ch-2 Notes on Components of Food | class 6th science

Nutrients: A nutrient can be defined as components that are needed by our body to grow, survive and carry on different daily activities.

Components of Food

Our food contains mainly five major kinds of nutrients namely vitamins, minerals, carbohydrates, proteins and fats. Additionally, food also contains water and dietary fibres/roughage and water which are also required by our bodies.

Major nutrients of food

Fig. 1: Major nutrients of food

Carbohydrates: Carbohydrates main function is providing energy to the body. These are found in our food in the form of sugar and starch i.e. simple and complex carbohydrates. Example It is found in bread, potatoes etc.

Simple Carbohydrates: These are also referred to as simple sugars, containing single monosaccharide units and found in natural sources of food i.e. milk, fruits and vegetables. These carbohydrates add certain sweetness to the food. They raise the level of blood glucose quickly but are easier to break down.

Complex Carbohydrates: These are also referred to as polysaccharides, meaning they contain hundreds or thousands of such monosaccharide units. These are typically found in wheat grain, white bread, kernel and cakes. They are relatively less sweet than simple carbohydrates and also raise blood glucose level rather slowly. However, these are tougher to break down. Cellulose is present in plant cell wall. It is a complex carbohydrate. Humans cannot digest cellulose.

Test for carbohydrates: We can test whether a particular food item contains carbohydrates by pouring 2 to 3 drops of dilute iodine solution on it. If the iodine changes its colour to blueblack, then we can ascertain that the food item does, in fact, contain carbohydrates.

The Two Types of Carbohydrates

Fig.2: The Two Types of Carbohydrates

Proteins: Proteins performs the very essential function of helping our body grow and repair itself. These are found in food items such as milk, pulses, eggs, meat etc. Foods containing proteins are called ‘body-building’ foods.

Test for proteins: To test whether a food item contains proteins, first we need to grind it into a paste or powder form and add 10 drops of water. To this mixture when we add 2 drops of copper sulphate solution and 10 drops of caustic soda solution. After a few minutes, if the mixture turns violet, it is indicative of the presence of protein.

Fats: Fats are also responsible for providing energy to our body. In fact, they provide more energy than carbohydrates. The body uses fat as a fuel source. Fats are essential for the absorption of vitamins A, D, E and K in the body.  Butter, cheese, oil are all examples of fat-rich foods.

Test for fats: To test for fats, take a small quantity of food item and wrap a piece of paper around it and proceed to crush it. After removing the paper, allow it to dry. If you see a film of an oily patch on the paper when holding it against light, it is proof that the food contains fats.

Vitamins: Vitamins help in protecting our bodies from various kinds of diseases. They also help in keeping our eyes, gums, bones and teeth in good shape. Different types of vitamins and their uses:

Vitamin TypeSourcesFunctionsDeficiency Diseases
Vitamin ALeafy green vegetables, oranges, carrots, Pumpkin, Soy, Sweet potatoesForms and helps maintain bones, skin, tissue and teethColor blindness, night blindness- poor visibility at night.
Vitamin B1 (thiamine)Dried herbs, sunflower seeds, whole grain cereals, sesame seeds, brown riceEnables cells to turn carbohydrates into energyBeriberi-  loss of appetite, loss of weight.
Vitamin B2 (riboflavin)Almonds, Asparagus, bananas, green beans, wheat bran, dried spicesMaintains Body growth and RBCs i.e. Red Blood CellsSkin disorders, Cheilosis-breaking of lips
Vitamin B12 (cyanocobalamin)Mutton, fish, beef, lobster, clams, eggs, oysters, crabHelps in maintenance of central nervous system and RBCsPale skin, lack of RBC, Less stamina and less appetite.
Vitamin CFresh herbs, cauliflower, papaya, oranges, strawberries, guavaPromotes healthy gums and teethScurvy i.e. gum disease (gingivitis).
Vitamin DSunshine, Mushrooms, liver, fish and eggsNecessary for the healthy development of bones and teethRickets and Osteomalacia – weakening and softening of bones.
Vitamin ESoyabean oil, red chilli powder, pine nuts, apricots, green olives and cooked spinachHelps in processing vitamin K and formation of RBCs Muscle weakness and transmission problems in nerve impulses
Vitamin KGreen leafy vegetables, Soyabean oil. Essential for blood coagulationExcessive bleeding from wound. 

Minerals: Minerals are used by the body to perform various functions like building strong bones, maintaining the heartbeat, making hormones etc. The major five minerals are Calcium, Phosphorus, Magnesium, Sodium and Potassium. Examples of mineral-rich foods include leafy vegetables, fish, beans etc.

Mineral TypeSourcesFunctionsDeficiency Disease
CalciumTofu, Dairy products, Salmon, Cabbage, Kale and BroccoliEssential for efficient functioning of nervous system and healthy bonesWeak bones,  lower than normal bone density and stunted growth
PhosphorousLean meats, grain and milkEssential for the maintenance of acid-base balance in bodyLoss of appetite, bone fragility, muscle weakness, poor physique
IodineGreen leafy vegetables, Seafood, iodised saltFormation of thyroid hormoneGoitre- Enlargement of thyroid gland, mental disability.
SodiumTable salt, celeryHelps keep control on blood pressureNausea, irritability
IronWhole grain, eggs, leafy vegetables and meatsEssential for haemoglobin formation in rbc.Anaemia – weakness, fatigue, shortness of breath
The five major minerals and their source foods

Fig.3: The five major minerals and their source foods

Dietary Fibres/Roughage: While dietary fibres do not provide any such nutrition to our bodies but nevertheless are an important component of food. They help in easy absorption of food, helps in movement of bowel and prevents constipation. It helps our body get rid of undigested food. Cereals, fruits and vegetables are some of the roughage rich foods.

Water: Water performs the essential function of absorbing nutrients from our food. It also helps in releasing waste from our body in form of sweat and urine.

Balanced Diet

A balanced diet is one that contains a variety of food items providing different types of nutrients in adequate amounts necessary for maintaining good health. The diet should contain a good amount of dietary fibre and water as well.

A balanced diet includes a combination of protein-rich pulses, sprouted seeds etc. with combinations of various flours and cereals for carbohydrates and fats along with fruits and vegetables which provide the necessary vitamins and minerals.

  • In addition to making sure that the right amount of food is eaten, it should also be ensured that food is properly cooked so that it does not end up losing its nutrients.
  • Repeated washing of fruits, pulses, rice and vegetables can result in the loss of essential vitamins and minerals.
  • Throwing away excess water which is used for cooking vegetables can result in the loss of considerable amounts of important proteins and minerals present in them.
  • It’s a well-known fact that vitamin C gets destroyed in the heat while cooking.

Obesity: Obesity is a medical condition that results from excess intake of fat-rich foods. The excess fat gets accumulated to such an extent that it starts negatively affecting one’s health, well-being and the ability to carry out certain activities.

Deficiency Diseases

Deficiency: Sometimes simply getting adequate amounts of food might not be enough if the food does not contain the required nutrients in the right amounts. Prolonged usage of such nutrient-less food may result in a condition known as Deficiency.

