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Answered on 21/08/2017 Exam Coaching/Foreign Education Exam Coaching/SAT Coaching

I am very weak in Maths. Will I be able to pass SAT?

Harideep Nidi

Yes, You can.

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Answers 3 Comments Lesson Posted on 07/07/2017 Exam Coaching/Foreign Education Exam Coaching/SAT Coaching

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Ratios: Ratio, a very basic arithmetic concept commonly tested on reasoning/aptitude tests in different levels and different formats is simple to understand and sometimes tricky to handle. "Ratios" - On the simplest level, by definition, it is a quantitative relation between 2 amounts showing the number... read more

**Ratios:**

Ratio, a very basic arithmetic concept commonly tested on reasoning/aptitude tests in different levels and different formats is simple to understand and sometimes tricky to handle.

"Ratios" - On the simplest level, by definition, it is a quantitative relation between 2 amounts showing the number of times one value contains or is contained within the other.

For example, the number of boys and girls in a class are in the ratio of 2:3 or 2/3, means that the number of boys is 2/3 of the number of girls. (There could be 20 boys and 30 girls)

Now, even if the actual number of boys and girls are not 2 and 3 respectively the ratio may still be 2:3.

For example, if the number of boys in a class were 12 and the number of girls were 18, then the ratio between the number of boys and girls still remains 12:18 simplified to 2:3.

On a bit higher level, the ratio can be expressed between more than 2 amounts too.

For example, ‘the ratio of number of roses to lilies to tulips in a bouquet of flowers is 1:3:5’, means that the number of roses and lilies are in the ratio of 1:3 while the number of lilies to tulips is 3:5. Again, the actual number of roses, lilies and tulips in the bouquet need not be exactly 1, 3, 5 respectively only. They could be in multiples. This means the flowers could be such that they are a set of 1 rose, 3 lilies and 5 tulips or 2 roses, 6 lilies and 10 tulips or 3 roses, 15 lilies and 18 tulips or so on. Generalising, they could be x, 3x and 5x respectively where x represents a positive integer.

The following 2 examples illustrate questions in which ‘ratio’ as a concept is tested in 2 different, commonly tested formats

**Example 1**:

If the ratio of the number of shirts, trousers, belts that Sam has is 2:5:13 and he has 120 accessories (shirts, trousers and belts) altogether, how many more trousers than Shirts does Sam have?

**Solution:**

Given the ratio between the shirts, trousers, belts is 2:5:13. The actual number of shirts, trousers and belts that Sam has can be assumed to be 2x, 5x, 13x (as explained above) where x is a positive integer. The sum of all of these equals 20x (2x+3x+15x) which is given in the question equivalent to 120. Hence, equating 20x to 120 we get to find the value of x as 6. Now, to know how many trousers than shirts did sam have, we can calculate it by subtracting 2x from 5x. We get it as 3x, which equals 3 times 6 = 18.

**Example 2**:

If the ratio of red colour marbles to green colour marbles in a bag of marbles is 3:5 and the ratio of green colour marbles to yellow colour marbles in the bag is 2:7, then what is the ratio of the number of red colour marbles to the number of yellow colour marbles?

**Solution:**

Given, the ratio between red colour marbles to green colour marbles = 3:5

Assuming, the number of red colour marbles = 3x

and hence the number of green colour marbles = 5x

Also, the ratio of the number of green colour marbles to yellow ones = 2:7

Assuming, the number of green colour marbles to equal = 2y

and hence the number of yellow colour marbles = 7y

Notice that the number of green colour marbles is assumed to be 5x as well as 2y. Equating them, 5x = 2y

Implying, x = 2y/5

The question asks us to find the ratio of number of red colour marbles to yellow ones.

The red coloured ones are 3x in number which also equals, 3x = 3 (2y/5) = 6y/5 (Substituting the x value)

Now, the required ratio of the number of red coloured marbles to number of yellow coloured marbles will be (6y/5) / (7y) = 6/35

**You may try doing this now:**

If the ratio of a to b to c is 2:3:5 and the ratio of b to c to d is 6:10:15, then what is the ratio of a to d?

