 fingers. This means there are 2 closed fingers

For 5, use your left hand and raise no finger. This means that there are 5 closed fingers

Sum of raised fingers = 3 + 0 = 3. This means we have 3 tens or 30

Product of closed fingers = 2 × 5 = 10. This means that we have 10 ones or 1 ten or 10

30 + 10 = 40

Squaring any two digit number ending in five

In this lesson, we will show you how straightforward squaring any two digit number ending in five can be


First look at the following multiplication

Did you make the following important observation?

The answer always ends in 25. This will be the case when you square any two digit number ending with 5

Now, how did we get the number(s) before 25? Easy!

Look for the number in the tens place.For the multiplications above, these are 2, 4, and 1

Multiply each number by its next higher digit

So,

2 × 3 = 6

4 × 5 = 20

1 × 2 = 2

Put these numbers on the left of 25

Other examples

1) 35 × 35

The digit in the tens place is 3

Multiply 3 by its next higher digit, which is 4

3 × 4 = 12

Write down 12 and put 25 next to it

The answer is 1225

2) 55 × 55

The digit in the tens place is 5

Multiply 5 by its next higher digit, which is 6

5 × 6 = 30

Write down 30 and put 25 next to it

The answer is 3025

3) 95 × 95

The digit in the tens place is 9

Multiply 9 by its next higher digit, which is 10

9 × 10 = 90

Write down 90 and put 25 next to it

The answer is 9025

Important note:

Before you can really apply these tricks, you must know your multiplication table by heart

Have you learned your muliplication table yet?

If having problems to learn the multiplication table by heart, have you tried finger muliplication ?

Multiplication by 11

To understand the multiplication by 11 trick, take a look at the following multiplication below:

Did you make the following two important observations?

First, notice that the digits of the number that is multiplied by 11 appear again in the answer

Second, the number in the middle of the answer is always found by adding the digits of the number that is multiplied by 11

This is the basic facts that you have to remember to quickly multiply numbers by 11

Therefore, if you want to use this trick, all you have to do is add the digits of the number and put the answer between the number

Other examples

1) 35 × 11

Just add 3 and 5 to get 8

Put 8 between 3 and 5

The answer is 385

2) 18 × 11

Just add 1 and 8 to get 9

Put 9 between 1 and 8

The answer is 198

3) 43 × 11

Just add 4 and 3 to get 7

Put 7 between 4 and 3

The answer is 473


Now, look at this example 48 × 11
Add 4 and 8 to get 12

Just put 12 between 4 and 8???

No, it does not work like this!

You can still put 2 in between. However, since the number is bigger than 9, you have to carry the 10 represented with a 1 over the 4

1 + 4 = 5. Thus the answer is 528

4) 85 × 11

Just add 8 and 5 to get 13

Add 1 to 8 to get 9

Put 3 between 9 and 5

The answer is 935


5) 78 × 11

Just add 7 and 8 to get 15

Add 1 to 7 to get 8

Put 5 between 8 and 8

The answer is 858

Multiplying by powers of ten

Follow the following shortcut when multiplying by powers of ten

Whole numbers multiplied by powers of 10

When multiplying a whole number by a power of ten, just count how many zero you have and attached that to the whole number

Examples:

1) 56 × 10

There is only one zero, so 56 × 10 = 560

2) 45 × 10,000

There are 4 zeros, so 45 × 10000 = 450000

3) 18 × 10,000,000

There are 7 zeros, so 18 × 10,000,000 = 180,000,000

Decimals multiplied by powers of 10

When multipying a decimal by a positive power of ten (positive exponent), move the decimal point one place to the right for each zero you see after the 1

Examples:

1) 0.56 × 10

There is only one zero, so move the decimal point one place to the right.

0.56 × 10 = 5.6

2) 0.56 × 100

There are 2 zeros, so move the decimal point two places to the right

0.56 × 100 = 56

3) 0.056 × 1000

There are three zeros, so move the decimal point 3 places to the right.