Deficiency Diseases: Diseases that occur from the lack of an element in the diet, usually a particular vitamin or mineral are known as deficiency diseases.

  • A diet lacking proteins may result in skin diseases, stunted growth, diarrhoea, swelling of face and discolouration of hair.
  • A diet deficient in both carbohydrates and protein may hinder the growth completely and the person becomes so frail and lean that he or she might not even be able to move.
  • Deficiency of certain vitamins and minerals can cause diseases like scurvy, goitre, anaemia etc. as mentioned in the tables above.

Hence one must always make sure to include all nutrients in their food and drink a good amount of water to maintain a healthy body that is free from diseases.

Malnutrition

Improper intake of nutrient-rich food may lead to another condition called malnutrition or undernutrition. Without adequate intake of food, our bones become brittle, our muscles weaken and our thinking becomes foggy. In such situations, our bodies are said to become malnourished.

When there is lack of protein in our bodies, repair of wounds and an injury becomes difficult. When this is combined with inadequate intake of calories, it leads to a condition known as Protein-energy Undernutrition or malnutrition.

There are two ways in which protein-energy malnutrition manifests itself:

Two types of protein-energy malnutrition diseases

Fig.4: Two types of protein-energy malnutrition diseases

Kwashiorkor: People with severe protein deficiency are often at high risk of developing this disease. People in rural areas are more likely to suffer from this disease, as there is lack of protein rich food. If one often indulges in diets that are high on carbohydrates and low on protein it can lead to them showing symptoms of kwashiorkor i.e. Edema (puffy appearance due to retention of fluid), inability to gain weight and bulging of abdomen.

Marasmus: Children and young adults are more likely to develop this disease. If there is inadequate intake of both energy and protein, it tends to manifest as marasmus. Chronic diarrhoea, weight loss, dehydration and stomach shrinkage are the major symptoms of marasmus.

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ch-1 Notes on Food: Where Does It Come From | class 6th science

Food Materials and Sources

Food Materials and Sources

Plants act as sources of food ingredients such as fruits, vegetables, grains, pulses etc.

Animals are sources of food ingredients such as milk, eggs, meat products etc.

These are shown in the figure above.

Sources of food

Some examples of edible plant parts are shown in the image above and their examples are given below:

  • Roots: beets, carrots, radishes, turnips, ginger,
  • Stems: broccoli stem, bamboo shoots, sugar cane, potato
  • Leaves: spinach, lettuce etc.
  • Fruits: apple, pear, tomatoes, grapes, cherries, oranges
  • Edible Flowers: broccoli heads, cauliflower heads
  • Seeds: sunflower seeds

What do Animals Eat?

Animals Eat

Animals can be classified into three broad categories in terms of what they eat as can be seen in the image above,

First we have the herbivores i.e. plant eaters. They only consume plant parts. Examples: Cows, goats, deer, giraffe etc.

Next we have the carnivores i.e. meat eaters. They only consume meat of other animals. Examples: Lions, tigers, vultures etc.

Last we have omnivores i.e. animals who eat both plant parts and meat products. Examples: Humans, bears etc.

There are five major components of food namely vitamins, minerals, proteins, carbohydrates and fats. While the diet of carnivores is rich in fats and proteins, there are some necessary vitamins and minerals in plant-based foods that their diet lacks. Similarly an all plant-based diet lacks in a good amount of protein and certain minerals.

It is important to know that there is difference between vegetarians and herbivores. While vegetarians make a conscious decision to not eat meat, herbivores on the other hand are incapable of eating meat and dairy products. Similarly carnivores lack the necessary enzymes in their stomachs to digest cellulose which is a major component of green food like grass. 

Important Definitions

Nectar: The juicy sweet liquid secreted by within flowers which is sucked by bees and is made into honey by them

Sprouting: The process by which seeds shoot out small white structures as way of growth is called Sprouting.

Cellulose: It is a substance that is found in the cell walls of a number of plants. It is an indigestible fibre and is found in grass.

Enzymes: It is defined as a chemical substance that helps in bringing about changes to certain other substances without undergoing any changes in themselves.

 A bowl of sprouts

A bowl of sprouts

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ch-14 Practical Geometry notes | class 6th mathematics

Geometry tells us about the lines, angles, shapes and practical geometry is about the construction of these shapes.

Geometry

Tools to be used for construction

1. The Ruler

A ruler is a straight edge which we sometimes call a scale. It is marked with the centimetres on one side and inches on the other side. It is used to draw line segments and also to measure them.

The Ruler

2. The Compass

The compass has two ends –one is a pointer and the other is for the pencil. It is used to draw arcs and circles and also to measure the line segment.

The Compass

3. The Divider

It is a pair of pointers on both the ends. It is used to compare the length of line and arcs.

The Divider

4. Set -squares

It is a set of two triangular pieces.

  • One is having 45°, 45° and 90° angles at the corners. If we join two same triangular pieces, we will get a square. It is used to draw parallel and perpendicular lines.
Set -squares
  • Other is having 30°, 60° and 90° angles at the corners. If we join two same triangular pieces, we will get a rectangle. It is also used to draw parallel and perpendicular lines.
Set -squares

5. The Protractor

It is a semi-circular device which is divided into 180-degree parts. It starts from the 0° from the right-hand side and ends with the 180° on the left-hand side and vice versa. It is used to measure and draw angles.

The Protractor

The Circle

It is a round shape in which all the points of its boundary are at equal distance from its centre.

The Circle

Construction of a circle if the radius is known

Draw a circle of radius 5 cm.

To draw a circle, we need a compass and a ruler to measure the length.

Step 1: Open the compass and measure the length of 5 cm using a ruler.

Step 2: Mark a point O, which we will use as the centre of the circle.

Step 3: Put the pointer on the point O.

Step 4: Turn the compass to make a complete circle. Remember to do it in one instance. 

Construction of a circle

A Line Segment

A line segment is a part of a line which has two endpoints. It has a fixed length so it can be measured using a ruler.

1. Construction of line segment if the length is known

Draw a line segment of 5 cm.

To draw a line segment of a particular length, we need a compass and a ruler.

Step 1: First of all draw a line l of any length and mark a point named A on it.

Step 2: Measure the length of 5 cm by putting the compass on the ruler. Put the pointer on 0 and open the compass till 5 cm on the ruler.

Step 3: Put the pointer at point A on the line and draw an arc of 5 cm which intersects the line at B.

Step 4: Hence the line segment of 5 cm is

Construction of line segment if the length

2. Construction of a copy of a given line segment

To make a copy of a line segment, there are three methods-

1. Measure the length of the given line segment using a ruler and draw the line segment of the same length with the ruler only.

2. Take a transparent sheet and trace the given line on another part of the paper.

3. These methods are not very accurate so we can use a ruler and compass to make an accurate line segment.

Draw a copy of the line segment

Step 1: is the given line segment whose length is known.

 line segment AB

Step 2: Use the compass and put the pointer on A. open the compass to place the pencil side on point B. Now the open compass is of the same length as given

compass

Step 3: Draw a new line l and mark a point C on it. Put the pointer on the point C.

