Category: Mathematics Questions

The number system mathematics consists of natural numbers, whole numbers, prime & composite numbers etc. which all are used to perform various calculations and arithmetic operations in our daily lives. We use numbers in our day to day activities as well such as for telling time (3 O’clock), for exchanging money (£ 50), match scores ( 6 goals), cooking recipes (measurement of ingredients), counting objects (9 spoons) etc. These all fall under the category of whole numbers.

Let’s understand what are whole numbers?

Whole numbers are set of all natural numbers including zero.

In natural numbers, we do not include zero. Natural numbers start from 1 and go to infinity. While whole numbers start from 0 and go up to infinity.

To sum up we can say that whole numbers are a set of all natural numbers (positive numbers) with 0. But, this set does not include decimal numbers, fractions, or negative numbers.

So,  w = { 0,1,2,3,4,5,………….}

Here w denotes the set of whole numbers.

Things to remember:

• All natural numbers are whole numbers.
• All counting numbers are whole numbers.
• All positive integers including 0 are whole numbers.
• All whole numbers are real numbers.
• Whole numbers don’t include fractions, decimals, and negative integers.
• The smallest whole number is 0.
• There is no largest whole number.
• Except zero every whole number has an immediate predecessor.

Properties of Whole Numbers

The basic operations on whole numbers: addition, subtraction, division, and multiplication lead to some properties of whole numbers that are listed below:

1. Closure Property

This property states that if we add or multiply two whole numbers the result is always a whole number.

For example: 3+4 = 7 (whole number)

3 x 4 = 12 (whole number)

But one thing should be kept in mind that this property doesn’t hold true in case of subtraction and division

For example: 5 – 6 = -1(not a whole number)

1 / 4 = 0.25 (not a whole number)

2. Commutative Property

This property states that the sum and product of two whole numbers remain the same even after interchanging the order of the numbers.

If a and b are two whole numbers than a + b = b + a

Example:

14 + 3 = 17             and           3 + 14 = 17

We see that sum remains same after interchanging the order of the numbers. Same is the result in case of product.

Like Closure Property this property also doesn’t hold true in case of subtraction and division.

Example:

4 – 3 = 1 (whole number) but   3 – 4 is not equal to 1

We see the result is not the same after interchanging the order of digits. Same is in the case of division.

3. Associative Property

This property states that the sum or product of three whole numbers remains the same even irrespective of the order in which they are placed in the expression.

Say: x , y , z are three whole numbers then,

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

For example:

3 + (4 + 15 ) = 3 + 19 = 22

(3 + 4) + 15  = 7 + 15 = 22

Hence, we get the same result no matter how the numbers are grouped.

In case of subtraction and division, this property doesn’t holds true.

4. Additive Identity

When a whole number is added to 0, the sum remains the same i.e. the number itself.

For example: 4 + 0 = 4

So, 0 is also known as additive identity.

5. Multiplicative Identity

When a whole number is multiplied by 1, the product remains the same i.e. the number itself

For example: 1 4 x 1 = 14

So, 1 is also known as multiplicative identity.

6. Distributive Property

This property states that when two numbers, take for example: a and b are multiplied with the same number c and are then added Or, if we first do the sum of a and b and then multiply it by c we will get the same answer.

The expression becomes: (a  x c) + (b x c)   = c x ( a + b)

For example:

Let a = 12  ,   b = 13 ,       c =1 4

Now,  (12 x 14) + (13 x 14) =  168 + 182 = 350

14 x (12 + 13) = 14 x25 = 350

This proves distributive property.

 

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Chord: Chord is the line segment that joins the two points on the circumference of the circle or we can say on the boundary of circle.

Let us take an example to make the concept more clear.

Example:

In the above figure, AB is the chord which joins the two points on the circumference of the circle. Not only AB is the chord of the circle but all the line segments that join any two points on the circumference or boundary of the circle will make the chords of that circle.

Longest Chord

Any line that passes through the center and whose end points lie on the circle will be of the maximum length. Longest chord is that chord which passes through the center of the circle that is also called the diameter of the circle. So, diameter is the longest chord of the circle.

We can say that all diameters are the chords of the circle but all chords are not the diameters of the circle. Only that chord is termed as a diameter which passes through the center of the circle.

