Class 7

Easy-to-read Ganita Prakash Class 7 Notes and Part 2 Chapter 7 Finding the Unknown Class 7 Notes save valuable study time during exam season.

Class 7 Maths Chapter 7 Finding the Unknown Notes

Class 7 Finding the Unknown Notes

The Weighing Scale and the Idea of Equality
A weighing scale compares weights placed on its two pans.
The number in the middle shows the total weight on both sides.

The scale is balanced only when both sides have equal total weights. The scale tilts in the direction of the heavier weight.

This is similar to the mathematical principle of equality (a = b) which states:
“Lets assume two expressions represent the same value. To maintain this equality, any operation applied to one side must be exactly applied to the other.”

Meaning of an Equation
A statement of equality between two algebraic expressions is called an equation.
We write an equation as two algebraic expressions with an equal sign ‘= ’ between them.
Here are some examples of equations:
2x + 7 = 11, 5 – 3y = 2y + 12, \(\frac{m+5}{3}\) = 3p + 2 etc.
An equation may contain one or more letter numbers (called variables) representing unknown quantities.

Solving Equations Systematically
The process of finding the value(s) of the letter-numbers for which the equality holds, or for which the value of LHS of the equation becomes equal to the value of RHS of the equation, is called solving the equation.

The specific value that satisfies the equation is known as its solution.
For example, x = 5 is the solution of equation 3x – 5 = 10 as x = 5 satisfies the equation.

There are three methods of solving an equation:

1. Trial and error method
2. Balancing / Systematic method
3. Transposition method

Trial and error method
To solve an equation using trial and error method, we follow the given steps:
Step 1: Obtain the LHS and RHS of the given equation.
Step 2: Check the different values of the variable in LHS and RHS one by one until LHS = RHS.
Step 3: The value of the variable for which LHS = RHS is the solution of the given equation.

The trial and error method is considered a crude approach because it involves testing many possible solutions one by one.

Balancing / Systematic method
An equation remains balanced when we perform the same operation (addition/subtraction/ multiplication/division) on both sides.
There are four rules which must be followed while balancing an equation.

1. If a number is added to one side of the equation, it must also be added to the other side.
2. If a number is subtracted from one side of the equation, it must also be subtracted from the other side.
3. If one side of an equation is multiplied by a number, the other side must also be multiplied by the same number.
4. If one side of an equation is divided by a non-zero number, the other side must also be divided by the same non-zero number.

Transposition Method
Transposition method is an efficient version of the balancing method.
If a term is transposed from one side of the equation to the other side, its sign changes.

If a number in multiplication (or division) is transposed from one side of the equation to the other side, its operation changes to division (or multiplication).

Solving Real Life Situations Using Equations
The word problems are first translated into an equation containing variables and constants.
Then, the equation is solved using the methods discussed earlier.

Follow the steps given below to solve a word problem:
Step 1: Obtain the given word problem.
Step 2: Study the problem carefully and identify what information is provided and what is the “unknown” or required to be found.
Step 3: Denote the unknown quantities by variables x,y, z, etc.
Step 4: Convert the language statements (or the practical situation) into a mathematical statement. For example, twice of a number is written as 2x.
Step 5: Accordingly, apply all the conditions which are given in the problem to form the equation.
Step 6: Solve the equation obtained in step 5.

The Weighing Scale and Equality

1. A weighing scale compares weights on two pans.
2. The scale is balanced when both sides have equal weight.
3. If one side is heavier and other side is lighter, then the heavier side goes down and lighter side goes up.
4. Adding (or removing) the same weight to (or from) both sides keeps the scale balanced.

Idea of an Equation
An equation shows equality between two expressions.
Example: 5x = 2x + 21, 3m – 4 = 11, 8y = 4 + 7y, etc.
It uses the equals sign (=).
Value of both sides of an equation are equal.
The value of variable that makes LHS = RHS is called the solution of the equation.

Variables and Expressions
Letters like x, y, n represent unknown numbers.
Expressions are formed using numbers and variables.
The same variable always represents the same value.

Methods of Solving Equations
The process of finding the solution of the equation is called “solving the equation’
Important methods of solving equations are:

Balancing Method
An equation remains balanced when we perform the same operation (addition / subtraction / multiplication / division) on both sides.
Rules:
Anything added to one side of the equation must also be added to the other side.

Anything subtracted from one side of the equation must also be subtracted from the other side.

If one side of an equation is multiplied by a number, the other side must also be multiplied by the same number.

If one side of an equation is divided by a non-zero number, the other side must also be divided by the same number.

Transposition Method
Transposition method is an efficient version of balancing method.

If a term is transposed from one side of the equation to the other side, its sign changes. For example, 5x + 2 = 4 ⇒ 5x = 4 – 2 If a number in multiplication (or division) is transposed from one side of the equation to the other side, its operation changes to division (or multiplication).
For example, 3x = 15 ⇒ x = \(\frac{15}{3}\)

Solving Word Problems
Follow the path given below to solve a word problem:

Easy-to-read Ganita Prakash Class 7 Notes and Part 2 Chapter 6 Constructions and Tilings Class 7 Notes save valuable study time during exam season.

Class 7 Maths Chapter 6 Constructions and Tilings Notes

Class 7 Constructions and Tilings Notes

A geometric construction is a step-by-step method of drawing lines, angles and shapes with accuracy, without guessing or drawing freehand.
Each construction is based on mathematical reasoning.

Perpendicular Bisector
A line that bisects a given line and is perpendicular to it, is called the perpendicular bisector.

Construction of Perpendicular Bisector:

Step 1: Draw the given line segment, say AB.
Step 2: Open the compass such that its radius is greater than half of AB. Place the compass’s pointer on point A and draw two arcs, one above and one below the line segment.
Step 3: Keeping the same radius (do not change the compass setting), place the pointer on B. Draw another two arcs intersecting the previous arcs at points P and Q.
Step 4: Using a ruler, draw a line passing through P and Q. The line PQ is the required perpendicular bisector of AB, intersecting it at point R. Here, R is the midpoint of AB.

Angle Bisector
An angle bisector is a ray drawn from the vertex of an angle that makes the two parts of the angle exactly of the same measure.

For example, in the figure, ray KM divides the ∠LKJ measuring 60° into two equal parts, the measure of each smaller angle is equal to 30°, i.e. ∠LKM = ∠JKM = 30°.

Additionally, ray KM is called angle bisector of ∠LKJ.

Construction of an Angle Bisector:

Step 1: Draw the given angle, say ∠ABC.
Step 2: Place compass pointer on B. Open it to an arbitrary radius and draw an arc intersecting both arms of ∠ABC at points P and Q. This ensures BP = BQ.
Step 3: With P as the centre and a radius more than half of PQ, draw an arc. With the same radius and Q as the centre, draw another arc intersecting the previous arc at point X.
Step 4: Join BX. The ray BX is the required angle bisector of ∠ABC.

Construction of an Angle Equal to Given Angle
Suppose we have to copy an angle ∠BAC i.e. to make an angle equal to ∠BAC. However, we do not know the measure of ∠BAC. For this we follow the given steps:

Step 1: Draw a ray PQ with initial point P.
Step 2: Place the compass pointer at vertex A. Open it to a convenient/arbitrary radius and draw an arc intersecting both arms of ∠BAC at points J and K.
Step 3: Taking caution and making sure that the opening (or the radius) of the compass does not change, place the pointer on P. Draw an arc intersecting ray PQ at point M.

Step 4: Place the compass pointer on K and adjust the width so that pencil lead rests exactly on measuring the distance KJ).
Step 5: Again, taking caution that the radius of the compass has not changed, place the pointer on M and draw a new arc that intersects the previous arc at point L.

Step 6: Draw a ray starting from point P and passing through point L using a ruler.
Step 7: Label a point R on ray PL. ∠QPR is required angle, which is an exact copy of ∠BAC.

Construction of an Angle Measuring 60°

To draw 60° angle, we follow the given steps:
Step 1: Draw a ray OQ with initial point O.
Step 2: Taking 0 as the centre and with the convenient radius, draw an arc intersecting the ray OQ at point A.
Step 3: With A as the centre and using the same radius as step 2, draw another arc to intersect the first arc at point B.
Step 4: Draw ray OB. ∠AOB is required angle of measure 60°.

Construction of a line parallel to a Given Line
If a transversal intersects two lines and the corresponding angles formed are equal, then the two lines are parallel.

To draw a line through B which is parallel to line m, we follow the given steps:
Step 1: Take any point A on line m. Join B and A to form a transversal line l.
Step 2: With A as centre and any convenient radius, draw an arc intersecting line m at point
C and line l at point D.
Step 3: Using the same radius as in step 2, draw another arc with B as centre. Let this arc intersect line l at point E.
Step 4: Place the pointed tip of the compass at D and adjust the opening so that the pencil lead is at C. This measures the length CD.
Step 5: With E as centre and the radius equal to length CD, draw an arc. This arc will cut the previous arc (drawn in step 3) at point F.
Step 6: Join B and F with a ruler to get a new line n.
The line n drawn through point B is parallel to line m, because the corresponding angles formed are equal, i.e. ∠EBF = ∠DAC.

