Question: If 10, 12 and ‘x’ are sides of an acute angled triangle, how many integer values of ‘x’ are possible?

(A) 7
(B) 12
(C) 9
(D) 13
(E) 11

Solution: The question is very interesting. It asks you for an acute triangle i.e. a triangle with all angles less than 90 degrees. It’s a little hard to wrap your head around it, isn’t it? We know that the third side of a triangle can take many values. Right from a little more than the difference of the other two sides to a little less than the sum of the other two sides (Since we know that the sum of any two sides of a triangle is always greater than the third side). So x can be anything from a little more than 2 to a little less than 22. But how do we find out the values for which all the angles will be less than 90?

We want no obtuse or right angles. An obtuse angled triangle has one angle more than 90. So the thought here is that before one of the angles reaches 90, find out all the values that x can take.

Look at the figure given above. The value of x in the first figure is very small – slightly more than 2 – minimum required to make a triangle. There is an obtuse angle in that triangle. We keep making x bigger and bigger and the angle keeps becoming smaller till it reaches 90 (Fig III). We use Pythagorean theorem to get the value of x in that case:

x = √(12^2 – 10^2)
x = √44 which is 6.something
x should be greater than 6.something because the angle cannot be 90.

We further keep increasing x and all the angles are acute now. We reach Fig V where we hit another right triangle. We use Pythagorean theorem again to get the value of x (the hypotenuse) in this case:

x = √(12^2 + 10^2)
x = √244 which is 15.something
x should be less than 15.something so that the angle is not 90.

Further on, in Fig VI, we obtain an obtuse angle again.

We only need integral values of x so values that x can take range from 7 to 15 which is 9 values.

Answer (C).

Note: We made two angles 90 and found the values of x in between those two angles. The third angle cannot be 90 because that will make 10 the hypotenuse but hypotenuse is always the greatest side.

Karishma, a Computer Engineer with a keen interest in alternative Mathematical approaches, has mentored students in the continents of Asia, Europe and North America. She teaches the GMAT for Veritas Prep and regularly participates in content development projects such as this blog!

Let me explain with the help of an official Data Sufficiency question.

Question:

In the figure above, is the area of the triangle ABC equal to the area of the triangle ADB?

Statement 1: (AC)^2=2(AD)^2

Statement 2: ∆ABC is isosceles.

Solution:

When presented with this question, people see right triangles and jump to Pythagorean theorem, isosceles triangles and then wage a war on AC, AB, CB and AD relations. Well, that is our traditional approach. But what do we do if making equations and solving for relations isn’t our style?

We make diagrams and figure out the relations! One thing that is apparent the moment we read statement 1 is that the figure is not to scale. From the figure it looks as if AD is greater than or at best, equal to AC. That itself is an indication that if you draw the figure on your own, you could see something that will make this question very simple. The question setter doesn’t want to show you that and hence he made the distorted figure.

Anyway, let’s first analyze the question. Then we will look at the statements.

We need to compare areas of ABC and ADB. Both are right angled triangles.

Area of ABC = (1/2) * AC * BC

Area of ADB = (1/2) * AD * AB

We need to figure out whether these two are the same.

Think about it this way – we are given a triangle ABC with a particular area. So the length of AD must be defined. If AD is very small, (shown by the dotted lines in the diagram given below) the area of ADB will be very close to 0. If AD is very large, the area will be much larger than the area of ABC. So for only one value of AD, the area of DAB will be equal to the area of ABC.

We need to figure out whether for the given relations, the triangles have equal area.

Statement 1: (AC)^2=2(AD)^2
This gives us AD = AC/√2. Let’s draw AC and AD such that AD is somewhat shorter than AC. Now can we say that the areas of the two triangles are the same? No. The area of ABC is decided by AC and BC both not just AC. We can vary the length of BC to see that the relation between AC and AD is not enough to say whether the areas will be the same (see the diagrams given below).

So this statement alone is not sufficient.

(2) ∆ABC is isosceles.
This means that AB = BC. Notice that the triangle is right angled so the hypotenuse must be the largest side. If ABC is isosceles, it means that the two legs of the triangle must be equal. Hence sides of ABC must be in the ratio 1:1:√2 = AC:BC:AB. Since we only need to consider relative length of the sides, let’s say that AC = 1, BC = 1 and AB = √2 or some multiple thereof.