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SAT Writing Standard English Conventions Shifts In Verb Tense, Voice And Mood

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Verb Tenses: After subject-verb agreement, verb tenses are among the most tested rules in the SAT. Let’s begin by reviewing what a verb is and what its tense signifies. A verb is one of the main parts of the sentence that indicates activity either through an action, a happening or a state of being. And... read more

**Verb Tenses:**

After subject-verb agreement, verb tenses are among the most tested rules in the SAT. Let’s begin by reviewing what a verb is and what its tense signifies.

A verb is one of the main parts of the sentence that indicates activity either through an action, a happening or a state of being.

And verb tenses describe the time that the action or occurrence took place. Without the tenses, it would be impossible to know when the event (action) took place.

There are typically 12 tense forms to a verb, which we will look at in the table below (through the verb ‘drive’ as an example):

But it is important to remember that in the context of the SAT, you don’t need to know the names of the tenses. All you need to be able to do is to identify them and to understand how they are used in sentences so you can spot the errors and replace them with the correct forms of the tense (when needed).

Let’s understand the different forms of the three tenses: past, present and future

**Simple Tense:**

This is the simplest form of the tenses used to describe an action that was completed in the past; is a daily occurrence or that is currently happening (present); or will happen tomorrow (future).

Here it is important to remember that the simple past tense form is used to describe a completed action. Here are three examples that use the verb ‘drive’:

He drives the car every day to work (Present).

I drove the truck to the beach last night (Past).

She will drive to the mall tomorrow (Future).

**Progressive ****Tense:**

This form of verb tense is used to show something that was happening in the past, is happening at present or will happen for a period of time. Here the tense is reflected by the past/present/future forms of the helping verb ‘to be’ i.e. was/is/will be which is used along with a gerund (the -ing form of the main verb).

**Note: The past progressive tense refers to an action that remains incomplete**.

Here are three more examples:

He was swimming in the river.

He is swimming in the rooftop pool.

He will be swimming in the race tonight.

**Perfect Tense:**

Let’s consider these sentences:

Men have walked the Earth for millennia (Present Perfect).

By the time the sun rose, the frost had evaporated from the trees (Past Perfect).

Two years from now, she will have graduated from high school (Future Perfect).

The present perfect tense is formed with the verbs has/have along with the past participle of the action verb. It is used to indicate actions that began in the past but are continuing in the present. While the past perfect verb tense form is used to indicate an action that happened first in a sentence that features two completed actions. Often (but not always), sentences that take the present perfect tense feature words such as ‘for’ or ‘since’.

Future perfect tense forms highlight an action that will take place at some point in the future.

**Perfect Progressive Tense:**

In the world of verb tenses, ‘perfect’ signifies the completion of an action and ‘progressive’ signifies an incomplete action. Verb forms in the perfect progressive tenses reflect the time taken to complete an action that was unfolding.

The verb form is made up of the helping verb ‘to have’, used along with the verb ‘to be’ and finally the action verb in the progressive tense.

Consider these examples to better understand this verb tense form:

I have been waiting to complete the examination for a year (Present Perfect Progressive).

I had been waiting to complete the examination for a year (Past Perfect Progressive).

I will have been waiting to complete the examination for a year (Future Perfect Progressive).

The present perfect progressive form is used to indicate the relationship of the action to the present. Sometimes it is also used without a reference to time, in the form of questions relating to recently completed actions (For instance, ‘Have you been working out?’)

The past perfect progressive form is usually used to highlight the duration of a action that took place in the past before another action happened.

While the future perfect progressive tense is used to indicate an action in connection with an event in the future. These tenses form rare, complex sentences.

