Mukhlis Tanveer Jan writer.

Mathematics With Tanveer Kurd.

Multiplication and division of fractions

Multiplying fractions:

Three simple steps are required to multiply two fractions:

Step 1: Multiply the numerators from each fraction by each other (the numbers on top). The result is the numerator of the answer.

Step 2: Multiply the denominators of each fraction by each other (the numbers on the bottom). The result is the denominator of the answer.

Step 3: Simplify or reduce the answer.

Examples of multiplying fractions:

fraction

In the first example, you can see that we multiply the numerators 2 x 6 to get the numerator for the answer, 12. We also multiply the denominators 5 x 7 to get the denominator for the answer, 35. In the second example, we use the same method. In this problem, the answer we get is 2/12 which can be further reduced to 1/6.

Multiplying different types of fractions:

The examples above multiplied proper fractions. The same process is used to multiply improper fractions and mixed numbers. There are a couple of things to watch out for with these other types of fractions.

Improper fractions – With improper fractions (where the numerator is greater than the denominator) you may need to change the answer into a mixed number. For example, if the answer you get is 17/4, your teacher may want you to change this to the mixed number 4 ¼.

Mixed numbers – Mixed numbers are numbers that have a whole number and a fraction, like 2 ½. When multiplying mixed numbers, you need to change the mixed number into an improper fraction before you multiply. For example, if the number is 2 1/3, you will need to change this to 7/3 before you multiply.

You may also need to change the answer back to a mixed number when you are done multiplying.

Example:

fraction

In this example, we had to change 1 ¾ to the fraction 7/4 and 2 ½ to the fraction 5/2. We also had to convert the multiplied answer to a mixed number at the end.

Dividing fractions:

Dividing fractions is very similar to multiplying fractions; you even use multiplication. The one change is that you have to take the reciprocal of the divisor. Then you proceed with the problem just as if you were multiplying.

Step 1: Take the reciprocal of the divisor.

Step 2: Multiply the numerators.

Step 3: Multiply the denominators.

Step 4: Simplify the answer.

Taking the reciprocal: To get the reciprocal, invert the fraction. This is the same as taking 1 divided by the fraction. For example, if the fraction is 2/3 then the reciprocal is 3/2.

Examples:

fraction

In our next class, we will be talking more about Estimation. We hope you enjoyed the class.

Should you have any further question, feel free to ask in the comment section below and trust us to respond as soon as possible.

Welcome to class!

In today’s class, we will be talking about estimation. Enjoy the class!

Estimation

Estimation (or estimating) is the process of finding an estimate, or approximation, which is a value that is usable for some purpose even if input data may be incomplete, uncertain, or unstable. The value is nonetheless usable because it is derived from the best information available

Example 1:

(a) The length of our classroom is 18 paces or 21 meters.

(b) The distance of this school from my house is

Some estimations are as follows:

(a) The length of our class is 18 paces or 21 meters

(b) A bottle of wine is 2 1/2 l****s or 25 deciliters

(c) The distance of this school from my house is 3 kilometres.

(d) The mass of my friend is 20 kilograms.

(e) The weight of our teacher’s table is 500 grimes.

These are only estimating. People make estimates in units they are familiar with. There are many types of instruments used for confirming estimates.

A long time ago, people used the distance from a man’s elbow to the tip of his fingers, called ‘the cubit ‘to measure distances. Some use a man’s pace or foot. Some use the man’s hand palm stretch for measurements. These and many others are known as crude instruments

for measurement. They cannot give accurate answers. Later on, standard measurements were developed and used in most countries of the world. These include the meters and the kilometres for distances, the gram for mass or weight and the litre for capacity etc.

Example 2:

Estimate the length of your teacher’s table with a crude measurement.

Measure the top part of the teacher’s table with the hand palm stretch.

Various answers as 8,9. 10, 11, 12 palms.

Example 3:

Estimate the inside width of our classroom with cubit measure.

Various results such as 6, 7,7,8 or 10 cubits.

Average is about 8 cubits.

Estimation of distances

Like estimation of length, estimation of distances involves longer lengths.

Activity:

Every student should estimate his or her height in meters and then measure the actual height.

Estimate the length of the classroom and later find the accurate length.

Let the students be in groups of five. Each group should go to the football field. Measure 10 meters and place a student every 10 meters apart.

