ANGLE

When two sides or line-segments are inclined and meet at a point, then this inclination is called an angle.

A polygon has as many angles as number of sides or number of vertices. The triangle has 3 angles, quadrilateral has 4 angles, pentagon has 5 angle, etc.

The angles have different measures.

Right Angle – an angle whose measure is exactly 90 degrees.

Right Angle

Acute Angle – an angle whose measure is less than 90 degrees..

Acute Angle

Obtuse Angle- an angle whose measure is greater than 90 degrees but less than 180 degrees.

Obtuse Angle

Reflex Angle – an angle whose measure is greater than 180 degrees but less than 360 degrees.

Straight Angle – measures exactly 180 degrees.

TRIANGLES

A simple closed curve or a polygon formed by three line-segments (sides) is called a triangle.

The above shown shapes are triangles. The symbol of a triangle is ∆.

a. What are the parts of a triangle?

(The parts of a triangle are the sides, the vertices and the angles)

b. What are the kinds of triangles according to sides?

An equilateral triangle has three equal sides. An isosceles triangle has two equal sides. A scalene triangle has no sides equal.

c. What are the kinds of triangles according to angles?

An acute triangle has 3 acute angles. A right triangle has one right angle. An obtuse triangle has one obtuse angle.

QUADRILATERAL

A simple closed curve or a polygon formed by four line-segments is called a quadrilateral.

Rhombus -It has 4 equal sides, but has no right angles.

Trapezoid- It has exactly one pair of parallel sides.

Rectangle -It has 2 pairs of parallel sides and 4 right angle

Parallelogram - It has 2 pairs of parallel sides.

Square - It has 4 equal sides and 4 right angles.

CIRCLEA circle is a set of all points on a plane that are of equal distance from a given point called center.

A diameter is a line segment that passes through the center of a circle and has both endpoints on the circle.

A radius is a segment that has one endpoint at the center and the other on the circle.

Circumference is the distance around the circle.

Names of Polygons

   

Name Sides Shape

Triangle (or Trigon) 3

Quadrilateral (or Tetragon) 4

Pentagon 5

Hexagon 6

Heptagon (or Septagon) 7

Octagon 8

Nonagon (or Enneagon) 9

Decagon 10

Hendecagon (or Undecagon) 11

Dodecagon 12

Triskaidecagon 13  

Tetrakaidecagon 14  

Pentadecagon 15  

Hexakaidecagon 16  

Heptadecagon 17  

Octakaidecagon 18  

Enneadecagon 19  

Icosagon 20  

Triacontagon 30  

Tetracontagon 40  

Pentacontagon 50  

Hexacontagon 60  

Heptacontagon 70  

Octacontagon 80  

Enneacontagon 90  

Hectagon 100  

Chiliagon 1,000  

Myriagon 10,000  

Megagon 1,000,000  

Googolgon 10100  

n-gon n

Perimeter

Perimeter is the distance around a two-dimensional shape.

To find the perimeter of a polygon, take the sum of the length of each side. The polygons below are much smaller than a fenced-in yard. Thus, we use smaller units in our examples, such as centimeters and inches.

Example 1: Find the perimeter of a triangle with sides measuring 5 centimeters, 9 centimeters and 11 centimeters.

FORMULA: s1+s2+s3

Solution: P = 5 cm + 9 cm + 11 cm = 25 cm

Example 2: A rectangle has a length of 8 centimeters and a width of 3 centimeters. Find the perimeter.

FORMULA: ( Length + Width)x2 or 2L + 2W

Solution 1: P = 8 cm + 8cm + 3 cm + 3 cm = 22 cm

Solution 2: P = 2(8 cm) + 2(3 cm) = 16 cm + 6 cm = 22 cm

In Example 2, the second solution is more commonly used. In fact, in mathematics, we commonly use the following formula for perimeter of a rectangle:

,  where   is the perimeter,   is the length and   is the width.

In the next few examples, we will find the perimeter of other polygons.

Example 3: Find the perimeter of a square with each side measuring 2 inches.

FORMULA: 4s or 4 x s

Solution: = 2 in + 2 in + 2 in + 2 in = 8 in

Example 4: Find the perimeter of an equilateral triangle with each side measuring 4 centimeters.

 

Solution:

   = 3(4 cm) = 12 cm

Example 5: Find the perimeter of a regular pentagon with each side measuring 3 inches.

Solution:  = 5(3 in) = 15 in

Example 6: The perimeter of a regular hexagon is 18 centimeters. How long is one side?