**Verb tense concepts tested in the SAT:**

**1) Consistency in Tense:**

This is one of the most tested concepts in the SAT. Here all you have to do is to ensure that all the verbs in the sentence follow the same tense, this means that sentences that begin with a particular tense must maintain the same throughout, unless the verbs are in different clauses.

See if you can spot the error in the following sentence:

I ate the cake and drinks the tea.

Here the sentence begins in the past tense, with the verb ‘ate’ but suddenly switches to the present with the verb ‘drinks’. This goes against the rule that verbs in a sentence clause must follow the same tense. So the correct version will be:

I ate the cake and drank the tea

or

I eat the cake and drink the tea.

The correct version would also depend on the context in which it is placed.

Observe this sentence:

He listened to the song because he wants to sing it later to the audience.

Here the verbs are in different tenses but the sentence is still correct because they are in different clauses.

**2) G****erund Vs. Infinitive:**

Gerunds (verbs which take the -ing form) and infinitives (verbs which are preceded by the word ‘to’, but not as a proposition) sometimes take the form of nouns. They can be replaced by infinitives, but not always. Some sentences require that you use either the gerund form or the infinitive form of the verb. The best way to decide which one to use, when either is underlined in an SAT question, is to simply see which sounds better.

I want learning Spanish this year.

I want to learn Spanish this year.

Which sentence do you think sounds better? The second sentence, naturally.

**3) Choosing the right verb form:**

**a) between the simple past and the present perfect:**

In some SAT questions, you might have to replace a verb carrying the simple past tense for the present perfect. Remember, the present perfect tense is used for an action that began in the past but continues in the present while the simple past tense is used to indicate a completed action.

If such a verb is underlined as part of the question in the SAT, check what purpose the verb in the sentence serves.

Observe this sentence:

For the last six months, I went to the gym.

Here the beginning of the sentence does not indicate completion, it seems to indicate that the action is ongoing. The correct way to write the sentence would be:

For the last six months, I have gone to the gym.

**b) between the simple past and the past perfect:**

Remember, verbs carrying the simple past tense are used to signify completed actions while verbs carrying the past perfect tense are used to signify the first of two completed actions.

Can you correct the following sentence?

I walked ten miles before I realised I forgot my water bottle

The correct version of the sentence is:

I had walked ten miles before I realised I forgot my water bottle.

The sentence features two completed actions, and so the first action (of walking ten miles) will take the past perfect tense.

Let’s review what we have learnt through these sample SAT questions:

Choice B is the best answer because it provides a grammatically standard preposition that connects the verb “serves” and noun “digestive aid” and accurately depicts their relationship. Choice A is incorrect because the infinitive form “to be” yields a grammatically incorrect verb construction: “serves to be.” Choices C and D are incorrect because both present options that deviate from standard English usage.

Choice C is the best answer because it presents a verb tense that is consistent in the context of the sentence. The choice is also free of the redundant “it.” Choice A is incorrect because the subject “it” creates a redundancy. Choices B and D are incorrect because they present verb tenses that are inconsistent in the context of the sentence.

Choice A is the best answer because the verb tense is consistent with the preceding past tense verbs in the sentence, specifically “produced” and “drifted.” Choices B, C, and D are incorrect because each utilizes a verb tense that is not consistent with the preceding past tense verbs in the sentence.

Choice B is the best answer because it provides a verb that creates a grammatically complete, standard, and coherent sentence. Choices A, C, and D are incorrect because each results in a grammatically incomplete and incoherent sentence.

Choice D is the best answer because the gerund “waiting” corresponds with the preposition “for” and the present tense used in the rest of the sentence. Choices A, B, and C are incorrect because each contains a verb form not used with the preposition “for.”

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Lesson Posted on 06/07/2017 Exam Coaching/Foreign Education Exam Coaching/SAT Coaching

Linear Equations - Wordy Problems

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Wordy Problems: The concepts of linear equations in 1 variable and 2 variables generally are tested in wordy questions on the SAT. The equations are generally translations of wordy situations in the questions. Translating the wordy questions to mathematical equations is generally tricky on the SAT. Example:... read more

**Wordy Problems:**

The concepts of linear equations in 1 variable and 2 variables generally are tested in wordy questions on the SAT. The equations are generally translations of wordy situations in the questions. Translating the wordy questions to mathematical equations is generally tricky on the SAT.

