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]]>We have often found that one strategy, which is very helpful in other question types too, helps sort out most questions of this type, though not in the same way. That strategy is – ‘find the conclusion(s)’

The conclusion of the argument is the position taken by the author.

Boldface questions (and others too) sometimes have more than one conclusion – One would be the conclusion of the argument i.e. the author’s conclusion. The argument could mention another conclusion which could be the conclusion of a certain segment of people/ some scientists/ some researchers/ a politician etc. We need to segregate these two and how each premise supports/opposes the various conclusion. Once this structure is in place, we automatically find the answer. Let’s see how with an example.

**Question:**** Recently, motorists have begun purchasing more and more fuel-efficient economy and hybrid cars that consume fewer gallons of gasoline per mile traveled. There has been debate as to whether we can conclude that these purchases will actually lead to an overall reduction in the total consumption of gasoline across all motorists.** The answer is no, since motorists with more fuel-efficient vehicles are likely to drive more total miles than they did before switching to a more fuel-efficient car, negating the gains from higher fuel-efficiency.

Which of the following best describes the roles of the portions in bold?

(A)The first describes a premise that is accepted as true; the second introduces a conclusion that is opposed by the argument as a whole.

(B)The first states a position taken by the argument; the second introduces a conclusion that is refuted by additional evidence.

(C)The first is evidence that has been used to support a position that the argument as a whole opposes; the second provides information to undermine the force of that evidence.

(D)The first is a conclusion that is later shown to be false; the second is the evidence by which that conclusion is proven false.

(E)The first is a premise that is later shown to be false; the second is a conclusion that is later shown to be false.

**Solution**: As our first step, let’s try to figure out the conclusion of the argument:

The author’s view is that “purchases of fuel efficient vehicles will NOT lead to an overall reduction in the total consumption of gasoline across all motorists.”

This is the position the argument (and author) takes.

The argument gives us another conclusion: these purchases will actually lead to an overall reduction in the total consumption of gasoline across all motorists.

Some people take this position (implied by the use of “there has been debate”)

This is our second bold statement. It introduces the opposing conclusion.

Let’s look at our options now.

(A) The first describes a premise that is accepted as true; the second introduces a conclusion that is opposed by the argument as a whole.

The first bold statement: Recently, motorists have begun purchasing more and more fuel-efficient economy and hybrid cars that consume fewer gallons of gasoline per mile traveled.

This is a premise and has been accepted as true. We know it has been accepted as true since the last line ends with – “…negating the gains from higher fuel-efficiency”

We have seen above that the second bold statement tells us about a conclusion that the argument opposes.

So (A) is correct. We have found our answer but let’s look at the other options too.

(B) The first states a position taken by the argument; the second introduces a conclusion that is refuted by additional evidence.

The first bold statement is a premise. It is not the position taken by the argument. Let’s move on.

(C) The first is evidence that has been used to support a position that the argument as a whole opposes; the second provides information to undermine the force of that evidence.

This option often confuses test-takers.

The evidence is – “Recently, motorists have begun purchasing more and more fuel-efficient economy and hybrid cars that consume fewer gallons of gasoline per mile traveled.”

That is, “the motorists have begun purchasing fuel efficient cars that give better mileage.”

The second bold statement does not undermine this evidence at all. In fact, it builds up on it with – “This brings up a debate on whether it will lead to overall decreased fuel consumption?”

Hence (C) is not correct.

(D)The first is a conclusion that is later shown to be false; the second is the evidence by which that conclusion is proven false.

The first bold statement is not a conclusion. So no point dwelling on this option.

(E)The first is a premise that is later shown to be false; the second is a conclusion that is later shown to be false.

The premise is taken to be true. The argument ends with “… the gains from higher fuel-efficiency”. Hence, this option doesn’t stand a chance either.

We hope you see how easy it is to break down the options once we identify the conclusion(s).

Keep practicing!

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*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!*

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]]>*Twenty-four men can complete a job in sixteen days. Thirty-two women can complete the same job in twenty-four days. Sixteen men and sixteen women started working on the job for twelve days. How many more men must be added to complete the job in 2 days?*

*(A) 16*

* (B) 24*

* (C) 36*

* (D) 48*

* (E) 54*

Here, we are dealing with two groups of people: men and women. These two groups have different rates of completing a job. We are also told that a certain number of men and women do a part of the job, and we are asked to find the number of additional “men” required to finish the job in a shorter amount of time.