Solution:  = 18 cm

  Let   represent the length of one side. A regular hexagon has 6 sides, so we can divide the perimeter by 6 to get the length of one side ( ).

   = 18 cm ÷ 6

   = 3 cm

Summary: To find the perimeter of a polygon, take the sum of the length of each side. The formula for perimeter of a rectangle is:  . To find the perimeter of a regular polygon, multiply the number of sides by the length of one side.

Circumference of a Circle

A circle is a shape with all points the same distance from the center. It is named by the center. The circle to the left is called circle A since the center is at point A. If you measure the distance around a circle and divide it by the distance across the circle through the center, you will always come close to a particular value, depending upon the accuracy of your measurement. This value is approximately 3.14159265358979323846... We use the Greek letter   (pronounced Pi) to represent this value. The number   goes on forever. However, using computers,   has been calculated to over 1 trillion digits past the decimal point.

The distance around a circle is called the circumference. The distance across a circle through the center is called the diameter.   is the ratio of the circumference of a circle to the diameter. Thus, for any circle, if you divide the circumference by the diameter, you get a value close to  . This relationship is expressed in the following formula:

where   is circumference and   is diameter. You can test this formula at home with a round dinner plate. If you measure the circumference and the diameter of the plate and then divide   by  , your quotient should come close to . Another way to write this formula is:   where · means multiply. This second formula is commonly used in problems where the diameter is given and the circumference is not known (see the examples below).

The radius of a circle is the distance from the center of a circle to any point on the circle. If you place two radii end-to-end in a circle, you would have the same length as one diameter. Thus, the diameter of a circle is twice as long as the radius. This relationship is expressed in the following formula:  , where   is the diameter and   is the radius.

Example 1: The radius of a circle is 2 inches. What is the diameter?

Solution:

   = 2 · (2 in)

   = 4 in

Example 2: The diameter of a circle is 3 centimeters. What is the circumference?

Solution:

   = 3.14 · (3 cm)

   = 9.42 cm

Example 3: The radius of a circle is 2 inches. What is the circumference?

Solution:

   = 2 · (2 in)

   = 4 in

 

   = 3.14 · (4 in)

   = 12.56 in

Example 4: The circumference of a circle is 15.7 centimeters. What is the diameter?

Solution:

  15.7 cm = 3.14 · 

  15.7 cm ÷ 3.14 = 

   = 15.7 cm ÷ 3.14

   = 5 cm

Summary: The number   is the ratio of the circumference of a circle to the diameter. The value of   is approximately 3.14159265358979323846...The diameter of a circle is twice the radius. Given the diameter or radius of a circle, we can find the circumference. We can also find the diameter (and radius) of a circle given the circumference. The formulas for diameter and circumference of a circle are listed below. We round   to 3.14 in order to simplify our calculations.

AREA Area is measured in square units such as square inches, square feet or square meters.

Rectangle

A= Length x Width

Example 1: A rectangle has a length of 8 centimeters and a width of 3 centimeters. Find the area.

Solution:

 = (8 cm) · (3 cm) = 24 cm2

Example 2 : The area of a rectangle is 12 square inches and the width is 3 inches. What is the length?

Solution:

12 in2 =   · 3 in

Since 4 · 3 = 12, we get (4 in) · (3 in) = 12 in2. So   must equal 4 in.

 = 4 in

Square

A= s2 or sxsA square is a rectangle with 4 equal sides. To find the area of a square, multiply the length of one side by itself. The formula is:

   or   ,  where A is the area, s is the length of a side, and · means multiply.

Example Find the area of a square with each side measuring 2 inches.

Solution:

 = (2 in) · (2 in) = 4 in2

Example 3: The area of a square is 9 square centimeters. How long is one side?

Solution:

9 cm2 =   · 

Since 3 · 3 = 9, we get 3 cm · 3 cm = 9 cm2. So   must equal 3 cm.

 = 3 cm

Area of a Parallelogram = Base x height

Example 1: Find the area of a parallelogram with a base of 12 centimeters and a height of 5 centimeters.

Solution:

 = (12 cm) · (5 cm)

 = 60 cm2

Example 2: Find the area of a parallelogram with a base of 7 inches and a height of 10 inches.

Solution:

 = (7 in) · (10 in)

 = 70 in2

Example 3: The area of a parallelogram is 24 square centimeters and the base is 4 centimeters. Find the height.