We have no idea about the length of AD so this statement alone is also not sufficient.

Let’s consider both statements together now:

AD = AC/√2 = 1/√2 (Since AC = 1)

Area of ABC = (1/2) * AC * BC = (1/2) * 1 * 1 = 1/2

Area of ADB = (1/2) * AD * AB = (1/2) *  (1/√2 ) * √2 = 1/2

Both triangles have the same area. Sufficient!

Answer (C)

Now compare this approach with your Pythagorean approach. Is this simpler?

Karishma, a Computer Engineer with a keen interest in alternative Mathematical approaches, has mentored students in the continents of Asia, Europe and North America. She teaches the GMAT for Veritas Prep and regularly participates in content development projects such as this blog!

Question: A certain square is to be drawn on a coordinate plane such that all the coordinates of its vertices are integers. One of the vertices must be at the origin, and the area of the square must be 25. How many different squares can be drawn?

(A) 4
(B) 6
(C) 8
(D) 10
(E) 12

Solution:

Since the question tells us that there are many different ways to draw the square, let’s draw it in one particular way. We will have something with which to proceed.

One vertex must be at (0, 0). Area must be 25 which means the side of the square must be 5. The easiest way to draw such a square would be the blue square shown in the figure.

How will we get many different squares? By turning the square around (0, 0) as shown (Push one side of the square – notice the vertex with the green dot at (0, 5) moving clockwise). Now the problem is that we need the coordinates of all the vertices to be integers. How do we assure that? One vertex is at (0, 0) and will always stay at (0, 0) so we don’t need to worry about that. What about the rest of the three vertices? When you turn the square, at any one position, they may or may not have integer coordinates – and we have three of them to worry about!

This is where our last two posts come in. First let’s establish that we only need to focus on one vertex – the other two will follow. Look at the blue square. Each one of its vertices is at integer coordinates: (0, 0), (0, 5) – the green dot vertex, (5, 5) and (5, 0). Let’s say, when you push the side joining the center and the green dot, the green dot moves by 1 unit down and 3 units to the right. What happens to the side perpendicular to this side (the one joining the green and the yellow dots)? Let’s think about it.

Pay attention to the diagram. When we turn the square, the coordinates of all three vertices change. The change is similar in all three vertices.

Hope it’s intuitive that if one vertex takes integral steps, all vertices will move in a similar fashion. The diagram shown above is just to reinforce this point. So if our first square has all integral coordinates and we move one vertex such that its coordinates remain integers, all the other vertices will follow suit.

Let’s catch hold of the green dot vertex with coordinates (0, 5). We want the x and y coordinates to stay integers such that

x^2 + y^2 = 5^2 (since the length of the side must stay 5)

As discussed above, x = 0 and y = 5 satisfies this. When x = 1, will y be integral? No.

What about when x = 2? No.

When x = 3? Yes, y will be 4.

When x = 4? Yes, y will be 3.

When x = 5? Yes, y will be 0.

So you have two pairs of numbers 0, 5 and 3, 4 (and their variations) which satisfy this equation. (Hope it reminded you of Pythagorean theorem and you jumped to this conclusion right away!)

Notice that coordinates can be negative too so 0, -5 and -3, -4 will also work. So how many total squares do we get?

So the green dot can take any of the following coordinates:

(0, 5), (0, -5)

(5, 0), (-5, 0)

(3, 4), (3, -4)

(4, 3), (4, -3)

(-3, 4), (-3, -4)

(-4, 3), (-4, -3)

A total of 12 values.

Alternatively, you have 3 different squares keeping the green dot in the first quadrant and when rotated, they will give you one square in each of the three other quadrants too.

Total 12 such squares.

The correct answer is (E).

If it seems difficult, that is because it is – 750 level.

But keep practicing. Most of it will become intuitive once you get the hang of it.

Karishma, a Computer Engineer with a keen interest in alternative Mathematical approaches, has mentored students in the continents of Asia, Europe and North America. She teaches the GMAT for Veritas Prep and regularly participates in content development projects such as this blog!