**Example: **If twice a number is three less than itself, what is nine more than three times the number?

**Solution:**

Such wordy questions are better dealt in parts:

Firstly, assume the number to be x

Then, twice the number would be 2x

Three less than itself would be x-3

Given both are equal 2x = x-3

Solving, x = -3

However, three times the number would be 3*-3 = -9

And, nine more than it would be 9+-9 = 0

Hence, the answer is 0

The math part on the right column of the above solution is all simple solving of a linear equation in 1 variable.

The wordy part on the left column of the above solution is all translating the wordy questions part by part into mathematical expressions/equations, which is what makes the simple math a bit more challenging on the SAT.

**Example: **If the price of 3 type A pencils less than the price of 2 type B pencils equals $1 and a set of 2 type A pencils and 3 type B pencils costs $8, what is the price of a type A pencil?

**Solution:**

Solving a pair of linear equations in 2 variables: 2y-3x=1, 3y+2x = 8 would give the answer.

How are the equations formed out of the wordy questions is what the SAT is testing one on.

So let’s understand the same:

Firstly, assuming the price of one type A, type B pencil to be $x, $y respectively, And then, the price of 3 type A pencils would be 3 times x = 3x

The price of 2 type B pencils would be 2 times y = 2y

Also, the price of 3 type A pencils less than the price of 2 type B pencils would be = 2y-3x

Given, equals $1. Hence, 2y-3x=1

Similarly converting the information into equations we get another equation in 2 variables, 2x+3y = 8.

To get the price of a type A pencil, we can simply solve for x using the pair of equations derived above. x, y turn out to be 1 and 2 respectively.

**You may try:**

Albert’s and Benjamin’s ages add up to 35 years now. If Albert is twice as old as Benjamin was when Albert was as old as Benjamin is now, then how old is Albert?

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GRAPHS: All of the linear equations can be graphed as lines on the coordinate plane (rectangular coordinate system). Example: x/3 = -x + 1 is a linear equation in 1 variable that can be simplified to x =3/4. On the coordinate plane, the graph of the above linear equation would look like below:Similarly,... read more

**GRAPHS:**

All of the linear equations can be graphed as lines on the coordinate plane (rectangular coordinate system).

**Example:**

x/3 = -x + 1 is a linear equation in 1 variable that can be simplified to x =3/4.

On the coordinate plane, the graph of the above linear equation would look like below:

Similarly, a linear equation in 2 variables, 2x+3y=6 can be plotted on the coordinate plane as shown below:

In general, the process to plot a graph manually would be to find the points that lie on the graph/equation and join them.

In the second example, if we substitute x with 3, y would equal 0 and if we substitute x with 0, y would equal 2. Hence, the paired coordinates (3,0) and (0,2) lie on the line which can be seen in the graph too.

A table of paired values of x, y which satisfy the equation is shown below:

x | -3 | -2 | -1 | 0 | 1 | 2 | 3 |

y | 4 | 10/3 | 8/3 | 2 | 4/3 | 2/3 | 0 |

All the points that lie on the line (graph) that can be derived from the above table would hence be (-3,4), (-2,10/3), (-1,8/3), (0,2), (1,4/3), (2,2/3) and (3,0) lie on the line shown in the graph.

A line containing points in which the value of y decreases as the value of x increases is negatively sloped (like the one graphed above) and shows a negative correlation between the 2 variables.

Plotting such a linear graph for another linear equation in 2 variables will help us understand the solution for the pair of linear equations if any.