Solution:

24 cm2 = (4 cm) · 

24 cm2 ÷ (4 cm) = 

 = 6 cm

Area of a Triangle = ½ x base x height or b x h /2

Example 1: Find the area of an acute triangle with a base of 15 inches and a height of 4 inches.

Solution:

 =   · (15 in) · (4 in)

 =  · (60 in2)

 = 30 in2

Example 2: Find the area of a right triangle with a base of 6 centimeters and a height of 9 centimeters.

Solution:

 =  · (6 cm) · (9 cm)

 =  · (54 cm2)

 = 27 cm2

Example 3: Find the area of an obtuse triangle with a base of 5 inches and a height of 8 inches.

Solution:

 =  · (5 in) · (8 in)

 =  · (40 in2)

 = 20 in2

Example 4: The area of a triangular-shaped mat is 18 square feet and the base is 3 feet. Find the height. (Note: The triangle in the illustration to the right is NOT drawn to scale.)

Solution: In this example, we are given the area of a triangle and one dimension, and we are asked to work backwards to find the other dimension.

18 ft2 =  · (3 ft) ·     

Multiplying both sides of the equation by 2, we get:

36 ft2 = (3 ft) · 

Dividing both sides of the equation by 3 ft, we get:

12 ft = 

Commuting this equation, we get:

 = 12 ft

Area of a Trapezoid = ½ x ( base1 +base2) x height

A trapezoid is a 4-sided figure with one pair of parallel sides. For example, in the diagram to the right, the bases are parallel. To find the area of a trapezoid, take the sum of its bases, multiply the sum by the height of the trapezoid, and then divide the result by 2, The formula for the area of a trapezoid is:

  or  

where  is  ,   is  ,   is the height. and · means multiply.

Each base of a trapezoid must be perpendicular to the height. In the diagram above, both bases are sides of the trapezoid. However, since the lateral sides are not perpendicular to either of the bases, a dotted line is drawn to represent the height.

Example 1: Find the area of a trapezoid with bases of 10 inches and 14 inches, and a height of 5 inches.

Solution:

  =   · (10 in + 14 in) · 5 in

 =   · (24 in) · (5 in)

 =   · 120 in2

A = 60 in2

Example 2: Find the area of a trapezoid with bases of 9 centimeters and 7 centimeters, and a height of 3 centimeters.

Solution:

  =   · (9 cm + 7 cm) · 3 cm

 =   · (16 cm) · (3 cm)

 =    · 48 cm2

= 24 cm2

Example 3: The area of a trapezoid is 52 square inches and the bases are 11 inches and 15 inches. Find the height.

Solution:

 52 in2 =    · (11 in + 15 in) · 

52 in2 =    · (26 in) · 

52 in2 = (13 in) · 

52 in2 ÷ (13 in) = 

= 4 in

Summary: To find the area of a trapezoid, take the sum of its bases, multiply the sum by the height of the trapezoid, and then divide the result by 2, The formula for the area of a trapezoid is:

   or  

  where

 is  ,   is  ,  and  is the height.

Area of a Circle = x r2 = 3.14

The distance around a circle is called its circumference. The distance across a circle through its center is called its diameter. We use the Greek letter   (pronounced Pi) to represent the ratio of the circumference of a circle to the diameter. In the last lesson, we learned that the formula for circumference of a circle is:  . For simplicity, we use   = 3.14. We know from the last lesson that the diameter of a circle is twice as long as the radius. This relationship is expressed in the following formula:  .

The area of a circle is the number of square units inside that circle. If each square in the circle to the left has an area of 1 cm2, you could count the total number of squares to get the area of this circle. Thus, if there were a total of 28.26 squares, the area of this circle would be 28.26 cm2 However, it is easier to use

one of the following formulas:

   or   

where   is the area, and   is the radius. Let's look at some examples involving the area of a circle. In each of the three examples below, we will use  = 3.14 in our calculations.

Example 1: The radius of a circle is 3 inches. What is the area?

Solution:

   = 3.14 · (3 in) · (3 in)

   = 3.14 · (9 in2)

   = 28.26 in2

Example 2: The diameter of a circle is 8 centimeters. What is the area?

Solution:

  8 cm = 2 ·   8 cm ÷ 2 = 

   = 4 cm

 

   = 3.14 · (4 cm) · (4 cm)

   = 50.24 cm2

Example 3: The area of a circle is 78.5 square meters. What is the radius?