You might wonder whether we can expect such a question in actual GMAT. The question we discussed in the last post was an official question and we could solve it easily using this concept. Of course there are many other ways of solving it but this is simplest (or trickiest depending on how you look at it), and it certainly is the fastest, no two ways about it! It is a very logical big-picture approach and people who get Q50-51 often use such methods. The question we will discuss today can also be solved in other ways but we will use the last week’s ‘turning minute hand 90 degrees’ approach.

Question: On the coordinate plane (6, 2) and (0, 6) are the endpoints of the diagonal of a square. What are the coordinates of the point on the corner of the square which is closest to the origin?

(A) (0, 1)
(B) (1, 0)
(C) (1, 1)
(D) (2, 0)
(E) (2, 2)

Solution:

First rule of coordinate geometry – draw what you can.

So we make the xy axis and plot the given points, (6, 2) and (0, 6) on it. Let’s say the square is denoted by points ABCD. Say, A is (6, 2) and C is (0, 6). We see that AC is a sloping line. Its two end points are two vertices of the square. We need to find the other two vertices of the square. One of them will lie closest to the origin. The other two vertices will be the end points of the diagonal BD. BD will be perpendicular to AC at the mid-point of AC since a square’s diagonals bisect each other and are perpendicular. So the question is, how do we obtain the end points, B and D? Let’s try to figure out what information we need to draw BD. BD must pass through the mid-point of AC.

How will we obtain the mid-point of AC? By averaging x and y co-ordinates of the points A and C:

x coordinate of mid-point is (0 + 6)/2 = 3

y coordinate of mid-point is (6 + 2)/2 = 4

So BD must pass through (3, 4). When AC turns by 90 degrees, with point (3, 4) as the axis, we get the diagonal BD. So how do the coordinates of AC change when it turns by 90 degrees? Go back to last week’s post and look at the clock face again.

Think of a horizontal line PQ passing through (3, 4).

P coordinate will be given by (0, 4) and Q coordinate will be given by (6, 4) since length of P to mid point is 3 and length of mid-point to Q will also be 3. P shifts up by 2 units to give the point A and Q shifts down by 2 units to give the point C.

Now rotate PQ by 90 degrees and you get RS. We know the coordinates of a line perpendicular to PQ. R will be (3, 1) and S will be (3, 7). This is because R and S will have the same x coordinates as the mid-point (3, 4) and S is 3 units above the mid-point and R is 3 units below the mid-point. We are assuming that you can intuitively see these values on the graph. If not, it may be too soon to spend time on this post.

Now, can we obtain the diagonal BD using RS as reference? If you move S two units to the right, you will get point B (just like A was obtained by moving P two units up) and if you move R two units to the left, you will get point D. Notice that we are using PQ and RS as reference lines. It is easy to calculate vertical/horizontal distances. So B will be given by (5, 7) and D will be given by (1, 1).

The closest co-ordinate to (0, 0) is (1, 1).

Answer (C).

Take a few minutes to review the logic discussed here. The ability to ‘see’ such symmetry makes GMAT Quant very simple.

Karishma, a Computer Engineer with a keen interest in alternative Mathematical approaches, has mentored students in the continents of Asia, Europe and North America. She teaches the GMAT for Veritas Prep and regularly participates in content development projects such as this blog!

Imagine a clock face. Think of the minute hand on 10. Ignore the hour hand for our discussion today. Say, the length of the minute hand is 2 cm. Its distance from the vertical and horizontal axis is shown in the diagram below (using the green and the red dotted lines). Let’s say the minute hand moves to 1. Can you say something about the lengths of the dotted black and dotted blue lines?

Isn’t it apparent that when the minute hand moved by 90 degrees, the dotted green line became the dotted black line and the dotted red line became the dotted blue line. So can we say that the dotted black line is √3 cm in length and the dotted blue line is 1 cm in length? The same thing will happen when the minute hand goes to 4 and to 7. We don’t think there is much explanation needed here, right? The diagram makes it all clear.