In the above circle AB is the chord as well as the diameter of the circle as it passes through the center of the circle while CD is only the chord of the circle not the diameter as it doesn’t passes through the center of the circle.

Properties of chord of circle:

1. The chords that are equal in length subtend equal angles at the center.
2. Perpendicular from the center of a circle to a chord bisects a chord.
3. The chords of a circle that are equal in length are always equidistant from the center.
4. Diameter is the longest chord of the circle.

So, chord is the line segment that joins the two points on the circumference of the circle.

Till now, we have understood the concept of chord, difference between chord & diameter and properties of chord.

Now, Let’s learn how to measure a chord’s length and discuss about the methods to find the length of chord.

Methods to find length of chord:

Method 1: using radius and perpendicular distance

Chord Length= 2 × √(r² – d²)

Where r is the radius and d is the perpendicular distance from the center of the circle to the chord.

Example:

Problem 1:  Find the length of chord of circle where radius is 5 cm and perpendicular distance from the center to the chord is 4cm?

Solution:

Given radius, r =5 cm

distance, d =4 cm

using the formula, Chord length =2 × √(r² – d²)

=2 × √(5² – 4²)

=2 × √25-16

=2 × √9

=2 x 3

= 6 cm

Length of the chord is 6 cm.

Method 2: Using trigonometry

Chord Length=2 x r x sin(c / 2)

Where, r is the radius and c is the angle subtended at the center of the circle by the chord.

Some Important Points

• The length of chord is inversely proportional to the perpendicular distance from center. It increases when the distance decreases.
• The diameter is the longest chord of circle, whereby the perpendicular distance from center to chord is zero.
• Two radii joining the ends of chord to the center of circle form an isosceles triangle.

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Expand in English means to make something larger or giving a fuller version of a thing.

Expand in math means opening the brackets ( ) or { } for arithmetic operations and simplifying the expression.

Tips to do that:

• Whatever is inside the parentheses or brackets should be considered as a single term.
• If we are performing multiplication every single term within the bracket would get multiplied with the expression written outside.

For example: on expanding 9(5+b)\text{ }\!\!\hat{\ }\!\!\text{ } we have to multiply 9 with both 5 and b. So, the expression becomes,

= 9 x 5 + 9 x b = 45 + 9b   (remember the formula 3 x a = 3a where a is a variable)

Example 2:  Expand {10 (7y+9z)}

Here, on opening the bracket we have to multiply 10 with both the terms inside.

We get  = {10 (7y+9z)}\,\text{=}\,\text{10 x 7y+10 x 9z}

= 70y + 90z is the answer

Now, Let’s understand expanding algebraic terms and brackets with powers.

Power or indices indicate that how many times a number has been multiplied by itself.

For example: 2 x 2 x 2 can be written as {{2}^{3}}

Similarly, when a is multiplied by n times,

a x a x a… x a (n times) = {{a}^{n}}

Always remember, when we add multiply powers of same variables they get added.

Such as {{r}^{4}}\,x\,{{r}^{5}} , Here the powers of r get added i.e. 4 + 5 = 9;

Or, ( r x r x r x r) x (r x r x r x r x r) = r x r x r x r x r x r x r x r x r or; r9

Example: Expand 5{{b}^{4}}\,(9{{b}^{2}}+8{{b}^{3}})

For that we need to open the bracket and multiply 5{{b}^{4}}   with 9{{b}^{2}}   and 8{{b}^{3}}   respectively.

On multiplying 5{{b}^{4}}   with 9{{b}^{2}}  we get

5{{b}^{4}}x\,9{{b}^{{2\,}}}\,=\,5x9x\,{{b}^{{4+2}}}\,=\,45{{b}^{6}}

And the On multiplying 5{{b}^{4}}   with 8{{b}^{3}}   we get

5{{b}^{4}}x\,8{{b}^{3}}\,=\,5x8\,{{b}^{{4+3}}}\,=\,40{{b}^{7}}

So, the answer becomes 45{{b}^{6}}\,+\,\,40{{b}^{7}}

Things to remember

• While simplifying and expanding any expression, you must follow the rules of BODMAS i.e. the order of mathematical operations which is brackets, indices, division, multiplication, addition and subtraction comes at last.
• Multiplying two negative numbers would yield a positive number.
• The powers or indices get added during multiplication and subtracted during division.
• Additional and multiplication of variables can only be done when they have same powers. These are called like terms. There is no such thing for multiplication and division.