Recall that we have already come across questions where workers start some work and then more workers join in to complete the work before time.

The problem with this question is that we have two types of workers, not just one. So let’s try to simplify the question to a form that we know how to easily solve.

We’ll start by finding the relation between the rate of work done by men and the rate of work done by women. Let’s make the number of men and women the same to find the number of days it will take each group to complete 1 job.

Given: 24 men complete 1 job in 16 days

Given: 32 women complete 1 job in 24 days

So how many days will 24 women take to complete 1 work? (Why 24 women? Because we know how many days 24 men take)

We know how to solve this problem. (It has already been discussed in a past post).

32 women ……………. 1 work ………………. 24 days

24 women ……………. 1 work ………………. ?? days

No. of days taken = 24 * (32/24) = 32 days

Now this is what we have: 24 men take 16 days while 24 women take 32 days

So women take twice the time taken by men to do the same work (32 days vs 16 days). This means the rate of work of women is half the rate of work of men. This means 2 women are equivalent to 1 man i.e. 2 women will do the same work as 1 man does in the same time.

So now, let us replace all women by men so that we have only one type of worker.

Now this is our regular work rate question –

Given: 24 men complete the work in 16 days

Given: 16 men and 16 women work for 12 days

This means that we have 16 men and 8 men work for 12 days

which implies 24 men work for 12 days

We know that 24 men complete the work in 16 days. If they work for 12 days, there are 4 more days to go. But the work has to be completed in 2 days.

24 men …………… 4 days

?? men ……………. 2 days

No of men needed = 24 * (4/2) = 48

So we need 24 additional men to complete the work in 2 days.

Or looking at it another way, 24 men need 16 days to complete the work, so they need another 4 days to complete. But if we want them to complete the work in half the time (2 days), we will need twice the work force. So we need another 24 men.

Answer (B)

Basically, the question involved solving two smaller work-rate problems. Doesn’t seem daunting now, right?

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*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!*

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]]>*In 2009, a private school spent $200,000 on a building which housed classrooms, offices, and a library. In 2010, the school was unable to turn a profit. Therefore, the principal should be fired.*

*Each of the following, if true, weakens the author’s conclusion EXCEPT:*

*(A) The principal was hired primarily for her unique ability to establish a strong sense of community, which many parents cited as a quality that kept children enrolled in the school longer.*

*(B) The new library also features a seating area big enough for all students to participate in cultural arts performances, which the head of school intends to schedule more frequently now.*

*(C) The principal was hired when the construction of the new building was almost completed.*

*(D) A significant number of families left the school in 2010 because a favourite teacher retired.*

*(E) More than half of the new families who joined the school in 2010 cited the beautiful new school facility as an important factor in their selection of the school.*

This is a weaken/exception question, so four of the five answer choices will weaken the argument, while the fifth option (which will be the correct answer) will either not have any impact on the argument or it might even strengthen it. As we know, such questions require a bit more effort to answer, since four of the five options will definitely be relevant to the argument. The important thing is to focus on what we are given and not assume what the various answer options may or may not lead to. Let’s understand this:

The gist of the argument:

- Last year, a lot of money was spent to construct a new building with many amenities.
- This year, the school did not see a profit.
- Hence, fire the principal.

Based on the two given facts – “a lot of money was spent to make the building in 2009” and “the school did not see a profit in 2010” – the author has decided to fire the principal. Many pieces of information could weaken his stance. For example:

- It was not the principal’s decision to construct the building.
- The school’s revenue in 2010 took a hit because of some other factor.
- The school’s losses reduced by a huge amount in 2010 and the probability of it seeing a profit in 2011 is high.

Information such as this could improve the principal’s case to stay. We know that for this particular question, there will only be one option that does not help the principal.

You will have to choose the answer choice which, with the given information, does not help the principal’s case. Let’s look at the options now:

*(A) The principal was hired primarily for her unique ability to establish a strong sense of community, which many parents cited as a quality that kept children enrolled in the school longer.*

With this answer choice, we see that the principal was hired not to increase school profits, but for another critical purpose. Perhaps the school’s finance department is in charge of worrying about profits, and so the head of that department needs to be fired! This answer choice makes a strong case for keeping the principal, and hence, weakens the author’s argument.