Let’s look at the clock from coordinate geometry perspective. Let’s say the center of the clock is the origin (0, 0). What are the x and y coordinates of the “10’o clock point” i.e. the tip of the minute hand before it moves to 1? Notice that the x coordinate will be -√3 (since the point is in the second quadrant, x coordinate will be negative) and y coordinate will be 1.

What are the x and y coordinates of the “1’o clock point” i.e. the tip of the minute hand after it moves to 1. Notice that the absolute values of x coordinate and y coordinate have switched because the hand has turned 90 degrees. The x coordinate is 1 now and the y coordinate is √3. Since it is the first quadrant, both the coordinates will be positive.

Now, think, what will be the coordinates of the “4’o clock point”, “7’o clock point”? What about the “11’o clock point”, “12’o clock point” etc?

Using this concept, we can solve a very tough GMAT Prep question in a few seconds.

Question 1:

In the figure above, points P and Q lie on the circle with center O. What is the value of s?

(A) 1/2

(B) 1

(C) √2

(D) √3

(E) 1/√2

Solution: You might be tempted to think on the lines of ‘slope of a line’ or ’30-60-90’ triangle (because of the presence of √3) etc. But we should be able to arrive at the answer without using any of those.

Point P is (-√3, 1). O is the center of the circle at (0, 0). When OP is turned 90 degrees to give OQ, the x and y co-ordinates get interchanged. Also both x and y co-ordinates will be positive in the first quadrant. Hence s, the x co-ordinate of Q will be 1 (and y co-ordinate of Q will be √3).

Answer (B)

The question doesn’t seem difficult now (after understanding the concept); actually, it is a 700+ level.

Try another question using the same concept:

Question 2: On the coordinate plane (6, 2) and (0, 6) are the endpoints of the diagonal of a square. What are the coordinates of the point on the corner of the square which is closest to the origin?

(A) (0, 1)

(B) (1, 0)

(C) (1, 1)

(D) (2, 0)

(E) (2, 2)

Karishma, a Computer Engineer with a keen interest in alternative Mathematical approaches, has mentored students in the continents of Asia, Europe and North America. She teaches the GMAT for Veritas Prep and regularly participates in content development projects such as this blog!

1.  If the given statements are true, which of the following must also be true?
2.  Which of the following must be true for the argument to hold?

These are two different questions – The first one asks you to find the option which must be true given that the statements in the argument are true i.e. what can you infer from the argument? Here you are looking for the inference of the argument.

The second one asks you to find the option which must be true for the argument i.e. conclusion of the argument to hold i.e. what is necessary for the conclusion to hold. The statement that is necessary for the conclusion to hold is called an assumption so basically we are looking for the assumption here.

Let’s look at a question which often raises the inference vs assumption debate.

Question: According to psychoanalytic theory, people have unconscious beliefs that are kept from becoming conscious by a psychological mechanism termed “repression.” Researchers investigating the nature of this mechanism observed occasions on which a patient undergoing therapy became aware of and expressed a previously unconscious belief. They found that such occasions were marked by an unusual decrease in the patient’s level of anxiety.

If the information above is true, and if the researchers’ investigation was properly conducted, then which of the following must also be true?

(A) Changes in the patient’s anxiety level during therapy can generally be used as an accurate measure of the extent to which the patient is becoming conscious of previously repressed beliefs.
(B) Even when one of a patient’s unconscious beliefs remains unconscious, researchers are sometimes able to discover this belief.
(C) If psychoanalytic theory is correct, then most conscious beliefs originate as unconscious beliefs.
(D) Researchers were able to distinguish expressed beliefs that had previously been unconscious from those that had long been conscious but that the patient had not previously expressed.
(E) Although the beliefs on which the mechanism of repression works are all unconscious, the operation of the mechanism itself is something of which patients are consciously aware.

Solution:

As suggested in our CR book, it might be a good idea to first read the question stem to figure out the type of question.

“If the information above is true, and if the researchers’ investigation was properly conducted, then which of the following must also be true? “

The word ‘also’ is a dead giveaway. The statements are true – which of the following must also be true. We are looking for an inference – since the statements are true, what else is true based on the statements? i.e. what is the inference?

So the question is an inference/conclusion question. Great!

Let’s focus on the stimulus now.