Name the different villages and towns near the Give a rough map and put the distances.

Example 1:

Estimate the following distances and then measure them to confirm your estimates.

(a) the length of the Assembly hall.

(b) the width of a football field.

(a) The length of the Assembly hall should involve a longer instrument of measurement, that is the meter, and not only the centimetre.

Estimation: 25 meters long.

Measurement: 21 meters 50 centimeters.

Written shortly as 21 m 50 cm

(b) Estimation: 60 meters.

Measurement: 65 meters.

Note: Assembly halls and football fields may vary from one institution to the other. There is no single correct answer.

Unit of Volume

Volume is the amount of space in a container or some given objects occupies. while capacity is usually referred to as the amount of space occupied by the liquid in a container.

The units of volume are derived from the units of length, and the basic units are the cubic centimetres for smaller containers and cubic meters for bigger ones.

The common units are:

1 000 cubic millimetres = l cubic centimetre

1 000 cubic centimetres = I cubic decimeter

1 000 cubic decimeters = l cubic meter

1 000 000 cubic centimeters = 1 cubic meter

The cubic millimetre is abbreviated as cu mm or mm³, cubic centimetre as cu cm or cm³ or cc, cubic meter as m³, cubic decimeter as dm³, and so on.

The basic unit of volume is the litre. Other units in Common use are the cubic centimetre (or cc) and the millilitre (or ml).

The conversion we can use are:

I liter = 1 000 milliliter (m)

1 liter = 1 000 cubic centimeter (cc)

1 centiliter (cl) = 1/10 liter

= 10 ml

I ml = l cc

The basic unit of capacity is the litre. A millilitre is abbreviated as ml. Centiliter as cl and cubic centimetre as cc.

Some common measures we can use for our estimation of containers are:

Cube of sugar, as l cu. cm or I cc. or 1 cm³

A box, which is 10 cm x 10 cm x 10 cm = 1000 ml

1 liter = 1 000 milliliters = 1 000 ml

1 liter = 1 000 cu. Cm = 1 000 cm³

I ml = 1 cc. Teaspoon = 5 ml

Tablespoon = 10 ml Normal bottle = liter.

Example

Estimate the content of:

(a) St. Louis packet of sugar.

(b) The volume of your classroom.

(c) Your teacup

(d) A kerosene tins.

(a) St. Louis packet of sugar is about

(4 x 6 x 3) cm³ = 72 cm³ or 70 cu cm

(b) Our classroom is about

(8 x 12 x 4) m³ = 384 m³ or 350 cu m

(c) Normal tea cup is about 15 tablespoons

i.e. (15 x 10) ml = 150 milliliters.

(d) A kerosene tin is about 2 liters.

We hope you enjoyed the class.

Welcome to class!

In today’s class, we will be talking about approximation. Enjoy the class!

APPROXIMATION

It is the degree of accuracy of numbers and how to determine it. Rounding up of numbers, significant figures, decimal places, nearest whole numbers, tens, hundreds, and thousand, rounding up of numbers to nearest tenths, hundredths and thousandths

What is approximation?

An approximation is anything similar, but not exactly equal, to something else. A number can be approximated by rounding. A calculation can be approximated by rounding the values within it before performing the operations.

Rounding Numbers to the nearest 10, 100, 1,000:

To approximate to the nearest ten, look at the digit in the ten’s column.

To approximate to the nearest hundred, look at the digit in the hundred’s column.

For the nearest thousand, look at the digit in the thousand’s column.

Then do the following:

draw a vertical line to the right of the place value digit that is required

look at the next digit

if it’s 5 or more, increase the previous digit by one

if it’s 4 or less, keep the previous digit the same

fill any spaces to the right of the line with zeros.

Examples:

Round 4,853 to the nearest 10, 100 and 1,000.

485|3to the nearest 10 is 4,850

48|53 to the nearest 100 is 4,900

4|853 to the nearest 1,000 is 5,000

Round 76,982 to the nearest 10, 100 and 1,000.

7698|2to the nearest 10 is 76,980

769|82 to the nearest 100 is 77,000

76|982 to the nearest 1,000 is 77,000

Notice that in some cases the answers for rounding are the same.