Solution:

  78.5 m2 = 3.14 ·   ·   78.5 m2 ÷ 3.14 =   ·   25 m2 =   ·    = 5 m

Summary Given the radius or diameter of a circle, we can find its area. We can also find the radius (and diameter) of a circle given its area. The formulas for the diameter and area of a circle are listed below:

   or 

PERCENT

A percent is a ratio whose second term is 100. Percent means parts per hundred. The word comes from the Latin phrase per centum, which means per hundred. In mathematics, we use the symbol % for percent.

Example 1: Write each ratio as a fraction, a decimal, and a percent:   4 to 100,   63 to 100,   17 to 100

  Solution

Ratio Fraction Decimal Percent

04 to 100 .04 04%

63 to 100 .63 63%

17 to 100 .17 17%

Example 2: Write each percent as a ratio, a fraction in lowest terms, and a decimal:   24%,   5%,   12.5%

  Solution

Percent Ratio Fraction Decimal

24% 24 to 100 .240

05% 05 to 100 .050

  12.5% 12.5 to 100 .125

Example 3: Write each percent as a decimal:   91.2%,   4.9%,   86.75%

 

Solution

Percent Decimal

91.2% .9120

04.9% .0490

86.75% .8675

FRACTIONSPERCENT

Problem: Last marking period, Ms. Jones gave an A grade to 15 out of every 100 students and Mr. McNeil gave an A grade to 3 out of every 20 students. What percent of each teacher's students received an A?

 

Solution

Teacher Ratio Fraction Percent

Ms. Jones 15 to 100 15%

Mr. McNeil 3 to 20 15%

Solution: Both teachers gave 15% of their students an A last marking period.

In the problem above, the fraction for Ms. Jones was easily converted to a percent. This is because It is easy to convert a fraction to a percent when the denominator is 100. If a fraction does not have a denominator of 100, you can convert it to an equivalent fraction with a denominator of 100, and then write the equivalent fraction as a percent. This is what was done in the problem above for Mr. McNeil. Let's look at some problems in which we use equivalent fractions to help us convert a fraction to a percent.

Example 1: Write each fraction as a percent:    

 

Solution

FractionEquivalent Fraction

Percent

50%

90%

80%

Example 2: One team won 19 out of every 20 games played, and a second team won 7 out of every 8 games played. Which team has a higher percentage of wins?

  Solution

Team FractionEquivalent

FractionPercent

1 95%    

2 87.5%

Solution: The first team has a higher percentage of wins.

In Examples 1 and 2, we used equivalent fractions to help us convert each fraction to a percent. Another way to do this is to convert each fraction to a decimal, and then convert each decimal to a percent. To convert a fraction to a decimal, divide its numerator by its denominator. Look at Example 3 below to see how this is done.

Example 3: Write each fraction as a percent:    

 

Solution

Fraction Decimal Percent

87.5%

95%    

 1.5%

Now that you are familiar with writing fractions as percents, do you see a pattern in the problem below?

Problem: If 165% equals  , and 16.5% equals  , then what fraction is equal to 1.65%?

 

Solution

Percent Fraction

165%

16.5%

1.65%

DECIMALS PERCENT

Problem: What percent of a dollar is 76 cents?

$1.00 = 1.0

76 cents = .76

.76 = 76%

Solution: 76% of a dollar is 76 cents.

The solution to the above problem is not surprising, since dollars, cents and percents are all based on the number 100. To convert a decimal to a percent, multiply the decimal by 100, then add on the % symbol. An easy way to multiply a decimal by 100 is to move the decimal point two places to the right. This is done in the example below.

Example 1: Write each decimal as a percent:    . 93,    .08,    .67,    .41

  Solution

Decimal Percent

.93 93%

.08 08%

.67 67%

.41 41%

Each decimal in Example 1 went out two places to the right of the decimal point. However, a decimal can have any number of places to the right of the decimal point. Let's look at Examples 2 and 3 below.

Example 2: Write each decimal as a percent:     .786,    .002,    .059,    .8719

 

Solution

Decimal Percent

.7860 78.6%  

.0020     .2%  

.0590   5.9%  

.8719 87.19%

Example 3: Write each decimal as a percent:     .1958,     .007,    .05623,    .071362

 

Solution

Decimal Percent

.195800  19.58%    

.007000       .7%        

.056230    5.623%  

.071362   7.1362%

Summary: To write a decimal as a percent, multiply it by 100, then add on the % symbol. To multiply a decimal by 100, move the decimal point two places to the right. Do not forget to include the percent symbol when writing a percent.