Premises:

• Psychoanalytic theory says that people have unconscious beliefs that are kept from becoming conscious by repression.
• Researchers investigating repression observed occasions on which a patient undergoing therapy became aware of and expressed a previously unconscious belief.
• They found that such occasions were marked by an unusual decrease in the patient’s level of anxiety.

We are looking for something we can infer from these statements. Let’s look at the options.

(A)   Changes in the patient’s anxiety level during therapy can generally be used as an accurate measure of the extent to which the patient is becoming conscious of previously repressed beliefs.

The stimulus says that the occasions when patients become conscious of previously repressed beliefs are marked by decreased anxiety. They don’t say that the level of decrease varies directly or is correlated with the level of becoming conscious. Hence we cannot infer this from the stimulus.

(B) Even when one of a patient’s unconscious beliefs remains unconscious, researchers are sometimes able to discover this belief. 

Not relevant to the given statements. Whether there are other mechanisms of identifying unconscious beliefs is out of scope.

(C) If psychoanalytic theory is correct, then most conscious beliefs originate as unconscious beliefs.

Again, not relevant. All we can say is that people can be made to express some unconscious beliefs through therapy. We cannot say that all conscious beliefs originate as unconscious.

(D) Researchers were able to distinguish expressed beliefs that had previously been unconscious from those that had long been conscious but that the patient had not previously expressed. 

The given stimulus tells us that the researchers have been able to make people express their unconscious beliefs through therapy. A valid point here is – how do researchers know that the beliefs being expressed by the patients are unconscious beliefs and not merely conscious beliefs that the patients did not express previously? Since the stimulus says, “Researchers investigating repression observed occasions on which a patient undergoing therapy became aware of and expressed a previously unconscious belief” and we have to take the statement to be true, it means they were able to distinguish between unconscious beliefs and conscious but unexpressed beliefs. Hence (D) is the answer.

(E) Although the beliefs on which the mechanism of repression works are all unconscious, the operation of the mechanism itself is something of which patients are consciously aware.

Focus on the first part of this option. The statements in the stimulus tell us that repression keeps unconscious beliefs unconscious. They don’t tell us whether repression works on conscious beliefs or not – perhaps it keeps people from expressing conscious beliefs or perhaps it doesn’t work on conscious beliefs at all – the statements don’t say. We cannot say that the beliefs on which repression works are ALL unconscious. Hence we cannot infer this. Not the answer.

The correct answer is (D). 

Hope you will keep this in mind next time you come across a ‘must be true’ question.

Karishma, a Computer Engineer with a keen interest in alternative Mathematical approaches, has mentored students in the continents of Asia, Europe and North America. She teaches the GMAT for Veritas Prep and regularly participates in content development projects such as the Quarter Wit, Quarter Wisdom column of this blog!

Let me explain this with an example:

Question: Exports of United States cotton will rise considerably during this year. The reason for the rise is that the falling value of the dollar will make it cheaper for cloth manufacturers in Japan and Western Europe to buy American cotton than to get it from any other source.

Which of the following is an assumption made in drawing the conclusion above?

(A) Factory output of cloth products in Japan and Western Europe will increase sharply during this year.
(B) The quality of the cotton produced in the United States would be adequate for the purposes of Japanese and Western European cloth manufacturers.
(C) Cloth manufacturers in Japan and Western Europe would prefer to use cotton produced in the United States if cost were not a factor.
(D) Demand for cloth products made in Japan and Western Europe will not increase sharply during this year.
(E) Production of cotton by United States companies will not increase sharply during this year.

Solution:

First, let’s analyze the given argument:

Premises:
- Dollar is falling.
- It will be cheaper for cloth manufacturers in Japan and Western Europe to buy American cotton than to get it from any other source.

Conclusion:
- Exports of United States cotton will rise considerably during this year.

The conclusion links ‘sale of cotton’ to ‘cost of cotton’. It says that since the cost of American cotton will be lower than the cost of cotton from any other source, American cotton will sell. We are assuming here that the American cotton is adequate in all other qualities that the cloth manufacturers look for while buying cotton or that lower cost is all that matters. We are assuming that lowest cost will automatically lead to sale.