0.056 × 1000 = 56

4) 0.056 × 100,000

0.056 × 100,000 = 0.056 × 1000 × 100 = 56 × 100 = 5600

When multipying a decimal by a negative power of ten (negative exponent), move the decimal point one place to the left for each zero you see before the 1

Note that 0.1 = 10^-1, 0.01 = 10^-2, 0.001 = 10^-3, and so forth....

We call 10^-1, 10^-2, and 10^-3 negative powers of 10 because the exponents are negative

Examples:

1) 56 × 0.1

There is only one zero, so move the decimal point one place to the left.

56 × 0.1 = 5.6

2) 560 × 0.01

There are 2 zeros, so move the decimal point two places to the left

560 × 0.01 = 5.6

2) 560 × 0.001

There are 3 zeros, so move the decimal point two places to the left

560 × 0.001 = 0.560

3) 0.56 × 0.1

There is only one zero, so move the decimal point one place to the left.

0.56 × 0.1 = 0.056

4) 0.56 × 0.01

There are 2 zeros, so move the decimal point two places to the left

0.56 × 0.01 = 0.0056

Any questions about multiplying by powers of ten? Let me know...

Basic math calculator

Multiplication table A multiplication table makes it easier for you to perform operations. I strongly recommend that you learn it and know it by heart.

The multiplication table contain some interesting patterns that can help you remember some of those multiplications

First,notice that any number times 0 will give 0.

2 times 0 = 0

5 times 0 = 0

Any number times 1 will give the number

6 times 1 = 6

9 times 1 = 9

When you multiply any number( the number must be between 1 and 12; It may not be true for other higher numbers) by 9, the sum of the digits for the answer add up to 9.

For instance,

9 times 3 = 27 and 2+7 is 9

9 times 7 = 63 and 6 + 3= 9

See if you can find other interesting patterns that will help you remember the multiplication table!

x	0	1	2	3	4	5	6	7	8	9	10	11	12
0	0	0	0	0	0	0	0	0	0	0	0	0	0
1	0	1	2	3	4	5	6	7	8	9	10	11	12
2	0	2	4	6	8	10	12	14	16	18	20	22	24
3	0	3	6	9	12	15	18	21	24	27	30	33	36
4	0	4	8	12	16	20	24	28	32	36	40	44	48
5	0	5	10	15	20	25	30	35	40	45	50	55	60
6	0	6	12	18	24	30	36	42	48	54	60	66	72
7	0	7	14	21	28	35	42	49	56	63	70	77	84
8	0	8	16	24	32	40	48	56	64	72	80	88	96
9	0	9	18	27	36	45	54	63	72	81	90	99	108
10	0	10	20	30	40	50	60	70	80	90	100	110	120

State real property tax

State real property tax is a tax you will pay if you own a house or a property. You need to understand the following two definitions

Market value:

The selling price of a property

Assessed value:

The assessed value is a percent of the market value

The state uses the assessed value of the property to find out how tax you will pay

Consider the following example

A House has a market value of 220,000. The assessed value of the house is 70%. If the tax rate is $2 per $100, how much is the property tax?

Assessed value = Market value × rate of assessment = 220,000 × 0.70 = 154000

The assessed value in hundreds is 154000/100

The assessed value is 1540 dollars

Property tax = assessed value × tax rate = 1540 × 2 = 3080 dollars

Understanding mortgage loans

Understanding mortgage loans while you sharpen your basic mathematics skills is what I will show in this lesson

When buying a house most people take mortgage loans from a bank for the amount they finance, or still unpaid.

When a loan is given, it is repaid with interest in equal monthly installments over a period of time, usually from 15 to 30 years.

Ever wondered about some simple basic math involved in that type of loan?

Say for instance you buy a house for 250,000. Then, you make a down payment of 15% of the purchase price and take a 30-year mortgage for the balance.

What is your down payment?

What is your mortgage?