If the lines graphed intersect, the point of intersection itself is the solution of the pair of linear equations. If the lines so plotted turn out to be parallel to each other, there exists no point of intersection and if the lines so plotted turn out to be coinciding(on over another), there exist infinitely many solutions.

A pair of linear equations which does not have a solution (parallel lines) is shown below:

The equations 2x+3y=6 and 4x+6y=10 are plotted as shown and are hence parallel and do not have a solution (point of intersection).

The equations 2x+3y=6 and 4x+6y=12 are plotted as shown and are hence coinciding and have infinitely many solutions.

In general, the standard format of a linear equation is ax+by+c=0 where x, y are variables and a, b, care constants.

If ax+by+c=0 and px+qy+r=0 are a pair of linear equations in 2 variables, then, they would have a unique solution if a/b is not equal t p/q; no solution if a/b = p/q but not equal to c/r; infinitely many solutions if a/p=b/q=c/r.

**You may try:**

If 2x-3y=5 and px-6y=r are two linear equations which have infinitely many solutions what is the value of r-p?

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SAT Maths: Linear Equations In 2 Variables: Wordy Problems

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Linear Equations In 2 Variables: Wordy Problems A linear equation in 2 variables again is an equation of degree 1 but has 2 variables. Example: 2x+3y=5, 3x/2 –y = 0.5 Generally, these are equations that one would have seen simultaneously (2 at a time), where solving for the values of the 2 variables... read more

**Linear Equations In 2 Variables: Wordy Problems**

A linear equation in 2 variables again is an equation of degree 1 but has 2 variables.

**Example:**

2x+3y=5, 3x/2 –y = 0.5

Generally, these are equations that one would have seen simultaneously (2 at a time), where solving for the values of the 2 variables involved processes like elimination or substitution.

**Example:**

Solve for x and y given the above 2 linear equations in 2 variables (in the example above)

**Solution:**

Elimination process involves multiplying the equation (2x+3y=5) with 3/4 so that the coefficients of x in both the equations equal each other.

3/4 (2x + 3y =5) implies 3x/2 + 9y/4 = 15/4. Now, as the coefficients of x in both the equations is the same, we can subtract one equation from another to eliminate x from them.

(3x/2 + 9y/4 = 15/4) – (3x/2 – y = 0.5) implies (9y/4+y = 15/4 -0.5) from which it can be implied that 13y/4 = 13/4. Hence y = 1. Substituting the value of y back into any of the equation we also get the value of x as 1.

The challenges in questions on the SAT math on this concept are generally about taking care of the signs in the equations. Most of the questions that test takers go wrong at are those in which care has to be taken in handling the signs (-, +) when doing the operations explained above.

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SAT Maths: System Of Linear Equations

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Linear Equations - System Of Equations A system of linear equations can be represented by lines on the coordinate plane. Unique Solution / Intersecting Lines: Graphically, a pair of linear equations having a Unique solution would be two intersecting lines on the coordinate plane. In such a case, the... read more

**Linear Equations - System Of Equations**

A system of linear equations can be represented by lines on the coordinate plane.

**Unique Solution / Intersecting Lines:**

Graphically, a pair of linear equations having a Unique solution would be two intersecting lines on the coordinate plane. In such a case, the pair would be such that the ratio of the coefficients of x is not equal to that of y.

**Example 1**:

2x + 3y = 5 and 5x + 6y = 11 are two linear equations that form a system of equations.

2/5 is 0.4 and is not equal to 3/6 which is 0.5. Hence the above equations have a unique solution.

**No Solution / Parallel Lines:**

Let ax+by+c = 0 and px+qy+r=0 are two linear equations in x and y with a, b, p, q being constants. If (a/p), which is the ratio of coefficients of x = (b/q), which is the ratio of coefficients of y is not equal to (c/r), the ratio of the constants, then the pair of linear equations has No solution. On the coordinate plane, they would be represented by parallel lines.

**Example 2:**

2x + 3y = 5 and 4 x + 6y = 11

2/4 which is ratio of coefficients of x is equal to 3/6 which is the ratio of coefficient of y but not equal to the ratio of the constants 5/11. Hence the above equations have no solution.