Arch Designs

Arch designs have a symmetrical and geometric structure.

To create these arches, one must first draw accurate supporting lines on a flat surface like paper or stone.

A trefoil arch is a type of arch that has three rounded curves joined together, forming a shape like three small overlapping circles. The word “trefoil” means three leaves, so the arch looks a bit like a three-leaf clover.

Symmetry is essential for a trefoil arch. It must posses line symmetry to ensure the structure is balanced.

A pointed arch is formed by two arcs that meet at a sharp top point.

It uses two equal line segments placed at an angle. By locating their midpoints, arcs are drawn using the wavy wave method

Regular Hexagons

A regular polygon is a shape whose all sides are equal in length and all angles are equal in measure.

A regular hexagon can be visualised as being made up of six identical equilateral triangles.

The six triangles form a closed hexagonal shape.

The outer boundary becomes a regular hexagon.

Each interior angle of the hexagon is 120° (since two 60° angles meet at each vertex of hexagon).

Lines joining opposite vertices (diagonals) of the hexagon pass straight through the centre.

Tiling
Tiling means covering a region completely with given shapes, without any gaps or overlaps. One important type of tiling problem is to check whether the total number of unit squares in a given region can be fully covered by a given type of tile.

In any tiling problem, the first and most important step is a counting check, i.e. the total number of unit squares in the grid must be divisible by the area of one tile.

In general, an m × n rectangular grid can be tiled using 2×1 dominoes if and only if m × n is even, which means at least one of m orn must be even.

Certain regular shapes, like squares, equilateral triangles, and regular hexagons, can tile the entire plane without leaving gaps or overlaps. For example, regular hexagons form a natural tiling pattern, commonly seen in bee hives.

Other regular polygons (like Pentagon) result in unavoidable gaps or overlaps, so they cannot tile a flat surface (plane) perfectly.

Types of Bisector
Perpendicular bisector:
A line that bisects a given line and is perpendicular to it, is called the perpendicular bisector

PQ is perpendicular bisector of AB.

Angle bisector: A ray drawn from the vertex of an angle that divides the angle into two equal parts is called the angle bisector.

Here, BX is angle bisector of ∠ABC.

Copying An Angle
Copying an angle means constructing an angle that is exactly equal in measure to a given angle. We can copy a given angle using geometric instruments (a ruler and a compass), without measuring the angle with a protractor.
Consider the given angle to be ∠BAC.
Steps of construction to copy an angle:

Step 1: Draw a ray PQ with P as the initial point.
Step 2: With the compass at A and any radius, draw an arc cutting the arms of ∠BAC at J and K.

Step 3: Without changing the compass opening, place it at P and draw an arc cutting PQ at M.
Step 4: Place the compass at K and open it to reach J (measure KJ).

Step 5: Keeping the same opening, place the compass at M and draw an arc intersecting the previous arc at L.
Step 6: Using a ruler, draw a ray from P through L.
Step 7: Mark any point R on this ray. ∠RPQ is the required angle, equal to ∠BAC.

Tilings
Tiling (or tessellation) is the process of covering a flat region using a set of shapes so that there are absolutely no gaps or overlaps.

In general, a regular polygon can tile the plane only if the interior angle divides 360° exactly (whole number of times).

The hexagonal tiling is highly efficient; it is famously used by bees in honeycombs because it covers the area without wasting space and uses the minimum amount of wax to store the maximum amount of honey.

Tangram
A tangram is a mathematical puzzle made by cutting a square into seven geometric pieces, called tans.

Go through BSE Odisha Class 7 Science Solutions and Class 7 Science Chapter 6 Adolescence: A Stage of Growth and Change Question Answer to understand textbook questions more clearly.

Class 7 Science Curiosity Chapter 6 Question Answer

Science Curiosity Class 7 Chapter 6 Question Answer

Adolescence: A Stage of Growth and Change Class 7 Questions and Answers

Let Us Enhance Our Learning

Question 1.
Ramesh, an 11-year-old boy, developed a few pimples on his face. His mother told him that this is because of ongoing biological changes in his body.
(i) What could be the possible reasons for the development of these pimples on his face?
(ii) What can he do to get some relief from these pimples?
Answer:
(i) The development of pimples on Ramesh’s face could be due to the hormonal changes and oily secretions from the skin that occur during adolescence.
(ii)

• Avoid squeezing or picking the pimples.
• Maintain good body hygiene.
• Drink plenty of water.
• Use a gentle cleanser for the face.

Question 2.
Which of the following food groups would be a better option for adolescents and why?

Answer:
The food group (B), would be a better choice for adolescents because it contains sources of all essential nutrients, i.e., carbohydrates, proteins, fats, vitamins, and minerals that are necessary for supporting the rapid growth and development occurring during this period. On the other hand, food group (A) contains various junk foods that are high in fats and sugar but low in essential nutrients.

Question 3.
Unscramble the underlined word in the following sentences:
(i) The discharge of blood in adolescent girls which generally occurs every 28-30 days is nstmnoiaretu.
(ii) The hoarseness in the voice of adolescent boys is due to enlarged iceov xob.
(iii) Secondary sexual characteristics are natural signs that the body is preparing for adulthood and mark the onset of urtypeb.
(iv) We should say NO to lahoclo and srugd as they are addictive.
Answer:
(i) nstmnoiaretu – menstruation
(ii) iceov xob – voice box
(iii) urtypeb – puberty
(iv) lahoclo – alcohol, srugd – drugs

Question 4.
Shalu told her friend, “Adolescence brings only physical changes, like growing taller or developing body hair.” Is she correct? What would you change in this description of adolescence?
Answer:
No, Shalu is partially correct because adolescence not only brings physical changes like growing taller or the appearance of body hair, but also includes internal (biological) changes indicating reproductive capability, as well as emotional and behavioural changes.

Question 5.
During a discussion in the class, some of the students raised the following points. What questions would you ask them to check the correctness of these points?
(i) Adolescents do not need to worry about behavioural changes.
(ii) If someone tries a harmful substance once, they can stop anytime they want.
Answer:
The following questions can be asked to check the correctness of points made by the students:
(i)

• Can ignoring certain behavioural changes affect mental health or social relationships later on?
• Why do you think it is important for adolescents to adapt to changes in a positive way rather than worrying about them?
• Are behavioural changes during adolescence linked to hormonal changes, and how can one handle these changes?

(ii)

• Can harmful substances like tobacco and alcohol cause addiction, and is it easy to stop using them once a person becomes dependent?
• How does peer pressure or emotional state influence the ability to stop using a substance?
• Has any research shown that people can easily quit using harmful substances after trying them just once?

Question 6.
Adolescents sometimes experience mood swings. On some days, they feel very energetic and happy, while on other days, they may feel low. What other behavioural changes are associated with this age?
Answer:
Adolescents undergo several behavioural changes, such as:

• A strong desire for independence, developing their own identity, and personal decision making.
• A desire to explore new interests and engage in varied activities like music, dance, or sports.
• Increased influence of peers, attraction to them, and trying to mimic their behaviour.
• Intense emotions, frequent mood swings and increased sensitivity.
• Getting involved in social work and initiatives to support the needy and the disadvantaged.

Question 7.
While using a toilet, Mohini noticed that used sanitary pads were scattered near the bin. She got upset and shared her feelings with her friends. They discussed the importance of menstrual hygiene and healthy sanitary habits. What menstrual hygiene and sanitary habits would you suggest to your friends?
Answer:
The following menstrual hygiene and sanitary habits should be followed:

• Properly disposing used sanitary pads by wrapping them in paper and disposing them in a dustbin.
• Using clean, reusable cloth pads or sanitary napkins.
• Washing hands with soap and water before and after handling sanitary products.
• Cleaning the pubic area regularly to maintain hygiene and prevent bacterial or fungal infections.

Question 8.
Mary and Manoj were classmates and good friends. On turning 11, Mary developed a little bulge on the front of her neck. She visited the doctor who gave her medication and asked to take iodine-rich diet. Similarly, a bump was developed on the front of Manoj’s neck when he turned 12. However, the doctor told him that it was a part of growing up. According to you, what could be the possible reason for advising Mary and Manoj differently?
Answer:
The doctor advised Mary and Manoj differently because the causes of their neck bulges were different.

• In Mary’s case, the bulge on her neck was likely due to goitre, a condition caused by iodine deficiency. Iodine is essential for performing physical and mental activities efficiently. When the body lacks iodine, it leads to a disease called goitre and a prominent bulge or swelling on the front of the neck is a characteristic symptom of this disease. Therefore, the doctor advised her to take an iodine-rich diet and prescribed required medication.
• On the other hand, in Manoj’s case, the bulge on his neck was due to the natural growth of the voice box during puberty. This results in the formation of the Adam’s apple, which is a normal part of male development and also causes the voice to become deeper and hoarse.