Rounding to decimal places:

When rounding using decimal places(d.p), the degree of accuracy that is required is usually given. However, there are certain calculations where the degree of accuracy may be more obvious. For example, calculations involving money should be given to two decimal places to represent the pence.

To round to a decimal place:

look at the first digit after the decimal point if rounding to one decimal place or the second digit for two decimal places

draw a vertical line to the right of the place value digit that is required

look at the next digit

if the next digit is 5 or more, increase the previous digit by one

if it’s 4 or less, keep the previous digit the same

remove any numbers to the right of the line

Examples:

Round 248.561 to 1 decimal place, then round it to 2 decimal places:

5|61 to 1 decimal place is 248.6

56|1to 2 decimal place is 248.56

Notice that your answer should have the same number of decimal places as the approximation asked for.

Round 0.08513 to 1 decimal place and then to 2 decimal places:

0|8513 to 1 decimal place is 0.1

08|513 to 2 decimal places is 0.09

Rounding to significant figures:

The method of rounding to a significant figure is often used as it can be applied to any kind of number, regardless of how big or small it is. When a newspaper reports a lottery winner has won £3 million, this has been rounded to one significant figure. It rounds to the most important figure in the number.

To round to a significant figure:

look at the first non-zero digit if rounding to one significant figure

look at the digit after the first non-zero digit if rounding to two significant figures

draw a vertical line after the place value digit that is required

look at the next digit

if the next digit is 5 or more, increase the previous digit by one

if it is 4 or less, keep the previous digit the same

fill any spaces to the right of the line with zeros, stopping at the decimal point if there is one

Example:

Round 53,879 to 1 significant figure, then 2 significant figures.

5|3879 to 1 significant figure is 50,000

53|879 to 2 significant figures is 54,000

Notice that the number of significant figures in the question is the maximum number of non-zero digits in your answer.

Round 0.005089 to 1 significant figure, then 2 significant figures.

005|089 to 1 significant figure is 0.005

0050|89 to 2 significant figures is 0.0051

Question

What is 98,347 rounded to 1 significant figure, and 2 significant figures?

In our next class, we will be talking more about Approximation. We hope you enjoyed the class.

Welcome to class!

In today’s class, we will be talking more about approximation. Enjoy the class!

APPROXIMATION

APPROXIMATION classnotes.ng

Approximating values of addition, subtraction, multiplication and division

Exercise on the degree of accuracy round up numbers

Problems solving on quantitative reasoning and approximating things in everyday activities

When adding or subtracting approximate numbers, the result should have the precision of the least precise number.

Example: When adding 2.3, 5.704 and 12.67, our final answer should be correct to one decimal place.

2.3 + 5.704 + 12.67 = 20.674 ≈ 20.7

Accuracy when multiplying or dividing

When multiplying or dividing approximate numbers, the result should have the accuracy of the least accurate number.

Example: When multiplying 3.564 and 2.37, our final answer should have three significant digits.

3.564 × 2.37 = 8.44668 ≈ 8.45

Accuracy when finding the square root

When finding the square root of a number, the result has the same accuracy as the number.

Example: √22.97​ should be written correctly to 4 significant digits:

√22.97 ​ ≈ 4.793

Both numbers have the same accuracy.

In our next class, we will be talking about Number Bases. We hope you enjoyed the class.

Binary Subtraction:

The subtraction of the binary digit depends on the four basic operations

0 – 0 = 0

1 – 0 = 1

1 – 1 = 0

10 – 1 = 1

The above first three operations are easy to understand as they are identical to decimal subtraction. The fourth operation can be understood with the logic two minus one is one.

For a binary number with two or more digits, the subtraction is carried out column by column as in decimal subtraction. Also, sometimes one has to borrow from the next higher column. Consider the following example.

The above subtraction is carried out through the following steps.

0 – 0 = 0

For 0 – 1 = 1, taking borrow 1 and then 10 – 1 = 1

For 1 – 0, since 1 has already been given, it becomes 0 – 0 = 0

1 – 1 = 0

Therefore, the result is 0010.