*(B) The new library also features a seating area big enough for all students to participate in cultural arts performances, which the head of school intends to schedule more frequently now.*

If true, this statement would have no impact on whether or not the principal should be fired. It describes an amenity provided by the new building and how it will be used – it neither strengthens nor weakens the principal’s case to stay, hence, this is the correct answer choice. But let’s look at the rest of the options too, just to be safe:

*(C) The principal was hired when the construction of the new building was almost completed.*

This tells us that the new building was not her decision. So if it did not have the desired effect, she cannot be blamed for it. So it again helps her case.

(D) A significant number of families left the school in 2010 because a favourite teacher retired.

This answer choice shows that there was another reason behind the school’s loss in profit. The construction of the building could still be a good idea that leads to future profits, which the principal’s case and weakens the author’s argument.

*(E) More than half of the new families who joined the school in 2010 cited the beautiful new school facility as an important factor in their selection of the school.*

For some reason, this is the answer choice that often trips up students. They feel that it doesn’t help the principal’s case – that because the new building attracts students, if there are losses, it means that the loss is due to a fault with the new building, and thus, the principal is at fault. But note that we are assuming a lot to arrive at that conclusion. All we are told is that the new building is attracting students. This means the new building is serving its purpose – it is generating extra revenue. The fact that the school is still experiencing losses could be explained by many different reasons.

Since the author’s decision to fire the principal is based solely on the premise that a lot of money was spent to construct the new building, which now seems to serve no purpose (because the school experienced losses), this answer choice certainly weakens the argument. The option tells us that the principal’s decision to make the building was justified, so it helps her case to stay with the school.

After examining each answer choice, we can see that the answer is clearly B. Remember, in Critical Reasoning questions it is crucial to come to conclusions only based on the facts that are given – creating assumptions based on information that is not given can lead you to fall in a Testmaker trap.

*Getting ready to take the GMAT? We have free online GMAT seminars running all the time. And, be sure to follow us on Facebook, YouTube, Google+, and Twitter!*

*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!*

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]]>The post Quarter Wit, Quarter Wisdom: When Can You Divide by a Variable? appeared first on Veritas Prep Blog.

]]>For example:

Is division by x allowed here: x^2 = 10x?

Is division by x allowed here: y = 4x?

Is division by x allowed here: x^2 < 4x?

Let’s take a detailed look at all these questions today.

The basic guidelines:

- Division by 0 is not allowed, hence you cannot divide by a variable until and unless we know that it cannot be 0.
- In the case of an inequality, when you divide by a negative number, the sign of the inequality flips. So we cannot divide by a variable until and unless we know that it cannot be 0 AND whether it is positive or negative.

Let’s look at the three questions given above and try to solve them using these guidelines:

*Is division by x allowed here: x^2 = 10x?*

The first thing to find out here is whether or not x can equal 0.

Case 1: If no other information has been given, then x can be 0 and we cannot divide by it. This is how we proceed in that case:

x^2 – 10x = 0

x(x – 10) = 0

x = 0 or 10

Case 2: If the question stem tells us that x is not 0, then we can divide by x.

x^2/x = 10x/x

x = 10

Obviously, we don’t get the second solution (x = 0) in this case, as we already know that x cannot be 0. Now let’s look at the second problem:

*Is division by x allowed here: y = 4x?*

Again, this is an equation and we need to know whether or not x can equal 0.

Case 1: If x can be 0, you cannot divide by it. In this case, x = 0 and y = 0 is one of the infinite possible solutions.

Case 2: If the question stem states that x cannot be 0, then we can do the following:

y/x = 4

Now let’s look at the final question:

*Is division by x allowed here: x^2 > -4x?*

Here, we have an inequality. Before deciding whether we can divide by x or not, we need to know not only whether x can be equal to 0, but also whether x is positive or negative.

Case 1: If we know nothing about the possible values that x can take, then this is how we proceed:

x^2 + 4x > 0

x(x + 4) > 0

Now we can use the method discussed in the first problem to arrive at the range of x.

x > 0 or x < -4

Case 2: If we know that x is positive, then we can proceed like this:

x^2/x > -4x/x

x > -4

Since we are given that x is positive, we know that that x > 0 (looking at the two options above).

Case 3: If we know that x is negative, then this is how we will proceed:

x^2/x < -4x/x (we flip the sign of the inequality because we divide by x, which is negative)

x < -4

The results obtained are logical, right? When x can be anywhere on the number line, we get the range as x > 0 or x < -4.

If x has to be positive, the range is x > 0.