Question 9.
During adolescence, the boys and girls undergo certain physical changes, a few of which are given below.
(i) Change in voice
(ii) Development of breasts
(iii) Growth of moustache
(iv) Growth of facial hair
(v) Pimples on the face
(vi) Growth of hair in the pubic region
(vii) Growth of hair in armpits
Categorise these changes in the table given below:

Answer:

Physical changes during adolescence
Observed only in boys	Observed only in girls	Common in boys and girls
Change in voice	Development of breasts	Pimples on the face
Growth of moustache		Growth of hair in the pubic region
Growth of facial hair		Growth of hair in armpits

Question 10.
Prepare a poster mentioning the tips for adolescents to live a healthy lifestyle.
Answer:
The students should prepare the poster themselves.

Class 7 Curiosity Chapter 6 Question Answer

InText Questions

Question 1.
What changes are most commonly seen during adolescence? [Page 75]
Answer:
Commonly observable changes during adolescence are:

• Increase in height and weight
• Appearance of hair in different parts of the body, like the armpits and pubic region
• Change in voice (more prominent in boys)
• Oily secretions from the skin causing pimples on the face

Question 2.
Is adolescence just about physical changes or changes associated with reproductive capability? [Page 78]
Answer:
No, adolescence also brings about significant emotional and behavioural changes along with other changes.

Question 3.
Does an ideal nutritious diet for adolescents mean consuming only protein and carbohydrate-rich foods? [Page 79]
Answer:
For proper growth and to perform well in studies and sports, we not only require proteins and carbohydrates, but also an adequate amount of fats, vitamins, and minerals, along with a proper intake of dietary fibre and water.

Question 4.
In what ways do nutrients support our growth and development? [Page 79]
Answer:
Nutrients such as proteins, fats, carbohydrates, vitamins and minerals play a crucial role in keeping our body healthy. These nutrients are obtained from foods like milk, curd, cheese, spinach, millets, kidney beans, and dried fruits such as raisins and figs.
These nutrients help our body in the following ways:

• Support the growth and development of bones and muscles
• Increase strength and help build a healthy body
• Aid in the formation of blood
• Improve energy levels

Question 5.
Adolescents, especially girls, may suffer from blood-related health problems due to a deficiency of iron or vitamin B12. Name and describe such health problem(s). [Page 80]
Answer:
Anaemia is a blood-related health problem caused by a deficiency of iron or vitamin B12. It leads to low haemoglobin levels, resulting in weakness, fatigue, and pale skin. Adolescent girls are more prone to anaemia due to rapid growth and blood loss during menstruation.

Question 6.
How can we manage iron deficiency in our body? [Page 80]
Answer:
Iron deficiency can be prevented and managed by including iron-rich foods in our diet, such as beans, nuts, dried apricots, spinach, lentils, legumes, broccoli, and fortified cereals. In some cases, doctors may also recommend iron supplements to maintain healthy iron levels in the body.

Question 7.
What are some sources of vitamin B12? [Page 80]
Answer:
Vitamin B12 is mainly obtained from foods such as eggs, meat, liver, fish, cheese, milk, fortified cereals, nutritional yeast, and green leafy vegetables.

Question 8.
List some government schemes that aim to prevent deficiency diseases like anaemia. [Page 80]
Answer:
In India, the Adolescent Anaemia Control Programme focuses on reducing the prevalence and severity of anaemia among adolescents and supports the government’s Anaemia Mukt Bharat initiative under the National Health Mission.
The Weekly Iron and Folic Acid Supplementation (WIFS) programme is a key component of the Anaemia Mukt Bharat strategy. It involves the supervised weekly consumption of Iron and Folic Acid (IFA) tablets by both school-going and outof-school adolescents to prevent iron-deficiency anaemia.

Question 9.
Why do changes occur during the adolescent life stage? [Page 84]
Answer:
Many changes during adolescence, including menstruation and other signs of puberty, are primarily due to the production of certain chemicals in our bodies called hormones.

Easy-to-read Ganita Prakash Class 7 Notes and Part 2 Chapter 5 Connecting the Dots Class 7 Notes save valuable study time during exam season.

Class 7 Maths Chapter 5 Connecting the Dots Notes

Class 7 Connecting the Dots Notes

Statistical Questions and Statements
A statistical question is a question that can be answered by collecting and analysing data.

A statistical statement is a claim or summary about some phenomenon, expressed in terms of numerical values, proportions, probabilities or predictions.

A non-statistical question does not need data collection as it has usually one, fixed answer.

Asking a statistical question, collecting data, analysing it and making a statistical statement together form the statistical process.

Visualisation of data with a Dot Plot
A dot plot is a simple way of visualising data using dots, where each dot represents one observation.
In a dot plot:

1. Spacing between the horizontal and the vertical lines must be equal.
2. The dots are placed in a line or a row, usually along a number line.
3. If the same observation occurs more than once, place the dots one above the other.

For example, the given data shows how the prices of potatoes change across different months of the year in Jaipur.

Month	Jan	Feb	Mar	Apr	May	Jun	Jul	Aug	Sep	Oct	Nov	Dec
Price in Jaipur (in ₹/ kg)	25	24	43	28	30	35	39	26	49	56	59	44

The dot plot for the given data is drawn as:

Range of the given data is difference between its extremes, i.e. the difference between the highest and the lowest values in the dataset.
Range = Highest Value (Maximum) Lowest Value (Minimum)

Representative Value of the Data
A representative value of a dataset is a single number, often called a measure of central tendency, that summarises the entire dataset.

Average or Arithmetic mean (or simply Mean) is the value we get when we add up all the observations in the group and divide the total by the count of observations in the group.
\(\text { Mean }=\frac{\text { Sum of all values in the data }}{\text { Number of values in the data }}\)

The value of the Arithmetic mean depends on the total of all values and number of values, so more observations do not guarantee a higher mean.

The Median of a data set is the value in the middle, when all observations are arranged in order. It splits the data into two groups of equal size.

If n is odd, Median = value of the \(\left(\frac{n+1}{2}\right)^{\text {th }}\) observation
If n is even, \(\text { Median }=\frac{\text { value of the }\left(\frac{n}{2}\right)^{\text {th }} \text { observation }+ \text { value of }\left(\frac{n}{2}+1\right)^{\text {th }} \text { observation }}{2}\)

An outlier is a data value which significantly deviates from the rest of the values in the data set.

Effect of Outliers on Mean and Median:
The mean may move up (or down), depending on the presence of an outlier to the right (or left).

The median usually remains near the centre of the main cluster of data and is more stable than the mean (especially in presence of outliers).

Mean is sensitive to outliers, while the median is more stable and often better represents the central tendency of the complete data set.

While working with data, it is important to understand the difference between a zero value and no value. For example, in a cricket match:

A score of 0 means the player played and scored 0 runs. Therefore, it must be counted to find mean or median.

A blank (no score) means the player did not bat in that match. Therefore, that match should not be counted while finding the average.

Visualisation of data with a Double Bar Graph
A clustered bar graph is a type of bar graph in which two or more bars are placed side by side to compare related quantities.

Like the single bar graphs, clustered bar graphs can be of two types: Horizontal and Vertical, depending on the orientation of bars.

A double bar graph is a special case of a clustered bar graph in which exactly two bars are drawn for each category such as a team, a place, a time period, a type of item etc.

To draw a double bar graph for a given data, follow the steps given below:
Step 1: Draw axes and choose a scale: Draw vertical and horizontal axes, label them, and select a common scale for the vertical axis (or horizontal axis in case of horizontal double bar graph).
Step 2: Draw bars for each data set: For each category, draw bars of equal width side by side, using different colours or patterns.
Step 3: Add title and key: Give the graph a proper title and include a key (legend) to identify each data set.
Step 4: Check and label carefully: Ensure all bars are correctly labelled and spaced evenly.

For example, for the given data:

	Special Abilities Chosen by Students
	Invisibility (I)	Super Speed (S)	Time Travel (T)	None(N)
Class 6	7	9	5	3
Class 8	6	7	9	2

The double graph is drawn as:

Interpretation of Double Bar Graphs
Interpreting a graph means studying it carefully to understand the information shown, comparing values, and drawing meaningful conclusions from the data.

To make sense of the data easily, we can follow a simple two-step process:
Step 1: Observe and identify the given data

1. To do this, first read the title of the graph to know what is being measured and compared.
2. Next, observe the x-axis and y-axis to identify the categories shown and the unit of measurement.
3. Finally, check the scale of the graph so that the values and differences can be interpreted correctly.

Step 2: Analyse the data and make inferences

1. To do this, analyse the values shown in the graph.
2. Compare the data for different categories.
3. Draw conclusions based on your observations.

A well-drawn graph must be clear and accurate: It should have a proper title, correctly labelled axes, a common scale, equal spacing, and a key so that the data can be understood easily.

Introduction
Statistical questions are questions that involve variability in data and require data collection and analysis to answer.

A data set is an organised collection of related information or numerical values,
Range is the difference between the maximum and minimum values in a data set.

Representative Value
A representative value of a data set is a single number, often called a measure of central tendency, that summarises the entire data set. For example, mean and median.