Multiplication of two digits binary numbers:

Problem solving on quantitative aptitude related to conversion and application in real life situations

Binary multiplication is actually much simpler to calculate than decimal multiplication. In the case of decimal multiplication, we need to remember 3 x 9 = 27, 7 x 8 = 56, and so on. In binary multiplication, we only need to remember the following,

0 x 0 = 0

0 x 1 = 0

1 x 0 = 0

1 x 1 = 1

Note that since binary operates in base 2, the multiplication rules we need to remember are those that involve 0 and 1 only. As an example of binary multiplication, we have 101 times 11,

101

x11

Firstly, we multiply 101 by 1, which produces 101. Then we put a 0 as a placeholder as we would in decimal multiplication, and multiply 101 by 1, which produces 101.

101

x11

101

1010 <– the 0 here is the placeholder

The next step, as with decimal multiplication, is to add. The results from our previous step indicates that we must add 101 and 1010, the sum of which is 1111.

101

x11

101

1010

1111

In our next class, we will be talking about Basic Operations. We hope you enjoyed the clasbasic.Welcome to class!

In today’s class, we will be talking about basic operations. Enjoy the class!

BASIC OPERATIONS

BASIC OPERATIONS classnotes.ng

Addition and subtraction of numbers (Emphasis on place values using spike or abacus) Addition and subtraction of numbers (emphasis on the use of number line)

BASIC

The abacus above, represents the number 374. In the image, we see that:

We have 3 discs in hundreds’ place

We have 7 discs in tens’ place

We have4 discs in ones’ place

Therefore, the place value of 3 is hundred, the place value of 7 is ten and that of 4 is one. To simplify this in mathematical terms, we can write:

3 hundreds + 7 tens + 4 ones

= 300 + 70 + 4

= 374

Now let’s look at a few examples to understand how the spike abacus really helps:

Addition using abacus:

As we can see from the image, 12 is represented as 2 disks in the ones’ place and 1 disk in the tens’ place. Similarly, 25 is 5 disks in the ones’ place and 2 disks in the tens’ place. We know that addition is a combination of numbers. On combining the two numbers, we get 7 disks in the ones’ place and 3 disks in the tens’ place and this can be read as 37.

basic

Subtraction using abacus:

Let’s consider subtracting 12 from 25 in a similar way as shown in the image. In this case we take away the second number from the first, which is the basic definition of subtraction. So we take away 2 discs from 5 and are left with 3 discs. Then we take away 1 disc from 2 to get 1 disc. This number is 13.

basic

Addition and subtraction of positive and negative integers using number line and collection of terms. The everyday application of positive and negative integers solving problems on quantitative reasoning in basic operation.

Numbers Can be positive or negative

This is the Number line:

Negative Numbers (−) Positive Numbers (+)

“−” is the negative sign. “+” is the positive sign

If a number has no sign it usually means that it is a positive number. No Sign Means Positive

Example: 5 is really +5

Balloons and Weights

Let us think about numbers as balloons (positive) and weights (negative):

This basket has balloons and weights tied to it:

The balloons pull up (positive)

And the weights drag down (negative)

Adding a positive number:

Adding positive numbers is just a simple addition.

We can add balloons (we are adding positive value)

the basket gets pulled upwards (positive)

Example: 2 + 3 = 5 is really saying “Positive 2 plus Positive 3 equals Positive 5”

We could write it as (+2) + (+3) = (+5)

Subtracting a positive number:

Subtracting positive numbers is just simple subtraction.

We can take away balloons (we are subtracting positive value)

the basket gets pulled downwards (negative)

Example: 6 − 3 = 3 is really saying “Positive 6 minus Positive 3 equals Positive 3”

We could write it as (+6) − (+3) = (+3)

Adding a negative number:

Now let’s see what adding and subtracting negative numbers looks like:

We can add weights (we are adding negative values)

the basket gets pulled downwards (negative)

Example: 6 + (−3) = 3 is really saying “Positive 6 plus Negative 3 equals Positive 3”

We could write it as (+6) + (−3) = (+3)

The last two examples showed us that taking away balloons (subtracting a positive) or adding weights (adding a negative) both make the basket go down.

So, these have the same result:

(+6) − (+3) = (+3)

(+6) + (−3) = (+3)

In other words, subtracting a positive is the same as adding a negative.

Subtracting a negative number:

Lastly, we can take away weights (we are subtracting negative values)

the basket gets pulled upwards (positive)

Example: What is 6 − (−3)?

6−(−3) = 6 + 3 = 9

Yes indeed! Subtracting a negative is the same as adding!