Down payment = Purchase Price × Percent Down

Down payment = 25,000 × 0.15 = 37500

Amount of Mortgage = Purchase Price − Down Payment

Amount of Mortgage = 250,000 − 37500 = 212500

If your monthly payment is 1200 dollars, what is the total interest charged over the life of the loan?

Total Monthly Payment = Monthly payment × 12 Months per year × Number of years

Total Monthly Payment = 1200 × 12 × 30 = 432000

Total Interest Paid = Total Monthly Payment − Amount of Mortgage

Total Interest Paid = 432000 − 212500 = 219500

Percentage word problems

Before you learn about percentage word problems, review Formula for percentage or you can use the approach that I use here.

Example #1:


A test has 20 questions. If peter gets 80% correct, how many questions did peter missed?

The number of correct answers is 80% of 20 or 80/100 × 20

80/100 × 20 = 0.80 × 20 = 16

Recall that 16 is called the percentage. It is the answer you get when you take the percent of a number

Since the test has 20 questions and he got 16 correct answers, the number of questions he missed is 20 − 16 = 4

Peter missed 4 questions



Example #2:


In a school, 25 % of the teachers teach basic math. If there are 50 basic math teachers, how many teachers are there in the school?

I shall help you reason the problem out:

When we say that 25 % of the teachers teach basic math, we mean 25% of all teachers in the school equal number of teachers teaching basic math

Since we don't know how many teachers there are in the school, we replace this with x or a blank

However, we know that the number of teachers teaching basic or the percentage = 50

Putting it all together, we get the following equation:

25% of ____ = 50 or 25% × ___ = 50 or 0.25 × ____ = 50

Thus, the question is 0.25 times what gives me 50

A simple division of 50 by 0.25 will get you the answer

50/0.25 = 200

Therefore, we have 200 teachers in the school

In fact, 0.25 × 200 = 50



Example #3:


24 students in a class took an algebra test. If 18 students passed the test, what percent do not pass?

Set up the problem like this:

First, find out how many student did not pass.

Number of students who did not pass is 24 − 18 = 6

Then, write down the following equation:

x% of 24 = 6 or x% times 24 = 6

To get x%, just divide 6 by 24

6/24 = 0.25 = 25/100 = 25%

Therefore, 25% of students did not pass

If you really understand the percentage word problems above, you can solve any other similar percentage word problems.

If you still do not understand them, I strongly encourage you to study them again and again until you get it. The end result will be very rewarding!

Calculating percent error

When calculating percent error, just take the ratio of the amount of error to the accepted value or true value, or real value. Then, convert the ratio to a percent.

We can expresss the percent error with the following formula shown below:

The amount of error is a subtraction between the measured value and the accepted value

Keep in mind that when computing the amount of error, you are always looking for a positive value.

Therefore, always subtract the smaller value from the bigger. In other words, amount of error = bigger − smaller

Percent error word problem #1

A student made a mistake when measuring the volume of a big container. He found the volume to be 65 liters.

However, the real value for the volume is 50 liters. What is the percent error?

Percent error = (amount of error)/accepted value

amount of error = 65 - 50 = 15

The accepted value is obviously the real value for the volume, which 50

So, percent error = 15/50

Just convert 15/50 to a percent. We can do this multiplying both the numerator and the denominator by 2

We get (15 × 2)/(50 × 2) = 30/100 = 30%

Notice that in the problem above, if the true value was 65 and the measured value was 50, you will still do 65 − 50 to get the amount of error, so your answer is still positive as already stated

However, be careful! The accepted value is 65, so your percent error is 15/65 = 0.2307 = 0.2307/1 = (0.2307 × 100)/(1 × 100) = 23.07/100 = 23.07%

Percent error word problem #2

A man measured his height and found 6 feet. However, after he carefully measured his height a second time, he found his real height to be 5 feet.

What is the percent error the man made the first time he measured his height?