Step 4: Using the same radius draw an arc on line l which cuts the line at the point D. 

Using the same radius draw an arc on line l
line CD

Perpendiculars

If two lines intersect in such a way that they make a right angle at the point of intersection then they are perpendicular to each other.

Perpendiculars

1. Perpendicular to a line if a point is given on it (using a ruler and a set square)

Step 1: A line l is given and a point P on it.

A line l is given and a point P on it.

Step 2: Put the ruler with one of its side next to l.

Put the ruler with one of its side next to l.

Step 3: Put the set square with one of its sides by the side of the already aligned edge of the ruler so that the right-angled corner comes in contact with the ruler.

Right-angled corner comes in contact with the ruler

Step 4:  Slide the set-square so that the right-angled corner coincides with P.

Right-angled corner coincides with P

Step 5: Hold the set-square. Draw PQ along the edge of the set-square.

PQ is the required perpendicular to l from the given point P.

PQ is the required perpendicular to l from the given point P

2. Perpendicular to a line if a point is given on it(using a ruler and compass)

Draw a perpendicular of the line at a given point A.

Step 1: Take A as the centre and draw a big arc with any radius so that it intersects the line at point B and C.

intersects the line at point B and C

Step 2: Take B as the centre and draw an arc with the radius more than AC and then take C as center and draw an arc so that they intersect each other at point D.

intersect each other at point D.

Step 3: Join AD. AD is perpendicular to CB.

AD ⊥ CB.

AD ⊥ CB.

3. Perpendicular to a line from a point which is not on it(using ruler and a set square)

Step 1: P is the point outside the given line l.

P is the point outside the given line l

Step 2: Put the set-square on l so that one side of its right angle comes on l.

Right angle comes on l.

Step 3: Put a ruler on the hypotenuse of set square.

Hypotenuse of set square

Step 4: Hold the ruler and slide the set-square until it touches the point P.

set-square until it touches the point P

Step 5: Join PM.

Now PM ⊥ l.

PM ⊥ l.

4. Perpendicular to a line from a point not on it (using ruler and a compass)

Step 1: P is the point outside the given line l.

Step 2: Take P as the centre and draw a big arc of any radius which intersects line l at two points A and B.

Two points A and B.

Step 3: Take A and B as radius and draw arcs from both points with the same radius as taken before so that they intersect with each other.

intersect with each other.

Step 4: Join PQ.

PQ is the perpendicular to line l.

PQ is the perpendicular to line l.

5. The perpendicular bisector of a line segment (using transparent tapes)

A line which divides the given line into two equal parts is known as a perpendicular bisector.

Step 1: Draw a line segment AB.

line segment AB.

Step 2: Take a strip of transparent rectangular tape and put it diagonally on the line AB so that the endpoints of the line segment lie on the edges of the tape.

The edges of the tape.

Step 3: Take another strip and put it over A and B diagonally across the previous tape so that the two tapes intersect at point M and N.

Two tapes intersect at point M and N.

Step 4: Join MN. MN is the perpendicular of AB.

MN is the perpendicular of AB.

6. The perpendicular bisector of a line segment (using ruler and compass)

Step 1: Draw a line segment AB.

Line segment AB

Step 2: Take A as the centre and draw two arcs – one upside and one downside with the radius of more than half of the length of AB. Or you can draw a circle taking A as the centre for the convenience.

Again make the arcs with taking B as the centre so that they intersect the previous arcs.

Two arcs – one upside and one downside

Step 3: Join the intersections of the arcs and name them as C and D.

Join the intersections of the arcs

Step 4: The required perpendicular bisector of AB is CD. Hence, AO = OB.

Angles

Angle is a stature formed by two rays. The two rays have a common endpoint which is called the Vertex of the Angle.

Angles

1. Construction of an angle with a given measure

Draw an angle of 60 ° using a protractor.

Step 1: Draw a line BA.

Line BA

Step 2: Put the protractor on the line in such a way that the centre of the protractor lays on point B and the zero edge comes on the line segment BA.

Step 3: Start from 0° and mark the point C at 60°.

Step 4: Join BC.

Required angle

∠CBA is the required angle.

2. Construction of a copy of an angle of unknown measure (using a ruler and a compass).

Draw a copy of ∠A.

Angle A

Step 1: P is a point on the line l.

P is a point on the line l.

Step 2: Take A as the centre in the given angle and draw an arc of any radius which cuts the two rays at B and C.

Two rays at B and C.

Step 3: In the line l, take P as the centre and draw an arc with the same radius as above which cuts line l at Q.

cuts line l at Q

Step 4: Open your compass to take the length of the arc BC.

arc BC

Step 5: Take Q as the centre and draw an arc with the same radius, to cut the arc drawn earlier, at point R. 

cut the arc drawn earlier, at point R

Step 6: Join PR. It will make the angle of the same measure as given.

Join PR

Hence, ∠P = ∠A

3. The bisector of an angle

An angle bisector is the line segment which divides a particular angle into two equal parts. It is also called the line of symmetry of the angle.

Construction of angle bisector (using a ruler and a compass)

Draw the angle bisector of ∠O.

Angle bisector of ∠O

Step 1: Put the pointer on O and draw an arc of any radius so that it cut the rays at point A and B.

cut the rays at point A and B.

Step 2: Put the pointer on point A and draw an arc of the radius of more than half of AB.

Arc of the radius of more than half of AB.

Step 3: While taking B as the centre we will draw an arc of the same radius so that it cut the previous arc at point C.

Arc at point C

Step 4: Join OC.OC is the required angle bisector of ∠O.

The required angle bisector of ∠O

Hence, ∠BOC = ∠COA.

4. Angles of special measures

There are some angles which we can construct accurately with the help of a compass without using a protractor.

a. Construction of 60° angle.

Step 1: Draw a line m and mark a point C on it.

A line m and mark a point C

Step 2: Take C as the centre and draw an arc of any radius to cut the line at point D.

Step 3: While taking D as the centre we need to draw an arc of the same radius to cut the previous arc.

cut the previous arc.

Step 4: Join CE. ∠C = 60°.

Angle C = 60°

b. Construction of 120° angle.

It is the twice of 60° angle.

Step 1: Draw a line m and mark a point C on it.

 a line m and mark a point C

Step 2: Put the pointer on point C and draw an arc of any radius which cuts the line m at point D.

cuts the line m at point D.

Step 3: While taking D as the centre we need to draw an arc with the same radius to cut the previous arc at E.

Again take E as the centre and draw an arc of the same radius to cut the first arc at point F.

radius to cut the previous arc at E

Step 4: Join CF. ∠FCD = 120°.

Join CF

c. Construction of 30° angle.

To construct an angle of 30°, we need to draw an angle of 60° as above then bisect it with the process of an angle bisector.

Construction of 30° angle

d. Construction of 90° angle

It can be made by two methods-

i. Draw a perpendicular bisector of 180° i.e. a straight line.

Construction of 90° angle

ii. Draw a bisector of 60° and 120°.

Draw a bisector of 60° and 120°.

e. Construction of 45° angle.

Draw an angle of 90° then bisect it to make an angle of 45°.