Let’s look at each of the options now:

(A)   Factory output of cloth products in Japan and Western Europe will increase sharply during this year.

Notice that we don’t NEED the factory output to increase. Even if it stays the same or in fact, even if it falls, as long as the manufacturers find American cotton suitable, the cotton exports of US could rise. Hence this is not the assumption.

(B)   The quality of the cotton produced in the United States would be adequate for the purposes of Japanese and Western European cloth manufacturers.

This option says that the quality is adequate and hence this is an assumption. Notice that it is necessary for our conclusion. If the quality is not adequate, no matter what the cost, US cotton sale may not increase. Hence, answer is (B).

(C)   Cloth manufacturers in Japan and Western Europe would prefer to use cotton produced in the United States if cost were not a factor.

This is the tricky non-correct option! Many people will swing between (B) and (C) for a while and then choose (C).  This option says that Japanese and Europeans prefer to use US cotton if cost does not matter. Do we NEED this to be true? No. It is good if it is true because it means that if cost of US cotton goes down, US cotton will sell more (hence, it is sufficient for the conclusion to be true – assuming all else stays constant). But do we NEED them to prefer US cotton? No. It is not necessary for our conclusion to be true. Even if the manufacturers don’t particularly prefer US cotton, US cotton exports could still increase if the price is the lowest.

Beware of this difference between ‘necessary’ and ‘sufficient’ conditions. Remember that assumptions are NECESSARY conditions; they don’t need to be sufficient. We end up incorrectly choosing sufficient conditions because they seem to be all encompassing and hence more attractive for our conclusion. If the sufficient condition is satisfied, then the conclusion has to be true. But mind you, that is not what the question is asking you. The question is looking for only a necessary condition, not a sufficient one. Also notice that sufficient conditions may not be necessary.

(D)   Demand for cloth made in Japan and Western Europe will not increase sharply during this year.

This is incorrect. We are not assuming that the demand for their cloth will not increase.

(E)    Production of cotton by United States companies will not increase sharply during this year.

It doesn’t matter what happens to the production of cotton in US. All we care about is that exports should rise.

The correct answer is (B). 

Hope you will be careful next time when you come across such a question.

Karishma, a Computer Engineer with a keen interest in alternative Mathematical approaches, has mentored students in the continents of Asia, Europe and North America. She teaches the GMAT for Veritas Prep and regularly participates in content development projects such as this blog!

You really don’t need to know many formulas for GMAT – just the basic ones e.g. Distance = Speed*Time, Work = Rate*Time (which are actually the same if you look at them closely) etc. If a Time-Distance-Speed question pertains to GMAT, rest assured it can be solved using just the formula given above and that too, within 1-2 mins. Then, do you need to learn the many formulas that people claim speed up question solving? No! In fact, the more specific the formula, the more constraints it has. It can be used in only particular circumstances and hence when the situation differs even a little bit from the ideal, you could end up using the formula incorrectly. Therefore, we recommend our students to stay away from the umpteen, less generic formulas until and unless they have already used them extensively. Let’s discuss this point with an example:

Question: A and B start from Opladen and Cologne respectively at the same time and travel towards each other at constant speeds along the same route. After meeting at a point between Opladen and Cologne, A and B proceed to their destinations of Cologne and Opladen respectively. A reaches Cologne 40 minutes after the two meet and B reaches Opladen 90 minutes after their meeting. How long did A take to cover the distance between Opladen and Cologne?

(A) 1 hour
(B) 1 hour 10 minutes
(C) 2 hours 30 minutes
(D) 1 hour 40 minutes
(E) 2 hours 10 minutes

Solution: People often like to use a formula for this situation. Let’s quickly discuss that first.

If two objects A and B start simultaneously from opposite points and, after meeting, reach their destinations in ‘a’ and ‘b’ hours respectively (i.e. A takes ‘a hrs’ to travel from the meeting point to his destination and B takes ‘b hrs’ to travel from the meeting point to his destination), then the ratio of their speeds is given by:

Sa/Sb = √(b/a)

i.e. Ratio of speeds is given by the square root of the inverse ratio of time taken.