Percent error = (amount of error)/accepted value

amount of error = 6 - 5 = 1

The accepted value is the man's real height or the value he found after he carefully measured his height, or 5

So, percent error = 1/5

Just convert 1/5 to a percent. We can do this multiplying both the numerator and the denominator by 20

We get (1 × 20)/(5 × 20) = 20/100 = 20%

I hope what I explained above was enough to help you understand what to do when calculating percent error

Any questions? Contact me.

Calculating percent error

Multiplication table

A multiplication table makes it easier for you to perform operations. I strongly recommend that you learn it and know it by heart.

x	0	1	2	3	4	5	6	7	8	9	10	11	12
0	0	0	0	0	0	0	0	0	0	0	0	0	0
1	0	1	2	3	4	5	6	7	8	9	10	11	12
2	0	2	4	6	8	10	12	14	16	18	20	22	24
3	0	3	6	9	12	15	18	21	24	27	30	33	36
4	0	4	8	12	16	20	24	28	32	36	40	44	48
5	0	5	10	15	20	25	30	35	40	45	50	55	60
6	0	6	12	18	24	30	36	42	48	54	60	66	72
7	0	7	14	21	28	35	42	49	56	63	70	77	84
8	0	8	16	24	32	40	48	56	64	72	80	88	96
9	0	9	18	27	36	45	54	63	72	81	90	99	108
10	0	10	20	30	40	50	60	70	80	90	100	110	120

The multiplication table contain some interesting patterns that can help you remember some of those multiplications

First,notice that any number times 0 will give 0.

2 times 0 = 0

5 times 0 = 0

Any number times 1 will give the number

6 times 1 = 6

9 times 1 = 9

When you multiply any number( the number must be between 1 and 12; It may not be true for other higher numbers) by 9, the sum of the digits for the answer add up to 9.

For instance,

9 times 3 = 27 and 2+7 is 9

9 times 7 = 63 and 6 + 3= 9

See if you can find other interesting patterns that will help you remember the multiplication table!

Then, strengthen your skill with the following multiplication calculator

Enter your first number:       

Enter your second number:

       

The result is :  

Check also:

Percents

Percents are used to communicate ideas to people everyday. All my lessons are designed to help you deeply explore the topic with real life applications.

You may have heard people say that they will give you a 5 % discount if you purchase two T-shirts, or 5 pants.

Some jobs such as car sales pay based on a percentage of sales

If you have a saving account, the bank may give you a 3 % interest every year.

Credit cards company charge high interest of 10 % to 20 % depending on your credit.

This unit will teach you many real life applications of the topic.

In the end, you should feel comfortable with the following topics: discount, sales tax, simple interest, commission, and tips.

Converting repeating decimals to fractions

When converting repeating decimals to fractions, just follow the two steps below carefully.

Step 1:

Let x equal the repeating decimal you are trying to convert to a fraction

Step 2:

Examine the repeating decimal to find the repeating digit(s)

Step 3:

Place the repeating digit(s) to the left of the decimal point

Step 4:

Place the repeating digit(s) to the right of the decimal point

Step 5:

Subtract the left sides of the two equations.Then, subtract the right sides of the two equations

As you subtract, just make sure that the difference is positive for both sides

Now let's practice converting repeating decimals to fractions with two good examples

Example #1:

What rational number or fraction is equal to 0.55555555555

Step 1:

x = 0.5555555555

Step 2:

After examination, the repeating digit is 5

Step 3:

To place the repeating digit ( 5 ) to the left of the decimal point, you need to move the decimal point 1 place to the right

Technically, moving a decimal point one place to the right is done by multiplying the decimal number by 10.

When you multiply one side by a number, you have to multiply the other side by the same number to keep the equation balanced

Thus, 10x = 5.555555555

Step 4:

Place the repeating digit(s) to the right of the decimal point

Look at the equation in step 1 again. In this example, the repeating digit is already to the right, so there is nothing else to do.

x = 0.5555555555

Step 5:

Your two equations are:

10x = 5.555555555

   x = 0.5555555555

10x - x = 5.555555555 − 0.555555555555

9x = 5

Divide both sides by 9

x = 5/9

Example #2:

What rational number or fraction is equal to 1.04242424242

Step 1:

x = 1.04242424242

Step 2:

After examination, the repeating digit is 42

Step 3:

To place the repeating digit ( 42 ) to the left of the decimal point, you need to move the decimal point 3 place to the right

Again, moving a decimal point three place to the right is done by multiplying the decimal number by 1000.