Construction of 45° angle.
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ch-13 Symmetry notes | class 6th mathematics

When we fold a paper in such a way that the picture is divided into two equal halves then the line which divides the picture into two halves is called a Line of Symmetry.

Symmetry

Here the line divides the star into two halves so it is the line of symmetry. It is also called the Mirror Line because if we place the mirror on that line then one side of the picture will fall exactly on the other side of the picture.

Non-symmetrical Figure

Non-symmetrical Figure

This figure is not symmetrical as if we fold the image from the dotted line then it does not divide it into two equal halves.

Making Symmetric Figures: Ink-blot Devils

To make an ink-blot pattern-

  • Take a piece of paper and fold it in half.
  • Put some drops of ink on one side of the paper.
  • Then press the halves together.
  • It will make a symmetric pattern with the fold as the line of symmetry.

Inked-string pattern

To make an inked string pattern-

  • Take a piece of paper and fold it in half.
  • Dip a string in different colours and arrange it on the one side of the paper.
  • Press the two halves together and pull the string.
  • It will make a symmetric inked string pattern with the fold as the line of symmetry.
Inked-string pattern

Two Lines of Symmetry

Some figures have two lines of symmetry.

1. A Rectangle

Take a rectangular sheet and fold it horizontally in two equal halves and then again fold it vertically in two equal halves. After opening it, we get two lines of symmetry of the rectangular sheet.

Rectangle

2. More Figures with two Line of Symmetry

If we take a rectangular piece of paper and double fold it to make two lines of symmetry and cut it in some new shape then after opening it we will get a new image that too with the two lines of symmetry.

More Figures with two Line of Symmetry

Construction of figure with two Lines of Symmetry

1. To draw a figure with two lines of symmetry, take one figure.

Two lines of symmetry

2. Let L and M be the two lines of symmetry.

L and M be the two lines of symmetry

3. Draw the figure in such a way that L is the line of symmetry,

L is the line of symmetry,

4. Now complete the figure by drawing the remaining part so that M will also become the line of symmetry.

Line of symmetry

Hence this is the final figure with two lines of symmetry.

Multiple Lines of Symmetry

Take a square sheet of paper and fold it in two halves vertically and again horizontally .open it and fold it in two equal halves diagonally then again open it and fold it along another diagonal.

When you will open the paper you will see four imaginary lines and these lines are the lines of symmetry.

Multiple Lines of Symmetry

Some more images with more than two lines of symmetry

Some more images with more than two lines of symmetry
  • Equilateral triangle will have three lines of symmetry.
  • Square will have four lines of symmetry.
  • Regular pentagon will have five lines of symmetry.
  • Regular hexagon will have six lines of symmetry.

Some Real-life Examples of Symmetry

In Taj Mahal and the butterfly there is one line of symmetry and there are so many other things also in our daily life which are having one or more line of symmetry.

Reflection and Symmetry

The line of symmetry is also called Mirror Line because the mirror image of an object is symmetrical to the image. When we see an object in the mirror then there is no change in the length and angles of the object except one thing i.e. the image is opposite to the original image.

Reflection and Symmetry

Some Examples of Reflection Symmetry

1. Paper Decoration

We can use a rectangular sheet to fold and create some intricate patterns by cutting paper.

Paper Decoration

2. Kaleidoscope

In Kaleidoscope, mirrors are used to create pictures having various lines of symmetry. Two mirrors strips forming a V-shape are used. The angle between the mirrors determines the number of lines of symmetry.

Kaleidoscope

Example

Which alphabet will remain same after reflection symmetry? Check for R, C, N, A and T.

Solution

In the alphabet reflection symmetry, the alphabets look opposite in the mirror i.e. the alphabet written from right to left will appear as written from left to right.

Alphabet reflection symmetry

Hence C, N and R will not look the same after reflection.

Reflection symmetry

Hence A and T will look same after reflection symmetry.

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ch-12 Ratio and Proportion notes | class 6th mathematics

If we compare two quantities using division then it is called ratio. It compares quantities in terms of ‘How many times’. The symbol to represent ratio is “:”.

Ratio

It reads as “4 is to 3”

It can also be written as 4/3.

Example

If there are 35 boys and 25 girls in a class, then what is the ratio of

  • Number of boys to total students
  • Number of girls to total students.

Solution

In the ratio, we want the total number of students.

Total number of students = Number of boys + Number of girls

35 + 25 = 60

  • Ratio of number of boys to total number of students
  • The ratio of the number of girls to the total number of students

The unit must be same to compare two quantities

If we have to compare two quantities with different units then we need to convert them in the same unit .then only they can be compared using ratio.

Example

What is the ratio of the height of Raman and Radha if the height of Raman is 175 cm and Radha is 1.35 m?

Solution

The unit of the height of Raman and Radha is not same so convert them in the same unit.

Height of Radha is 1.35 m = 1.35 × 100 cm = 135 cm

The ratio of the height of Raman and Radha 

Equivalent Ratios

If we multiply or divide both the numerator and denominator by the same number then we get the equivalent ratio. There could be so many equivalent ratios of the same ratio.

In the case of equivalent ratios only their value changes but they represent the same portion of the quantity.

Example

Find two equivalent ratios of 2/4.

Solution

Equivalent Ratios

To get the equivalent ratio we multiply both the numerator and denominator with 2.

To get another equivalent ratio we divide both the numerator and denominator with 2.

From the above figure, we can see that in all the equivalent ratios only the number of equal parts is changing but all the ratios are representing the half part of the circle only.

The Lowest form of the Ratio

If there is no common factor of numerator and denominator except one then it is the lowest form of the ratio.

Example

Find the lowest form of the ratio 25: 100.

Solution

The common factor of 25 and 100 is 25, so divide both the numerator and denominator with 25.

Hence the lowest ratio of 25: 100 is 1: 4.

Proportion

If we say that two ratios are equal then it is called Proportion.

Proportion

We write it as a: b : : c: d or a: b = c: d

And reads as “a is to b as c is to d”.

Example

If a man runs at a speed of 20 km in 2 hours then with the same speed would he be able to cross 40 km in 4 hours?

Solution

Here the ratio of the distances given is 20/40 = 1/2 = 1: 2

And the ratio of the time taken by them is also 2/4 = 1/2 = 1: 2

Hence the four numbers are in proportion.

We can write them in proportion as 20: 40 : : 2: 4

And reads as “20 is to 40 as 2 is to 4”.
 

Extreme Terms and Middle Terms of Proportion

Extreme Terms and Middle Terms of Proportion

The first and the fourth term in the proportion are called extreme terms and the second and third terms are called the Middle or the Mean Terms.

In this statement of proportion, the four terms which we have written in order are called the Respective Terms.

If the two ratios are not equal then these are not in proportion.

Example 1

Check whether the terms 30,99,20,66 are in proportion or not.

Solution 1.1

To check the numbers are in proportion or not we have to equate the ratios.

As both the ratios are equal so the four terms are in proportion.

30: 99 :: 20: 66

Solution 1.2

We can check with the product of extremes and the product of means.

In the respective terms 30, 99, 20, 66

30 and 66 are the extremes.

99 and 20 are the means.

