Last Digit of Base:

0 – Last digit of expression with any power will be 0.

1 – Last digit of expression with any power will be 1.

2 – 2, 4, 8, 6, 2, 4, 8, 6… Cyclicity is 4.

3 – 3, 9, 7, 1, 3, 9, 7, 1… Cyclicity is 4.

4 – 4, 6, 4, 6, 4, 6, 4, 6… Cyclicity is 2.

5 - Last digit of expression with any power will be 5.

6 – Last digit of expression with any power will be 6.

7 – 7, 9, 3, 1, 7, 9, 3, 1… Cyclicity is 4.

8 – 8, 4, 2, 6, 8, 4, 2, 6… Cyclicity is 4.

9 – 9, 1, 9, 1, 9, 1, 9, 1… Cyclicity is 2.

Cyclicity is nothing but pattern recognition. You see that when you multiply 2 by itself, there is a pattern of last digit which goes 2, 4, 8, 6, 2, 4, 8, 6 and so on. We can use the same principle for when a question asks us for the last two digits of the expression. Let me remind you first that here at QWQW, we sometimes flirt with the lines that define GMAT scope. Obviously, we do point out whenever we are indulging and that’s exactly what we are going to do in this post. We are carrying on for the love of Math and the Q51 score.

The last two digits of the base decide the last two digits of the expression. For example,

**Example 1: **Let’s look at powers of 11.

11^1 = 11

11^2 = 121

11^3 = 1331

11^4 = …41 (we should just multiply the last two digits together and ignore the rest)

11^5 = …51

11^6 = …61

11^7 = …71

Note that the last two digits are displaying a pattern depending on the power. So we expect the cyclicity here to be 10.

11^8 = …81

11^9 = …91

11^(10) = …01

11^(11) = …11

11^(12) = …21

and so on. So the last two digits should go from 11, 21 to 91, 01 and then go to 11 again. The cycle of 10 starts from power of 1, 11, 21 etc. This means that 11^(46) should have last two digits as 61, 11^(92) should have last two digits as 21 and 11^(168) should have last two digits as 81.

Let’s look at some other numbers now:

**Example 2:** Say, we need the last two digits of 6^{58}

6^1 = 6 (No second last digit)

6^2 = 36

6^3 = 216

6^4 = …96 (Just multiply the last two digits)

6^5 = …76

6^6 = …56

6^7 = …36

and hence starts the cycle again:

3, 1, 9, 7, 5, 3, 1, 9, 7, 5 and so on.

The new cycle with tens digit of 3 begins at the powers of 2, 7, 12, 17, 22, 27 etc. So the new cycle will also begin at power of 57 and 6^58 will have 1 as the tens digit.

**Example 3:** How about the last two digits of 7^102?

7^1 = 7 (No second last digit)

7^2 = 49

7^3 = 343

7^4 = …01

7^5 = …07

7^6 = …49

7^7 = …43

We see a cyclicity of 4 here: 49, 43, 01, 07, 49, 43, 01, 07 … and so on. The new cycle begins at 2, 6, 10, 14 i.e. even powers which are not multiples of 4. So a new cycle will begin at 102 too. So the last two digits of 7^(102) will be 49.

Now there can be many variations in the questions asking us to find the last two digits. We will use different concepts for different question types. Today we saw how to use pattern recognition. We will look at some other methods next week.

*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!*

Mark is playing poker at a casino. Mark starts playing with 140 chips, 20% of which are $100 chips and 80% of which are $20 chips. For his first bet, Mark places chips, 10% of which are $100 chips, in the center of the table. If 70% of Mark’s remaining chips are $20 chips, how much money did Mark bet?

You can view this as a word problem where you assume the number of chips and then go splitting them up or you can view this as a mixtures problem even though it doesn’t use words such as ‘mixture’, ‘solution’, ‘combined’ etc. As we have seen enough number of times, our mixture problems are solved in seconds using the weighted average concept.

The question discussed here also belongs to the same category – looks super tricky but can be easily solved with weighted averages formula. But we have seen plenty and more of such questions in our blog posts. Today we will take a look at a different type of sinister question and I suggest you to think about the concept being tested in that before trying to solve it.

**Question**: Mark owns four low quality watches. Watch1 loses 15 minutes every hour. Watch2 gains 15 minutes every hour relative to watch1 (that is, as watch1 moves from 12:00 to 1:00, watch2 moves from 12:00 to 1:15). Watch3 loses 20 minutes every hour relative to watch2. Finally, watch4 gains 20 minutes every hour relative to watch3. If Mark resets all four watches to the correct time at 12 noon, what time will watch4 show at 12 midnight that day?