Sa/Sb = √(90/40) = 3/2

This gives us that the ratio of the speed of A : speed of B as 3:2. We know that time taken is inversely proportional to speed. If ratio of speed of A and B is 3:2, the time taken to travel the same distance will be in the ratio 2:3. Therefore, since B takes 90 mins to travel from the meeting point to Opladen, A must have taken 60 (= 90*2/3) mins to travel from Opladen to the meeting point

So time taken by A to travel from Opladen to Cologne must be 60 + 40 mins = 1 hr 40 mins

Now let’s see how we can solve the question without using the formula.

Think of the point in time when they meet:

A starts from Opladen and B from Cologne simultaneously. After some time, say t mins of travel, they meet. Since A covers the entire distance of Opladen to Cologne in (t + 40) mins and B covers it in (t + 90) mins, A is certainly faster than B and hence the Meeting point is closer to Cologne.

Now think, what information do we have? We know the time taken by A and B to reach their respective destinations from the meeting point. We also know that they both traveled the same distance i.e. the distance between Opladen and Cologne. So let’s try to link distance with time taken. We know that ‘Distance’ varies directly with ‘Time taken’. (Check out this post if you don’t know what we are talking about here.)

Distance between Opladen and Meeting point /Distance between Meeting point and Cologne = Time taken to go from Opladen to Meeting point/Time taken to go from Meeting point to Cologne = t/40 (in case of A) = 90/t (in case of B)
t = 60 mins

So A takes 60 mins + 40 mins = 1 hr 40 mins to cover the entire distance.

Answer (D)

We could easily solve the question without using any specific formula. So stick to your basics and kick those little grey cells to get to the answer!

Karishma, a Computer Engineer with a keen interest in alternative Mathematical approaches, has mentored students in the continents of Asia, Europe and North America. She teaches the GMAT for Veritas Prep and regularly participates in content development projects such as this blog!

Last week

Say, the question stem of a DS question asks you to find the value of n, the number of people in the room. Statement 1 of the question gives you the following equation:

60/(n – 5) – 60/n = 2

We can easily figure out that a value of n that satisfies this equation is 15. Now, is that enough to say that statement 1 is sufficient alone? No! It could be a trap! The equation, when manipulated, gives us a quadratic. It is important to find out whether the second solution of the quadratic works for us. When n is the number of people, it must be positive. So one extra step that we should take is re-arrange the equation to get the quadratic. If the constant term i.e. the product of the roots is negative, it means one root is positive and one is negative. Since we have already found the positive root, it is the only answer and hence we can say that the statement 1 is sufficient alone.

60/(n – 5) – 60/n = 2

60*n – 60*(n – 5) = 2*n*(n – 5)

n^2 – 5n – 150 = 0

The constant term, -150, is negative so the product of the roots must be negative. This means one root must be negative and the other must be positive. Since we have already found the positive root i.e. the number of people in the room, we can say that statement 1 is sufficient alone.

Let’s look at an example where we could fall in the trap.

Say statement 1 gives us an equation which looks like this:

60/(n +5) – 10/(n – 5) = 2

As discussed last week, we will easily see that n = 10 satisfies this equation. So should we move on now and say that statement 1 is sufficient alone? No, not so fast! Let’s try to manipulate the equation to get the quadratic.

60/(n +5) – 10/(n – 5) = 2

60*(n – 5) – 10*(n + 5) = 2*(n – 5)(n + 5)

n^2 – 25n + 150 = 0

n = 10 or 15

So actually, there are two values of n that satisfy this equation. In PS questions, since we have a single answer, there would be only one solution so once you get one, you are done. In DS questions, you need to be certain that only one value satisfies. There is a possibility that both values satisfy your constraints in which case your answer would change.

Therefore, it may not be necessary to solve the equation for the PS question, but it is certainly necessary to solve it for DS. That’s counter intuitive, isn’t it? We hope you understand the reason.

Another related trap in DS questions: Statement 1 gives you a quadratic and asks you for the value of x (no constraints that x must be an integer or positive number etc). You know that it is a quadratic and it will give you two values of x so you say that statement 1 is not sufficient alone and move on. But hold it! What if both the roots of the equation are same? It may not apparent to you when you look at the equation. When you solve it, you realize that the roots are the same. Hence, ensure that you solve the equation in DS questions before you decide on the sufficiency.