If x has to be negative, the range is x < -4.

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]]>The post Quarter Wit, Quarter Wisdom: When a Little Information is Enough to Solve a GMAT Problem appeared first on Veritas Prep Blog.

]]>Let’s explore this idea through an example GMAT data sufficiency question:

*What is the standard deviation of a set of numbers whose mean is 20?*

*Statement 1: The absolute value of the difference of each number in the set from the mean is equal.*

*Statement 2: The sum of the squares of the differences from the mean is greater than 100.*

We need to determine whether the information we have been given is sufficient to get us the exact value of the standard deviation of a particular set of numbers. To find the standard deviation of a set, we need to know the deviation of each term from the mean so that we can square those deviations, sum the squares, divide them by the number of terms, and then find the square root.

Essentially, to find the standard deviation we either need to know each element of the set, or we need to know the deviation of each element from the mean (which will also give us the number of terms), or we need to know the sum of the square of deviations and the number of terms in the set.

The question stem here tells us that the mean of the set is 20. We have no other information about any of the actual elements of the set or the number of elements. With this in mind, let’s examine each of the statements:

Statement 1: The absolute value of the difference of each number in the set from the mean is equal.

With this statement, we don’t actually know what the absolute value of the difference is. We also don’t know how many elements there are. The set could be something like:

19, 21 (each term is exactly 1 away from the mean 20)

or

18, 18, 22, 22 (each term is exactly 2 away from the mean 20)

etc.

The standard deviation in each case will be different. We don’t know the elements of the set and we don’t know the number of elements in the set. Because of this, there is no way for us to know the value of the standard deviation – this statement alone is not sufficient.

Statement 2: The sum of the squares of the differences from the mean is greater than 100.

“Greater than 100” encompasses a large range of numbers – it could be any value larger than 100. Again, we cannot find the exact standard deviation of the set, so this statement is also not sufficient alone.

Using both statements together, we still do not have any idea of what the elements of the set are or what the sum of the squares of the differences from the mean is. We also still don’t know the number of elements. Hence, both statements together are not sufficient, so the answer is E.

Now, let us add just one more piece of information to the problem in this similar question:

*What is the standard deviation of a set of 7 numbers whose mean is 20?*

*Statement 1: The absolute value of the difference of each number in the set from the mean is equal.*

*Statement 2: The sum of the squares of the differences from the mean is greater than 100.*

What would you expect the answer to be? Still E, right? The sum of the deviations are still unknown and the exact elements of the set are still unknown – all we know is the number of elements. Actually, this information is already too much. All we need to know is that the number of elements is odd and suddenly we can find the standard deviation.

Here is why:

Statement 1 is quite tricky.

If we have an odd number of elements, in which case can the absolute values of the differences of each number in the set from the mean be equal?

Think about it – the mean of the set is 20. What could a possible set look like such that the mean is 20 and the absolute values of the differences of each number in the set from the mean are equal. Try to think of such a set with just 3 elements. Can you come up with one?

19, 19, 21? No, the mean is not 20

19, 20, 21? No, the absolute value of the difference of each number in the set from the mean is not equal. 19 is 1 away from mean but 20 is 0 away from mean.

Note that in this case, the only possible set that could fit the given criteria is one consisting of just an odd number of 20s (all elements in this set must be 20). Only then can each number be equidistant from the mean, i.e. each number would be 0 away from mean. If the numbers of the set all have equal elements, then obviously the standard deviation of the set is 0. It doesn’t matter how many elements it has; it doesn’t matter what the mean is! In this case, Statement 1 alone is sufficient so the answer would be A.

**Takeaway**:

If a set has an even number of distinct terms, the absolute values of the distances of each term from the mean could be equal. But if a set has an odd number of terms and the absolute values of the distances of each term from the mean are equal, all the terms in the set must be the same and will be equal to the mean.

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]]>The post Quarter Wit Quarter Wisdom: How to Read GMAT Questions Carefully appeared first on Veritas Prep Blog.

]]>*Alice’s take-home pay last year was the same each month, and she saved the same fraction of her take-home pay each month. The total amount of money that she had saved at the end of the year was 3 times the amount of that portion of her monthly take-home pay that she did NOT save. If all the money that she saved last year was from her take-home pay, what fraction of her take-home pay did she save each month?*

*(A) 1/2*

*(B) 1/3*

*(C) 1/4*

*(D) 1/5*

*(E) 1/6*

Let’s consider the question stem sentence by sentence:

“Alice’s take-home pay last year was the same each month, and she saved the same fraction of her take-home pay each month.”