**Infinite Solutions / Coinciding Lines:**

A pair of linear equations has infinitely many solutions when (a/p), the ratio of the coefficients of x is equal to (b/q), the ratio of the coefficients of y is equal to (c/r), the ratio of the constants. On the coordinate plane, they would be represented by two lines one over another.

**Example 3:**

2x + 3y = 5 and 4x + 6y = 10

2/4 is equal to 3/6 is equal to 5/10. Hence, the above pair of linear equations has infinite solutions.

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Rates: "Rates" - A common topic (quant/math aptitude section) to almost all competitive/admission tests! Broadly, it constitutes of 2 sub-topics: Distance - Speed - Time, Work - Time. Distance - Speed - Time: Though limited to just 2-3 formulae, the question structures can vary a lot. The formulae are... read more

**Rates**:

"Rates" - A common topic (quant/math aptitude section) to almost all competitive/admission tests! Broadly, it constitutes of 2 sub-topics: Distance - Speed - Time, Work - Time.

**Distance - Speed - Time**:

Though limited to just 2-3 formulae, the question structures can vary a lot. The formulae are very basic ones, which one is expected to have learnt during school days:

**Distance = Speed * Time,****Average Speed = Total Distance / Total Time,****Relative Speed = Sum (or) Difference of the speeds based on the directions of travel**.

Let’s understand usage of relevant formulae using some examples:

**Example 1**:

If a superfast train travels at a speed of 120km/hr for 3.5 hours, how far would it travel? Similarly, If a superfast train travels at a speed of 120km/hr, how long would it take to cover 420 km? Or, If a superfast train could travel a distance of 420 km in 3.5 hours, what was the speed at which travelled.

**Solution:**

Solving the above questions requires the usage of the formula listed above "Distance = Speed * Time".

**Example 2**:

If a car starts its journey from Ahmedabad travelling at a speed of 60 km/hr and reaches Mumbai (480 km from Ahmedabad) and then starts its journey to Bangalore(320 km from Mumbai) at a speed of 80 km/hr and reaches there. What was the average speed of the car’s journey from Ahmedabad to Bangalore?

**Solution:**

This would require the usage of the second formula listed above "Average Speed = Total Distance / Total Time" & also the first formula "Distance = Speed*Time too".

**Example 3**:

If a police (travelling at a speed of 80 km/hr) chases a thief (travelling at a speed of 60 km/hr) who is 300 km away from the police, how long would it take the police to catch the thief?

**Solution:**

This would require the usage of the third formula "Relative Speed = Sum (or) Difference of the speeds" based on the directions of the travel along with the first formula "Distance = Speed * Time".

(Note: in this example the relative speed would be the difference of the speeds of the police and the thief as one of them is chasing another)

Now, there may a variety of designs/structure of the questions based on the above three formulae. Based on the designs/structure, the question sub-topics (boats & streams, trains/lamppost) are named.

Let’s understand the structure of the question using some examples:

**Example 4**:

If a boat travels up-stream at a speed of 10 km/hr and down-stream at a speed of 20 km/hr, what is the speed of the boat in still water?

**Solution:**

The third formula of Relative speed can be applied to solve the above example.

**Example 5**:

If a Train A (length of 1 km) travels at a speed of 140 km/hr and a train B (length of 2 km), 70 km away from Train A travels in the opposite direction on parallel tracks at a speed of 210 km/hr, how long would it take train A to cross Train B completely.

The lengths of the trains are additional considerations towards distance parameter, while solving such questions.

Let’s understand all of the above discussed using an example:

**Example 6****:**

Train A travelled from Cochin to Ooty (distance of 320 km) with a speed of 80 km/hr and then from Ooty to Bangalore (distance of 180 km) with a speed of 45 km/hr and then from Bangalore to Hyderabad (distance of 600 km) with a speed of 150 km/hr. Train B travelled from Hyderabad to Bangalore with a speed of 75 km/hr and then from Bangalore to Ooty with a speed of 20 km/hr and from Ooty to Cochin at a speed of 40 km/hr.