Arithmetic Mean (Average)	Median
\(\text { Mean }=\frac{\text { Sum of all observations }}{\text { Number of observations }}\)	Median is the middle value of an ordered data.
Mean can be understood as fair share or equal distribution.	If number of observations i.e., n is even, \(\text { Median }=\left(\frac{n+1}{2}\right)^{\text {th }} \text { value }\)
Mean is widely used in daily life (rainfall, height, income, prices, sports performance)	If number of observations i.e., n is even, \(\text { Median }=\frac{\left(\frac{n}{2}\right)^{\text {th }} \text { value }+\left(\frac{n}{2}+1\right)^{\text {th }} \text { value }}{2}\)

Note:

1. An outlier is a number that is very different from the others in a set of data — it is much bigger or much smaller than most of the values.
2. The mean can be easily affected by outliers, but the median usually stays the same and gives a better idea of the middle of the data set.

Visualisation of Data
Data visualisation helps in understanding data better using tools like dot plots and clustered bar graphs.

Dot Plots	Double Bar Graphs
A dot plot is a simple, visual graph used to represent numerical data by placing dots above a labeled number line.	Double bar graphs are used to compare two related quantities by showing a pair of bars for each category using the same scale.
In a dot plot: Each dot represents one value from the data set. Spacing between the horizontal and vertical lines must be equal. The dots are placed in a line or row, usually along a number line. If the same number appears more than once, we stack the dots on top of each other.	To draw a double bar graph for a given data, follow the steps given below: Step 1: Draw axes and choose a scale Step 2: Draw bars for each data set Step 3: Add title and key Step 4: Check and label carefully

 

Interpreting the Graphs
Interpreting a graph means studying it carefully to understand the information shown, comparing values and drawing meaningful conclusions from the data.

To make sense of the data easily, we can follow a simple two-step process:
Step 1: Observe and identify the given data
Step 2: Analyse the data and make inferences

Easy-to-read Ganita Prakash Class 7 Notes and Part 2 Chapter 4 Another Peek Beyond the Point Class 7 Notes save valuable study time during exam season.

Class 7 Maths Chapter 4 Another Peek Beyond the Point Notes

Class 7 Another Peek Beyond the Point Notes

Decimals are another way to show parts of a whole. Instead of writing fractions like \(\frac{1}{4}\) or \(\frac{1}{5}\), we use a point or period (‘.’) as a separator called a decimal point.
A decimal number has 3 parts.

A decimal point indicates where the whole number part ends and the fractional part begins in a decimal number.

Also, decimals such as 0.1 and 0.01 can be expressed in the form of fractions as \(\frac{1}{10}\) and \(\frac{1}{100}\), respectively.

Place Value in a Decimal: Each digit in a number denotes a value depending on where it is placed. This is called the place value.

Place value chart for decimals is given in the figure.
For example,
137.85 = 1 × 100 + 3 × 10 + 7 × 1 + 8 × \(\frac{1}{10}\) + 5 × \(\frac{1}{100}\)

Converting Fractions to Decimals (When Denominators are 10, 100, 1000, …): A fraction represents the division of the numerator by the denominator.

Write the dividend as it is and put a decimal point at the end.
Count the zeroes in the divisor.

Move the decimal point left by the same number of places as the count of zeroes in the divisor. Add zeroes in front if needed.

Multiplication of a Decimal Number with a Whole Number
Repeated Addition Method: Multiplying a decimal by a whole number can be done by adding the decimals as many times as the whole number indicates.
For example, multiplying 6.5 by 7 is same as adding 6.5 seven times,
i. e. 6.5 × 7 = 6.5 + 6.5 + 6.5 + 6.5 + 6.5 + 6.5 + 6.5 = 45.5

Multiplication of Decimal by 10, 100 and 1000: When a decimal number is multiplied by 10, 100, or 1000, the decimal point shifts to the right by as many places as there are zeros in the multiplier.
For example, (i) 24.576 × 100 = 2457.6
(ii) 0.3251 × 1000 = 325.1

If there are not enough digits to shift the decimal point, zeroes are added to the right.
For example, 1.34 × 1000 = 1340.0 = 1340

Multiplication of a Decimal Number with a Decimal Number
Fraction Method (Converting Decimals to Fractions): Multiplication of decimals is same as multiplication of their corresponding fractions.
Step 1: Change decimals to corresponding fractions
Step 2: Multiply Numerator and Denominator, respectively
Step 3: Turn the final fraction back into a decimal

Decimal Shifting Method (Making them Whole Number)

Step 1: Ignore the decimal points and multiply the numbers as whole numbers.
Step 2: Count the total number of digits after the decimal point in both numbers.
Step 3: From the right of the product, shift the decimal point to the left by the count obtained in step 2.

Relation between Product and Numbers
If both numbers are greater than 1 (e.g., 3.4 X 6.5), the product (22.1) is greater than both the numbers.

If both number are between 0 and 1 (e.g., 0.75 x 0.4), the product (0.3) is less than both the numbers.

If one number is between 0 and 1 and one number is greater than 1 (e.g., 0.75 X 5), the product (3.75) is less than the number greater than 1 and greater than the number between 0 and 1.

Division of Decimals by 10, 100, 1000, …: When a number is divided by 10, 100, 1000, and so
on, move the decimal point to the left by as many places as there are zeros in the divisor.

Number	÷10	÷100	÷1000
12.02	1.202	0.1202	0.01202

If we run out of digits while moving the decimal point, we add zeros at the start of the number.

Decimal Division Using Place Value

Step 1: Divide as you do with counting numbers, starting from the highest place value.
Step 2: If there is a remainder, regroup it into the next smaller place value digit and continue dividing.
Step 3: When you move from the Ones place to the Tenths place in the division, place a decimal point in the quotient.
Step 4: If necessary, add zeros to the right of the decimal point in the dividend and continue the division until the remainder becomes zero or the digits in the quotient begin to repeat.

When the divisor is a decimal, we can convert it into a counting number by suitably multiplying it by 10, 100, 1000, and so on. We must also multiply the dividend by the same number.

Once the divisor is converted into a counting number, we proceed with the division by following the same place-value procedure (long division) to find the quotient.

A recurring decimal is a decimal that follows a repeating pattern of digits after the decimal point.
For example, \(\frac{1}{3}\) = 0.333… and \(\frac{12}{99}\) = 0.121212…

Relation between Dividend, Divisor and Quotient
When the divisor is between 0 and 1.
For example, 135 ÷ 0.5 (0 < 0.5 <1) The quotient (270) is greater than the dividend (135). When the divisor is greater than 1. For example, 135 ÷ 2.5 (> 1)
The quotient (54) is smaller than the dividend (135).

When the divisor is exactly 1.
For example, 135 ÷ 1
The quotient (135) is equal to the dividend (135).

Leap Year
A leap year is a year with 366 days. In the Gregorian calendar, years divisible by 4 are leap years, but century years are leap years only if they are divisible by 400.

Process to determine whether a year is a leap year:

Multiplication of Decimal Numbers
For small whole numbers, multiplication can be viewed as adding the decimal number multiple times.

While multiplying a number by 10, 100, 1000, move the decimal point to the right by as many places as there are zeros in the multiplier.
If you run out of digits while moving the decimal point, add zeros at the end.

Multiplication of decimals is same as multiplication of their corresponding fractions.

Division of Decimal Numbers
When the divisor is a decimal, we can convert it into a counting number by suitably multiplying it by 10, 100, 1000, and so on. We must also multiply the dividend by the same number.

Once the divisor is converted into a counting number, we proceed with the division by following the same place-value procedure (long division) to find the quotient.

A recurring decimal is a decimal that follows a repeating pattern of digits after the decimal point.

Relationship between Dividend, Divisor and Quotient
When the divisor is between 0 and 1 [e.g., 135 ÷ 0.5 (0 < 0.5 <1)], the quotient (270) is greater than the dividend (135). When the divisor is greater than 1 [e.g., 135 ÷ 2.5 (> 1)], the quotient (54) is smaller than the dividend (135).

When the divisor is exactly 1 (e.g., 135 ÷ 1), the quotient (135) is equal to the dividend (135).

Relation between Product and Numbers
If both numbers are greater than 1 (e.g., 3.4 × 6.5), the product (22.1) is greater than both the numbers.

If both numbers are between 0 and 1 (e.g., 0.75 × 0. 4), the product (0.3) is less than both the numbers.

If one number is between 0 and 1 and one number is greater than 1 (e.g., 0.75 × 5), the product (3.75) is less than the number greater than 1 and greater I than the number between 0 and 1.

Leap Year
A leap year is a year with 366 days. In the Gregorian calendar, years divisible by 4 are leap years, but century years are leap years only if they are divisible by 400.

Process to determine whether a year is a leap year:

Easy-to-read Ganita Prakash Class 7 Notes and Part 2 Chapter 3 Finding Common Ground Class 7 Notes save valuable study time during exam season.

Class 7 Maths Chapter 3 Finding Common Ground Notes

Class 7 Finding Common Ground Notes

The Highest Common Factor (HCF) of a group of numbers is the greatest number that divides each number in the group completely, without leaving any remainder.
For example: HCF of 12 and 18 is 6.

We use HCF when we need to find the largest possible size that fits perfectly into different measurements, such as the largest tile for a floor.

A prime number is a number greater than 1 that has exactly two factors: 1 and the number itself. For example, 11 has only two factors 1 and 11, so it is a prime number.