To be in proportion the product of extremes must be equal to the product of means.

30 × 66 = 1980

99 × 20 = 1980

The product of extremes = product of means

Hence, these terms are in proportion.
 

Example 2

Find the ratio 30 cm to 4 m is proportion to 25 cm to 5 m or not.

Solution 2

As the unit is different so we have to convert them into the same unit.

4 m = 4 × 100 cm = 400 cm

The ratio of 30 cm to 400 cm is

5 m = 5 × 100 cm = 500 cm

Ratio of 25 cm to 500 cm is

Here the two ratios are not equal so these ratios are not in proportion.

3: 40 ≠ 1: 20

Unitary Method

If we find the value of one unit then calculate the value of the required number of units then this method is called the Unitary method.

Example 1

If the cost of 3 books is 320 Rs. then what will be the cost of 6 books?

Books

Solution 1

Cost of 3 books = Rs. 320

Cost of 1 book = 320/3 Rs.

Cost of 6 books = (320/3) × 6 = 640 Rs.

Hence, the cost of 6 books is Rs. 640.
 

Example 2

If the cost of 20 toys is Rs. 4000 then how many toys can be purchased for Rs. 6000?

Solution 2

In Rs. 4000, the number of toys can be purchased = 20

In Rs. 1, the number of toys can be purchased = Rs. 20/4000

Therefore, in Rs. 6000, the number of toys can be purchased = (20/4000) × 6000 = 30

Hence, 30 toys can be purchased by Rs. 6000.

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ch-11 Algebra notes | class 6th mathematics

Introduction to Algebra

There are so many branches of mathematics-

  • The study of numbers is called Arithmetic.
  • The study of shapes is called Geometry.
  • The study to use the letters and symbols in mathematics is called Algebra.

Algebra

Algebra is a part of mathematics in which the letter and symbols are used to represent numbers in equations. It helps us to study about unknown quantities.

Matchstick Patterns

Matchstick Patterns

No. of matchsticks used to make 1st square = 4

No. of matchsticks used to make 2nd square = 7

No. of matchsticks used to make 3rd square = 10

So, the pattern that we observe here is 3n + 1

With this pattern, we can easily find the number of matchsticks required in any number of squares.

Example

How many matchsticks will be used in the 50th figure?

Solution

3n + 1

3 × 50 + 1

= 151 matchsticks

The Idea of a Variable

Variable refers to the unknown quantities that can change or vary and are represented using the lowercase letter of the English alphabets.

One such example of the same is the rule that we used in the matchstick pattern

3n + 1

Here the value of n is unknown and it can vary from time to time.

More Examples of Variables

  • We can use any letter as a variable, but only lowercase English alphabets.
  • Numbers cannot be used for the variable as they have a fixed value.
  • They can also help in solving some other problems.

Example: 1

Karan wanted to buy story books from a bookstall. She wanted to buy 3 books for herself, 2 for her brother and 4 for 2 of her friends. Each book cost Rs.15.how much money she should pay to the shopkeeper?

Solution: 1

Cost of 1 book = Rs.15

We need to find the cost of 9 books.

No. of notebooks1234  …….a  ……..
Total cost 15304560   …….     15 a   …….

In the current situation, a (it’s a variable) stands for 9

Therefore,

Cost of 9 books = 15 × 9

= 135

Therefore Karan needs to pay Rs.135 to the shopkeeper of the bookstall.

The variable and constant not only multiply with each other but also can be added or subtracted, based on the situation.

Example: 2

Manu has 2 erasers more than Tanu. Form an expression for the statement.

Solution: 2.1

Erasers that Tanu have can be represented using a variable (x)

Erasers that Manu have = erasers that Tanu have + 2

Erasers with Manu = x + 2

Solution: 2.2

Erasers that Manu have can be represented using a variable (y)

Erasers that Tanu has = erasers that Manu have – 2

Erasers with Tanu = y – 2

Use of Variables in Common Rules (Geometry)

1. Perimeter of Square

The perimeter of a square = Sum of all sides

= 4 × side

= 4s

Thus, p = 4s

Here s is variable, so the perimeter changes as the value of side change.

2. Perimeter of Rectangle

Perimeter of rectangle = 2(length + breadth)

= 2 (l + b) or 2l + 2b

Thus, p + 2 × (l + b) or 2l + 2b

Where, l and b are variable and the value of perimeter changes with the change in l and b.

Use of Variables in Common Rules (Arithmetic)

1. Commutativity of Addition

5 + 4 = 9

4 + 5 = 9

Thus, 5 + 4 = 4 + 5

This is the commutative property of addition of the numbers, in which the result remains the same even if we interchanged the numbers.

a + b = b + a

Here, a and b are different variables.

Example

a = 16 and b = 20

According to commutative property

16 + 20 = 20 + 16

36 = 36

2. Commutativity of Multiplication

8 × 2 = 16

2 × 8 = 16

Thus, 8 × 2 = 2 × 8

This is the commutative property of multiplication, in which the result remains the same even if we interchange the numbers.

a × b = b × a

Here, a and b are different variables.

Example

18 ×12 = 216, 12 ×18 = 216

Thus, 18 × 12 = 12 × 18

3. Distributivity of Numbers

6 × 32

It is a complex sum but there is an easy way to solve it. It is known as the distributivity of multiplication over the addition of numbers.

6 × (30 + 2)

= 180 + 12

= 192

Thus, 6 × 32 = 192

A × (b + c) = a × b + a × c

Here, a, b and c are different variables.

4. Associativity of Addition

This property states that the result of the numbers added will remain same regardless of their grouping.

(a + b) + c = a + (b + c)

Example

(4 + 2) + 7 = 4 + (2 + 7)

6 + 7 = 4 + 9

13 = 13

Expressions

Arithmetic expressions may use numbers and all operations like addition, subtraction, multiplication and division

Example

2 + (9 – 3), (4 × 6) – 8 etc…

(4 × 6) – 8 = 24 – 8

= 16

Expressions with variable

We can make expressions using variables like

2m, 5 + t etc…..

An expression containing variable/s cannot be analyzed until its value is given.

Example

Find 3x – 12 if x = 6

Solution

(3 × 6) – 12

= 18 – 12

= 5

Thus,

3x – 12 = 5

Formation of Expressions

StatementExpression 
y subtracted from 1212 – y
x multiplied by 66x
t Multiplied by 4, and then subtract 5 from the product.4t – 5 


Practical use of Expressions

Example

3 boys go to the theatre. The cost of the ticket and popcorn is $33 and $15 respectively. What is the cost per person?

Solution

Let’s say,

 x = cost of ticket per person

 y = cost of popcorn/person

Total cost of the movie (ticket + popcorn) per person = x + y

Total cost of ticket + popcorn for 3 boys = 3(x + y)

= 3 (33 + 15)

= 3 (48)

= 144

Hence the total cost of movie ticket and popcorn for 3 boys is $144.

Equation

If we use the equal sign between two expressions then they form an equation.

An equation satisfies only for a particular value of the variable.

The equal sign says that the LHS is equal to the RHS and the value of a variable which makes them equal is the only solution of that equation.