PERCENT DECIMALS

Example 1: Write each percent as a decimal:     18%,    7%,    82%,    55%

 

Solution

Percent Fraction Decimal

18% .18

07% .07

82% .82

55% .55

Let's look at another example of writing percents as decimals.

Example 2: Write each percent as a decimal:     12.5%,    89.19%,    39.2%,    71.935%

 

Solution

Percent Decimal

12.5%       .12500

89.19%    .89190

39.2%       .39200

71.935% .71935

PERCENT FRACTIONS

Example 1: Write each percent as a fraction in lowest terms:    55%,    41%,    36%

 

Solution

Percent Fraction ReducingLowest Terms

55%

41%

36%

In Example 1, the GCF of 55 and 100 is 5; the GCF of 41 and 100 is 1; and the GCF of 36 and 100 is 4. Note that when the GCF is 1, this means that the fraction is already in lowest terms. Let's look at some more examples.

Example 2: Write each percent as a fraction in lowest terms:    7%,    12.5%,    62.5%

 

Solution

Percent Fraction    Reducing   Lowest Terms

 7%      

12.5% 

62.5%     

Example 3: Write each percent as a fraction in lowest terms:    67.5%,    56.25%,    13.1%

 

Solution

Percent Fraction    Reducing   Lowest Terms

67.5%  

56.25%

13.1%  

Percents Less Than 1or Greater Than 100

Problem 1: Glosser Grade School has an enrollment of 400 students. Only  % of students ride their bicycles to school each day. How many students is this?

  We know from previous lessons how to write a percent as a decimal by moving the decimal point two places to the left. So we get the following:

 

If   = 0.5, then  % = 0.5%, and 0.5% =.005. Thus we get  % = .005

  Going back to our original problem, we get: .005 of 400 = .005 x 400 = 2

Solution: 2 students ride their bicycle to school each day.

Problem 2: This year, enrollment at Glosser Grade school is 150% of last year. How many students are enrolled now?

  Writing 150% as a decimal, we get: 150% = 1.50 = 1.5

  1.5 of 400 = 1.5 x 400 = 600

Solution: 600 students are now enrolled at Glosser Grade School.

In Problem 1, we were asked to work with a percent less than 1, and in Problem 2, we were asked to work with a percent greater than 100. When working with percents less than 1 and greater than 100, we follow the same rules that we learned for percents between 1 to 100. Let's look at some examples.

Example 1: Write each percent as a decimal:      %,    1.8%,    0.32%,    235%

  Since   = 0.4, we get  % = 0.4%

Solution

Percent Decimal

    0.4% 0.004  

    1.8% 0.018  

      0.32% 0.0032

235%     2.35    

In example 1, note that instead of writing .32%, we used 0.32%. The leading zero reminds us that this number is between 0 and 1 percent. Let's look at some more examples.

Example 2: Write each percent as a fraction in lowest terms:    0.42%,    395%,    0.07%

 

Solution

Percent Decimal FractionLowest Terms

    0.42% 0.0042

395%        3.95      3  3

    0.07% 0.0007

Example 3: Write each decimal as a percent:    12.2,    0.00459,    1.2765

 

Solution

Decimal Percent

12.21,220%.000

0

0.00459      0.459%

1.2765    127.65%0

TIME CONVERSION

The standard units of time are: 

1 hour = 60 minutes

1 minute = 60 seconds

1 hour = 60 minutes = 3600 seconds (60 × 60)

1 day = 24 hours

1 week = 7 days

1 year = 365 days

1 year = 12 months

1year = 52 weeks

These are the units of time conversion table.

For example:

1. How many minutes are there in a year?

Solution:

We know, 

1 year = 365 days.

1 day = 24 hours.

1 hour = 60 minutes.

So one year = (365 × 24 × 60) minutes.

= (8760 × 60) minutes.

= 525600 minutes.

2. How many hours are there in a year?

Solution:

We know,

1 year = 365 days.

1 day = 24 hours.

So in one year = (365 × 24) 

= 7860 hours. 

3. How many minutes in 6 hours?

Solution:

We know,

1 hour = 60 minutes.

So 6 hours = (6 × 60) minutes.

= 360 minutes.

4. Convert 220 minutes to hours and minutes.

Solution:

We know that 60 minutes = 1 hour 

220 minutes = (220/60) hours 

= 3 hours 40 minutes.