Say Alice’s take-home pay last year was $100 each month. She saves a fraction of this every month – let the amount saved be x.

“The total amount of money that she had saved at the end of the year was 3 times the amount of that portion of her monthly take-home pay that she did NOT save.”

What would be “the total amount of money that she had saved at the end of the year”? Since Alice saves x every month, she would have saved 12x by the end of the year.

What would be “the amount of that portion of her monthly take-home pay that she did NOT save”? Note that this is going to be (100 – x). Many test takers end up using (100 – x)*12, however this equation is not correct. The key word here is “monthly” – we are looking for how much Alice does not save each month, not how much she does not save during the whole year.

The total amount of money that Alice saved at the end of the year is 3 times the amount of that portion of her MONTHLY take-home pay that she did not save. Now we know we are looking for:

12x = 3*(100 – x)

x = 20

“If all the money that she saved last year was from her take-home pay, what fraction of her take-home pay did she save each month?”

From our equation, we have determined that Alice saved $20 out of every $100 she earned every month, so she saved 20/100 = 1/5 of her take-home pay.

Therefore, the answer is D.

Often, test-takers make the mistake of writing the equation as:

12x = 3*(100 – x)*12

x = 300/4

However, this will give them the fraction (300/4)/100 = 3/4, and that’s when they will wonder what went wrong.

Be extra careful when reading GMAT questions so that precious minutes are not wasted on such avoidable errors.

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]]>The post Quarter Wit, Quarter Wisdom: Solving the Fuel-Up Puzzles appeared first on Veritas Prep Blog.

]]>(Before we tackle today’s puzzle, first take a look at our posts on how to solve pouring water puzzles, weighing puzzles, and hourglass puzzles.)

Another variety of puzzle involves distributing fuel among vehicles to reach a destination. Let’s look at this type of question today:

*A military car carrying an important letter must cross a desert. There is no petrol station in the desert, and the car’s fuel tank is just enough to take it halfway across. There are other cars with the same fuel capacity that can transfer their petrol to one another. There are no canisters to carry extra fuel or rope to tow the cars.*

*How can the letter be delivered?*

Here, we are given that a single car can only reach the midpoint of the desert on its own tank of gas. Since there are no canisters, the car cannot carry extra fuel, so it will need to be fueled up by other cars traveling along with it.

Let’s fill up 4 cars and get them to start crossing the desert together. By the time they cover a quarter of the desert, half of their fuel tanks will be empty. Hence, we will have 4 cars with half tanks, and the status of their fuel tanks will be:

(0.5, 0.5, 0.5, 0.5)

If we transfer the fuel from two of the cars into two other cars, we will have:

(1, 1, 0, 0)

The two cars with fuel in their tanks will continue to cross the desert and cover another quarter of it. Now both of the cars will have half tanks again, and they will have reached the middle of the desert:

(0.5, 0.5, 0, 0)

Now one car will transfer all of its fuel to the other car, allowing that car to have one full tank:

(1, 0, 0, 0)

That car can then carry the letter through the remaining half of the desert.

For this problem, we didn’t really care about the stalled cars in the middle of the desert since we are not required to bring them back. The only important thing is to get the letter completely across the desert. Now, how do we handle a puzzle that asks us to get all of the vehicles back, too? Let’s look at an example question with those constraints:

*A distant planet “X” has only one airport located at the planet’s North Pole. There are only 3 airplanes and lots of fuel at the airport. Each airplane has just enough fuel capacity to get to the South Pole (which is diametrically opposite the North Pole). The airplanes can land anywhere on the planet and transfer their fuel to one another.*

*The mission is for at least one airplane to fly completely around the globe and stay above the South Pole; in the end, all of the airplanes must return to the airport at the North Pole.*

For this problem, we are given that a plane with a full tank of fuel can only reach the South Pole, i.e. cover half the distance it needs to travel for the mission. We need it to take a full trip around the planet – from the North Pole, to the South pole, and back again to North Pole. Obviously, we will need more than one plane to fuel the plane which will fly above the South pole.

Let’s divide the distance from pole to pole into thirds (from the North Pole to the South Pole we have three thirds, and from the South Pole to the North Pole we have another three thirds).