(A)10:00

(B)10:34

(C)11:02

(D)11:48

(E)12:20

Before we look at the solution, think about the concept being tested here – clocks? Circular motion?

Neither!

**Solution**: Note that when giving data about watch1, you are told how it varies with the actual time. Data about all other watches tells us about the time they show relative to the incorrect watches. The concept being tested here is Relative Speed!

What do we mean by “gains 15 mins” or “loses 20 mins” etc? When a watch gains 15 mins every hour, it means that even though it should show that one hour has passed, it shows that 1 hr 15 mins have passed. So the watch runs faster than it should. Hence the speed of the watch is more than the speed of a correct watch. Now the question is how much more? The minute hand of the correct watch travels one full circle in one hour. The minute hand of the incorrect watch travels one full circle and then a quarter circle in one hour (to show that 1 hour 15 mins have passed even when only an hour has passed). So it is 5/4 times the speed of a correct watch. On the same lines, let’s analyze each watch.

Say the speed of a correct watch is s.

- “Watch1 loses 15 minutes every hour. “

Watch1 covers only three quarters of the circle in an hour.

Speed of watch1 = (3/4)*s

- “Watch2 gains 15 minutes every hour relative to watch1 (that is, as watch1 moves from 12:00 to 1:00, watch2 moves from 12:00 to 1:15).”

Now we have the speed of watch2 relative to speed of watch1. Speed of watch2 is (5/4) times the speed of watch1.

Speed of watch2 = (5/4)*(3/4)s = (15/16)*s

- “Watch3 loses 20 minutes every hour relative to watch2.”

Watch3 loses 20 mins every hour means its speed is (2/3)rd the speed of watch2

Speed of watch3 = (2/3)*(15/16)*s = (5/8)*s

- “Finally, watch4 gains 20 minutes every hour relative to watch3.”

Speed of watch4 = (4/3)*Speed of watch3 = (4/3)*(5/8)*s = (5/6)*s

So the speed of watch4 is (5/6)th the speed of a correct watch. So if a correct watch shows that 6 hours have passed, watch4 will show that 5 hours have passed. If a correct watch shows that 12 hours have passed, watch4 will show that 10 hours have passed. From 12 noon to 12 midnight, a correct watch would have covered 12 hours. Watch4 will cover 10 hours and will show the time as 10:00.

Answer (A)

*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!*

*While traveling from Detroit to Novi, a car averaged 10 miles per gallon, and while traveling from Novi to Lapeer, it averaged 18 miles per gallon. If the distance between Detroit and Novi is half the distance between Novi and Lapeer, what is the average miles per gallon for the entire journey?*

We have two figures for mileage given here – 10 miles per gallon and 18 miles per gallon. We need to find the average mileage. So we can use the weighted average formula but what will the weights be? Will they be 1:2 since the distance between the two cities is given to be in the ratio 1:2? If you think that taking the distance to be the weights in this problem is correct, then you fell for the trap in this question.

To explain the concept, let us use a simpler example first:

When talking about average speed, what are the weights? We know that the weight given to each speed is the time for which that speed was maintained, right? Yes! But why?

Let’s review our weighted average formula:

Cavg = (C1*w1 + C2*w2)/(w1 + w2)

Average Speed = (Speed1*Time1 + Speed2*Time2)/(Time1 + Time2)

Average Speed = (Distance1 + Distance2)/(Time 1 + Time2)

Average Speed = Total Distance/Total Time

This is an accurate representation of average speed.

Now see what happens when you use distance as the weights.

Cavg = (C1*w1 + C2*w2)/(w1 + w2)

Average Speed = (Speed1*Distance1 + Speed2*Distance2)/(Distance1 + Distance2)

Speed*Distance doesn’t represent any physical quantity. So this doesn’t make sense. The units of the quantities will help you see the relation clearly.

Cavg = (C1*w1 + C2*w2)/(w1 + w2)

Average Speed = (Speed1*Time1 + Speed2*Time2)/(Time1 + Time2)

Average Speed = (miles/hour * hour + miles/hour * hour)/(hour + hour)

Average Speed = (miles + miles)/(hour + hour)

Average Speed = Total miles/Total hours

What happens when you take distance as the weights?