CONVERSION OF LENGTHS

In math when we use length, we know that the standard unit of length is ‘Metre’ which is written in short as ‘m’.

A metre length is divided into 100 equal parts. Each part is named centimetre and written in short as ‘cm’.Thus, 1 m = 100 cm and 100 cm = 1 m

The long distances are measured in kilometre. This kilometre equals to 1000 metres. The kilometre is written in short as km.

1 kilometre (km) = 1000 metres (m)

Or

1 km = 1000 m

The different units of length conversion charts and their equivalents are given here:

1 kilometre (km) = 10 Hectometres (hm) = 1000 m

1 Hectometre (hm) = 10 Decametres (dcm) = 100 m

1 Decametre (dcm) = 10 Metres (m)

1 Metre (m) = 10 Decimetres (dm) = 100 cm = 1000 mm

1 Decimetre (dm) = 10 Centimetres (cm)

1 decimeter = 0.1 meter

1 Centimetre (cm) = 10 Millimetres (mm)

1 centimeter = 0.01 meter

1 millimeter = 0.001 meter

Mostly we use Kilometre (km), Metre (m) and Centimetre (cm) as the units of length measurement or unit of length conversion chart.

In customary units of length are discussed her in the units of length conversion table:

1 mile = 1760 yards

1 mile = 5280 feet

1 yard = 3 feet

1 foot = 12 inches

CONVERSION OF MASS AND WEIGHT

In math when we use the units of mass and weight conversion chart, we normally use gram to represent as the standard unit of mass and weight, and the other standard units are as follows:

1 milligram = 0.001 gram

1 centigram = 0.01 gram

1 decigram = 0.1 gram

1 kilogram = 1000 grams

1 gram = 1000 milligrams

These are the general units of mass and weight measurement conversion which are very important and useful to know while solving the questions.

Customary units of mass and weight conversion chart are:

1 ton = 2000 pounds

1 pound = 16 ounces

CONVERSION OF CAPACITY AND VOLUME

In math when we use capacity and volume, we normally use liter to represent as the standard unit, and the other standard units are in the units in capacity and volume conversion chart.

1 milliliter = 0.001 liter

1 centiliter = 0.01 liter

1 deciliter = 0.1 liter

1 kiloliter = 1000 liters

Customary units of capacity and volume conversion table are:

1 gallon = 4 quarts

1 gallon = 128 ounces

1 quart = 2 pints

1 pint = 2 cups

1 cup = 8 ounces

DIVISIBILITY RULES

Dividing by 2

1. All even numbers are divisible by 2. E.g., all numbers ending in 0,2,4,6 or 8.

Dividing by 3

1. Add up all the digits in the number.2. Find out what the sum is. If the sum is divisible by 3, so IT IS DIVISIBLE BY 3.3. For example: 12123 (1+2+1+2+3=9) 9 is divisible by 3, therefore 12123 is DIVISIBLE BY 3.

Dividing by 4

1. Are the last two digits in your number divisible by 4?2. If so, the number is too!3. For example: 358912 ends in 12 which is divisible by 4, thus so is 358912.

Dividing by 5

1. Numbers ending in a 5 or a 0 are always divisible by 5.

Dividing by 6

1. If the Number is divisible by 2 and 3 it is divisible by 6 also.

Dividing by 7 (2 Tests)

Take the last digit in a number. Double and subtract the last digit in your number from the rest of the digits. Repeat the process for larger numbers. Example: 357 (Double the 7 to get 14. Subtract 14 from 35 to get 21 which is divisible by 7 and we can

now say that 357 is divisible by 7.

NEXT TEST Take the number and multiply each digit beginning on the right hand side (ones) by 1, 3, 2, 6, 4, 5.

Repeat this sequence as necessary Add the products. If the sum is divisible by 7 - so is your number. Example: Is 2016 divisible by 7? 6(1) + 1(3) + 0(2) + 2(6) = 21 21 is divisible by 7 and we can now say that 2016 is also divisible by 7.

Dividing by 8

1. This one's not as easy, if the last 3 digits are divisible by 8, so is the entire number.2. Example: 6008 - The last 3 digits are divisible by 8, therefore, so is 6008.

Dividing by 9

1. Almost the same rule and dividing by 3. Add up all the digits in the number.2. Find out what the sum is. If the sum is divisible by 9, so is the number.3. For example: 43785 (4+3+7+8+5=27) 27 is divisible by 9, therefore 43785 is too!

Dividing by 10

1. If the number ends in a 0, it is divisible by 10.