**Step #1:** 2 airplanes will fly to the first third. A third of their fuel will be used, so the status of their fuel tanks will be:

(2/3, 2/3)

One airplane will then fuel up the other plane and go back to the airport. Now the status of their tanks is:

(3/3, 1/3)

**Step #2:** 2 airplanes will again fly from the airport to the first third – one airplane will fuel up the other plane and go back to the airport. So the status of these two airplanes is this:

(3/3, 1/3)

**Step #3:** Now there are two airplanes at the first third mark with their tanks full. They will now fly to the second third point, giving us:

(2/3, 2/3)

One of the airplanes will fuel up the second one (until its tank is full) and go back to the first third, where it will meet the third airplane (which has just come back from the airport to support it with fuel) so that they both can return to the airport.

In the meantime, the airplane at the second third, with a full tank of fuel, will fly as far as it can – over the South Pole and towards the North pole, to the last third before the airport.

**Step #4:** One of the two airplanes from the airport can now go to the first third (on the opposite side of the North pole as before), and share its 1/3 fuel so that both airplanes safely land back at the airport.

And that is how we can have one plane travel completely around the planet and still have all airplanes arrive back safely!

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]]>The post Quarter Wit Quarter Wisdom: Solving the Weighing Puzzle (Part 2) appeared first on Veritas Prep Blog.

]]>Today, we will look at some puzzles that require the use of a traditional weighing scale. When we put an object on this scale, it shows us the weight of the object.

This is what such a scale looks like:

Puzzles involving a weighing scale can be quite tricky! Let’s take a look at a couple of examples:

*You have 10 bags with 1000 coins in each. In one of the bags, all of the coins are forgeries. A true coin weighs 1 gram; each counterfeit coin weighs 1.1 grams. *

*If you have an accurate weighing scale, which you can use only once, how can you identify the bag with the forgeries?*

We are allowed only a single weighing, so we cannot weigh all 10 bags on the scale individually to measure which one has counterfeit coins. We need to find the bag in only one weighing, so we need to somehow make the coins in the bags distinctive.

How do we do that? We can take out one coin from the first bag, two coins from the second bag, three coins from the third bag and so on. Finally, we will have 1 + 2 + 3 + … + 10 = 10*11/2 = 55 coins.

Let’s weigh these 55 coins now.

If all coins were true, the total weight would have been 55 grams. But since some coins are counterfeit, the total weight will be more. Say, the total weight comes out to be 55.2 grams. What can we deduce from this? We can deduce that there must be two counterfeit coins (because each counterfeit coin weighs 0.1 gram extra). So the second bag must be the bag of counterfeit coins.

Let’s try one more:

*A genuine gummy bear has a mass of 10 grams, while an imitation gummy bear has a mass of 9 grams. You have 7 cartons of gummy bears, 4 of which contain real gummy bears while the others contain imitation bears. *

*Using a scale only once and the minimum number of gummy bears, how can you determine which cartons contain real gummy bears?*

Now this has become a little complicated! There are three bags with imitation gummy bears. Taking a cue from the previous question, we know that we should take out a fixed number of gummy bears from each bag, but now we have to ensure that the sum of any three numbers is unique. Also, we have to keep in mind that we need to use the minimum number of gummy bears.

So from the first bag, take out no gummy bears.

From the second bag, take out 1 gummy bear.

From the third bag, take out 2 gummy bears (if we take out 1 gummy bear, the sum will be the same in case the second bag has imitation gummy bears or in case third bag has imitation gummy bears.

From the fourth bag, take out 4 gummy bears. We will not take out 3 because otherwise 0 + 3 and 1 + 2 will give us the same sum. So we won’t know whether the first and fourth bags have imitation gummy bears or whether second and third bags have imitation gummy bears.

From the fifth bag, take out 7 gummy bears. We have obtained this number by adding the highest triplet: 1 + 2 + 4 = 7. Note that anything less than 7 will give us a sum that can be made in multiple ways, such as:

0 + 1 + 6 = 7 and 1 + 2 + 4 = 7

or

0 + 1 + 5 = 6 and 0 + 2 + 4 = 6

But we need the sum to be obtainable in only one way so that we can find out which three bags contain the imitation gummy bears.

At this point, we have taken out 0, 1, 2, 4, and 7 gummy bears.