Cavg = (C1*w1 + C2*w2)/(w1 + w2)

Average Speed = (Speed1*Distance1 + Speed2*Distance2)/(Distance1 + Distance2)

Average Speed = (miles/hour * miles + miles/hour * miles)/(miles + miles)

miles^2/hour doesn’t represent a physical quantity and hence doesn’t make sense here. Therefore, whenever you are confused what the weights should be, look at the units.

Let’s go back to the original question now. Average required is miles per gallon. So you are trying to find the weighted average of two quantities whose units must be miles/gallon.

Cavg = (C1*w1 + C2*w2)/(w1 + w2)

The unit of Cavg, C1 and C2 is miles/gallon so w1 and w2 should be in gallons to get

miles/gallon = (miles/gallon * gallon + miles/gallon * gallon)/(gallon + gallon)

miles/gallon = Total miles/Total gallons

So how will we actually solve this question?

**Question**: While traveling from Detroit to Novi, a car averaged 10 miles per gallon while traveling from Novi to Lapeer, it averaged 18 miles per gallon. If the distance between Detroit and Novi is half the distance between Novi and Lapeer, what is the average miles per gallon for the entire journey?

**Solution**:

Let the distance between Detroit and Novi be D. So the distance between Novi and Lapeer must be 2D.

Amount of fuel used to cover distance D = D/10

Amount of fuel used to cover distance 2D = 2D/18 = D/9

So the two weights used must be D/10 and D/9

Average miles/gallon = (10*D/10 + 18*D/9)/(D/10 + D/9) = 3D*90/19D = 270/19 = 14.2 miles/gallon

Or simply, Average miles/gallon = Total miles/Total gallons = 3D/(D/10 + D/9) = 14.2 miles/gallon

Food for thought: Which one of the following can you solve?

- If a vendor sold 10 apples at a profit of 10% and 15 oranges at a profit of 20%, what was his overall profit%?

- If a vendor sold apples at a profit of 10% and oranges at a profit of 20%, what was his overall profit% if cost price of each apple was $0.20 and the cost price of each orange was $.06?

*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!*

Myth 3: Use of ‘being’ is always wrong on GMAT!

Often, the way we use ‘being’ in our day-to-day communication, is incorrect. For example,

Being a doctor, he is very well respected.

But there are correct ways of using ‘being’. Since most students believe that ‘being’ is wrong, don’t trust the GMAC to not use this nugget of information to misdirect the test takers. The correct answers of questions at higher ability are worded in such a way that they make the test takers uncomfortable!

So how is ‘being’ used correctly?

‘Being’ is used to express a temporary state.

The little boy started screaming when he saw his dog being impounded.

‘Being impounded’ is a temporary state and would be over – unlike being a doctor. So the use of being is correct here.

Let’s look at one of our own sentence correction questions now:

**Question**: The data being collected in the current geological survey are providing a strong warning for engineers as they consider the new dam project, but their greatest importance might lie in how they influence the upcoming decision by those same engineers on whether to retrofit 75 bridges in the survey zone.

A. The data being collected in the current geological survey are providing a strong warning for engineers as they consider the new dam project, but their greatest importance

B. The data being collected in the current geological survey provide a strong warning for engineers as they consider the new dam project, but its greatest importance

C. The data collected in the current geological survey is providing a strong warning for engineers as they consider the new dam project, but their greatest importance

D. The data collected in the current geological survey provides a strong warning for engineers in consideration of the new dam project, but its greatest importance

E. The data collected in the current geological survey provide a strong warning for engineers in consideration for the new dam project, but the greatest importance

**Solution: **Let’s find the decision points:

First decision point: being collected vs collected

‘The data being collected’ is a temporary state here. Data won’t always be collected, but are being collected for a short time right now, so ‘being’ is used properly here. And the sentence makes it very clear that this is temporary; look at the word ‘current’ before ‘geological survey.’ If you were skeptical of the word ‘being’ before, that is understandable, but the word ‘current’ should serve as a clear warning that this is a temporary, ongoing event. (And furthermore, the non-underlined portion talks about an upcoming decision, even further cementing the idea that this is a temporary survey with data ‘being’ collected for a short period of time). Alas, even with that temporary state, there isn’t really anything wrong with ‘The data collected in …’ either so we retain all answer options.