Every number has a unique way of being broken down into its prime building blocks or prime factors. We call this process Prime Factorisation.
For example: 6 = 2 × 3; Here 2 and 3 are primes.
50 = 2 × 5 × 5; Here, 2 and 5 are primes.

Any composite number can be written as a product of prime numbers.

Methods of finding HCF
Prime Factorisation Method
Step 1: Find the prime factorisation of each of the given numbers.
Step 2: Identify the common factors in the prime factorisation of all the given numbers.
Step 3: Multiply the common factors to obtain the HCF.
For example: Consider numbers 36, 42 and 64. Their prime factorisation are:
36 = 2 × 2 × 3 × 3
42 = 2 × 3 × 7
64= 2 × 2 × 2 × 2 × 2 × 2
∴ HCF of 36, 42 and 64 = 2

Common Factor Method
Step 1: Arrange all the given numbers in a row, separated by commas, for which we need to find the HCF.
Step 2: Divide all the numbers by any factor i.e., common to all of the given numbers. ;
Step 3: If there are still any common factors, divide the quotients by them and keep dividing until there is no common factor for all the given numbers.
Step 4: The product of these common factors will give the highest common factor.
For example: Consider numbers 45 and 63. Using common factor method, we get
HCF of 45 and 63 = 3 × 3 = 9

Repeated Division Method
Step 1: Divide the larger number by the smaller number and find the remainder.
Step 2: Take the remainder as the new divisor and divisor of step 1 as the new dividend, then divide.
Step 3: Repeat until the remainder is 0. The last divisor thus obtained is the HCF.
For example: Consider numbers 26 and 455. Using repeated division method, we get
HCF of 26 and 455 = 13

For numbers that share no common factors other than 1 (called co-primes, like 7 and 15), the F1CF is always 1. Therefore, the HCF of two co-prime numbers is always 1.

The Least Common Multiple (LCM) is the smallest non-zero number that is a multiple of all the given numbers.

For example, LCM of 15 and 20 is 60, as 60 is the smallest non-zero number that is a multiple of both 15 and 20.

We use LCM to find when events that happen at different intervals will align or meet again, like bells ringing or joggers crossing a start line.

Methods of finding LCM
Prime Factorisation Method
Step 1: Write the prime factorisation of each given number and list all the prime factors involved.
Step 2: For each prime factor, take the greatest number of times it appears in any of the numbers.
If two numbers are multiplied
Step 3: Multiply these factors to obtain the LCM.
For example: Consider numbers 36, 42 and 64. Their prime factorisation are:
36 = 2 × 2 × 3 × 3
42 = 2 × 3 × 7
64 = 2 × 2 × 2 × 2 × 2 × 2
∴ LCM of 36, 42 and 64 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 7 = 4032

Common Division Method
Step 1: Write the given numbers in a row, separated by commas.
Step 2: Divide these numbers by the least prime number which divides at least one of the numbers.
Step 3: Write the quotients below, and bring down the numbers that are not divisible by the prime number.
Step 4: Repeat Steps 2 and 3 with the new row. Continue until all numbers in a row become 1.
Step 5: Find the product of all the divisors used in the process. This gives the LCM.
For example: Consider numbers 45 and 75. Using common division method, we get
LCM of 45 and 75 = 3 × 5 × 3 × 5 = 225

Properties of HCF and LCM
The HCF of a set of given numbers is always either smaller than all the numbers or equal to the smallest number.
For example, HCF (18, 24) = 6 → smaller than both the numbers
Also, HCF (8, 16) = 8 → equal to the smallest number

The LCM of a set of given numbers is always either greater than all the numbers or equal to the greatest number.
For example, LCM (6, 8) = 24 → greater than both the numbers
Also, LCM (8, 16) = 16 → equal to the greatest number

The HCF of a set of co-prime numbers is 1 and the LCM of co-prime numbers is the product of the co-primes.
For example, 8 and 15 are co-prime numbers as their common factor is 1.
Here, LCM (8, 15) = 120 = 8 × 15 and HCF (8, 15) = 1

The HCF of a set of given numbers is a factor of their LCM. In other words, the LCM of given numbers is a multiple of their HCF.
For example, HCF and LCM of 12 and 18 are 6 and 36, respectively. Here, 6 is a factor 36.

If two numbers are multiplied by the same number, then both their HCF and LCM are also multiplied by that number.

If the HCF of two numbers a and b is H and their LCM is L, then for any number k:
HCF (ka, kb) = kH and LCM (ka, kb) = kL

For any two given numbers, if the first number is a factor of the second number, then the first number is their HCF and the second number is their LCM.
For example, consider the numbers 8 and 48. As 8 is a factor of 48,
HCF (8, 48) = 8 and LCM (8, 48) = 48

Relationship Between HCF, LCM and the Product of Two Numbers
The product of LCM and HCF of two numbers is equal to the product of the numbers, i.e.
For any two numbers a and b, LCM (a, b) × HCF (a, b) = a × b

Highest Common Factor (HCF)
The largest positive integer that divides two or more given numbers exactly.

Least Common Multiple (LCM)
The smallest positive integer that is a multiple of two or more given numbers.

Properties of HCF and LCM
The HCF of given numbers is always either smaller than all the numbers or equal to the smallest number.

The LCM of given numbers is always either greater than all the numbers or equal to the greatest number.

HCF of co-prime numbers is 1 and the LCM of co-prime numbers is the product of the co-primes.

The HCF of a set of given numbers is a factor of their LCM. In other words, the LCM of given numbers is a multiple of their HCF.

If two numbers are multiplied by the same number, then both their HCF and LCM are also multiplied by that number.
HCF(ka, kb) = kH and LCM(ka, kb) = kL, where, H and L are HCF (a, b) and LCM (a, b) respectively.

Given any two numbers, if the first number is a factor of the second number, then the first number is their HCF and the second number is their LCM.

The product of LCM and HCF of two numbers is equal to the product of the numbers, i.e. LCM × HCF = Product of two numbers.

Easy-to-read Ganita Prakash Class 7 Notes and Part 2 Chapter 2 Operations with Integers Class 7 Notes save valuable study time during exam season.

Class 7 Maths Chapter 2 Operations with Integers Notes

Class 7 Operations with Integers Notes

Integers are a set of numbers that include natural numbers, their additive inverses (negative integers) and zero.

Two numbers are said to be additive inverses of each other if their sum is zero.
For example, — 3 is the additive inverse of 3 and 5 is the additive inverse of – 5 as 3 + (— 3) = 0 and (— 5) + 5 = 0.

Addition and Substraction of Integers
Sum of two positive integers is always positive.
Sum of two negative integers is always negative.

When two integers with the same sign are added, their magnitudes are added and the sign of the result remains the same as the given integers.

When two integers with different signs are added, the result is obtained by finding the difference of their magnitudes (greater magnitude – smaller magnitude), and the sign of the answer is the sign of the integer with greater magnitude.

“Subtraction” is same as “adding the opposite” or “adding the additive inverse”.

Multiplication of Integers
In a × b = p, numbers a, b and p are called multiplier, multiplicand and product respectively.

The magnitude of the product does not change Multiplier Multiplicand with the change in the signs of both the multiplier and the multiplicand.

When both numbers (multiplier and multiplicand) are positive, the product is positive. When both numbers (multiplier and multiplicand) are negative, the product is positive.

When one of the multiplier or the multiplicand is positive and the other is negative, their product is negative.

Magic Grid of Integers

Each number in the grid is formed by multiplying one row factor and one column factor.

So, each cell contains the product of one row factor and one column factor.

When a number is circled, its row and column are crossed out.
This ensures that no row or column is used more than once.
As a result, each row factor and each column factor is included exactly once in final multiplication.

No matter which valid numbers are chosen—provided only one number is selected from each row and each column—the product remains the same.

Division of Integers
The magnitude of the quotient does not change with the change in the signs of the dividend and the divisor.

When a positive integer is divided by another positive integer, a positive result is obtained.
When a negative integer is divided by a positive integer, a negative result is obtained.

When a positive integer is divided by a negative integer, a negative result is obtained.

When a negative integer is divided by another negative integer, a positive result is obtained.

Properties Related to Multiplication and Division of Integers
Commutative Property of Multiplication (Swapping)
For any two integers, a and b, we can say that a × b = b × a

Associative Property of Multiplication (Grouping)
For any three integers, a, b and c, we can say that (a × b) × c = a × (b × c).

Distributive Property of Multiplication Over Addition
For any three integers, a, b and c, we can say that a × (b + c) = a × b + a × c

The sign of a product is positive if the number of negative integers in the product is even.
For example, (- 3) × (- 2) × (- 1) × (- 4) = 24 [Number of negative integers = 4 (even)]

The sign of a product is negative if the number of negative integers in the product is odd.
For example, (- 5) × (- 1) × (- 3) = – 15 [Number of negative integers = 3 (odd)]

Division of two numbers is not commutative. For example, 4 ÷ 2 ≠ 2 ÷ 4

Division of three numbers is not associative. For example, (16 ÷ 4) ÷ 2 ≠ 16 ÷ (4 ÷ 2)

Identify the Pattern
Imagine a “number machine” or “machine”, which performs a defined set of arithmetic operations on the numbers it receives as inputs.

