Do you know you use a lot of machines every day without even knowing about it? Remember those scissors that you used to cut paper for your art class? And the stapler that you used to staple extra sheets together? They are all simple machines.
Every day you use machines without even thinking about it. A machine is anything that helps make work easier. Basic tools like staplers, screwdrivers and scissors are simple machines. These machines are all based on simple inventions like levers, planes, pulleys or wheels.
Why do we need machines?
With the help of machines, a small force can be used to overcome a large force. For example, a screw jack is used to lift an object as heavy as a car to change its tyres.
We need machines to:
Lift heavy loads with a small effort.
Carry out unsafe and dangerous tasks.
To increase the speed of a moving object.
To move, lift or perform an action in the required direction.
To reduce the risk in performing hazardous tasks.
Simple Machines
Simple Machine:
A machine is a device that helps us to do work with less effort in less time. A simple machine has few or no moving parts.
Simple machines are basically classified into two groups, levers and inclined planes.
Simple machines like the pulley, screw, wheel and axle and wedge come under these two categories.
The see-saw in the playground is a simple machine.
You can use these simple machines to build a complex machine. For example, a bicycle is a complex machine that is made using of nearly every kind of simple machine.
Pulley
A pulley is a flat circular disc having a groove in its edge and capable of rotating around a fixed point passing through its central axis, called an axle.
The two commonly used types of pulleys are single fixed pulley system and the single movable pulley system.
Single Fixed Pulley System:
A single fixed pulley system or a simple pulley consists of a grooved wheel, made of wood or metal, with a rope passing through it.
The pulley rotates about an axle passing through its centre.
The axle is fixed to a frame or a block.
The pulley is normally fixed to a support above the load.
The load is tied to one end of the rope and the effort is applied at the other end.
Such a pulley makes our work easier by simply changing the direction of the force, i.e. a load is lifted up using a downward effort. It is easier to lift a load up by pulling it down rather than by pulling it up directly.
Note that a simple pulley does not reduce the effort required to lift a load.
A simple pulley is used to hoist a flag, to draw water from a well, etc.
Single Movable Pulley System
This pulley system has a block of two pulleys.
In this system, a load of W kgf can be lifted with an effort of W/2 kgf only.
In a block and tackle arrangement, the pulleys are assembled together to form blocks and then blocks are paired so that one is fixed and the other moves with the load.
The rope is threaded, or reeved, through the pulleys to provide a mechanical advantage that amplifies the effort applied to the rope.
An example of a practical application of pulleys and levers working together is the crane which is used to lift heavy loads.
The Wheel and Axle
Wheel and axle essentially consist of two cylinders of different radii joined together, such that if one is made to rotate the other also rotates. The cylinder with the larger radius is called the wheel and that with a smaller radius is called the axle.
Examples of wheel and axle system are a doorknob, a knob of water tap, a screwdriver, egg beater, a hand drill, etc.
Inclined Plane
An inclined plane is any sloping flat surface along which a load can be pushed or pulled with less effort. A hospital ramp on which a wheelchair can be pushed easily, a wooden plank used to load heavy boxes into the rear of a truck, winding roads around a hill, winding staircases, etc. are all inclined planes.
Wedge
If two inclined planes are put together to form a sharp edge, it is called a wedge. So a wedge is a double inclined plane. It works on the principle of inclined planes.
A wedge is used to tear apart or cut through objects. Knives, needles, axes, chisel, etc. are examples of a wedge.
Screw
A screw is a rod or nail with grooves on its circular curved surface and is used to hold two objects firmly together.
In simple terms, a rotating (winding) inclined plane is called a screw. The winding edge of a screw is called a thread. The grooved part of the screw is an inclined plane.
The head of the screw has a groove for the tip of a screwdriver. When the screw is held against a wooden block and its head is turned using a screwdriver, the tip of the screw moves into the wood. Because of the grooves, a screw holds the wood more firmly than a nail. Also, less force is required to insert a screw into wood than a nail because of the inclined edge.
Uses of Screws
Screws are used to join two pieces of wood or metal. As the screw is a winding inclined plane, it cannot be pulled out easily from the attached pieces.
A nut and bolt arrangement has two winding inclined planes. One inclined plane is on the external side of a metal cylinder and is called the bolt. The other inclined plane is on the inner side of a hollow metal cylinder and is called the nut. When the nut is given a circular motion over the bolt, it moves up or down without slipping and can withstand a lot of load.
A cork screw is used for pulling out the cork from the bottles of ketchup or wine.
The screw jack used to lift automobiles for repairs works on the principle of a screw.
Maintenance and Care of Machines
The following points should be remembered about the maintenance of machines.
Machines should be protected from dust to prevent their wear and tear. So when not in use, machines should be kept covered.
To avoid rusting due to exposure to moisture, the non-movable iron parts of a machine should be painted.
To reduce friction, the moving parts of a machine should be regularly lubricated.
Example
Why do we have three types of levers?
A rigid bar resting on a pivot that is used to move a heavy or firmly fixed load with one end when pressure is applied to the other is called alever. For example, imagine the incline in your parking lot that will take to your parking space in the garage.
Depending on the location of the load, the fulcrum, and the force levers are classified into three classes. They are:
First class lever: It has a fulcrum between the force and the load, like a crowbar.
Second class lever: It has a load between the fulcrum and the force, like a wheelbarrow.
Third class lever: This lever has the force between the fulcrum and the load, like the arm on a human.
Have you ever wondered why refrigerator magnets stick to the refrigerator door easily?
Magnets stick to the refrigerator door because, beneath the paint, the door is made of steel. A magnet is attracted to steel and hence sticks to the door.
A long time ago ancient people knew about the rocks that attracted metals. They were called lodestones.
The Greeks were the first to discover the phenomenon of magnetism about 4000 thousand years ago. A Greek shepherd named Magnes discovered a natural magnetic rock. This rock which had a compound of iron called magnetite was able to attract metals.
Ancient Chinese and Indians also knew about magnets.
In this chapter, you will learn about magnets, types and features of a magnet You will also learn about the properties and uses of magnets.
Magnetic Materials
Magnetic materials are those that are attracted by a magnet.
Examples: Steel, iron etc.
Non-Magnetic Materials
Non-magnetic materials are those that are not attracted by magnets. Do you know why the coins that we use are not attracted by a magnet? That s because, various metals are mixed together to make coins that are non-magnetic.
Examples: Paper, leather etc.
Magnetic Poles
Magnetic poles are the tips of a magnet and contain the highest magnetic strength. When a magnet is freely suspended, the tips point towards the north and south.
The tip pointing towards the geographical south is known as its south pole. The tip pointing towards the geographical north is known as the north pole.
Types of Magnets
Magnets are of the following two types, natural magnets and artificial magnets.
A natural magnet is a naturally occurring substance with magnetic properties. Example: Magnetite.
An artificial magnet is a substance into which magnetic properties are artificially induced.
Examples: Magnets made of iron, cobalt, nickel etc.
Artificial magnets can come in various shapes such as a dumb-bell shaped magnet, bar magnet, U-shaped magnet, cylindrical magnet, magnetic needle, etc.
Artificial magnets are beneficial compared to natural magnets as they can be made in any desired shape.
In addition to this, artificial magnets can be made very powerful that is not possible with natural magnets.
Magnets made of an alloy that consists of aluminium, nickel, and cobalt (ALNICO) added to iron are the strongest magnets.
Magnets also classified as temporary or permanent based on their capacity to retain magnetism.
Temporary magnets are those magnets which cannot retain their magnetism for a long time.
Examples: Pure iron (Soft iron) and electromagnets that are made by passing an electric current through an iron piece.
Permanent magnets are those magnets which retain their magnetism long after removal of the magnetising force.
Example: Magnets made from steel (carbon + pureiron).
Uses of a magnet:
Magnets can be made into different size, shape and strength, based on their use. Devices such as TVs, loudspeakers, radios, telephones etc. make use of magnets.
Properties of Magnets
A magnet attracts other magnetic material towards itself.
Like poles repel each other. and unlike poles attract each other.
A freely suspended bar magnet always aligns in the north-south direction.
When a bar magnet is rubbed on an iron bar, the iron bar is converted into a magnet.
There is no magnet with a single pole. Even when a magnet is cut into two pieces, each piece will behave as an independent magnet, with two poles, a north pole and a south pole.
Compass
Have you seen a compass? There must be one in the physics lab of your school. Most smartphones also have a compass. Ask your parent or neighbour to show you how the compass works.
As the needle in a traditional compass always points in the direction of the north, it is easy to find other directions easily. Let us see how the compass works.
A compass is an instrument with a thin magnetic needle supported from a pivot.
There is a round dial on the outer edge of the compass that marks the directions of North, South, East and West.
The needle is positioned on a dial with marked directions.
The north pole of the magnetic needle is painted with red colour.
The magnetic needle in the compass always points towards the north-south direction.
With the proper alignment of the dial, the directions can be found.
An airtight box contains the entire assembly.
In the olden days, an old pointing device namely the south-pointing fish was used to find the directions. In this instrument, the fish head was pointed towards the south.
Storage of Magnets
A magnet gets demagnetised when left by itself for a long period of time. In other words, the magnet loses its magnetic property. This can be avoided by storing them between soft iron pieces also known as keepers when a magnet is not used. The Arrangement of bar magnets in pairs such that the opposite poles face each other and keeping two soft iron pieces at the two tips of the pair of magnets avoids demagnetisation of bar magnets.
How to Protect Magnets from Losing Their Magnetic Properties
Do not:
Drop magnets from a height.
Heat a magnet.
Hammer a magnet.
Keep certain items such as DVD’s, debit cards, CD’s, audio and video cassettes, credit cards or ATM cards, and mobile phones which contain magnetic material, away from magnets to prevent damage.
Uses of Magnets
Magnets are used in:
In stickers, magnetic toys, refrigerator doors, etc.
Making magnetic compasses that help sailors and navigators to know directions.
For separating iron from ores containing other non-magnetic substances.
Removing tiny iron pieces that have accidentally fallen into the patient’s eye by eye doctors.
Electromagnets are used in:
Generators, motors, loudspeakers, telephones, TV sets, fans, mixers, electric bells, etc.
Cranes to lift heavy iron bars and to separate iron objects from scrap.
In mathematics, a percentage is a number or ratio that can be expressed as a fraction of 100. If we have to calculate percent of a number, divide the number by whole and multiply by 100. Hence, the percentage means, a part per hundred. The word per cent means per 100. It represented by the symbol “%”.
Examples of percentages are:
10% is equal to 1/10 fraction
20% is equivalent to ⅕ fraction
25% is equivalent to ¼ fraction
50% is equivalent to ½ fraction
75% is equivalent to ¾ fraction
90% is equivalent to 9/10 fraction
Percentages have no dimension. Hence it is called a dimensionless number. If we say, 50% of a number, then it means 50 per cent of its whole.
Percentages can also be represented in decimal or fraction form, such as 0.6%, 0.25%, etc. In academics, the marks obtained in any subject are calculated in terms of percentage. Like, Ram has got 78% of marks in his final exam. So, this percentage is calculated on account of total marks obtained by Ram, in all subjects to the total marks.
Percentage Formula
To determine the percentage, we have to divide the value by the total value and then multiply the resultant to 100.
Percentage formula = (Value/Total value)×100
Example: 2/5 × 100 = 0.4 × 100 = 40 per cent
How to calculate the percentage of a number?
To calculate the percentage of a number, we need to use a different formula such as:
P% of Number = X
where X is the required percentage.
If we remove the % sign, then we need to express the above formulas as;
P/100 * Number = X
Example: Calculate 10% of 80.
Let 10% of 80 = X
10/100 * 80 = X
X = 8
Percentage Difference Formula
If we are given with two values and we need to find the percentage difference between the two values, then it can be done using the formula:
For example, if 20 and 30 are two different values, then the percentage difference between them will be:
% difference between 20 and 30 =
Percentage Increase and Decrease
The percentage increase is equal to the subtraction of original number from a new number, divided by the original number and multiplied by 100.
% increase = [(New number – Original number)/Original number] x 100
where,
increase in number = New number – original number
Similarly, percentage decrease is equal to subtraction of new number from original number, divided by original number and multiplied by 100.
% decrease = [(Original number – New number)/Original number] x 100
Where decrease in number = Original number – New number
So basically if the answer is negative then there is percentage decrease.
Solved Example
Two quantities are generally expressed on the basis of their ratios. Here, let us understand the concepts of percentage through a few examples in a much better way.
Examples: Let a bag contain 2 kg of apples and 3kg of grapes. Find the ratio of quantities present, and percentage occupied by each.Solution: The number of apples and grapes in a bag can be compared in terms of their ratio, i.e. 2:3.The actual interpretation of percentages can be understood by the following way:The same quantity can be represented in terms of percentage occupied, which is given as:Total quantity present = 5 kgRatio of apples (in terms of total quantity) =
From the definition of percentage, it is the ratio that is expressed per hundred,
Thus, Percentage of Apples
Percentage of Grapes
Percentage Chart
The percentage chart is given here for fractions converted into percentage.
Fractions
Percentage
1/2
50%
1/3
33.33%
1/4
25%
1/5
20%
1/6
16.66%
1/7
14.28%
1/8
12.5%
1/9
11.11%
1/10
10%
1/11
9.09%
1/12
8.33%
1/13
7.69%
1/14
7.14%
1/15
6.66%
Converting Fractions to Percentage
A fraction can be represented by
Multiplying and dividing the fraction by 100, we have
From the definition of percentage, we have
Thus equation (i) can be written as:
Thus fraction can be converted to percentage simply by multiplying the given fraction by 100. Also, read: Ratio To Percentage
Percentage Questions
Q.1: If 16% of 40% of a number is 8, the number is?
Solution: Let the required number be X.
Therefore, as per the given question,
(16/100) x (40/100) x X = 8
So, X = (8 x 100 x 100) / (16 x 40)
= 125
Q.2: What percentage of 2/7 is 1/35 ?
Solution: Suppose X% of 2/7 is 1/35
∴ (2/7 x X) / 100 = 1/35
⇒ X = 1/35 x 7/2 x 100
= 10%
Q.3: Which number is 40% less than 90 ?
Solution: Required number = 60% of 90
= (90 x 60)/100
= 54
Therefore, the required number is 54.
Q.4: The sum of (16% of 24.2) and (10% of 2.42) is equal to what value?
Solution: As per the given question ,
Sum = (16% of 24.2) + (10% of 2.42)
Required value = (24.2 × 16)/100 + (2.42 × 10)/100
Required value = 3.872 + 0.242
Therefore, required value = 4.114
Word Problems
Q.1: A fruit seller had some apples. He sells 40% apples and still has 420 apples. Originally, he had how many apples?
Solution: Let he had N apples, originally.
Now as per the given question,
(100 – 40)% of N = 420
⇒ (60/100)x N = 420
⇒ N = (420 x 100/60) = 700
Q.2: Out of two numbers, 40% of the greater number is equal to 60% of the smaller. If the sum of the numbers is 150, then the greater number is?
Solution: Let us assume, greater number be X.
∴ Smaller number = 150 – X
According to the question,
(40 x X)/100 = 60(150 – X)/100
⇒ 2p = 3 × 150 – 3X
⇒ 5X = 3 × 150
⇒ X = 90
Difference between Percentage and Percent
The word percentage and percent are related closely to each other.
Percent ( or symbol %) is accompanied by a specific number.
E.g., More than 75% of the participants responded with their positive response to abjure.
The percentage is represented without a number.
E.g., The percentage of the population affected by malaria is between 60% and 65%.
Fractions, Ratios, Percents and Decimals are interrelated with each other. Let us look on to the conversion of one form to other:
S.no
Ratio
Fraction
Percent(%)
Decimal
1
1:1
1/1
100
1
2
1:2
1/2
50
0.5
3
1:3
1/3
33.333
0.3333
4
1:4
1/4
25
0.25
5
1:5
1/5
20
0.20
6
1:6
1/6
16.667
0.16667
7
1:7
1/7
14.285
0.14285
8
1:8
1/8
12.5
0.125
9
1:9
1/9
11.111
0.11111
10
1:10
1/10
10
0.10
11
1:11
1/11
9.0909
0.0909
12
1:12
1/12
8.333
0.08333
13
1:13
1/13
7.692
0.07692
14
1:14
1/14
7.142
0.07142
15
1:15
1/15
6.66
0.0666
Percentage in Maths
Every percentage problem has three possible unknowns or variables :
Percentage
Part
Base
In order to solve any percentage problem, you must be able to identify these variables.
Look at the following examples. All three variables are known:
Example: 70% of 30 is 21
70 is the percentage.
30 is the base.
21 is the part.
Example: 25% of 200 is 50
25 is the percent.
200 is the base.
50 is the part.
Example: 6 is 50% of 12
6 is the part.
50 is the percent.
12 is the base.
Percentage Tricks
To calculate the percentage, we can use the given below tricks.
x % of y = y % of x
Example- Prove that 10% of 30 is equal to 30% of 10.
Solution- 10% of 30 = 3
30% of 10 = 3
Therefore they are equal i.e. x % of y = y % of x holds true.
Marks Percentage
Students get marks in exams, usually out of 100. The marks are calculated in terms of per cent. If a student has scored out of total marks, then we have to divide the scored mark from total marks and multiply by 100. Let us see some examples here:
Marks obtained
Out of Total Marks
Percentage
30
100
30%
10
20
50%
23
50
46%
13
40
32.5%
90
120
75%
Problems and Solutions
Example– Suman has a monthly salary of $1200. She spends $280 per month on food. What percent of her monthly salary does she save?Solution– Suman’s monthly salary = $1200Savings of Suman = $(1200 – 280) = $ 920 Fraction of salary she saves =
Percentage of salary she saves
Example- Below given are three grids of chocolate. What percent of each White chocolate bar has Dark chocolate bar?Solution- Each grid above has 100 white chocolate blocks. For each white chocolate bar, the ratio of the number of dark chocolate boxes to the total number of white chocolate bars can be represented as a fraction.(i) 0 dark and 100 white.i.e. 0 per 100 or 0%.(ii) 50 dark and 50 white.I.e. 50 per 100 or 50%.(iii) 100 dark and 0 white.I .e., 100 per 100 or 100%.
Frequently Asked Questions – FAQs
What do you mean by percentage?
In maths, a percentage is a value or ratio that shows a fraction of 100. Percent means per 100. It does not have any unit.
What is the symbol of percentage?
Percentage is denoted by ‘%’ symbol. It is also termed as per cent.
What is the percentage formula?
The formula to calculate percentage of a number out of another number is: Percentage = (Original number/Another number) x 100
The idea of speed, time, distance continues the same, however, the type of problems asked in the examinations may have variations in terms of data given and asked.
Frequently, one-two word problems are proposed based on speed, time, and distance with variation but candidates must also retain themselves ready to see questions on data sufficiency and data interpretation that are based on TDS (i.e time, distance and speed) topic. Let us start our discussion with the definition of speed, time and distance.
Speed
It refers to the rate at which a particular distance is covered by an object in motion.
Time
It refers to an interval separating two events.
Distance
It refers to the extent of space between two points.
Units of Speed Time & Distance
Each of the speed, distance and time can be represented in different units:
Time can be generally expressed in terms of seconds(s), minutes (min) and hours (hr).
Whereas the distance is generally expressed in meters (m), kilometres (km), centimetres, miles, feet, etc.
Speed is commonly expressed in m/s, km/hr.
For example, if the distance is given in km and time in hr, then as per the formula:
Speed = Distance/ Time; the units of speed will become km/ hr.
Relationship Between Speed, Time & Distance
Now that we are well aware of the definition of speed, distance and time let us understand the relationship between them. It is said that an object attains motion or movement when it changes its position with respect to any external stationary point. Speed, Time and Distance are the three variables that represent the mathematical model of motion as, s x t = d.
Time is directly proportional to distance. It means that speed remains constant, if we have two vehicles moving two distances for two different time duration then the time is directly proportional to the distance.
Speed is directly proportional to distance. It means that time remains constant if we have two vehicles moving two distances at two different speeds respectively.
Speed is inversely proportional to time. It means that distance remains constant if we have two vehicles moving at two different speeds and taking times respectively.
In mathematical format:
The formula for speed calculation is:
Speed = Distance/Time
This shows us how slow or fast a target moves. It represents the distance covered divided by the time needed to cover the distance.
Speed is directly proportional to the given distance and inversely proportional to the proposed time. Hence,
Distance = Speed x Time, and
Time = Distance / Speed, since as the speed grows the time needed will decrease and vice versa.
Speed, Time & Distance Conversions
Another important concept is the conversion of speed, distance and time into various units as discussed below:
To convert a given data from km/hour to m/sec, we multiply by 5/18. As, 1 km/hour = 5/18 m/sec.
To convert a given data m/sec to km/hour, we multiply by 18/5. As, 1 m/sec = 18/5 km/hour = 3.6 km/hour.
In terms of formula, we can list it as:
x km/hr=x×518m/sec x km/hr=x×518m/sec
x m/sec =x×185 km/hrx m/sec =x×185 km/hr
Similarly some other conversion are listed below:
1 km/hr = 5/8 miles/hour
1 yard = 3 feet
1 kilometer= 1000 meters
1 mile= 1.609 kilometer
1 hour= 60 minutes= 3600 seconds
1 mile = 1760 yards
1 yard = 3 feet
1 mile = 5280 feet
Application of Speed, Time & Distance
The topic speed, time and distance topics tell us about the basis of the variation of questions asked in the examination. Let us understand some of their major application:
Average Speed
The average speed is determined by the formula = (Total distance travelled)/(Total time taken)
Average speed=d1+d2+d3⋯dnt1+t2+t3⋯tnAverage speed=d1+d2+d3⋯dnt1+t2+t3⋯tn
Sample 1 – When the distance travelled is constant and two speed is given then:
Average speed = 2xyx+y2xyx+y;
Where x and y are the two speeds at which the corresponding distance has been reached.
Sample 2 – When the time taken is constant average speed is calculated by the formula:
Average speed =(x+y)2(x+y)2;
Where x and y are the two speeds at which we covered the distance for the identical time.
Solved Example: An individual drives from one place to another at 40 km/hr and returns at 160 km/hr. If the complete time needed is 5 hours, then obtain the distance.
Solutions:
Here the distance is fixed, so the time taken will be inversely proportional to the speed. The ratio of speed is given as 40:160, i.e. 1:4.
Therefore the ratio of time taken will be 4:1.
Total time is practised = 5 hours; therefore the time taken while travelling is 4 hours and returning is 1 hour.
Hence, distance = 40x 4 = 160 km.
If the first part of any given distance is covered at a rate of v1 in time t1 and the second part of the distance is covered at a rate v2 in time t2 then the average speed is given by the formula:
Average speed = (v1t1+v2t2)t1+t2(v1t1+v2t2)t1+t2
Relative Speed
As the name suggests the idea is about the relative speed between two or more things. The basic concept in relative speed is that the speed gets combined in the case of objects moving in the opposite direction to one another. And the speed gets subtracted for the case when objects are moving in the identical direction.
For example, if two passenger trains are moving in the opposite direction with a speed of X km per hour and Y kilometre per hour respectively. Then their relative speed is given by the formula:
Relative speed=X + Y
On the other hand, if the two trains are travelling in the same direction with the speed of X km per hour and Y kilometre per hour respectively. Then their relative speed is given by the formula:
Relative speed=X -Y
For the first case time taken by the train in passing each other is given by the formula:
Relative speed=X + Y
Time taken= L1+L2X+YL1+L2X+Y
For the second case the time taken by the trains in crossing each other is given by the formula:
Relative speed=X -Y
Time taken= L1+L2X−YL1+L2X−Y
Here L1, L2L1, L2 are the lengths of the trains respectively.
Inverse Proportionality of Speed & Time
Speed is said to be inversely proportional to time when the distance is fixed. In mathematical format, S is inversely proportional to 1/T when D is constant. For such a case if the speeds are in the ratio m:n then the time taken will be in the ratio n:m.
There are two approaches to solve questions:
Applying Inverse Proportionality
Applying Constant Product Rule
Example:. After moving 100km, a train meets with an accident and travels at (34)th(34)th of the normal speed and reaches 55 min late. Had the accident occurred 20 km further on it would have arrived 45 min delayed. Obtain the usual Speed?
Solutions:
Applying Inverse Proportionality Method
Here there are 2 cases
Case 1: accident happens at 100 km
Case 2: accident happens at 120 km
The difference between the two incidents is only for the 20 km between 100 km and 120 km. The time difference of 10 minutes is just due to these 20 km.
In case 1, 20 km between 100 km and 120 km is covered at (34)th(34)th speed.
In case 2, 20 km between 100 km and 120 km is reached at the usual speed.
So the usual time “t” taken to cover 20 km, can be found as follows. 4/3 t – t = 10 mins = > t = 30 mins, d = 20 km
so the usual speed = 20/30min = 20/0.5 = 40 km/hr
Using Constant Product Rule Method
Let the actual time taken be equal to T.
There is a (1/4)th reduction in speed, this will result in a (1/3)rd increase in time taken as speed and time are inversely proportional to one another.
A 1/x increment in one of the parameters will result in a 1/(x+1) reduction in the other parameter if the parameters are inversely proportional.
The delay due to this reduction is 10 minutes
Thus 1/3 T= 10 and T=30 minutes or 0.5 hour
Also, Distance = 20 km
Thus Speed = 40 kmph
Meeting Point Question
If two individuals travel from two locations P and Q towards each other, and they meet at point X. Then the total distance traversed by them at the meeting will be PQ. The time taken by both of them to meet will be identical.
As the time is constant, the distances PX and QX will be in the ratio of their speed. Assume that the distance between P and Q is d.
If two individuals are stepping towards each other from P and Q respectively, when they meet for the first time, they collectively cover a distance “d”. When they meet each other for the second time, they mutually cover a distance “3d”. Similarly, when they meet for the third time, they unitedly cover a distance of “5d” and the process goes on.
Take an example to understand the concept:
Example:. Ankit and Arnav have to travel from Delhi to Hyderabad in their respective vehicles. Ankit is driving at 80 kmph while Arnav is operating at 120 kmph. Obtain the time taken by Arnav to reach Hyderabad if Ankit takes 9 hrs.
Solutions:
As we can recognise that the distance covered is fixed in both cases, the time taken will be inversely proportional to the speed. In the given question, the speed of Ankit and Arnav is in ratio 80: 120 or 2:3.
Therefore the ratio of the time taken by Ankit to that taken by Arnav will be in the ratio 3:2. Hence if Ankit takes 9 hrs, Arnav will take 6 hrs.
Speed, Time and Distance Formulas
Below are some important speed, distance and time formulas that would be helpful to ease the calculation in the various exams.
Terms
Formula
Speed
Speed=DistanceTimeSpeed=DistanceTime
Time
Time=DistanceSpeedTime=DistanceSpeed
Distance
D = (Speed x Time)
Average Speed
Average Speed=Total distance travelledTotal time takenAverage Speed=Total distance travelledTotal time taken
Average Speed(when the distance travelled is constant)
2xyx+y2xyx+y
Relative speed(If two trains are moving in the opposite direction)
Relative speed=X + YTime taken= L1+L2X+YL1+L2X+YHere L1, L2L1, L2 are the lengths of the trains.
Relative speed(If two trains are moving in the same direction)
Relative speed=X -YTime taken= L1+L2X−YL1+L2X−YHere L1, L2L1, L2 are the lengths of the trains.
Some additional formulas are:
If the ratio of the speeds of P and Q is p: q, then the ratio of the times used by them to reach the same distance is 1/p:1/q or q: p.
If two individuals or automobiles or trains start at the exact time in the opposing direction from two points say A and B and after crossing each other they take time a and b respectively to finish the journey then the speed ratio is provided by the formula:
Speed of firstSpeed of second=ba−−√Speed of firstSpeed of second=ba
If two individuals with two different speeds x and y cover the same distance and travel in opposite directions. Where the total time is given and distance is asked then the formula is:
Distance=xyx+y×Total TimeDistance=xyx+y×Total Time
Types of Questions from Speed, Time and Distance
There are some specific types of questions from Speed, time and distance that usually come in exams. Some of the important types of questions from speed, distance and time are as follows.
(a) Problems related to Trains
Please note that, in the case of the train problems, the distance to be covered when crossing an object is equal to, Distance to be covered = Length of train + Length of object.
Remember that, in case the object under consideration is a pole or a person or a point, we can consider them to be point objects with zero length. It means that we will not consider the lengths of these objects. However, if the object under consideration is a platform (non point object), then its length will be added to the formula of the distance to be covered.
(b) Boats and Streams
In such problems boats travel either in the direction of stream or in the opposite di- rection of stream. The direction of boat along the stream is called downstream and the direction of boat against the stream is called upstream.
If the speed of a boat in still water is u km/hr and the speed of the stream is v km/hr, then:
1) Speed downstream = (u + v) km/hr
2) Speed upstream = (u – v) km/hr
Once you’ve mastered Speed, Time and Distance, Also, learn more about Ratio and Proportion concepts in depth!
How to Solve Question Based on Speed, Time and Speed- Know all Tips and Tricks
Students can find different tips and tricks from below for solving the questions related to speed, time and distance.
Tip # 1: Relative speed is defined as the speed of a moving body with respect to another body. The possible cases of relative motion are, same direction, when two bodies are moving in the same direction, the relative speed is the difference between their speeds and is always expressed as a positive value. On the other hand, the opposite direction is when two bodies are moving in the opposite direction, the relative speed is the sum of their speeds.
Tip # 2: Average speed = Total Distance / Total Time
Tip # 3: When train crossing a moving body,
When a train passes a moving man/point object, the distance travelled by the train while passing it will be equal to the length of the train and relative speed will be taken as
1) If both are moving in same direction then relative speed = Difference of both speeds
2) If both are moving in opposite direction then relative speed = Addition of both speeds
Tip # 4: Train Passing a long object or platform, when a train passes a platform or a long object, the distance travelled by the train, while crossing that object will be equal to the sum of the length of the train and length of that object.
Tip # 5: Train passing a man or point object, when a train passes a man/object, the distance travelled by the train while passing that object, will be equal to the length of the train.
When you’ve finished with Speed, Time and Distance, you can read about Algebraic Identities concepts in depth here!
Speed, Time and Distance Solved Questions
Some of the solved questions regarding the topic for more practice are as follows:
Question 1: The speed of three cars are in the ratio 5 : 4 : 6. The ratio between the time taken by them to travel the same distance is
Solution: Ratio of time taken = ⅕ : ¼ : ⅙ = 12 : 15 : 10
Question 2: A truck covers a distance of 1200 km in 40 hours. What is the average speed of the truck?
Solution: Average speed = Total distance travelled/Total time taken
⇒ Average speed = 1200/40
∴ Average speed = 30 km/hr
Question 3: A man travelled 12 km at a speed of 4 km/h and further 10 km at a speed of 5 km/hr. What was his average speed?
Solution: Total time taken = Time taken at a speed of 4 km/h + Time taken at a speed of 5 km/ h
⇒ 12/4 + 10/5 = 5 hours [∵ Time = Distance/Speed] Average speed = Total distance/Total time
⇒ (12 + 10) /5 = 22/5 = 4.4 km/h
Question 4: Rahul goes Delhi to Pune at a speed of 50 km/h and comes back at a speed of 75 km/h. Find his average speed of the journey.
Question 5: Determine the length of train A if it crosses a pole at 60km/h in 30 sec.
Solution: Given, speed of the train = 60 km/h
⇒ Speed = 60 × 5/18 m/s = 50/3 m/s
Given, time taken by train A to cross the pole = 30 s
The distance covered in crossing the pole will be equal to the length of the train.
⇒ Distance = Speed × Time
⇒ Distance = 50/3 × 30 = 500 m
Question 6: A 150 m long train crosses a 270 m long platform in 15 sec. How much time will it take to cross a platform of 186 m?
Solution: In crossing a 270 m long platform,
Total distance covered by train = 150 + 270 = 420 m
Speed of train = total distance covered/time taken = 420/15 = 28 m/sec In crossing a 186 m long platform,
Total distance covered by train = 150 + 186 = 336 m
∴ Time taken by train = distance covered/speed of train = 336/28 = 12 sec.
Question 7: Two trains are moving in the same directions at speeds of 43 km/h and 51 km/h respectively. The time taken by the faster train to cross a man sitting in the slower train is 72 seconds. What is the length (in metres) of the faster train?
Solution: Given: The speed of 2 trains = 43 km/hr and 51 km/hr Relative velocity of both trains = (51 – 43) km/hr = 8 km/hr Relative velocity in m/s = 8 × (5/18) m/s
⇒ Distance covered by the train in 72 sec = 8 × (5/18) × 72 = 160 Hence, the length of faster train = 160 m
Question 8: How long will a train 100m long travelling at 72km/h take to overtake another train 200m long travelling at 54km/h in the same direction?
Solution: Relative speed = 72 – 54 km/h (as both are travelling in same direction)
= 18 km/hr = 18 × 10/36 m/s = 5 m/s
Also, distance covered by the train to overtake the train = 100 m + 200 m = 300 m Hence,
Time taken = distance/speed = 300/5 = 60 sec
Question 9: A boat takes 40 minutes to travel 20 km downstream. If the speed of the stream is 2.5 km/hr, how much more time will it take to return back?
Solution: Time taken downstream = 40 min = 40/60 = 2/3 hrs. Downstream speed = 20/ (2/3) = 30 km/hr.
As we know, speed of stream = 1/2 × (Downstream speed – Upstream speed)
An Algebraic Expression is the combination of constant and variables. We use the operations like addition, subtraction etc to form an algebraic expression.
Variable
A variable does not have a fixed value .it can be varied. It is represented by letters like a, y, p m etc.
Constant
A constant has a fixed value. Any number without a variable is a constant.
Example
1. 2x + 7
Here we got this expression by multiplying 2 and x and then add 7 to it.
In the above expression, the variable is x and the constant is 7.
2. y2
We get it by multiplying the variable y to itself.
Terms of an Expression
Terms
To form an expression we use constant and variables and separate them using the operations like addition, subtraction etc. these parts of expressions which we separate using operations are called Terms.
In the above expression, there are three terms, 4x, – y and 7.
Factors of a Term
Every term is the product of its factors. As in the above expression, the term 4x is the product of 4 and x. So 4 and x are the factors of that term.
We can understand it by using a tree diagram.
Coefficients
As you can see above that some of the factors are numerical and some are algebraic i.e. contains variable.The numerical factor of the term is called the numerical coefficient of the term.
In the above expression,
-1 is the coefficient of ab
2 is the coefficient of b2
-3 is the coefficient of a2.
Parts of an Expression
Here in the above figure, you can identify the terms, variables, constants and coefficients.
Like and Unlike Terms
Like Terms are the terms which have same algebraic factors. They must have the same variable with the same exponent.
Unlike Terms are the terms which have different algebraic factors.
2x2 + 3x – 5 does not contain any term with same variable.
2a2 + 3a2 + 7a – 7 contains two terms with same variable i.e. 2a2 and 3a2.so these are like terms.
Monomials, Binomials, Trinomials and Polynomials
Expressions
Meaning
Example
Monomial
Any expression which has only one term.
5x2, 7y, 3ab
Binomial
Any expression which has two, unlike terms.
5x2 + 2y, 2ab – 3b
Trinomial
Any expression which has three, unlike terms.
5x2 + 2y + 9xy, x + y – 3
Polynomial
Any expression which has one or more terms with the variable having non-negative integers as an exponent is a polynomial.
5x2 + 2y + 9xy + 4 and all the above expressions are also polynomial.
Remark: All the expressions like monomial, binomial and trinomial are also a polynomial.
Addition and Subtraction of Algebraic Expression
1. Addition of Like Terms
If we have to add like terms then we can simply add their numerical coefficients and the result will also be a like term.
Example
Add 2x and 5x.
Solution
2x + 5x
= (2 × x) + (5 × x)
= (2 + 5) × x (using distributive law)
=7 × x = 7x
2. Subtraction of Like Terms
If we have to subtract like terms then we can simply subtract their numerical coefficients and the result will also be a like term.
Example
Subtract 3p from 11p.
Solution
11p – 3p
= (11-3) p
= 8p
3. Addition of unlike terms
If we have to add the unlike terms then we just have to put an addition sign between the terms.
Example
Add 9y, 2x and 3
Solution
We will simply write it like this-
9y + 2x + 3
4. Subtraction of Unlike Terms
If we have to subtract the unlike terms then we just have to put minus sign between the terms.
Example
Subtract 9y from 21.
Solution
We will simply write it like this-
21 – 9y
5. Addition of General Algebraic Expression
To add the general algebraic expressions, we have to arrange them so that the like terms come together, then simplify the terms and the unlike terms will remain the same in the resultant expression.
While subtracting the algebraic expression from another algebraic expression, we have to arrange them according to the like terms then subtract them.
Subtraction is same as adding the inverse of the term.
Example
Subtract 4ab– 5b2 – 3a2 from 5a2 + 3b2 – ab
Solution
Finding the Value of an Expression
1. Expressions with One Variable
If we know the value of the variable in the expression then we can easily find the numerical value of the given expression.
Example
Find the value of the expression 2x + 7 if x = 3.
Solution
We have to put the value of x = 3.
2x + 7
= 2(3) + 7
= 6 + 7
= 13
2. Expressions with two or more variables
To find the value of the expression with 2 variables, we must know the value of both the variables.
Example
Find the value of y2 + 2yz + z2 if y = 2 and z = 3.
Solution
Substitute the value y = 2 and z = 3.
y2 + 2yz + z2
= 22 + 2(2) (3) + 32
= 4 + 12 + 9
= 25
Formula and Rules using Algebraic Expression
There are so many formulas which are made using the algebraic expression.
Perimeter Formulas
1. The perimeter of an equilateral triangle = 3l where l is the length of the side of the equilateral triangle by l and l is variable which can be varied according to the size of the equilateral triangle.
2. The perimeter of a square = 4l where l = the length of the side of the square.
3. The perimeter of a regular pentagon = 5l where l = the length of the side of the Pentagon and so on.
Area formulas
1. The area of the square = a2 where a is the side of the square
2. The area of the rectangle = l × b = lb where the length of a rectangle is l and its breadth is b
3. The area of the triangle = 1/2 × b × h where b is the base and h is the height of the triangle. Here if we know the value of the variables given in the formulas then we can easily calculate the value of the quantity.
Example
What is the perimeter of a square if the side of the square is 4 cm?
Solution
The perimeter of a square = 4l
l = 4 cm
4 × 4 = 16 cm
Rules for the Number Pattern
1. If we denote a natural number by n then its successor will always be (n + 1). If n = 3 then n + 1 will be 3 + 1 = 4.
2. If we denote a natural number by n then 2n will always be an even number and (2n + 1) will always be an odd number. If n = 3 then 2n = 2(3) = 6(even number), n = 3 then 2n + 1 = 2(3) + 1 = 7 (odd number)
3. If we arrange the multiples of 5 in ascending order then we can denote it by 5n. If we have to check that what will be the 11th term in this series then we can check it by 5n. n = 11 so 5n = 5(11) = 55.
Pattern in geometry
The number of diagonals which we can draw from one vertex of any polygon is (n – 3) where n is the number of sides of the polygon.
How many diagonals can be drawn from the one vertex of a hexagon?
The number of diagonals will be (n -3).
The number of sides in a hexagon is 8 so (n – 3) = (8 – 3) = 5
The main feature of algebra is the use of letters, which allow us to write rules and formulas in the general ways and one can talk about any number and not just a particular number. Letters may stand for unknown quantities, numbers, operation can be performed on them as numbers. Example : Find the value of y in the equation. 𝑦 7
3 = 0 Solution : 𝑦 7
3 = 0 Multiply by 7 in both sides, =7 ( 𝑦 7
3 ) = 0 x 7 =7 x 𝑦 7
7 x 3 = 0 =Y + 21 = 0 Subtract 21 in both sides = Y + 21 – 21 = 0 – 21 =Y = – 21 Hence , the value of y = -21
Solution- Each grid above has 100 white chocolate blocks. For each white chocolate bar, the ratio of the number of dark chocolate boxes to the total number of white chocolate bars can be represented as a fraction.(i) 0 dark and 100 white.i.e. 0 per 100 or 0%.(ii) 50 dark and 50 white.I.e. 50 per 100 or 50%.(iii) 100 dark and 0 white.I .e., 100 per 100 or 100%.