Let the two trains be facing each other (at a distance of 452 km) in opposite directions on parallel tracks. If they both start travelling at their average speeds of the above described journeys respectively and train A is 1 km long while train B is 2 km long. How long would it take each of them to cross other completely?

**Solution:**

Firstly, we calculate the Average Speed of each of the train’s journeys:

Train A:

Total Distance travelled = 320 + 180 + 600 = 1100 km

Total Time taken = (320/80) + (180/45) + (600/150) = 4 + 4 + 4 = 12 hrs

Average Speed = 1100/12 = 650/6 = 108.33 km/hr

Similarly, for Train B:

Average Speed = 1100/(8+9+8) = 1100/25 = 44 km/hr

Now, the Relative Speed of two trains travelling in opposite directions would be the sum of their speeds = 108.33+44 = 152.33 km/hr

The distance they are apart at the beginning of the show = 452 km.

The distance they have to cover together in order to cross each other completely = 452+1+2=457km

Now, the Time taken = Total Distance to be covered / Relative Speed = 457 / 152.33333 = 3 hrs.

In this example, we have tried to illustrate the usage of all the three formulae mentioned earlier.

Note: Also, in this case, the relative speed turned out to be the sum of the speeds as the trains are travelling towards each other and not one behind another.

Generally, relative speed would be used in cases where there are 2 bodies moving either towards each other or against each other or one behind another. When they are moving towards/against each other, the relative speed is the sum of their speeds while when they are moving in the same direction one behind another, the relative speed is the difference between their speeds.

Similar approach can be used to solve questions in which boats are travelling up-stream/down-stream where the sum of the speeds of the boat and the stream would be the relative speed if down-stream and the difference of their speeds, the relative speed if up-stream.

**Work-Speed / Rate -Time:**

This is a similar sub-topic to Distance-Speed-Time. The formula **Work = Speed * Time** demonstrates the similarity. There could be more than one body (person/machine) doing certain same work. Relative Speed in this case would also be either the sum of the speeds or the difference between the speeds based on their contribution towards the work.

Let’s understand using some simple examples first,

**Example 1**: Gadget 1 downloads data at a speed of 18mb/sec and gadget 2 downloads at a speed of 12 mb/sec. If they both download data together, how long will it take them to download 330 mb of data?

**Solution:**

The formula used to solve this would be "Work=Speed * Time", where Work = the amount of data to be downloaded, the Speed (relative speed) would be the sum of their speeds since both the gadgets are working towards the completion of the work.

**Example 2**: Gadget 1 downloads data at a speed of 32mb/sec and gadget 2 uploads data at a speed of 14 mb/sec. How long will it take an empty hard disk of capacity 9 gb to be completely filled with data when both gadgets are working. (1gb=1024mb).

**Solution:**

The formula again would be the same "Work=Speed*Time", where work equals 9 gb, the speed (relative speed) would be the difference between their speeds since one of them is uploading and the other is downloading.

However, one key differentiator between Distance-Speed-Time questions and Work-Time questions is the format in which the Speed of the bodies is mentioned.

In distance-speed-time questions, the speeds are generally given directly as distance travelled per unit time. For example, 80 km/hr, 25 miles/second. In work-rate-time questions, the speeds are to be derived as work done per unit time from the information given. For example, A does a complete job in 2 hours implies, the speed of A is ½ job/hour, B does the same job in 5 hours implies, the speed of B is 1/5 job/hour. A work-rate-time question would generally be solved by calculating the speeds in such a manner.

**Example 3**: A pipe fills a tank in 2 hours. Another pipe empties the tank in 4 hours. How long will it take the same tank to get filled when both the pipes are doing their job?