From the sixth bag, take out 13 gummy bears. We have obtained this number by adding the highest triplet: 2 + 4 + 7 = 13. Note that anything less than 13 will, again, give us a sum that can be made in multiple ways, such as:

12 + 1 + 0 = 13 and 2 + 4 + 7 = 13

or

0 + 1 + 9 = 10 and 1 + 2 + 7 = 10

…etc.

Note that this way, we are also ensuring that we measure only the minimum number of gummy bears, which is what the question asks us to do.

From the seventh bag, take out 24 gummy bears. We have obtained this number by adding the highest triplet again: 4 + 7 + 13 = 24. Again, anything less than 24 will give us a sum that can be made in multiple ways, such as:

0 + 1 + 15 = 16 and 1 + 2 + 13 = 16

or

0 + 1 + 19 = 20 and 0 + 7 + 13 = 20

or

0 + 1 + 23 = 24 and 4 + 7 + 13 = 24

…etc.

Thus, this is the way we will pick the gummy bears from the 7 bags: 0, 1, 2, 4, 7, 13, 24.

In all, 51 gummy bears will be weighed. Their total weight should be 510 grams (51*10 = 510) but because three bags have imitation gummy bears, the weight obtained will be less.

Say the weight is less by 8 grams. This means that the first bag (which we pulled 0 gummy bears from), the second bag (which we pulled 1 gummy bear from) and the fifth bag (which we pulled 7 gummy bears from) contain the imitation gummy bears. This is because 0 + 1 + 7 = 8 – note that we will not be able to make 8 with any other combination.

We hope this tricky little problem got you thinking. Work those grey cells and the GMAT will not seem hard at all!

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]]>The post Quarter Wit, Quarter Wisdom: Solving the Hourglass Puzzle appeared first on Veritas Prep Blog.

]]>First, understand what an hourglass is – it is a mechanical device used to measure the passage of time. It is comprised of two glass bulbs connected vertically by a narrow neck that allows a regulated trickle of sand from the upper bulb to fall into the lower one. The sand also takes a fixed amount of time to fall from the upper bulb to the lower bulb. Hourglasses may be reused indefinitely by inverting the bulbs once the upper bulb is empty.

This is what they look like:

Say a 10-minute hourglass will let us measure time in intervals of 10 minutes. This means all of the sand will flow from the upper bulb to the lower bulb in exactly 10 minutes. We can then flip the hourglass over – now sand will start flowing again for the next 10 minutes, and so on. We cannot measure, say, 12 minutes using just a 10-minute hourglass, but we can measure more time intervals when we have two hourglasses of different times. Let’s look at this practice problem to see how this can be done:

*A teacher of mathematics used an unconventional method to measure a 15-minute time limit for a test. He used a 7-minute and an 11-minute hourglass. During the whole time, he turned the hourglasses only 3 times (turning both hourglasses at once counts as one flip). **Explain how the teacher measured out 15 minutes.*

Here, we have a 7-minute hourglass and an 11-minute hourglass. This means we can measure time in intervals of 7 minutes as well as in intervals of 11 minutes. But consider this: if both hourglasses start together, at the end of 7 minutes, we will have 4 minutes of sand leftover in the top bulb of the 11-minute hourglass. So we can also measure out 4 minutes of time.

Furthermore, if we flip the 7-minute hourglass over at this time and let it flow for that 4 minutes (until the sand runs out of the top bulb of the 11-minute hourglass), we will have 3 minutes’ worth of sand leftover in the 7-minute hourglass. Hence, we can measure a 3 minute time interval, too, and so on…

Now, let’s see how we can measure out 15 minutes of time using our 7-minute and 11-minute hourglasses.

First, start both hourglasses at the same time. After the top bulb of the 7-minute hourglass is empty, flip it over again. At this time, we have 4 minutes’ worth of sand still in the top bulb of the 11-minute hourglass. When the top bulb of the 11-minute hourglass is empty, the *bottom bulb* of 7-minute hourglass will have 4 minutes’ worth of sand in it. At this point, 11 minutes have passed

Now simply flip the 7-minute hourglass over again and wait until the sand runs to the bottom bulb, which will be in 4 minutes.

This is how we measure out 11 + 4 = 15 minutes of time using a 7-minute hourglass and an 11-minute hourglass.