Example

3 + 2x = 13

5m – 7 = 3

p/6 = 18

If there is the greater then or less than sign instead of the equal sign then that statement is not an equation.

Some examples which are not an equation

23 + 6x  > 8

6f – 3 < 24

The Solution of an Equation

The value of the variable which satisfies the equation is the solution to that equation. To check whether the particular value is the solution or not, we have to check that the LHS must be equal to the RHS with that value of the variable.

Trial and Error Method

To find the solution of the equation, we use the trial and error method.

Example

Find the value of x in the equation 25 – x = 15.

Solution

Here we have to check for some values which we feel can be the solution by putting the value of the variable x and check for LHS = RHS.

Let’s take x = 5

25 – 5 = 15

20 ≠ 15

So x = 5 is not the solution of that equation.

Let’s take x = 10

25 – 10 = 15

15 = 15

LHS = RHS

Hence, x = 10 is the solution of that equation.

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ch-10 Mensuration notes | class 6th mathematics

The closed 2-D shapes are referred to as plane figures.

The closed 2-D shapes

Here “C” is the boundary of the above figure and the area inside the boundary is the region of this figure. Point D comes in the area of the given figure.

Perimeter

If we go around the figure along its boundary to form a closed figure then the distance covered is the perimeter of that figure. Hence the Perimeter refers to the length of the boundary of a closed figure.

If a figure is made up of line segments only then we can find its perimeter by adding the length of all the sides of the given figure.

Example

Find the Perimeter of the given figure.

Perimeter

Solution

Perimeter = Sum of all the sides

= (12 + 3 + 7 + 6 + 10 + 3 + 15 + 12) m

= 68 m

The Perimeter of a Rectangle

A rectangle is a closed figure with two pairs of equal opposite sides.

The Perimeter of a Rectangle

Perimeter of a rectangle = Sum of all sides

= length + breadth + length + breadth

Thus, Perimeter of a rectangle = 2 × (length + breadth)

Example: 1

The length and breadth of a rectangular swimming pool are 16 and 12 meters respectively .find the perimeter of the pool.

Solution: 

Perimeter of a rectangle = 2 × (length + breadth)

Perimeter of the pool = 2 × (16 + 12)

= 2 × 28

= 56 meters

Example: 2

Find the cost of fencing a rectangular farm of length 24 meters and breadth 18 meters at 8/- per meter.

Solution:

Perimeter of a rectangle = 2 × (length + breadth)

Perimeter of the farm = 2 × (24 + 18)

= 2 × 42

= 84 meter

Cost of fencing = 84 × 8

= Rs. 672

Thus the cost of fencing the farm is Rs. 672/-.

Regular Closed Figure

Figures with equal length of sides and an equal measure of angles are known as Regular Closed Figures or Regular Polygon.

Perimeter of Regular Polygon = Number of sides × Length of one side

Perimeter of Square

Square is a regular polygon with 4 equal sides.

Square

Perimeter of square = side + side + side + side

Thus, Perimeter of a square = 4 × length of a side

Example

Find the perimeter of a square having side length 25 cm.

Solution

Perimeter of a square = 4 × length of a side

Perimeter of square = 4 × 25

 = 100 cm

Perimeter of an Equilateral Triangle

An equilateral triangle is a regular polygon with three equal sides and angles.

Equilateral Triangle

Perimeter of an equilateral triangle = 3 × length of a side

Example

Find the perimeter of a triangle having each side length 13 cm.

Solution

Perimeter of an equilateral triangle = 3 × length of a side

Perimeter of triangle = 3 × 13

= 39 cm

Perimeter of a Regular Pentagon

A regular pentagon is a polygon with 5 equal sides and angles.

Regular Pentagon

Perimeter of a regular pentagon = 5 × length of one side

Example

Find the perimeter of a pentagon having side length 9 cm.

Solution

Perimeter of a regular pentagon = 5 × length of one side

Perimeter of a regular pentagon = 5 × 9

= 45 cm

Perimeter of a Regular Hexagon

A regular hexagon is a polygon with 6 equal sides and angles.

Regular Hexagon

Perimeter of a regular hexagon = 6 × Length of one side

Example

Find the perimeter of a hexagon having side length 15cm.

Solution

Perimeter of a regular hexagon = 6 × Length of one side

Perimeter of a regular hexagon = 6 × 15

= 90 cm

Perimeter of a Regular Octagon

A regular octagon is a polygon with 8 equal sides and angles.

Regular Octagon

Perimeter of a regular octagon = 8 × length of one side

Example

Find the perimeter of an octagon having side length 7cm.

Solution

Perimeter of a regular octagon = 8 × length of one side

Perimeter of a regular octagon = 8 × 7

= 56 cm

Area

Area refers to the surface enclosed by a closed figure.

Area

To find the area of any irregular closed figure, we can put them on a graph paper with the square of 1 cm × 1 cm .then estimate the area of that figure by counting the area of the squares covered by the figure.

Here one square is taken as 1 sq.unit.

Example

Find the area of the given figure. (1 square = 1 m2)

Square

Solution

The given figure is made up of line segments and is covered with some full squares and some half squares.

Full squares in figure = 32

Half squares in figure = 21

Area covered by full squares = 32 × 1 sq. unit = 32 sq. unit.

Area covered by half squares = 21 × (1/2) sq. unit. = 10.5 sq. unit.

Total area covered by figure = 32 + 10.5 = 42.5 sq. unit.

Area of a Rectangle

Rectangle

Area of a rectangle = (length × breadth)

Example

Find the area of a rectangle whose length and breadth are 20 cm and 12 cm respectively.

Solution

Length of the rectangle = 20 cm

Breadth of the rectangle = 12 cm

Area of the rectangle = length × breadth

= 20 cm × 12 cm

= 240 sq cm.

To find the length of a rectangle if breadth and area are given:

Example

What will be the length of the rectangle if its breadth is 6 m and the area is 48sq.m?

Solution

Length = 48/6

 = 8 m

To find the breadth of the rectangle if length and area are given:

Example

What will be the breadth of the rectangle if its length is 8 m and the area is 81 sq.m?

Solution

Breadth = 81/8

= 9 m

Area of a Square

Area of a square is the region covered by the boundary of a square.

Square

Area of a square = side × side

Example

Calculate the area of a square of side 13 cm.

Solution

Area of a square = side × side

= 13 × 13

= 169 cm2.

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ch-9 Data Handling notes | class 6th mathematics

Data

Data is a collection of raw facts and figures that give you information.

Recording Data

Recording of data depends upon the requirement of the data. Everybody has different ways to record data.

If we have to compare the choice of the people about certain movies then we have to collect the data of the survey which tells the choice of the people about those movies.

Organization of Data

Raw data is difficult to read, so we have to organize it in such a way so that we can use it in need.

Data can be organized in a tabular form.

Data is represented in a tabular form using frequency distribution and the tally marks.

Frequency tells the number of times the particular observation happened.

Tally marks are used to show the frequency of the data.

Tally marks are represented as

Tally marks

Example

There are 30 students in a class. They have to choose one sport each for the sports period.5 took badminton, 10 took cricket, 4 took football, 1 took hockey, 3 took tennis and 7 went for volleyball. Represent this data in the frequency distribution table.