**Solution:**

The pipe filling the tank takes 2 hours to fill it completely, implying that it fills ½ of the tank in 1 hour. Similarly, the second pipe emptying the tank takes 4 hours to empty it completely, implying that it empties ¼ of the tan in 1 hour. The relative speed with which the tank is getting filled would then be the difference between the individual speeds (½ – ¼) of the tank per hour. The formula Work =Speed * Time can then be applied (1 = ¼ * T) to calculate the time (4 hours) taken to fill the tank.

**Example 4**: A machine manufactures 24 tablets in 10 seconds while another machine does 30 tablets in 50 seconds. How long does it take both the machines working together to manufacture 100 strips of 24 tablets each?

**Solution:**

The speed at which the first machine is manufacturing (24 tablets in 10 seconds) = 2.4 tablets per second.

The speed at which the second machine is doing it (30 tablets in 50 seconds) = 0.6 tablets per second.

If both work together they will be able to do the job at a combined speed of (2.4+0.6) 3 tablets per second.

Work to be done = 100 strips of 24 tablets each = 100*24 = 2400 tablets.

Time taken = Work/Speed = 2400/3 = 800 seconds = 13 minutes 20 seconds.

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Comments Lesson Posted on 24/06/2017 Exam Coaching/Foreign Education Exam Coaching/GRE Coaching Exam Coaching/Foreign Education Exam Coaching/GMAT Coaching Exam Coaching/Foreign Education Exam Coaching/SAT Coaching

How To Determine If Points Are Collinear In Coordinate Geometry?

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How to prove if points are Collinear in coordinate geometry? Collinear points definition: Three or more points that lie on a same straight line are called collinear points. Consider a straight line L in the above Cartesian coordinate plane formed by x axis and y axis. This straight line L is passing... read more

How to prove if points are Collinear in coordinate geometry?

Collinear points definition:

- Three or more points that lie on a same straight line are called collinear points.
- Consider a straight line L in the above Cartesian coordinate plane formed by x axis and y axis.
- This straight line L is passing through three points A, B and C whose coordinates are (2, 4), (4, 6) and (6, 8) respectively.
- {We may also say, alternatively, that the three points A (2, 4), B (4, 6) and C (6, 8)are lying on a same straight line L}
- Three or more points which lie on a same straight line are called collinear points.

How to find if three points are collinear?:

- There are two methods to find if three points are collinear.
- One is slope formula method and the other is area of triangle method.
- Slope formula method to find that points are collinear.
- Three or more points are collinear, if slope of any two pairs of points is same.
- With three points A, B and C, three pairs of points can be formed, they are:
*AB, BC and AC.* - If Slope of AB = slope of BC = slope of AC, then A, B and C are
*collinear points.*

Example

Show that the three points A (2, 4), B (4, 6) and C (6, 8) are collinear.

Solution:

- If the three points A (2, 4), B (4, 6) and C (6, 8) are collinear, then
- slopes of any two pairs of points will be equal.
- Now, apply slope formula to find the slopes of the respective pairs of points:
- Slope of AB = (6 – 4)/ (4 – 2) = 1,
- Slope of BC = (8 – 6)/ (6 – 4) = 1, and
- Slope of AC = (8 – 4) /(6 – 2) = 1
- Since slopes of any two pairs out of three pairs of points are same, this proves that A, B and C are collinear points.
- Area of triangle to find if three points are collinear.
- Three points are collinear if the value of area of triangle formed by the three points is
*zero.* - Apply the coordinates of the given three points in the area of triangle formula. If the result for area is zero, then the given points are said to be collinear.
- First of all, recall the formula for area of a triangle formed by three points.

It is

In the formula above, the two vertical bars enclosing the variables represent a determinant.

Let us apply the coordinates of the above three points A, B and C in the determinant formula above for area of a triangle to check if the answer is zero.

Since the result for area of triangle is zero, therefore A (2, 4), B (4, 6) and C (6, 8) are collinear points.

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