Karishma, a Computer Engineer with a keen interest in alternative Mathematical approaches, has mentored students in the continents of Asia, Europe and North America. She teaches the GMAT for Veritas Prep in Detroit, Michigan, and regularly participates in content development projects such as this blog!

Let’ see whether we can get a value of n which satisfies this equation without going the whole nine yards. We will not use any options and will try to rely on our knowledge that GMAT questions don’t take much time.

60/(n – 5) – 60/n = 2

So, the difference between the two terms of the left hand side is 2. Try to look for values of n which give us simple numbers i.e. try to plug in values which are factors of the numerator.
Say, if n = 10, you get 60/5 – 60/10 = 12 – 6 = 6. The difference between them is much more than 2. 60/n and 60/(n – 5) need to be much closer to each other so that the difference between them is 2. The two terms should be smaller to bring them closer together. So increase the value of n.

Put n = 15 since it is the next number such that (15 – 5 =) 10 as well as 15 divide 60 completely. You get 60/10 – 60/15 = 6 – 4 = 2. It satisfies and you know that a value that n can take is 15. Usually, you will get a solution within 2-3 iterations. This is enough for a PS question. Notice that this equation gives us a quadratic so be careful while working on DS questions. You might need to manipulate the equation a little to figure out whether the other root is a possible solution as well. Anyway, today we will focus on the application of such equations in PS questions only. Let’s take a question now to understand the concept properly:

Question: Machine A takes 2 more hours than machine B to make 20 widgets. If working together, the machines can make 25 widgets in 3 hours, how long will it take machine A to make 40 widgets?

(A) 5
(B) 6
(C) 8
(D) 10
(E) 12

Solution: We need to find the time taken by machine A to make 40 widgets. It will be best to take the time taken by machine A to make 40 widgets as the variable x. Then, when we get the value of x, we will not need to perform any other calculations on it and hence the scope of making an error will reduce. Also, value of x will be one of the options and hence plugging in to check will be easy.

Machine A takes x hrs to make 40 widgets.

Rate of work done by machine A = Work done/Time taken = 40/x

Machine B take 2 hrs less than machine A to make 20 widgets hence it will take 4 hrs less than machine B to make 40 widgets. Think of it this way: Break down the 40 widgets job into two 20 widget jobs. For each job, machine B will take 2 hrs less than machine A so it will take 4 hrs less than machine A for both the jobs together.

Time taken by machine B to make 40 widgets = x – 4

Rate of work done by machine B = Work done/Time taken = 40/(x – 4).

We know the combined rate of the machines is 25/3

So here is the equation:

40/x + 40/(x – 4) = 25/3

The steps till here are not complicated. Getting the value of x poses a bit of a problem.

Notice here that that the right hand side is not an integer. This will make the question a little harder for us, right? Wrong! Everything has its pros and cons. The 3 of the denominator gives us ideas for the values of x (as do the options).  To get a 3 in the denominator, we need a 3 in the denominator on the left hand side too.

x cannot be 3 but it can be 6. If x = 6, 40/(6 – 4) = 20 i.e. the sum will certainly not be 20 or more since we have 25/3 = 8.33 on the right hand side.

The only other option that makes sense is x = 12 since it has 3 in it.

40/12 + 40/(12 – 4) = 10/3 + 5 = 25/3

Answer (E)

If we did not have the options, we might have tried x = 9 too before landing on x = 12. Nevertheless, these calculations are not time consuming at all since you can get rid of the incorrect numbers orally. Making a quadratic and solving it is certainly much more time consuming.

Another method could be to bring 3 to the left hand side to get the following equation:

120/x + 120/(x – 4) = 25

This step doesn’t change anything but it helps if you face a mental block while working with fractions. Try to practice such questions using these techniques – they will save you a lot of time.

Karishma, a Computer Engineer with a keen interest in alternative Mathematical approaches, has mentored students in the continents of Asia, Europe and North America. She teaches the GMAT for Veritas Prep and regularly participates in content development projects such as this blog!