It’s worth noting that they put the ‘being collected’ vs. ‘collected’ decision early in the sentence/answer-choices as a great place to put a ‘false decision point.’ The authors of these questions know that they want to reward emphases on logical meaning and core grammar rules, and they also know that students like to study quick tips and tricks, so they leave that bait there for the tips/tricks folks while they hold off the bigger reward for those willing to prioritize decision points from most important to least and not just from left to right in order of appearance.

Second decision point: are/is

Technically, data is plural of datum. In academic writing it is almost always treated as plural. It is treated as singular in informal writing but GMAT favors treating it as plural.

Even if you do not know this, the use of “*they* influence the upcoming” – in the portion of the sentence that is not underlined – should tell you that ‘data’ is used in plural form here.

Hence the use of ‘are’ is appropriate. Hence, options (C) and (D) are eliminated.

Third decision point: Pronouns

There are many pronouns used here. Antecedent of each pronoun is present in the sentence. The usage clarifies which pronoun refers to data and which refers to engineers.

Original Sentence: The data being collected in the current geological survey are providing a strong warning for engineers as **they** consider the new dam project, but **their** greatest importance might lie in how they influence the upcoming decision by those same engineers on whether to retrofit 75 bridges in the survey zone.

**they** – only engineers can consider the new dam project so ‘they’ refers to engineers

**their/its** - greatest importance will be of data, which is plural, so ‘their’ would be the correct usage. Eliminate option (D)

There is no ambiguity in the use of pronouns. The nouns are present and the usage clarifies the antecedent.

Now we are left with options (A) and (E). In option (E), “in consideration for the new dam project” is bad diction. Also, it doesn’t tell us ‘whose greatest importance?’.

Answer is (A)

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

Today, we bring another tip for you to help get that dream score of 51 – if you must write down the data given, write down all of it! Let us explain.

If you think that you will need to jot down the data given in the question and then solve it on your scratch pad (instead of in your mind), you must jot down every single detail. It is easy to overlook small things which are difficult to express algebraically such as ‘x is an integer’. These details are often critical and could make all the difference between an ‘unsolvable’ question and a ‘solvable within 2 minutes’ one. Once you start solving the question on your scratch pad, you will not refer back to the original question again and again and hence might forget these details. Have them along with the rest of the data. Read every word of the question carefully, and ensure that it is consolidated on your scratch pad. For example, look at this question:

A set of five positive integers has an arithmetic mean of 150. A particular number among the five exceeds another by 100. The rest of the three numbers lie between these two numbers and are equal. How many different values can the largest number among the five take?

It is a difficult question because it incorporates statistics as well as max-min – both tricky topics. On top of it, people often overlook the ‘are equal’ part of the question here. The reason for that is that they are actively looking for implications of the sentences and the moment they read “The rest three numbers lie between these two numbers”, they go back to the previous sentence which tells us “A particular number among the five exceeds another by 100”. They then make a note of the fact that 100 is the range of the five positive integers. In all this excitement, they miss the three critical words “and are equal”. Ensure that when you go to the sentence above, you pick the next sentence from the point where you left it. Another thing to note here is that all numbers are positive integers. This information will be critical to us.

Let’s demonstrate how you will solve this question after incorporating all the information given.

**Question**: A set of five positive integers has an arithmetic mean of 150. A particular number among the five exceeds another by 100. The rest of the three numbers lie between these two numbers and are equal. How many different values can the largest number among the five take?

(A) 18

(B) 19

(C) 21

(D) 42

(E) 59

Solution:

Let’s assume that the 5 natural numbers in increasing order are: a, b, b, b, a+100

We are given that a < b < a+100.

Also, we are given that a and b are positive integers. This information is critical – we will see later why.

The average of the 5 numbers is (a+b+b+b+a+100)/5 = 150

(a+b+b+b+a+100) = 5*150

2a+3b = 650

We need to find the number of distinct values that a can take because a+100 will also take the same number of distinct values.

Now there are two methods to proceed. Let’s discuss both of them.

**Method 1: Pure Algebra** – Write b in terms of a and plug it in the inequality

b = (650 – 2a)/3

a < (650 – 2a)/3 < a+100

3a < 650 – 2a < 3a + 300

Now split it into two inequalities: 3a < 650 – 2a and 650 – 2a < 3a + 300

Inequality 1: 3a < 650 – 2a

5a < 650

a < 130

Inequality 2: 650 – 2a < 3a + 300

5a > 350

a > 70

So we get that 70 < a < 130. Since a is an integer, can we say that a can take all values from 71 to 129? No. What we are forgetting is that b is also an integer. We know that

b = (650 – 2a)/3

For which values will be get b as an integer? Note that 650 is not divisible by 3. You need to add 1 to it or subtract 2 out of it to make it divisible by 3. So a should be of the form 3x+1.

b = (650 – 2*(3x+1))/3 = (648 – 6x)/3 = 216 – 2x

Here, for any positive integer x, b will be an integer.