Solution

To make a frequency distribution table-

  • Make a table with three columns.
  • Write the name of sports in the first column.
  • Write the respective frequencies in front of each sport.
  • Mark the tally marks according to the frequency given.
Frequency distribution table

Pictograph

If we represent the data with the pictures of objects instead of numbers then it is called Pictograph. Pictures make it easy to understand the data and answer the questions related to it by just seeing it.

Pictograph

We can easily answer the questions like who has a maximum number of toys, who has the least number of toys etc.

Interpretation of a Pictograph

In the pictograph, we have to understand it and get the information from the pictures given.

If we have to represent more number of items then we can use the key which represents more numbers with one picture.

Example

The number of cars parked in a parking lot every day is given in the pictograph.

Pictograph
  • Find the day when the highest number of cars are parked and how many?
  • When the least number of cars did park?

Solution

In the above pictograph one car represent 5 cars.

  • As there is the maximum number of cars is shown on Tuesday so the highest number of cars was parked on Tuesday.
pictograph one car

Hence, 40 cars were parked on Tuesday in the parking lot.

  • Least number of cars were parked on Monday as there are only 4 pictures of cars are shown on that day.

Drawing a Pictograph

Drawing a pictograph is an interesting task but it may be difficult to draw some difficult pictures repeatedly as we had used cars in the above example so we can use easy symbols to draw a pictograph.

We must use a proper key of the symbols so that it could be easily understood by anyone.

Example

The following table shows the choice of the fruits of the 35 students of class 3.represent the data in a pictograph.

Name of fruitNumber of students
Mango5
Apple12
Guava3
Litchi7
Grapes8

Solution

Table

Bar Graph

As the pictograph is a very time-consuming process, so we can use another way to represent data.

If we use the bars of the same width with equal spacing to represent the data in which the length of the bars represent the frequency is called Bar Graph or Bar Diagram.

Interpretation of a Bar Graph

Example

The following graph tells the favourite colours in a class of 30 students.

Interpretation of a Bar Graph

Answer the following questions:

  • Which colour is liked by the maximum number of students?
  • Which colour is liked by the same number of students?

Solution

  • The graph shows that the pink colour is liked by 9 students so it is the favourite colour of the maximum number of students.
  • Blue and green colour bars are equal in length and both are liked by 6 students.

Drawing a Bar Graph

Drawing a bar graph is an interesting task, but we must choose an appropriate scale to draw the bar graph. It depends upon our own choice that what we are taking for the scale.

Example

The daily sale of mobile phones in Vicky production is given below. Draw the bar graph to represent the data.

DaysMondayTuesdayWednesdayThursdayFridaySaturday
No. of phones sold251332144255

Solution

Steps to make a bar graph-

  • Draw two lines, one horizontal and one vertical in L shape.
  • Mark days on the horizontal line and no. of phones on the vertical line.
  • Take a suitable scale for the number of phones and mark on the vertical line. Let 1 unit = 10 phones.
  • Use the bars of equal width and draw them with the frequency given at the same distance.

The height of the bars tells the sale of the mobile phones in Vicky production.

The height of the bars

This same bar graph can be made by interchanging the positions of the days and the number of phones.

Bar graph
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ch-8 Decimals notes | class 6th mathematics

When we use dots to write some numbers then that dot is the decimal point. This is used to show the part of a whole number.

Decimals

Tenths

As we know that 1 cm = 10 mm, so if we have to find the opposite then

1mm = 1/10 cm or one-tenth cm or 0.1 cm.

Hence, the first number after the decimal represents the tenth part of the whole.

Tenths

This reads as “thirty-four point seven”.

Representation of Decimals on Number Line

To represent decimals on the number line we have to divide the gap of each number into 10 equal parts as the decimal shows the tenth part of the number.

Example

Show 0.3, 0.5 and 0.8 on the number line.

Solution

All the three numbers are greater than 0 and less than 1.so we have to make a number line with 0 and 1 and divide the gap into 10 equal parts.

Then mark as shown below.

Representation of Decimals on Number Line

Fractions as Decimals

It is easy to write the fractions with 10 as the denominator in decimal form but if the denominator is not 10 then we have to find the equivalent fraction with denominator 10.

Example

Convert 12/5 and 3/2 in decimal form.

Solution

Decimals as Fractions

Example

Write 2.5 in a fraction.

Solution

Hundredths

As we know that 1 m = 100 cm, so if we have to find the opposite then

1 cm = 1/100 m or one-hundredth m or 0.01 m.

Hence, the second numbers after the decimal represent the hundredth part of the whole.

Hundredths

It reads as “thirteen point nine five”.

Decimal in the hundredth form shows that we have divided the number into hundred equal parts.

Example

If we say that 25 out of 100 squares are shaded then how will we write it in fraction and decimal form?

Squares

Solution

25 is a part of 100, so the fraction will be 25/100.

In the decimal form we will write it as 0.25.

Place Value Chart

This is the place value chart which tells the place value of each digit in the decimal number. It makes it easy to write numbers in decimal form.

Place Value Chart

Example

With the given place value chart write the number in the decimal form.

Hundreds (100)Tens (10)Ones (1)Tenths (1/10)Hundredths (1/100)
46385

Solution

According to the above table-

Comparing Decimals

1. If the whole number is different.

If the whole numbers of the decimals are different then we can easily compare them .the number with the greater whole number will be greater than the other.

Example

Compare 532.48 and 682.26.

Solution:

As the whole numbers are different, so we can easily find that the number with a greater whole number is greater.

Hence 682.26 > 532.48

Compare 532.48 and 682.26.

2. If the whole number is the same

If the whole numbers of the decimals are same then we will compare the tenth and then the hundredth part if required.

The number with the greater tenth number is greater than the other.

Example

Compare 42.36 and 42.68.

Solution

As the whole number is the same in both the numbers so we have to compare the tenth part.

Hence 42.68 > 42.36

Compare 42.36 and 42.68.

Using Decimals

Generally, decimals are used in money, length and weight.

1. Money

Example: 1

Write 25 paise in decimals.

Solution:

100 paise = 1 Rs.

1 paise = 1/100 Rs. = 0.01 Rs.

25 paise = 25/100 Rs. = 0.25 Rs.

Example: 2

Write 7 rupees and 35 paise in decimals.

Solution:

7 rupees is the whole number, so

7 + 35/100 = 7 + 0.35 = 7.35 Rs.

2. Length

Example

If the height of Rani is 175 cm then what will be her height in meters?

Solution

100 cm = 1 m

1 cm = 1/100 m = 0.01 m

175 cm = 175/100 m

Hence, the height of Rani is 1.75 m.

3. Weight

Example

If the weight of a rice box is 4725 gram then what will be its weight in a kilogram?

Solution

1000 gm = 1 kg

1 gm = 1/1000 kg = 0.001 kg

Addition of Decimal numbers

To add the decimal numbers we can add them as whole numbers but the decimal will remain at the same place as it was in the given numbers. It means that we have to line up the decimal point in each number while writing them, and then add them as a whole number.