Problem1: Expand and simplify the expression: -4({{b}^{{(2)}}}-8)+7{{b}^{{(2)}}}

By following the rules of BODMAS, we will first open the brackets. So, the expression becomes

-4({{b}^{{(2)}}}-8)+7{{b}^{{(2)}}}=(-4x{{b}^{{(2)}}})-(-4x8)+7{{b}^{{(2)}}}

=\,-4{{b}^{2}}+32+7{{b}^{2}} ( two negative signs become plus)

=\,7{{b}^{2}}-4{{b}^{2}}+32 (collecting the like terms)

=\,3{{b}^{2}}+32 becomes the answer

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Prism is a 3D (3-dimensional) solid object with identical faces at both ends. It is thus, a combination of identical bases, flat faces along with equal cross-sections. A prism can be of various kinds which are defined based upon the shape of its base which could be a square, rectangle, triangle, or n-sided polygon.

A prism is an important member of the polyhedron family having congruent polygons at the base and top and the others as lateral faces.

Examples of prism

We find examples of Prism in everyday lives such as a fish aquarium, metallic nuts, Camping tents, boxes, tanks etc and many such real life examples in our surroundings.

Types of Prisms

The prisms can be classified based on the type of polygon base or the type of cross-section of the prism or the alignment of identical bases. A prism cannot have a curve. Therefore, a prism can have a defined base only such as a base of triangle, square, rectangle, pentagon and other polygon shapes except for circle or any other circular shape.

Now, what is a difference between a Prism and a Pyramid?

We know that both prism and pyramid are 3-D solids having flat faces and bases. The difference is that a pyramid has only one base while a prism has two identical bases.

Classification based on Type of Polygon base:

Based on the type of polygon base, prisms can be classified into the following:

1. Triangular Prism: A prism with bases triangle in shape
2. Square Prism: A prism with bases square in shape
3. Rectangular prism: A prism with bases rectangle in shape
4. Pentagonal Prism: A prism with bases pentagon in shape
5. Hexagonal Prisms: A prism with bases hexagon in shape
6. Octagonal Prism: A prism with bases octagon in shape
7. Trapezoidal Prism: A prism with bases  trapezoid in

Classification based on the type of cross section of Prism:

Based on the cross-section of the base, Prisms can be classified into regular and irregular prisms.

• Regular Prism: A regular prism is one with the bases of the prism in the shape of a regular polygon which means all sides are equal.

• Irregular Prism: An irregular prism is one with the bases of the prism in the shape of an irregular polygon. Here the sides of the base are unequal in length.

Classification based on the alignment of identical bases:

Based on the alignment of identical bases, Prisms can be classified into Right & Oblique Prisms.

1. Right Prism: In a right prism, all the edges as well as joining faces are at 90 degree to each other or are perpendicular to the base.
2. Oblique Prism: An oblique prism is one that appears to be tilted and the two flat ends are not aligned and the side faces are in the shape of parallelograms.

Geometrical formulas to calculate the surface area and volume of the prism.

 

Lateral Surface Area:

LSA = Base perimeter × height

Total Surface Area:

TSA = Lateral surface area of prism + area of the two bases

= (Base Area x 2) + LSA

Volume:

Volume = Area of Base x height

 

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In Maths, the term congruent refers to those shapes or figures which can be flipped or repositioned to coincide with the other shapes. In other words, congruent means identical in size and shape. There are different theorems which are associated with the concept of congruency.

For example, for line segments, congruency means that they are equal in length or measure. For angles, the two angles are congruent only if both have same angle measures. Similarly, two triangles can be congruent if their corresponding parts are equal.

Let us understand with a practical example.

Draw two circles of same radius on a piece of paper. Cut both the circles out of papers and match them completely one on another. We notice that since both are of same shape and size, they completely overlap each other and thus, can be considered as congruent. The congruent is normally depicted a symbol “≅” meaning that two things are congruent to each other.

Thus, the word congruence can be used to describe relation between two objects or figures that are of same shape & size. Two geometrical figures can be considered congruent if they exactly superimpose on each other.

In above figures, both rectangles are of equal shape and can exactly superimpose on one other, thus, the two rectangles defined as A&B are congruent or in other words, A≅B.