From 71 to 129, we have the following numbers which are of the form 3x+1:

73, 76, 79, 82, 85, … 127

This is an Arithmetic Progression. How many terms are there here?

Last term = First term + (n – 1)*Common Difference

127 = 73 + (n – 1)*3

n = 19

a will take 19 distinct values so the last term i.e. (a+100) will also take 19 distinct values.

**Method 2: Using Transition Points**

Note that a < b < a+100

Since a < b, let’s find the point where a = b, i.e. the transition point

2a + 3a = 650

a = 130 = b

But b must be greater than a. If we increase b by 1, we need to decrease a by 3 to keep the average same. But decreasing a by 3 decreases the largest number i.e. a+100 by 3 too; so we need to increase b by another 1.

We get a = 127 and b = 132. This give us the numbers as 127, 132, 132, 132, 227. Here the average is 150

Since b < a+100, let’s find the point where b = a+100

2a + 3(a+100) = 650

a = 70, b = 170

But b must be less than a+100. If we decrease b by 1, we need to increase a by 3 to keep the average same. But increasing a by 3 increases the largest number, i.e. a+100 by 3 too, so we need to decrease b by another 1.

We get a = 73 and b = 168. This gives us the numbers as 73, 168, 168, 168, 173. Here the average is 150

Values of a will be: 73, 76, 79, ….127 (Difference of 3 to make b an integer)

This is an Arithmetic Progression.

Last term = First term + (n – 1)*Common difference

127 = 73 + (n – 1)*3

n = 19

a will take 19 distinct values so the last term i.e. (a+100) will also take 19 distinct values.

Answer (B)

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

In this post, we discussed how to use graphing techniques to easily solve very high level questions on nested absolute values. We don’t think you will see such high level questions on actual GMAT. The aim of putting up the post was to illustrate the use of graphing technique and how it can be used to solve simple as well as complicated questions with equal ease. It was aimed at encouraging you to equip yourself with more visual approaches.

We gave you two questions at the end of that post to try on your own. We have seen quite a bit of interest in them and hence will discuss their solutions today.

The solutions involve a number of graphs and hence we have made pdf files for them.

Question 1: Given that y = |||x – 5| – 10| -5|, for how many values of x is y = 2?

Question 2: Given that y = |||x| – 3| – x|, for what range of x is y = 3?

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

Before you begin, you might want to review the post that discusses standard deviation: Dealing With Standard Deviation

So here goes the question.

Question: Given that set S has four odd integers and their range is 4, how** **many distinct values can the standard deviation of S take**?**

(A) 3

(B) 4

(C) 5

(D) 6

(E) 7

Solution: Recall what standard deviation is. It measures the dispersion of all the elements from the mean. It doesn’t matter what the actual elements are and what the arithmetic mean is – the standard deviation of set {1, 3, 5} will be the same as the standard deviation of set {6, 8, 10} since in each set there are 3 elements such that one is at mean, one is 2 below the mean and one is 2 above the mean. So when we calculate the standard deviation, it will give us exactly the same value for both sets. Similarly, standard deviation of set {1, 3, 3, 5, 6} will be the same as standard deviation of {10, 12, 12, 14, 15} and so on. But note that the standard deviation of set {25, 27, 29, 29, 30} will be different because it represents a different arrangement on the number line.

Let’s look at the given question now.

Set S has four odd integers such that their range is 4. So it could look something like this {1, x, y, 5} when the elements are arranged in ascending order. Note that we have taken just one example of what set S could look like. There are innumerable other ways of representing it such as {3, x, y, 7} or {11, x, y, 15} etc.

Now in our example, x and y can take 3 different values: 1, 3 or 5

x and y could be same or different but x would always be smaller than or equal to y.

- If x and y were same, we could select the values of x and y in 3 different ways: both could be 1; both could be 3; both could be 5

- If x and y were different, we could select the values of x and y in 3C2 ways: x could be 1 and y could be 3; x could be 1 and y could be 5; x could be 3 and y could be 5.