The operations done by the machine are hidden, and our task is to identify the rule by carefully observing the relationship between the input numbers and the output number.
Inputs → Machine → Outputs

Multiplication of Integers

The magnitude of the product does not change with the change in the signs of the multiplier and the multiplicand.
For example, 3 × 2 = 6; 3 × (- 2) = – 6; – 3 × 2 = – 6; -3 × (-2) = 6

Quick Rules: Multiplication of integers
(+) × (+) = (+); (-) × (-) = (+)
(+) × (-) = (-); (-) × (+) = (-)

Division of Integers

The magnitude of the quotient does not change with the change in the signs of the dividend and the divisor.
For example, 8 ÷ 4 = 2; 8 ÷ (- 4) = – 2;
– 8 ÷ 4 = – 2; – 8 ÷ (- 4) = 2

Quick Rules: Division of integers
(+) ÷ (+) = (+); (-) ÷ (-) = (+)
(+) ÷ (-) = (-); (-) ÷ (+) = (-)

Properties Related to Multiplication and Division
Commutative property of multiplication: a × b = b × a
Associative property of multiplication: (a × b) × c = a × (b × c)
Distributive property of multiplication over addition: a × (b + c) = a × b + a × c

The sign of a product is positive if the number of negative integers in the product is even.
For example, (- 3) × (- 2) × (- 1) × (- 4) = 24 [Number of negative integers = 4 (even)]

The sign of a product is negative if the number of negative integers in the product is odd.
For example, (-5) × (-1) × (-3) = – 15 [Number of negative integers = 3 (odd)]

Division of two numbers is not commutative. For example, 4 ÷ 2 ≠ 2 ÷ 4
Division of three numbers is not associative. For example, (16 ÷ 4) ÷ 2 ≠ 16 ÷ (4 ÷ 2)

Easy-to-read Ganita Prakash Class 7 Notes and Part 2 Chapter 1 Geometric Twins Class 7 Notes save valuable study time during exam season.

Class 7 Maths Chapter 1 Geometric Twins Notes

Class 7 Geometric Twins Notes

Two figures or objects that have same shape and same size, and can be superimposed on each other by sliding, turning or flipping, are called geometric twins or congruent figures.

Two line segments are said to be congruent if they have the same length.

Two angles are said to be congruent if they have the same measure.

Two polygons are congruent if they are of the same size and shape, i.e. if their corresponding angles and sides are equal.

Two triangles are congruent if their corresponding sides are equal in length and their corresponding angles are equal in measure.

Criteria for Congruence of Triangles
There are five criteria for congruence of triangles as given below:

SSS (Side – Side – Side) congruence criterion
If the three sides of one triangle are equal to corresponding three sides of another triangle, then the triangles are congruent.

In general, for any two triangles, ∆ABC and ∆PQR, if B AB = PQ, BC = QR and AC = PR, then according to SSS congruence criterion, ∆ABC ≅ ∆PQR.

SAS (Side – Angle – Side) congruence criterion

If two sides and the included angle of one triangle are equal to the corresponding two sides and the included angle of another triangle, then the triangles are congruent.

In general, for any two triangles ∆ABC and ∆PQR, if AB = PQ, ∠A = ∠P and AC = PR, then according to SAS congruence criterion, ∆ABC ≅ ∆PQR.

ASA (Angle – Side – Angle) congruence criterion

If two angles and the included side of one triangle are equal to the corresponding two angles and included side of another triangley. then the triangles are congruent.

In general, for any two triangles, ∆ABC and ∆PQR, if ∠B = ∠Q BC = QR and ∠C = ∠R, then according to ASA congruence criterion, ∆ABC ≅ ∆PQR.

AAS (Angle – Angle – Side) congruence criterion

If two angles and the non-included side of one triangle are equal to the corresponding two angles and the non-included side of another triangle, then the triangles are congruent.

In general, for any two triangles, ∆ABC and ∆PQR, if ∠A = ∠P, ∠B = ∠Q and BC = QR. then according to AAS congruence criterion,
∆ABC ≅ ∆PQR.

RHS (Right Angle – Hypotenuse – Side) congruence criterion

If the hypotenuse and one side of one right-angled triangle are equal to the hypotenuse and corresponding side of another right-angled triangle, then the triangles are congruent.

In general, if ∆ABC and ∆PQR are two right triangles g such that ∠B = ∠Q= 90°, AC = PR and BC = QR, then according to RHS congruence criterion, ∆ABC ≅ ∆PQR.

Angles of Isosceles and Equilateral Triangles
Angles opposite to equal sides of an isosceles triangle are equal. Conversely, sides opposite to equal angles of a triangle are equal.

An equilateral triangle has all sides equal and all the angles in an equilateral triangle are equal, i.e. 60°.

Important Points

1. The sum of all interior angles of a triangle is 180°, and that of a quadrilateral (polygon with 4 sides) is 360°.
2. Two congruent figures have the same area and same perimeter.
3. The angle that lies between two given sides of a triangle is called the included angle.
4. An angle of a triangle that does not lie between the two given sides is called a non-included angle.
5. The side of a triangle that lies between two given angles is called the included side.
6. A non-included side of a triangle is the side that does not lie between the two given angles.
7. AAA (Angle – Angle – Angle) and SSA (Side – Side – Angle) are not congruence criteria.

Concept of congruence

1. Two figures or objects that have same shape and same size and can be superimposed on each other by sliding, turning or flipping are called congruent figures.
2. Two line segments are said to be congruent if they have the same length.
3. Two angles are said to be congruent if they hat e the same measure.
4. Two squares are said to be congruent, if they have same length of the sides.
5. Two rectangles are said to be congruent if they have same length and breadth.
6. Two circles are said to be congruent if they have the same radius.

Congruency of Triangles
SSS Congruence Rule

If measure of all three sides of triangle ABC are equal to the measure of three corresponding sides of other triangle DEE then ∆ABC ≅ ∆DEF, by SSS (Side-Side-Side) congruence rule.

SAS Congruence Rule

If any two sides and the included angle of triangle ABC are equal to the corresponding two sides and the included angle in the another triangle DEF, then ∆ABC ≅ ∆DEF, bv SAS (Side-Angle-Side) congruence rule.

RHS Congruence Rule

In right-angled ∆ABC, if the hypotenuse and any one side of triangle are equal to the hypotenuse and corresponding side of another right angled ∆DEF, then ∆ABC ≅ ∆DEF, by RHS (Right angle-Hypotenuse- Side) congruence rule.

ASA Congruence Rule

If any two angles and the included side of triangle ABC are equal to the corresponding two angles and included side of the another triangle DEF, then ∆ABC ≅ ∆DEF, by ASA (Angle-Side-Angle) congruence rule.

AAS Congruence Rule

If any two angles and ,non- included side of triangle ABC are equal to the corresponding two angles and non-included side of the another triangle DEF, then ∆ABC ≅ ∆DEF, by AAS (Angle- Angle-Side) congruence rule.

Easy-to-read Ganita Prakash Class 7 Notes and Chapter 8 Working with Fractions Class 7 Notes save valuable study time during exam season.

Class 7 Maths Chapter 8 Working with Fractions Notes

Class 7 Working with Fractions Notes

Types of Fractions

Type of Fraction	What it means	Example
Proper Fraction	Numerator < Denominator {Numerator less than Denominator}	\(\frac{1}{3}, \frac{4}{7}, \frac{2}{5},\) etc.
Improper Fraction	Numerator ≥ Denominator {Numerator equal to or greater than Denominator}	\(\frac{1}{1}, \frac{3}{2}, \frac{7}{4},\) etc.
Mixed Fraction	A whole number + a proper fraction	\(1 \frac{1}{4}, 3 \frac{2}{5}, 5 \frac{3}{7},\) etc.
Equivalent Fractions	Look different but have the same value	\(\frac{1}{2}=\frac{2}{4}=\frac{3}{6}\) etc.

To multiply a whole number by a fraction, we simply multiply the numerator of the fraction by the whole number, keeping the denominator same.
For example, \(3 \times \frac{1}{2}=\frac{3 \times 1}{2}=\frac{3}{2}\)

Note: Fraction \(\frac{a}{b}\) is said to be in its lowest form, if a and b have no common factor other than 1.

Multiplication can be understood as repeated addition. It tells us how many times one number is added to itself.
For example, 3 × 4 = 12 because 4 + 4 + 4 = 12 (3 times).
This idea works with fractions too: \(3 \times \frac{1}{4}=\frac{3}{4}\) because \(\frac{1}{4}+\frac{1}{4}+\frac{1}{4}=\frac{3}{4}\).

Multiplication is the number that is to be multiplied by another number (called the multiplier).
For example, in \(8 \times \frac{3}{5}=\frac{24}{5},\) 8 is multiplier while \(\frac{3}{5}\) is multiplicand.

Sometimes ‘of’ means ‘multiplication’. For example, \(\frac{3}{4}\) of 20 = \(\frac{3}{4} \times 20\) = 15.