Let’s look at another problem:

*Having two hourglasses, a 7-minute one and a 4-minute one, how can you correctly time out 9 minutes?*

Now we need to measure out 9 minutes using a 7-minute hourglass and a 4-minute hourglass. Like we did for the last problem, begin by starting both hourglasses at the same time. After 4 minutes pass, all of the sand in the 4-minute hourglass will be in the lower bulb. Now flip this 4-minute hourglass back over again. In the 7-minute hourglass, there will be 3 minutes’ worth of sand still in the upper bulb.

After 3 minutes, all of the sand from the 7-minute hourglass will be in the lower bulb and 1 minute’s worth of sand will be in the upper bulb of the 4-minute hourglass.

This is when we will start our 9-minute interval.

The 1 minute’s worth of sand will flow to the bottom bulb of the 4-minute hourglass. Then we just need to flip the 4-minute hourglass over and let all of the sand flow out (which will take 4 minutes), and then flip the hourglass over to let all of the sand flow out again (which will take another 4 minutes).

In all, we have measured out a 1 + 4 + 4 = 9-minute interval, which is what the problem has asked us to find.

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]]>The post Quarter Wit, Quarter Wisdom: Solving the Weighing and Balancing Puzzle appeared first on Veritas Prep Blog.

]]>First of all, do we understand what a two-pan balance looks like?

Here is a picture.

As you can see, it has two pans that will be even if the weights in them are equal. If one pan has heavier objects in it, that pan will go down due to the weight. With this in mind, let’s try our first puzzle:

*One of twelve coins is a bit lighter than the other 11 (which have the same weight). How would you identify this lighter coin if you could use a two-pan balance scale only 3 times? (You can only balance one set of coins against another, i.e. you have no weight measurements.)*

There are various ways in which we can solve this.

We are given 12 coins, all of same weight, except one which is a bit lighter.

Let’s split the coins into two groups of 6 coins each and put them in the two pans. Since there is one lighter coin, one pan will be lighter than the other and will rise higher. So now we know that one of these 6 coins is the lighter coin.

Now split these 6 coins into another two groups of 3 coins each. Again, one pan will rise higher since it will have the lighter coin on it. Now we know that one of these three coins is the lighter coin.

Now what do we do? We have 3 coins and we cannot split them equally. What we can do is put one coin in each pan. What happens if the pans are not balanced? Then we know the pan that rises higher has the lighter coin on it (and thus, we have identified our coin). But what if both pans are balanced? The catch is that then the leftover coin is the lighter one! In any case, we would be able to identify the lighter coin using this strategy.

We hope you understand the logic here. Now let’s try another puzzle:

*One of 9 coins is a bit lighter than the other 8. How would you identify this lighter coin if you could use a two-pan balance scale only 2 times?*

Now we can use the balance only twice, and we are given an odd number of coins so we cannot split them evenly. Recall what we did in the first puzzle when we had an odd number of coins – we put one coin aside. What should we do here? Can we try putting 1 coin aside and splitting the rest of the 8 coins into two groups of 4 each? We can but once we have a set of 4 coins that contain the lighter coin, we will still need 2 more weighings to isolate the light coin, and we only have a total of 2 weighings to use.

Instead, we should split the 9 coins into 3 groups of 3 coins each. If we put one group aside and put the other two groups into the two pans of the scale, we will be able to identify the group which has the lighter coin. If one pan rises up, then that pan is holding the lighter coin; if the pans weight the same, then the group put aside has the lighter coin in it.

Now the question circles back to the strategy we used in the first puzzle. We have 3 coins, out of which one is lighter than the others, and we have only one chance left to weigh the coins. Just like in the first puzzle, we can put one coin aside and weigh the other two against each other – if one pan rises, it is holding the lighter coin, otherwise the coin put aside is the lighter coin! Thus, we were able to identify the lighter coin in just two weighings. Can you use the same method to answer the first puzzle now?

We will leave you with a final puzzle:

*On a Christmas tree there were two blue, two red, and two white balls. All seemed the same, however in each color pair, one ball was heavier. All three lighter balls weighed the same, just like all three heavier balls weighed the same. **Using a 2-pan balance scale only twice, identify the lighter balls.*

Can you solve this problem using the strategies above? Let us know in the comments!

**free online GMAT seminars** running all the time. And, be sure to follow us on **Facebook**, **YouTube**, **Google+**, and **Twitter**!

*GMAT** for Veritas Prep and regularly participates in content development projects such as this blog!*

The post Quarter Wit, Quarter Wisdom: Solving the Weighing and Balancing Puzzle appeared first on Veritas Prep Blog.

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