For clarification, let’s enumerate the different ways in which we can write set S:

{1, 1, 1, 5}, {1, 3, 3, 5}, {1, 5, 5, 5}, {1, 1, 3, 5}, {1, 1, 5, 5}, {1, 3, 5, 5}

These are the 6 ways in which we can choose the numbers in our example.

Will all of them have unique standard deviations? Do all of them represent different distributions on the number line? Actually, no!

Standard deviations of {1, 1, 1, 5} and {1, 5, 5, 5} are the same. Why?

Standard deviation measures distance from mean. It has nothing to do with the actual value of mean and actual value of numbers. Note that the distribution of numbers on the number line is the same in both cases. The two sets are just mirror images of each other.

For the set {1, 1, 1, 5}, mean is 2. Three of the numbers are distance 1 away from mean and one number is distance 3 away from mean.

For the set {1, 5, 5, 5}, mean is 4. Three of the numbers are distance 1 away from mean and one number is distance 3 away from mean.

The deviations in both cases are the same -> 1, 1, 1 and 3. So when we square the deviations, add them up, divide by 4 and then find the square root, the figure we will get will be the same.

Similarly, {1, 1, 3, 5} and {1, 3, 5, 5} will have the same SD. Again, they are mirror images of each other on the number line.

The rest of the two sets: {1, 3, 3, 5} and {1, 1, 5, 5} will have distinct standard deviations since their distributions on the number line are unique.

In all, there are 4 different values that standard deviation can take in such a case.

Answer (B)

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

Myth 1: Passive voice is always wrong.

Active voice is preferred over passive voice but that doesn’t make passive voice wrong.

Myth 2: The same pronoun cannot refer to two different antecedents in a sentence.

A pronoun, say ‘it’, can refer to two different objects in a single sentence, but it should not refer to two different objects in the same clause since that creates ambiguity.

We will explain these two points to you using a sentence correction question:

**Question**: Once the computer generates the financial reports, they are then used to program a company-wide balance sheet, so named because it demonstrates that every department’s accounting elements are in balance.

(A) Once the computer generates the financial reports, they are then used to program a company-wide balance sheet, so named because it demonstrates that every department’s accounting elements balance.

(B) Once the computer generates the financial reports, it is then used to program a company-wide balance sheet, named such because it demonstrated the balance of every department’s accounting elements.

(C) Once the computer generates the financial reports they are then used to program a company-wide balance sheet, which demonstrates the balance of every department’s accounting elements.

(D) Once the financial reports are generated by the computer, it is then used to program a company-wide balance sheet, so named because it demonstrates the balance of every department’s accounting elements.

(E) Once the financial reports are generated by the computer, they are then used to program a company-wide balance sheet, named such because it demonstrates that every department’s accounting elements are in balance.

**Solution**: Let’s split the sentence into clauses:

- Once the computer generates the financial reports,

- they are then used to program a company-wide balance sheet,

- so named because it demonstrates that every department’s accounting elements are in balance.

Find the decision points. The first clause is in active voice in the first three options and in passive in the other two. Both are correct.

The first decision point is they vs it. Should we use they or should we use it? The first clause talks about two things – computer (singular) and financial reports (plural). What do we want to refer to in the second clause? What do we use to program a balance sheet? A computer is used to program something. Reports cannot program anything. They can be used while programming but they cannot program. Hence, the use of ‘it’ would be correct here.

Only in options (B) and (D) do we use ‘it’ (singular) which refers back to the computer (singular). It cannot refer back to financial reports (plural). So eliminate options (A), (C) and (E).

Now comes our next decision point – we have to choose one of ‘named such’ and ‘so named’. ‘named such’ which is used in option (B) is awkward. Also, we use the past tense of the verb ‘demonstrate’ in option (B). This is not correct since a balance sheet is so called because is always demonstrates the balance of every element. It did not demonstrate it only in the past. Hence we need to use simple present tense.

This leads us to option (D). Everything is taken care of here.

Here are a couple of points about option (D):

(D) Once the financial reports are generated by the computer, it is then used to program a company-wide balance sheet, so named because it demonstrates the balance of every department’s accounting elements.

Sometimes, people eliminate it because it uses passive voice “the financial reports are generated by the computer”. Be aware that passive is not wrong. You have learned active passive in school. Passive is just a bit weaker form of writing than active and hence, given a choice, active is preferred but not at the expense of grammatical correctness! Using passive is not incorrect.