Unit fractions (like \(\frac{3}{n}\)) get smaller as the number in the bottom (denominator) gets bigger.
For example, \(\frac{1}{2}>\frac{1}{5}>\frac{1}{10}\)

Product ot two fractions = \(\frac{\text { Product of their numerators }}{\text { Product of their denominators }} \text {. For example, } \frac{2}{3} \times \frac{5}{11}=\frac{2 \times 5}{3 \times 11}=\frac{10}{33}\)

(i) When two proper fractions are multiplied, their product is less than each of the fractions.
For example, the product of \(\frac{1}{2} \text { and } \frac{3}{4} \text { is } \frac{3}{8} \text { and } \frac{3}{8}\) is less than both \(\frac{1}{2} \text { and } \frac{3}{4}\).

(ii) The product of a proper fraction and an improper fraction is greater than or equal to the proper fraction and less than the improper fraction.
For example, the product of \(\frac{5}{2} \text { and } \frac{1}{4} \text { is } \frac{5}{8} \text { and } \frac{1}{4}<\frac{5}{8}<\frac{5}{2}\).

(iii) When two improper fractions are multiplied, their product is greater than both the fractions or equal to either of them.
For example, the product of \(\frac{5}{2} \text { and } \frac{5}{4} \text { is } \frac{25}{8} \text { and } \frac{25}{8}\) is greater than both and \(\frac{5}{2} \text { and } \frac{5}{4}\).

The reciprocal of a fraction can be obtained by interchanging the numerator and denominator.
So, the reciprocal of a fraction \(\frac{a}{b} \text { is } \frac{b}{a}\).

Relation between the Area of a Rectangle and Multiplication of Fractions

In the given figure, a square of area 1 square unit is divided into 15 equal rectangles. So, the area of each small rectangle = \(\frac{1}{15}\) square units
Number of shaded rectangles = 4
Therefore, total shaded area = \(4 \times \frac{1}{15}=\frac{4}{15}\) square units

Now, let’s verify this mathematically:
Length of shaded region = \(\frac{4}{5}\); Breadth of shaded region = \(\frac{1}{3}\)
Area of shaded region = Length × Breadth = \(\frac{4}{5} \times \frac{1}{3}=\frac{4 \times 1}{5 \times 3}=\frac{4}{15}\)

In general, if we want to find the product of two fractions, we can find the area of the rectangle formed with the two fractions as its sides.

The area of the rectangle remains the same even if the length and the breadth are interchanged,
i. e. the order of multiplication does not matter. Thus,
\(\frac{a}{b} \times \frac{c}{d}=\frac{c}{d} \times \frac{a}{b}\)

Division of Fractions

Division of a Whole Number by a Fraction	Rule: \(a \div \frac{b}{c}=a \times \frac{c}{b}\)	For Example, \(5 \div \frac{5}{4}=5 \times \frac{4}{5}=4\)
Division of a Fraction by Another Fraction	Rule: \(\frac{a}{b} \div \frac{c}{d}=\frac{a}{b} \times \frac{d}{c}\)	For Example, \(\frac{3}{7} \div \frac{4}{7}=\frac{3}{7} \times \frac{7}{4}=\frac{3}{4}\)
Division of a Fraction by a Non-zero Whole Number	Rule: \(\frac{a}{b} \div c=\frac{a}{b} \times \frac{1}{c}\)	For Example, \(\frac{6}{13} \div 6=\frac{6}{13} \times \frac{1}{6}=\frac{1}{13}\)

Note:

1. When the divisor is greater than 1, the quotient is smaller than the dividend. For example, 15 ÷ 3 = 5. Here, 5 < 15 as 3 > 1.
2. When the divisor is equal to 1, the quotient is the same as the dividend. For example, 8 ÷ 1 = 8.
3. When the divisor is smaller than 1, the quotient is greater than the dividend. For example, 8 ÷ \(\frac{1}{2}\) = 16. Here, 16 > 8 as \(\frac{1}{2}\) < 1.

Important Points to Remember

1. Smaller divisor → Bigger quotient
2. Bigger divisor → Smaller quotient
3. When the whole number 0 is divided by a fraction, we get 0 as the quotient.

Introduction
Proper Fraction: Numerator < Denominator. For example, \(\frac{2}{7}, \frac{5}{8},\) etc. Value of proper fraction is always less than 1. Improper Fraction: Numerator > Denominator.
For example, \(1, \frac{8}{5}, \frac{13}{5},\) etc.
Value of an improper fraction is always greater than or equal to 1.

Mixed Fraction: A whole number + A proper fraction.
For example, \(1 \frac{1}{4}, 3 \frac{1}{2},\) etc.

Equivalent Fractions: Look different but have the same value.
For example, \(\frac{1}{2}=\frac{2}{4}=\frac{3}{6}\)

Multiplication of Fractions
Product of two fractions = \(\frac{\text { Product of their numerators }}{\text { Product of their denominators }} \text {. For example, } \frac{3}{5} \times \frac{5}{9}=\frac{5 \times 3}{5 \times 9}\)
Based on above formula we can conclude the following result:

Condition	Description of Product	Example	Comparison
Two Proper Fractions	The product is less than each of the fractions.	\(\frac{1}{3} \times \frac{5}{7}=\frac{5}{21}\)	\(\frac{5}{21}<\frac{1}{3} \text { and } \frac{5}{21}<\frac{5}{7}\)
Proper x Improper	The product is greater than or equal to the proper fraction but less than the improper fraction.	\(\frac{1}{3} \times \frac{7}{3}=\frac{7}{9}\)	\(\frac{1}{3}<\frac{7}{9}<\frac{7}{3}\)
Two Improper Fractions	The product is greater than both the fractions or equal to either of them.	\(\frac{7}{3} \times \frac{8}{5}=\frac{56}{15}\)	\(\frac{56}{15}>\frac{7}{3} \text { and } \frac{56}{15}>\frac{8}{5}\)

 

Division of Fractions
The reciprocal of a fraction can be obtained by interchanging the numerator and denominator.
So, the reciprocal of a fraction \(\frac{a}{b} \text { is } \frac{b}{a}\)

Whole Number ÷ Fraction	\(a \div \frac{b}{c}=a \times \frac{c}{b}\) For Example: \(5 \div \frac{3}{4}=5 \times \frac{4}{3}=\frac{20}{3}\)
Fraction ÷ Fraction	\(\frac{a}{b} \div \frac{c}{d}=\frac{a}{b} \times \frac{d}{c}\) For Example: \(\frac{3}{7} \div \frac{1}{5}=\frac{3}{7} \times \frac{5}{1}=\frac{15}{7}\)
Fraction ÷ Non-zero Whole Number	\(\frac{a}{b} \div c=\frac{a}{b} \times \frac{1}{c}\) For Example: \(\frac{1}{8} \div 3=\frac{1}{8} \times \frac{1}{3}=\frac{1}{24}\)

Go through BSE Odisha Class 7 Science Solutions and Class 7 Science Chapter 4 The World of Metals and Non-metals Question Answer to understand textbook questions more clearly.

Class 7 Science Curiosity Chapter 4 Question Answer

Science Curiosity Class 7 Chapter 4 Question Answer

The World of Metals and Non-metals Class 7 Questions and Answers

Let Us Enhance Our Learning

Question 1.
Which metal is commonly used to make food packaging materials as it is cheaper, and its thin sheets can be folded easily into any shape?
(a) Aluminium
(b) Copper
(c) Iron
(d) Gold
Answer:
(a) Aluminium is commonly used in food packaging materials as it is malleable, cheaper, and easily available.

Question 2.
Which of the following metals catches fire when it comes in contact with water?
(a) Copper
(b) Aluminium
(c) Zinc
(d) Sodium
Answer:
(d) Sodium, being highly reactive, catches fire when it comes in contact with water.

Question 3.
State with reason(s) whether the following statements are True [T] or False [F].
(i) Aluminium and copper are examples of nonmetals used for making utensils and statues.
Answer:
False Aluminium and copper are examples of metals.

(ii) Metals form oxides when combined with oxygen, the solution of which turns blue litmus paper to red.
Answer:
False Metals form basic oxides when combined with oxygen, and thus, the solution turns red litmus paper blue.

(iii) Oxygen is a non-metal essential for respiration.
Answer:
True

(iv) Copper vessels are used for boiling water because they are good conductors of electricity.
Answer:
False Copper vessels are used for boiling water because they are good conductors of heat.

Question 4.
Why are only a few metals suitable for making jewellery?
Answer:
To make jewellery, the material used must be lustrous, malleable, ductile, and resistant to corrosion. Generally, metals are lustrous, malleable, ductile, but not all metals are resistant to corrosion. Metals like gold and silver are the most malleable as well as resistant to corrosion and are therefore suitable for making jewellery.

Question 5.
Match the uses of metals and non-metals given in Column I with the jumbled names of metals and non-metals given in Column II.