Example: 1

Add 22.3 and 34.1

Solution:

Write the numbers as given below, and then add them.

Add 22.3 and 34.1

Example: 2

Add 1.234 and 4.1.

Solution:

There are three numbers after decimal in one number and one number after decimal in another number. So we should not get confused and write the numbers by lining up the decimal points of both the numbers, then add them.

Another way is to write the numbers in the place value chart, so that it will be easy to identify, how to write numbers.

 Ones (1)Tenths (1/10)Hundredths (1/100)Thousandths (1/1000)
+4100
=5334

Subtraction of Decimal Numbers

Subtraction is also done as normal whole numbers after lining up the decimals of the given number.

Example

Subtract 243.86 from 402.10.

Solution

  • Write the numbers in a line so that the decimal points of both the numbers lined up.
  • Then subtract and borrow as we do in whole numbers.
  • Line up the decimal point in the answer also.
Subtraction of Decimal Numbers
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Chapter 7 Fractions notes | class 6th mathematics

A Fraction is a part of whole. The ‘whole’ here could be an object or the group of objects. But all the parts of the whole must be equal.

Fractions
  1. The first one is the whole i.e. a complete circle.
  2. In the second circle, if we divide the circle into two equal parts then the shaded portion is the half i.e. ½ of the circle.
  3. In the third circle, if we divide the circle into four equal parts and shade only one part then the shaded part is the one fourth i.e. ¼ of the whole circle.
  4. In the fourth circle, if we divide the circle into four equal parts and shade three parts then the shaded part is the three fourth i.e. ¾ of the whole circle.

Numerator and Denominator

Fractions

The upper part of the fraction is called Numerator. It tells the number of parts we have.

The lower part of the fraction is called Denominator. It tells the total parts in a whole.

It reads as “three-fifths”.

Representation of fraction on Number line

Example

Fractions

Solution

  • Draw a number line.
  • We know that ½ is less than 1 and greater than 0, so we have to divide the gap between two equal parts and then mark the middle point as ½.

As the denominator is the whole and the numerator is the part, so we have to divide the gap between 0 and 1 in the number of parts as the denominator is given.

For 1/3, divide into 3 equal parts.

For ¼, divide into 4 equal parts and so on.

Fractions

Proper Fractions

If the numerator is less than the denominator then it is called proper fraction. If we represent a proper fraction on the number line than it will always lie between 0 and 1.

Examples

Fractions

Improper fractions and Mixed fractions

When the numerator is greater than the denominator then it is called Improper fraction.

Fractions

The above fraction is made by adding one whole part and one-fourth part.

Fractions

The fraction made by the combination of whole and a part is called Mixed fraction.

Convert Mixed fraction into Improper fraction

A mixed fraction is in the form of 

Fractions

We can convert it in the form of an improper fraction by

Fractions

Example

Fractions

Solution

Fractions

Equivalent Fractions

Equivalent fractions are those fractions which represent the same part of a whole.

All the above images are different but equivalent fractions as they represent the same i.e. half part of a whole circle.

Finding equivalent fractions

  1. Multiplying the same number

If we multiply the numerator and denominator of any fraction with the same number then we will get the equivalent fraction. There could be more than one equivalent fractions of one fraction.

Example

Find three equivalent fraction of ½.

Solution

Fractions
  1. Dividing the same number

If we divide the numerator and denominator of any fraction with the same number then we will get the equivalent fraction.

Example

Find the equivalent fraction of 18/27 with denominator 9.

Solution

To get the denominator 9 we need to divide it by 3.

So to find the equivalent fraction we need to divide the fraction by 3.

Fractions

Hence the equivalent fraction with denominator 9 is 6/9.

The simplest form of a Fraction

If the numerator and denominator do not have any other common factor than 1 then it is said to be the simplest or lowest form of that fraction.

Example

Fractions

To find the equivalent fraction which is the simplest form we have to find the HCF of numerator and denominator and then divide them both by that HCF.

Example

Reduce the fraction 18/27 in the simplest form.

Solution

HCF of 18 and 27 is 9.

Hence,

Fractions

2/3 is the lowest form of 18/27.

Like Fractions and Unlike Fractions

Fractions which have same denominators are known as Like fractions.

Example

Fractions

Fractions which have different denominators are known as unlike fractions.

Example

Fractions

Comparing fractions

Fractions

If we have to compare the above two fractions then it is easy as the first one is less than 3 and the second one is greater than 3. So we can clearly say that

Fractions

But sometimes it is not easy to compare it so easily. So we need some accurate procedure.

  1. Comparing like fractions

Like fractions are the fractions with the same denominator so we have to compare them with the numerator only. The fraction with greater numerator is greater.

In the above example, both are divided into 8 equal parts, so the fraction with seven shaded part is greater than the 5 shaded parts.

  1. Comparing unlike fractions

The fractions with different denominators are unlike fractions.

  1. Unlike fraction with the same numerator

If we have to compare the fractions with different denominator but same numerator, we have to compare with the denominator only.

In that case, the fraction with the small denominator is greater than the other.

Example

Here the numerator is same i.e.3 so we will compare with the denominator.

The fraction with small denominator i.e. ¾ is greater than the fraction with the large denominator i.e. 3/8.

  1. Unlike fraction with different numerators

If the numerator and denominator both are different then we have to make the denominator same by finding the equivalent fraction of both the fractions then compare the fractions as like fractions.

To find the equivalent fraction of both the fractions with the same denominator, we have to take the LCM of the denominator.

Example

Compare 6/7 and 3/5.

Solution

The product of 7 and 5 is 35.

So we will find the equivalent fraction of both the fractions with the denominator 35.

Fractions

Now we can compare them as like fractions.

Fractions

Addition and Subtraction of Fractions

  1. Adding like fractions

In case of like fractions, the denominator is same so we can add them easily.

Steps to add like fractions-

  1. Add the numerators.
  2. Leave the common denominator same. (Don’t add the denominator).
  3. Write the answer as
Fractions

Example

Fractions
Fractions

Solution

Fractions
  1. Subtracting like fractions

Steps to subtract the like fractions-

  1. Subtract the small numerator from the bigger one.
  2. Leave the common denominator same.
  3. Write the answer as
Fractions

Example

Fractions

Solution

Fractions
Fractions
  1. Adding unlike fraction

If we have to add the unlike fractions, first we have to find the equivalent fraction of the given fractions with the same denominator then add them.

Steps to add unlike fractions-

  1. Take the LCM of the denominator of the given fractions.
  2. Find the equivalent fractions of both fractions with LCM as the denominator.
  3. Add them as the like fractions.

Example

Fractions

Solution

Take the LCM of 5 and 8, which is 40.

Fractions
  1. Subtracting unlike fractions

Steps to subtract unlike fractions-

  1. Take the LCM of the denominator of the given fractions.
  2. Find the equivalent fractions of both fractions with LCM as the denominator.
  3. Subtract them as the like fractions.

Example

Fractions

Solution

LCM of 4 and 5 is 20.

Fractions

Remark: To add or subtract the mixed fractions, simply convert them in the improper fraction then add and subtract them directly

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