At other times, people have problems with the use of the pronoun ‘it’ for two different antecedents

- **it** (the computer) is then used to program a company-wide balance sheet,

- so named because **it** (the balance sheet) demonstrates the balance of every department’s accounting elements.

An OG problem has been pointed out here:

Starfish, with anywhere from five to eight arms, have a strong regenerative ability, and if one arm is lost **it** [animal] quickly replaces **it** [arm], sometimes by the animal overcompensating and growing an extra one or two.

The above answer is incorrect since the pronoun ‘it’ refers to two different antecedents in a single clause. Note that the pronoun ‘it’ refers to two different antecedents in the same clause. It is hard to understand what ‘it’ refers to.

But that is not the case in our option (D).

The first ‘it’ clearly refers to the computer since there is only one singular antecedent before it.

The second ‘it’ in the third clause clearly refers to the balance sheet because the clause talks about the balance sheet: … company wide balance sheet, so named because it …

There is no ambiguity of pronoun reference here.

We can’t re-iterate it enough – don’t try to learn up ‘rules’ for sentence correction. Every so called “rule” is not applicable in every situation. Use logic!

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

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The Meaning of Arithmetic Mean

Can You Solve these Mean GMAT Questions?

Finding Arithmetic Mean Using Deviations

Application of Arithmetic Means

This question needs you to apply all these concepts but can still be easily done in under two minutes. Now, without further ado, let’s go on to the question – there is a lot to discuss there.

**Question**: An automated manufacturing unit employs N experts such that the range of their monthly salaries is $10,000. Their average monthly salary is $7000 above the lowest salary while the median monthly salary is only $5000 above the lowest salary. What is the minimum value of N?

(A)10

(B)12

(C)14

(D)15

(E)20

**Solution**: Let’s first assimilate the information we have. We need to find the minimum number of experts that must be there. Why should there be a minimum number of people satisfying these statistics? Let’s try to understand that with some numbers.

Say, N cannot be 1 i.e. there cannot be a single expert in the unit because then you cannot have the range of $10,000. You need at least two people to have a range – the difference of their salaries would be the range in that case.

So there are at least 2 people – say one with salary 0 and the other with 10,000. No salary will lie outside this range.

Median is $5000 – i.e. when all salaries are listed in increasing order, the middle salary (or average of middle two) is $5000. With 2 people, one at 0 and the other at 10,000, the median will be the average of the two i.e. (0 + 10,000)/2 = $5000. Since there are at least 10 people, there is probably someone earning $5000. Let’s put in 5000 there for reference.

0 … 5000 … 10,000

Arithmetic mean of all the salaries is $7000. Now, mean of 0, 5000 and 10,000 is $5000, not $7000 so this means that we need to add some more people. We need to add them more toward 10,000 than toward 0 to get a higher mean. So we will try to get a mean of $7000.

Let’s use deviations from the mean method to find where we need to add more people.

0 is 7000 less than 7000 and 5000 is 2000 less than 7000 which means we have a total of $9000 less than 7000. On the other hand, 10,000 is 3000 more than 7000. The deviations on the two sides of mean do not balance out. To balance, we need to add two more people at a salary of $10,000 so that the total deviation on the right of 7000 is also $9000. Note that since we need the minimum number of experts, we should add new people at 10,000 so that they quickly make up the deficit in the deviation. If we add them at 8000 or 9000 etc, we will need to add more people to make up the deficit at the right.

Now we have

0 … 5000 … 10000, 10000, 10000

Now the mean is 7000 but note that the median has gone awry. It is 10,000 now instead of the 5000 that is required. So we will need to add more people at 5000 to bring the median back to 5000. But that will disturb our mean again! So when we add some people at 5000, we will need to add some at 10,000 too to keep the mean at 7000.

5000 is 2000 less than 7000 and 10,000 is 3000 more than 7000. We don’t want to disturb the total deviation from 7000. So every time we add 3 people at 5000 (which will be a total deviation of 6000 less than 7000), we will need to add 2 people at 10,000 (which will be a total deviation of 6000 more than 7000), to keep the mean at 7000 – this is the most important step. Ensure that you have understood this before moving ahead.

When we add 3 people at 5000 and 2 at 10,000, we are in effect adding an extra person at 5000 and hence it moves our median a bit to the left.