Answer:

Column I	Column II
(i) Used in electrical wiring	(C) COPPER
(ii) Most malleable and ductile	(E) GOLD
(iii) Living organisms cannot survive without it.	(A) OXYGEN
(iv) Plants grow healthy when fertilisers containing it are added to the soil.	(D) NITROGEN
(v) Used in water purification	(B) CHLORINE

Question 6.
What happens when oxygen reacts with magnesium and sulphur? What are the main differences in the nature of products formed?
Answer:
When oxygen reacts with magnesium (metal), a white ash of magnesium oxide is formed. When oxygen reacts with sulphur, sulphur dioxide gas is released.
Main difference: It is known that metals form basic oxides, while non-metals generally form acidic oxides. Hence, in the present case, the magnesium oxide is basic in nature, while sulphur dioxide is acidic in nature. This can also be confirmed via the litmus test. Magnesium oxide turns moist red litmus paper blue, while sulphur dioxide turns moist blue litmus paper red.

Question 7.
Complete the following flow chart:

Answer:
The answers from top to bottom in sequence are as follows:

• Magnesium
• Magnesium hydroxide (or solution of magnesium oxide)
• No change

Question 8.
You are provided with the following materials. Discuss which material would be your choice to make a pan that is most suitable for boiling water and why?

Answer:
Since metals are good conductors of heat, they are used to make utensils like a pan for boiling water or other cooking purposes. From the given materials, only iron and copper are metals. Here, copper is most suitable for making a pan because it does not react easily with water, and is a better conductor of heat.

Question 9.
You are provided with three iron nails, each dipped in oil, water and vinegar. Which iron nail will not rust, and why?
Answer:
The iron nail dipped in oil will not rust, as oil acts as a barrier that prevents the iron nail from coming in contact with air and water.

Question 10.
How do the different properties of metals and non-metals determine their uses in everyday life?
Answer:
Metals like copper and aluminium can be used to make electrical wires as these are ductile and good conductors of heat and electricity. Aluminium, being malleable, can be used to make foils to wrap food items. Gold and silver, being lustrous, are used to make jewellery.

Iron is strong and hard, and thus, used in construction to make buildings and tools like hammers, knives, and spades. Non-metals are also essential for life. For example, chlorine is used in water purification, nitrogen is used in the preparation of fertilisers, and oxygen is used for breathing.

Question 11.
One of the methods of protecting iron from getting rusted is to put a thin coating of zinc metal over it. Since sulphur does not react with water, can it be used for this purpose? Justify your answer.
Answer:
No, sulphur cannot be used to coat iron to prevent it from rusting. This is because sulphur reacts with oxygen to form sulphur dioxide gas which can easily leave the surface of iron and hence, cannot form a protective layer like zinc. Zinc forms a protective layer that prevents iron from coming in contact with air and moisture.

Question 12.
An ironsmith heats iron before making tools. Why is heating necessary in this process?
Answer:
It is important to heat the iron to make tools because heating makes the iron soft and easier to shape. Once it cools down, it becomes strong and hard again.

Class 7 Curiosity Chapter 4 Question Answer

InText Questions

Question 1.
Are all metals hard and solid (at room temperature)? Explain. [Page 43]
Answer:
No, some metals like sodium and potassium are so soft that they can be cut with a knife. Mercury is a metal that exists in its liquid state at room temperature.

Question 2.
Where do you find the use of metal wires? [Page 44]
Answer:
Metal wires are used in electrical fittings, fencing, ornaments like necklaces, and variety of stringed musical instruments such as veena, sitar, violin, and guitar.

Question 3.
Why can wires not be made up of coal or sulphur? [Page 45]
Answer:
Since ductility is mainly possessed by metals, while coal and sulphur are non-metals, they cannot be used to make wires.

Question 4.
How is the sound produced by a metallic object dropped on the floor different from the sound produced when a piece of coal or wood is dropped on the floor? [Page 45]
Answer:
A metallic object dropped on the floor produces a ringing sound, which is one of the characteristic properties of metals like sonority. Coal or wood produces a dull sound when dropped on the floor as they are non-sonorous.

Question 5.
What type of material is used for making the handle of a screwdriver, and why? [Page 47]
Answer:
A material that is a poor conductor of electricity is used to make the handle of a screwdriver to protect the electrician from electric shock.

Question 6.
When magnesium ribbon is burned in air, what can we observe? What effect does the aqueous solution of its product have on blue and red litmus papers? [Page 51]
Answer:
Magnesium ribbon burns in air with a dazzling white flame, and a white powder (magnesium oxide) is produced. The aqueous solution of it has no effect on the blue litmus paper, but it changes the colour of red litmus paper to blue.

Question 7.
Does sulphur behave in water the same way metals do? [Page 53]
Answer:
No, sulphur behaves differently from metals, as no reaction takes place when sulphur is added to water.

Easy-to-read Ganita Prakash Class 7 Notes and Chapter 7 A Tale of Three Intersecting Lines Class 7 Notes save valuable study time during exam season.

Class 7 Maths Chapter 7 A Tale of Three Intersecting Lines Notes

Class 7 A Tale of Three Intersecting Lines Notes

A triangle is a closed plane figure formed by three non-parallel line segments. The symbol “A” is used to denote a triangle (e.g., ∆ABC).

There are three types of triangles on the basis of the number of equal-length sides.

Equilateral Triangle	Isosceles Triangle	Scalene Triangle
All three sides are equal in length.	Any two sides of a triangle are equal in length.	Three sides of a triangle are different in length.
All three angles are equal, each measuring 60°.	Angles opposite to equal sides are equal in measure.	All three angles are of different measures and never equal.

In a triangle, an altitude is a line segment drawn from a vertex perpendicular to its opposite side. In the given figure, AD, BE and CF are the altitudes of the ∆ABC.

Each triangle has 3 altitudes.

Triangle Inequality Theorem:
The sum of any two sides of a triangle is greater than the third side.
In the given triangle ABC, we have b + c > a, c + a > b and a + b > c.

Angle Sum Property: The sum of the three interior angles of any triangle is always 180°.
i.e., ∠A + ∠B + ∠C = 180°

Exterior Angle Property:
If a side of a triangle is extended, the exterior angle formed is equal to the sum of the two interior opposite angles.
i.e., ∠a + ∠b = ∠f, ∠b + ∠c = ∠d and ∠a + ∠c = ∠e
The sum of exterior angles of a triangle is 360°.
i.e., ∠d + ∠e + ∠f = 360°

There are three types of triangles on the basis of measure of their interior angles.

Acute Angled Triangle	All three angles are less than 90°
Right Angled Triangle	One angle is 90° and other two are less than 90°.
Obtuse Angled Triangle	One angle is more than 90° and other two are less than 90°.

To construct a triangle, we need at least three elements. These can be:
Three sides, or
Two sides and the included angle, or
Two angles and the included side

Construction of triangles when three sides are given
To construct a triangle ABC with sides AB = 6 cm, BC = 7 cm andTC = 8 cm, follow the steps laid down below:

Step 1: Draw a line segment AB of 6 cm using a ruler.
Step 2: With B as centre, draw7 an arc with radius 7 cm.
Step 3: With A as centre, draw an arc with radius 8 cm to cut the previous arc at point C.
Step 4: Join C to A and B. Thus, AABC is required triangle.

Construction of triangles when two sides and the included angle are given
If two sides (say AB = 5 cm and AC = 7 cm) along with the included angle (say ∠A then we follow the steps for constructing triangle as follows:

Step 1: Using a ruler, drawn a line segment AB of length 5 cm.
Step 2: Using protractor, draw the given vertex angle, ∠A = 50°.
Step 3: On the new arm created from ∠A, mark a point C such that AC = 7 cm using ruler and compass.
Step 4: Using a ruler, connect points B and C to complete the triangle ABC.

Construction of triangles when two angles and the included side are given
If two angles (say ∠A = 35° and ∠B = 60°) along with the included side (say AB = 6 cm) are given, then the steps for constructing triangle are as follows:

Step 1: Using a ruler, draw a line segment.-Id of length 6 cm.
Step 2: Using protractor, draw the given vertex angle, ∠A = 35°, by placing protractor at vertex A.
Step 3: Using protractor, draw the given vertex angle, ∠B = 60°, by accurately placing the protractor at vertex B.
Step 4: If the two new lines (arms of angles) are intersecting, then mark the intersection point as C. If not, then extend the two new lines such that they intersect at a point. Mark that point as C.
Thus, the required triangle, ∆ABC is constructed.

Triangle
A triangle is a dosed plane figuie ioimed by three non-parallel line segments. T he symbol “A” is used to denote a triangle (e.g., ∆ABC).

Altitude: In a triangle, an altitude is a line segment drawn from a vertex perpendicular to its opposite side. In the given figure, AD,
BE and CF are the altitudes of the ∆ABC.
NOTE:
Each triangle has 3 altitudes.
Two perpendicular sides in a right angled triangle are its altitudes.

Based on Angles
Acute Angled Triangle
All three angles are less than 90°.

Right Angled Triangle
One angle is 90° and other two are less than 90°.

Obtuse Angled Triangle
One angle is more than 90° and other two are less than 90°.

Based on Sides
Equilateral Triangle
All three sides are equal in length.
All three angles are equal, each measuring 60°.

Isosceles Triangle
Any two sides of a triangle are equal in length.
Angles opposite to equal sides are equal in measure.

Scalene Triangle
Three sides of a triangle are different in length.