Two Negatives Make a Positive

What Did We Find? Adding a positive number is a simple addition …

Adding a Positive is Addition

Positive and Negative Together …

Subtracting a Positive or Adding a Negative is Subtraction

Example: What is 6 − (+3)?

6−(+3) = 6 − 3 = 3

Example: What is 5 + (−7)?

5+(−7) = 5 − 7 = −2

Subtracting a negative …

Subtracting a negative is the same as Adding

Example: What is 14 − (−4)?

14−(−4) = 14 + 4 = 18

The Rules:

It can all be put into two rules:

Rule

Example

+(+) Two like signs become a positive sign

3+(+2) = 3 + 2 = 5

−(−)

6−(−3) = 6 + 3 = 9

+(−) Two, unlike signs, become a negative sign

7+(−2) = 7 − 2 = 5

−(+)

8−(+2) = 8 − 2 = 6

In our next class, we will be talking about the Algebraic Process. We hope you enjoyed the class.

Welcome to class!

In today’s class, we will be talking about the algebraic process. Enjoy the class!

ALGEBRAIC PROCESSES

What Is the Algebraic Processes?

The algebraic processes refer to various methods of solving a pair of linear equations, including graphing, substitution and elimination.

What Does the Algebraic Method Tell You?

The graphing method involves graphing the two equations. The intersection of the two lines will be an x, y coordinate, which is the solution.

With the substitution method, rearrange the equations to express the value of variables, x or y, in terms of another variable. Then substitute that expression for the value of that variable in the other equation.

For example, to solve:

​8x+6y=16

−8x

−4y=−8​

First, use the second equation to express x in terms of y:

−8x = −8 + 4yx = −8 + 4y/​−8x = 1 − 0.5y

Then substitute 1 – 0.5y for x in the first equation:

8(1−0.5y) + 6y = 16

8 − 4y + 6y = 16

8 + 2y = 16

2y = 8

Y = 4

Then replace y in the second equation with 4 to solve for x:

8x + 6(4) = 16

8x + 24 = 16

8x = −8

X = −1

The second method is the elimination method. It is used when one of the variables can be eliminated by either adding or subtracting the two equations. In the case of these two equations, we can add them together to eliminate x:

8x + 6y = 16

−8x − 4y = −8

0 + 2y = 8

y = 4 ​

Now, to solve for x, substitute the value for y in either equation:

8x + 6y = 16

8x + 6(4) = 16

8x + 24 = 16

8x + 24− 24 = 16 − 24

8x =−8

x = −1​

We hope you enjoyed the class.

Adding and Subtracting Numbers using the Number Line

Adding numbers on a number line is a neat way to see how numbers are added using visual interpretations.

Steps on how to add numbers on the number Line

As indicated in the diagram below:

To add a positive number means that we move the point to the right of the number line.

Similarly, to add a negative number implies that we move the point to the left of the number line.

Examples of Adding Numbers on the Number Line

Example:

Simplify by adding the numbers, 2 + 4.

The first step is to locate the first number which is two (2) on the number line.

basic

Adding four (4) means we have to move the point, four (4) units to the right.

In our next class, we will be talking more about Basic Operations. We hope you enjoyed the class.

Welcome to class!

In today’s class, we will be talking about geometry plane shapes. Enjoy the class!

GEOMETRY PLANE SHAPES

Types of plain shapes and their properties

Similarities and differences between square, rectangles, triangles, trapeziums, parallelograms, and circles

The Circle:

shape

The circle is a shape that can be made by tracing a curve that is always the same distance from a point that we call the centre. The distance around a circle is called the circumference of the circle.

The Triangle:

shape

The triangle is a shape that is formed by 3 straight lines that are called sides. There are different ways of classifying triangles, according to their sides or angles.

According to their angles:

Right triangle: the largest of the 3 angles is a right angle.

Acute Triangle: the largest of the 3 angles is an acute angle (less than 90 degrees).

Obtuse Triangle: the largest of the 3 angles is an obtuse angle (more than 90 degrees).

According to their sides:

Equilateral Triangle: all 3 sides are the same length.

Isosceles Triangle: it has 2 (or more) sides that are of equal length. (An equilateral triangle is also isosceles.)

Scalene Triangle: no 2 sides are of equal measure.