Let’s try one such set of addition:

0 … 5000, *5000, 5000, 5000* … 10000, 10000, 10000, *10000, 10000*

The median is not $5000 yet. Let’s try one more set of addition.

0 … 5000, 5000, 5000, 5000, *5000, 5000, 5000* … 10000, 10000, 10000, 10000, 10000, *10000, 10000*

The median now is $5000 and we have maintained the mean at $7000.

This gives us a total of 15 people.

Answer (D)

Granted, the question is tough but note that it uses very basic concepts and that is the hallmark of a good GMAT question!

Try to come up with some other methods of solving this.

*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’s see how we decide that.

**Question 1**: In the wake of the global housing crisis, and amid dramatically changing demographics, it is likely that a widespread shift in thinking is ahead, which will reduce demand for large suburban homes, thus increasing demand for smaller urban apartments.

(A) it is likely that a widespread shift in thinking is ahead, which will reduce demand for large suburban homes, thus increasing demand for smaller urban apartments.

(B) it is likely that a widespread shift in thinking is ahead, which will reduce demand for large suburban homes, and thus increase demand for smaller urban apartments.

(C) it is not unlikely that a widespread shift in thinking is ahead, reducing demand for large suburban homes, thus creating an increase in demand for smaller urban apartments.

(D) it is not unlikely that a widespread shift in thinking is ahead, reducing demand for large suburban homes and increasing demand for smaller urban apartments.

(E) it is not unlikely that a widespread shift in thinking is ahead, reducing demand for large suburban homes, increasing demand for smaller urban apartments.

**Solution**: Let’s start looking for decision points – the first decision point is ‘it is likely’ vs ‘it is not unlikely’ – both have similar meanings and are grammatically correct so we cannot eliminate any option based on this right now. The next decision point is the beginning of the modifier. Options (A) and (B) use ‘which clauses’. Options (C), (D) and (E) use present participle modifiers.

‘which’ is a relative pronoun but there is no noun before it which can act as an antecedent. Hence, the use of which is incorrect here. On the other hand, the use of participle modifier is acceptable here. Last week, we discussed that present participle modifier after a comma will modify the preceding clause. It provides additional information about the preceding clause. ‘reducing …’ tells us more about ‘widespread shift in thinking‘. Hence, let’s focus on options (C), (D) and (E).

In (C), the “thus” used to introduce the second participle is incorrect: the two participles should be linked with a coordinating conjunction without a comma. One is not really leading to the other – they are both byproducts of the change in thinking – reducing demand for large homes and increasing demand for urban apartments. Lastly, in option (C), the “creating an…” is unnecessary and redundant – you just need “increasing demand.”

For option (E), you need something to link the two participle phrases together – without it, there is a comma splice error. Hence we eliminate (E) as well.

Option (D) gets the structure and meaning correct – “the shift in thinking is reducing … and increasing …”

Answer is (D).

Now, let’s look at an official GMAT question.

**Question 2**: In 1984, medical researchers at Harvard and Stanford universities concluded that sedentary life-styles lead to heart and lung diseases that shorten lives, strongly recommending middle-aged people to undertake some form of regular exercise.

(A) strongly recommending middle-aged people to

(B) strongly recommending that middle-aged people should

(C) and strongly recommended for middle-aged people to

(D) and their strong recommendation was for middle-aged people to

(E) and they strongly recommended that middle-aged people

**Solution**: The given sentence has two clauses:

Main clause – medical researchers at Harvard and Stanford universities concluded

That clause – that sedentary life-styles lead to heart and lung diseases that shorten lives

If we use a comma and the present participle ‘recommending’ here, it will modify the ‘that clause’. So ‘recommending’ will be done by ‘sedentary life-styles’. Obviously, this is incorrect since the researchers are the ones who recommend exercise. So we cannot use the participle here. Hence we eliminate options (A) and (B).

Options (C), (D) and (E) use ‘recommend’ in verb form.

Options (C) and (D) are unidiomatic in their usage of the verb recommend.

You recommend X for Y (say a person X for position Y)

or

You recommend that X do Y (say a person X do Y)

Option (C) says ‘recommended for X to do Y’ and option (D) says ‘recommendation was for X to do Y’ – both are incorrect.

Option (E) uses recommend properly – ‘recommended that X do Y’. Also, ‘… researchers concluded that … and recommended that …’ have parallel structure. Hence, option (E) is correct.

Answer (E)

Hope you now understand how participle phrases are used.

*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!