Properties of Congruence:

There are three properties of congruence as listed below:

• Reflexive property
• Symmetric property
• Transitive property

Reflexive Property:

In geometry, reflexive property means that a line segment or an angle or a shape is congruent to itself. For example, ∠Z ≅ ∠Z for any angle Z

Symmetric property:

This property means that if one geometrical figure is congruent to the other figure, then the second one is also congruent to the first figure. Consider that there are two angles ∠Z &∠Y. As per this property, if ∠Z ≅ ∠Y, then, the vice versa i.e. ∠Y ≅ ∠Z.

Transitive property:

This property states that if two shapes, angles or lines are congruent to the third one, then, the first two are congruent to each other. For example, If X ≅ Y and Y ≅ Z, then X ≅ Z.

Congruence of Triangles:

Two triangles are said to be congruent to each other if their sides are of equal length and their angles are equal in measures. Also, both these can be superimposed on each other.

As clear from the above figures, Δ ABC and Δ XYZ are the congruent triangles, as the corresponding sides and corresponding angles in both the triangles are equal.

The vertices:  A and X, B and Y, and C and Z are the same.
The sides:  XY = AB, YZ=BC and XZ=AC;
The angles: ∠X = ∠A, ∠Y = ∠B, and ∠Z = ∠C.

Thus, Δ ABC ≅ Δ XYZ

For finding two triangles are congruent or not, there are different ways as listed below:

In following listed ways, we can work upon to find if two triangles are congruent:

• SSS (Side, Side, Side)
• SAS (Side, Angle, Side)
• ASA (Angle, Side, Angle)
• AAS (Angle, Angle, Side)
• RHS (Right Angle-Hypotenuse-Side)

Similarly, we can identify congruency of any given geometrical shape.

 

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A factor of a number is defined as the number which is an exact divisor of the given number. Each factor is less than or equal to the given number i.e. it can not be greater than the number whose factors are being calculated.

Every number in maths has minimum two factors, 1 and the number itself. However, a given number may have more than two factors and thus, the number of factors of a given number can be finite. For example, 1,2,4,8 are the factors of number 8.

In Maths, a common factor is defined as the factor which is common to two or more numbers. In other words, a whole number which will divide exactly the two or more numbers without leaving a remainder is termed as a common factor.

How to find common factors?

To find the common factors:

Step1: List out all the factors of the numbers for which we have to calculate common factor separately

Step2: Now, do the comparison to find out the factors which are common to these numbers.

The factors that appear in both the numbers are called common factors.

Let us solve a simple problem to better understand the concept.

Problem 1: Find out the common factors of 8 & 12.

Solution:

The factors of 8 are: 1, 2, 4 and 8

The factors of 12 are: 1, 2, 3, 4, 6 and 12

1, 2 and 4 appear in while factorizing both the terms.

So, the common factors of 8 & 12 are: 1, 2 and 4.

Problem2: Find out the common factors of 15 & 25.

Solution:

On factorizing 15 we get: 1, 3, 5 and 15

On factorizing 25 we get: 1, 5 and 25

So, the common factors of 15 & 25 are: 1 and 5.

Similarly, we can find common factors of any of the given two or more numbers.

Real World applications:

Common factors are very helpful in solving various problems especially when simplifying the fractions. These may also be very useful in various other applications such as work problems, understanding the time-distance concepts and most importantly comparing the prices.

Let us now understand how to simplify the fractions using common factors.

Problem: Simplify the fraction 20/40

Factors of 20 are : 1, 2, 4, 5, 10, 20

Factors of 40 are: 1, 2, 4, 5, 10, 20, 40

Since, the common factors of 20 & 40 are: 1, 2, 4, 5, 10, 20

Pick the highest common factor and express the numbers of the fraction as a multiple of this greatest number.

Numerator becomes: 20 = 20 × 1

Denominator becomes: 40 = 20 × 2

20 in the numerator and denominator gets cancelled out. So, the equation automatically gets simplifies into ½ .

 

Problem 3: Find out the common factors of 45, 80 and 28?

The factors of 45 are: 1, 3, 5, 9, 15 and 45

The factors of 80 are: 1, 2, 4, 5, 8, 10, 16, 20, 40 and 80

The factors of 28 are: 1, 2, 4, 7, 14 and 28

Look for factors that are common to all three numbers.

So, the common factor of 45, 80 and 28 is 1.

 

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