The Rectangle:

shape

The rectangle is a shape that has 4 sides. The distinguishing characteristic of a rectangle is that all 4 angles measure 90 degrees.

The Rhombus:

shape

The rhombus is a shape formed by 4 straight lines. Its 4 sides measure the same length but, unlike the rectangle, any of all 4 angles measure 90 degrees.

The Square:

shape

The square is a type of rectangle, but also a type of rhombus. It has characteristics of both of these. That is to say, all 4 angles are right angles, and all 4 sides are equal in length.

The Trapezoid:

shape

The trapezoid also has 4 sides. It has two sides that are parallel but the other 2 are not.

Trapezium and Its Properties:

A trapezium or a trapezoid is a quadrilateral with a pair of parallel sides. A parallelogram may also be called a trapezoid as it has two parallel sides. The pair of parallel sides is called the base while the non-parallel sides are called the legs of the trapezoid. The line segment that connects the midpoints of the legs of a trapezoid is called the mid-segment.

Parallelograms:

shape

One special kind of polygons is called a parallelogram. It is a quadrilateral where both pairs of opposite sides are parallel.

In our next class, we will be talking more about Geometry Plane Shapes. We hope you enjoyed the class.

The perimeter of a regular polygon, square, rectangle, triangle, trapezium, parallelogram, and circle Area of plain shapes such as square, rectangle, triangle, trapezium, parallelogram, and circle

The different geometrical shapes formula of area and perimeter with examples are discussed below:

Perimeter and Area of Rectangle:

Welcome to class!

In today’s class, we will be talking about Wood. Enjoy the class!

THREE DIMENSIONAL SHAPES

THREE DIMENSIONAL SHAPES classnotes.ng

Identification of three-dimensional or 3D shapes Basic properties of cubes and cuboids, and cones Basic properties of cylinders and spheres Volume of cubes and cuboids

Surface Area and Volume of 3D shapes

The two distinct measures used for defining the 3D shapes are:

Surface Area

Volume

Surface Area is defined as the total area of the surface of the three-dimensional object. It is denoted as “SA”. The surface area is measured in terms of square units. The three different classifications of surface area are defined below. They are:

Curved Surface Area (CSA) is the area of all the curved regions

Lateral Surface Area (LSA) is the area of all the curved regions and all the flat surfaces excluding base areas

Total Surface Area (TSA) is the area of all the surfaces including the base of a 3D object

Volume is defined as the total space occupied by the three-dimensional shape or solid object. The volume is denoted as “V”. It is measured in terms of cubic units.

Faces, Edges, and Vertices of 3D Shapes

Three-dimensional shapes have many attributes such as vertices, faces and edges. The flat surfaces of the 3D shapes are called the faces. The line segment where two faces meet is called an edge. A vertex is a point where 3 edges meet.

shape

Faces, Edges, and Vertices

How to Make 3d Shapes for Maths Project

If you know what are three-dimensional shapes, it would ….be easy for you to build a 3d shape project for a house or a building. This would be easy for the students to make as they can measure the rooms easily. Rest all they need is cardboard, glue, scissors and art supplies to make it look exactly like a mini house or building. Here, we are going to discuss the list of different three-dimensional shapes with its properties and the formulas of different 3D shapes.

All edges are equal

8 vertices

12 edges

6 faces

The surface area and the volume of the cube are given below:

The Surface Area of a Cube = 6a2 square units

The volume of a Cube = a3 cubic units

Cuboid:

shape

A cuboid also called a rectangular prism, where the faces of the cuboid are a rectangle in shape. All the angle measures are 90 degree

8 vertices

12 edges

6 faces

The surface area and the volume of the cuboid are given below:

The Surface Area of a Cuboid = 2(lb+bh+lh) Square units

The Surface Area of a Prism =2(Base Area) + (Base perimeter × length) square units

The volume of a Prism = Base Area × Height Cubic units

Pyramid:

shape

A pyramid a solid shape which has a structure, whose outer faces are triangular and meet to a single point on the top. The pyramid base can be of any shape such as triangular, square, quadrilateral or in the shape of any polygon. The most commonly used type of a pyramid is the square pyramid i.e., it has a square base and four triangular faces. Consider a square pyramid, it has

5 vertices

8 edges

5 faces

The surface area and the volume of the pyramid are given below:

Math.