When you multiply one side by a number, you have to multiply the other side by the same number to keep the equation balanced

Thus, 1000x = 1042.42424242

Step 4:

Place the repeating digit(s) to the right of the decimal point

In this example, the repeating digit is not immediately to the right of the decimal point.

Look at the equation in step 1 one more time and you will see that there is a zero between the repeating digit and the decimal point

To accomplish this, you have to move the decimal point 1 place to the right

This is done by multiplying both sides by 10

10x = 10.4242424242

Step 5:

Your two equations are:

1000x = 1042.42424242

   10x = 10.42424242

1000x - 10x = 1042.42424242 − 10.42424242

990x = 1032

Divide both sides by 990

x = 1032/990

To master this lesson about converting repeating decimals to fractions, you will need to study the two examples above carefully and practice with other examples

Fractions to decimals

When changing fractions to decimals, look for an equivalent fraction that change the denominator to a power of ten

Power of 10 means that your denominator will look something like 10, 1000, 10000, and so forth...

Let us illustrate this with examples.

Example #1

Change 3/50 to a decimal

Notice that 50 was multiplied by 2 to get 100

Just remember that whatever you do for your denominator, you have to do it for your numerator



Example #2

Change 2/5 to a decimal



Example #3

Change 4/125 to a decimal

Notice here that 125 was multiplied by 8 to get 1000 because 125 × 8 = 1000

Therefore,when doing those problems,one of the big challenges is to find the number that you need to multiply the denominator to get 10, 1000, 10000, and so forth

You can find it by doing division. For instance 1000/125 = 8

Decimals to fractions

When converting decimals to fractions, it may be useful to understand the technique used to convert numbers to scientific notation. Review then scientific notation before starting this lesson.

By all means, do not start this lesson before reviewing scientific notation because that is the technique I am going to use.

Let's start with an example:

Convert 0.06 to a fraction

0.06 = 6 × 10^-2

Now, what is 10^-2 equal to ?

Pay close attention to the following demonstration:

Why 10⁰ = 1 ? Stay tune! I will put it in a different lesson. Just accept this fact for now. Since 0.06 = 6 × 10^-2 , we get the following:

Other examples:

1) 0.014

Follow all the steps above. In the end, you should get

2) 0.10252

Subtracting decimals

Subtracting decimals is similar to subtracting whole numbers I strongly encourage you to review subtraction of whole numbers before learning this lesson.

Before subtracting, line up the decimal points. It is also important to use 0 as a placeholder when a place value is blank

Look at the following subtractions problems. they are fairly easy to subtract.

Look at the following subtractions problems. they are fairly easy to subtract.



in the first problem on the left (0.5 - 0.2), just remove 2 tenths from 5 tenths to get 3 tenths. Then 0 take away 0 is 0. Finally write down your decimal point.

Now look at the following problem:



Since you cannot remove 8 hundredths from 5 hundredths, you need to take 1 tenth from 4 tenths and add that to 5 hundredths to get 15 hundredths.



Then, 15 hundredths - 8 hundredths = 7 hundredths

3 tenths - 2 tenths = 1 tenth

Finally, 0 - 0 = 0 and put your decimal point.

The following problem shows the importance of using 0 as a placeholder before subtracting:



Using 0 as a placeholder, we get:



Again, we cannot remove 5 hundredths from 0 hundredth, so we will borrow 1 tenth( 1 tenth = 10 hundredths) from 5 tenths and add that to 0 hundredth.

We get:



Study too the following example carefully:



Use 0 as placeholders



Then, do the following:



Study this last example too carefully:



Step 1:



Step 2:



Step 3: