DIRECT AND INVERSE VARIATIONS - PROBLEMS | EXAMPLES

Direct and inverse variations -  Unitary Method consists of two types of variations- DIRECT AND INVERSE. You can solve any question of the unitary method after reading my article-Problems/examples carefully.

Note- Inverse Variation is also called as Indirect Variation.

DIRECT AND INVERSE VARIATIONS - PROBLEMS & EXAMPLES

DIRECT AND INVERSE VARIATIONS

(PROBLEM / EXAMPLE )

TYPE - 1

Q1.) 15 stamps of equal value cost Rs. 9.00. How many stamps of same value can be bought for Rs              36.00?

Solution. Since the number of stamps and their cost are directly proportional to each other.
                Therefore, the ratio between the stamps=the ration between the corresponding costs.
                   Let the number of stamps=X
                  Stamps : Stamps = Cost:Cost
                     15 :   X             =      9 : 36
             
                    => 9 × x   =   15 × 36
                   =>  9X=15×36
                   =>  9X= 540
                 =>  x=540/9 = 60 STAMPS.
                    Hence the number of stamps=60.

Q2.) A machine takes 5 hours to cut 120 tools.How many tools will it cut in 20 hours? 
                     
Solution. Number of tools = 120 Time = 5 hours.
                Number of tools = ?     Time = 20 hours.
                 Since time and tools are directly proportional to each other.           
                  Therefore the ratio between the hours = ratio between the corresponding tools cut
                    by machines .
                    Let number of tools = X
                    Tools : Tools = Hours : Hours
                     120   :   X     =  5   :   20
                   OR
                   
                      =>  120 × 20  =  5 × x
                    =>   2400=5X
                    => 2400/5=X
                   =>   480=X
                          Thus the number of tools cut by the machine in 20 hours =480.

DIRECT AND INVERSE VARIATIONS

(PROBLEM / EXAMPLE )

 TYPE - 2

Q3.) Anupama takes 125 steps in walking a distance of 100metres. What distance would
          she covers in 315 steps?

Solution. Let the distance covered = X metres.
                                      STEPS            DISTANCE( IN METERS)
                                       125                          100
                                       315                           X
                 Hence the number of steps taken and distance covered is directly proportional
                  to each other.
      Hence, the ratio between steps taken = the ratio between the corresponding distance covered
                       Steps : Steps   =   Distance : Distance
                        125  :  315      =    100  :   X
                        OR      

                              By cross-multiply we have
                              125 ×  X = 100  × 315
                           125X= 31500
                                 X=31500/125
                                 X=252 metres.

Q4.) Suneeta types 1080 words in one hour. What is her GWAM(gross words a minute)                     rate?
 Solution: Let her GWAM = X words a minute.
                     MINUTES               WORDS
               one hour= 60min            1080
                     1min                            X 
                   Because the time and words typed are directly proportional to each other.
                 Therefore the ratio between minutes = the ratio between corresponding words
                       TIME-MINUTES: TIME-MINUTES =  WORDS: WORDS
                                   60 : 1     = 1080 : x     
                        OR
                                
                                   
                                  By cross multiplying we have,
                                 60 × x = 1080  × 1
                                  60X=1080
                                      X=1080/60
                                      X=18
                               Thus her GWAM = 18 words a minute.

   

REMEMBER:-   A) Pairs of terms above and below are in the same ratio.   B) The ratio of any two terms in the row is same as the ratio corresponding terms in the other row.

DIRECT AND INVERSE VARIATIONS

(PROBLEM / EXAMPLE )

TYPE - 3

Q5.) Shalu cycles to her school at an average speed of 12 km/h. It takes her 20 minutes to
       reach the school. If she wants to reach her school in 15 minutes, what should be
        her average speed?

Solution. Let her average speed = X km/h
                  SPEED( in km/h)        Time (in minutes)
                          12                                  20
                          X                                   15
              Here speed and time vary inversely.
                therefore this is the question of inverse variation.
                Hence, 
                 
                            By cross-multiplication, we have
                                   15 × X = 12 × 20
                                15X= 240
                                    X=240/15
                                    X=16
                          Thus her average speed = 16km/h.   

Q6.) A shopkeeper has just enough money to buy 52 cycles worth Rs.525.00 each. If 
       Each cycle cost Rs. 21.00 more, then how many cycles would he be 
      able to buy with that amount of money?

Solution: Let the required number of cycles be X.
                 NUMBER OF CYCLES                COST OF CYCLE(Rs)
                             52                                         525
                              X                                         546(new cost of cycles = 525+21=Rs546)
                Because number of cycles that can be purchased and cost of cycles 
               Vary inversely.
                        Therefore this is the question of inverse variation.
                  Hence, 
                     
                          546 ×  X= 525 × 52
                            546X=27300 
                                X=27300/546
                                X = 50
                     Thus the number of cycles that can be purchased =50.


Since this is my second article( First article link: Unitary Method in Maths)of the unitary method under- Direct and inverse variation- problems and examples.

For more problems and examples visit

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Different Types of Angles-Adjacent, Acute, Right, Obtuse, Straight, Reflex

Have you started learning lines and angles chapter read basics of Different types of angles-Acute, Right, Obtuse, Straight, Reflex, vertically opposite,co-interior, complementary and supplementary angles. To know more about lines and angles read my article carefully.

Lines and Angles Introduction

Ray:- It is a straight line which starts from a fixed point and moves in the same direction.The given figure shows a ray AB with initial fixed point A and moving in direction AB.

LINE SEGMENT:- It is a straight line with its both ends fixed. The given figure shows a line segment, whose both ends A and B are fixed.

ANGLE:- An angle is formed when two line segments or two rays have a common endpoint. The two line segments forming an angle are called the arms of the angle whereas their common endpoint is called the vertex of the angle. The adjacent figure represents an angle ABC.  AB and BC are the arms of the angle and their common point B is the vertex.

Different Types of Angles-Adjacent, Acute, Right, Obtuse, Straight, Reflex

     Angles About a Point

: If a number of angles are formed about a point, their sum is always 360.

In the given figure :

Angle AOB +BOC+COD+DOE+EOA=360

  Adjacent Angles

Two angles are said to be

Adjacent angles if:-

a)      They have common vertex,

b)      They have common arm and

c)       The other two arms of the two angles lie on opposite sides of the common arm.

The adjoining figure shows a pair of adjacent angles

Because (a) they have common vertex (O).

                (b) they have a common arm (OB) and

               (c) the other arms OA and OC of the two angles are

                     On opposite sides of the common arm OB.

         

  Vertically Opposite Angles

 When two straight lines intersect each other four angles are formed.

              The pair of angles which lie on the opposite sides of the point of intersection are called vertically opposite angles.

In the adjoining figure, two straight lines AB and CD intersect each other at point O. Angles AOD and BOC forms

one pair of vertically opposite angles, whereas angles AOC and BOD form another pair of vertically opposite angles.

    Vertically opposite angles are always equal.

     i.e. angle AOD= angle BOC

           angle AOC = angle BOD

Complementary And SupplementaryAngles

1.       Two angles are called complementary angles, if their sum is one right angle, i.e., 90. Each angle is called the complement of the other.

   e.g. 20 and 70 are complementary angles because 20 + 70 = 90

Clearly, 20 is the complement of 70 and 70 is the complement of 20

Thus the complement of angle 53= 90-53=37

2.       Two angles are called supplementary angles if their sum is two right angles, i.e. 180. Each angle is called the supplement of the other.

e.g. 30 and 150 are supplementary angles because 30 + 150 = 180

Clearly, 30 is the supplement of 150 and vice versa

Thus the supplement of 105 = 180 – 105 = 75.

You can also Read- Basics of Unitary Method.

This lecture is also available on Google YouTube This is all about the basics of Different Types of Angles-Adjacent, Acute, Right, Obtuse, Straight, Reflex.

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Angles Formed by Parallel Lines and Transversals

You are studying in any school grade !!! Then remember you can never get rid of yourself,  Angles/Parallel Lines/Transversal concepts are used in most of the chapters of mathematics. Even in various competition exams after completing school you will get questions on this topic. Let's discuss transversal first.

Concept of Transversal - Transversal Meaning

A straight line which cuts two or more given straight lines is called a transversal.

In the adjoining figure, PQ cuts straight lines AB and CD, and so it is a transversal.

When a transversal cuts two given straight lines, the following pairs of angles are formed.

Transversal Lines and Angles- Examples

1.       Two pairs of interior alternate angle:

Angle marked 1 and 2 form one pair of interior alternate angles, while angle marked 3 and 4 form another pair of interior alternate angle.

2.       Two pairs of exterior alternate angle:

Angles marked 5 and 8 form one pair, while angle marked 6 and 7 form the other pair of exterior alternate angles.

3.       Four pairs of the corresponding angle:

Angle marked 3 and 6, 1 and 5, 8 and 2, 7 and 4 form four pairs of corresponding angles.

4.       Two pairs of allied or co-interior or conjoined angle:

Angle marked 3 and 2 form one pair of allied angle and angle marked 1 and 4 form another pair of allied angles.

Parallel Lines and Angles

Two straight lines are said to be parallel, if they do not meet anywhere, no matter how long they are produced in any direction.

The adjacent figure shows two parallel lines AB and CD.

When two parallel lines AB and CD are cut by a transversal PQ:-

alternate interior angles definition geometry

Angles which makes Z-SHAPE are called alternate angles.

1.       Interior and exterior alternate angles are equal:

i.e.∠ 3 = ∠6 (interior alternate)

  ∠ 4 = ∠ 5(interior alternate)

   ∠1 = ∠8((exterior alternate )

   ∠2 = ∠ 7(exterior alternate )

Corresponding Angles

2.       Corresponding angles are equal:

Angles lying on the same side of the transversal are called corresponding angles.

i.e. ∠ 1 = ∠5, ∠ 2 = ∠6, ∠3 = ∠7 and ∠ 4 = ∠ 8.

3.       Co-interior or allied angles are supplementary:

i.e. ∠ 3 + ∠ 5=180 and  ∠4 +  ∠6 =180.

Co-interior angles make H-SHAPE. Or we can say a closed box open at one end.

Conditions of Parallelism

If two straight lines are cut by a transversal such that:

1.      A pair of alternate angles are equal, or

2.      A pair of corresponding angles are equal, or

3.      The sum of interior angles on the same side of transversal is 180, then two straight lines are parallel to each other.

Therefore, in order to prove that the given lines are parallel; show either alternate angles are equal or, corresponding angles are equal or, the co-interior angles are supplementary.

For more watch the video:

                   
This is all about the basics of  Angles Formed by Parallel Lines and Transversals.

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Unitary Method in Maths: Basic Concept | Formula | Example | Problems

The basic understanding of the concept of unitary method is very important because from school level to any type of competition level unitary concept questions are always on your way. This Method is important and always preferred.

What is Unitary Method?

The method in which the value of a unit quantity is first calculated to get the value of any quantity.

We come across two types of variations:-

Type of variations	Condition	Examples
Direct variation	Increase in one quantity causes an increase in the other quantity and decrease in one quantity causes decrease in the other quantity.	(I)With more money, more articles can be bought. (II)With a greater speed, the larger distance can be covered at the same time. (III)With less number of men, less work can be done in same time, etc.
Inverse variation	Increase in one quantity causes a decrease in the other quantity and decrease in one quantity causes an increase in the other quantity.	(I)With greater speed, less time will be taken to cover the same distance. (II)With less number of men, more days are required for the same work, etc.

Unitary Method in Maths

Consider the following examples

1. If the cost of  15m cloth= rs300 Cost of 1m cloth=rs 300/15 = rs20 And the cost of 10m cloth=10  x rs20= rs200

2. If 15 men can do some work in 300 days 1 man can do the same work in 300  x   15days=4500 days And 10 men will do it in 4500/10 days = 450 days.

• In part 1, given above, the cost of 1m cloth is first obtained from the given cost of 15m and then the cost of 10m is found.
• In the same way, in part 2, given above, the number of days taken by 1 man is first calculated from the number of days by 15 men and then the number of days taken by 10 men is obtained.

Unitary Method Formula

Example1: A man earns R 400 in 10 days. How much will he earn in 28 days? Answer: Note: For Solving problems, the sentences (statements) should be framed in Such a way that, whatever is to be found is Written at the end of the statement.

Arrow Method Formula:

Steps:

In the case of inverse variation, the arrow for this Column must be in the direction opposite to the direction of the first arrow. Since here we have the case of direct variation, therefore, for both the columns arrows must be in the same direction.

Unitary Method Example

Example2- 0.75-meter cloth costs rs 45. What will be the cost of 0.6m of the same cloth?

Example3- 4 men can do a piece of work in 5 Days.How many men will do it in 4 days?

Example4- With a speed of 60km/hr, it takes 4 hours to cover a certain distance. What should be the speed if the same journey is to be completed in 3 hours?

                

Problems For Practice

1. 6 books weigh 1.260kg.How many books will weigh 3.150kg?
2. 8 men complete a work in 6 hours. In how many hours will 12 men complete the same work?
3. A typist takes 80 minutes to type 24 pages. How long will he take to type 87 pages?
4. A truck uses 20 liters of diesel for 240 km.How many liters will be needed for 1200km?
5. If 3 dozen eggs costs 90rs find the cost of 3 score eggs?(1 score=20)

This is all about the Basic Concept, Formula, Example, Problems.

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