GMAT Quantitative Section: Probability Translation
August 8, 2012
Translating word problems into algebra is a staple skill of GMAT test-takers, one that underlies countless problems in practice and on Test Day. But some challenging translations occur as part of probability and combinatorics problems. That’s because a pair of the most basic words in the English language, “And” and “Or,” suddenly become overburdened with mathematical significance.
“And” is the simpler of the two. When “And” represents independent choices—cases in which one option or arrangement has no impact on the other choice—just multiply the outcomes. For instance:
“The number of ways to purchase three board games and two video games” is an independent choice. The board games we pick have no impact on the video games we pick. So, to translate: [The number of ways to purchase three board games] × [the number of ways to select two video games]. Of course, we’d need the combination formula to find actual values—but we’d know what to do with those values once we got them.
“Or” is a little more complicated. It’s confusing even in conversation, after all—if I say that you can have cake or ice cream for dessert, can you have both if you want? When you CAN have both, you can treat the problem similarly to an overlapping sets problem. But in most cases on the GMAT, the “Or”s will be mutually exclusive—for instance, if you want to know the odds of drawing a heart or a diamond out of a deck of cards, there is no card that is both a heart or diamond.
A mutually exclusive OR can be translated as a “plus.” That’s all you have to do. So:
“The probability of drawing a heart or a diamond from a deck of cards,” which is the odds of one of two mutually exclusive events occurring, translates to: [The probability of drawing a heart] + [The probability of drawing a diamond].
Today’s problem of the day hinges on those same ideas. Read carefully—you’re solving for the odds of one of two outcomes (an OR), but each of those two outcomes is the specific result of two independent events (an AND). Be systematic in your translation, and I’m sure you’ll get the right result.
Post your answers below before you read the solution, and we can go over them…
Question:
Each person in Room A is a student, and 1/6 of the students in Room A are
seniors. Each person in Room B is a student, and 5/7 of the students in Room
B are seniors. If 1 student is chosen at random from Room A and 1 student is
chosen at random from Room B, what is the probability that exactly 1 of the
students chosen is a senior?
(A) 5/42
(B) 37/84
(C) 9/14
(D) 16/21
(E) 37/42
Solution:
Step 1: Analyze the Question
This is a complex question, but it can be broken down into
simple steps. As with any probability question, we must first
consider all of the scenarios in which the desired outcome
can be true. In this question, there are two different ways
in which exactly one of two students chosen is a senior.
Either (i) a senior is chosen from Room A and a non-senior
is chosen from Room B or (ii) a non-senior is chosen from
Room A and a senior is chosen from Room B.
Step 2: State the Task
Determine the probabilities of the two scenarios above and
add them together.
Step 3: Approach Strategically
Let’s start with (i) and find the probability that a senior is
chosen from Room A and a nonsenior is chosen from Room B.
The probability that the student chosen from Room A is a
senior is 1/6 .
The probability that the student chosen from Room B is not
a senior is 1- 5/7=2/7
So the probability that the student chosen from Room A
is a senior and the student chosen from Room B is not a
senior is (1/6) x (2/7) = 2/42 .
Let’s not simplify this yet, because we can expect that the
probability we will find when working with (ii) will also
have a denominator of 42.
Now let’s work with (ii). Let’s find the probability that a
nonsenior is chosen from Room A and a senior is chosen
from Room B.
The probability that the student chosen from Room A is not
a senior is 1 – 1/6 = 5/6 .
The probability that the student chosen from Room B is a
senior is 5/7 .
So the probability that the student chosen from Room A
is a not a senior and the student chosen from Room B is a
senior is (5/6) x (5/7) = 25/42 .
Now we sum the total desired outcomes. The probability
that exactly one of the students chosen is a senior
is (2/42) + (25/42) = 27/42 = 9/14 .
(C) is correct.
GMAT Quantitative: Two Types of Mixture Problems
July 19, 2012
Mixture problems show up frequently on the quantitative section of the GMAT and fall into two basic categories. As each type of mixture question will be approached in fairly different ways, it is important that you know the difference between them.
First, there are mixture problems that ask you to alter the proportions of a single mixture. These questions could, for example, tell you that you have a 200 liter mixture that is 90% water and 10% bleach and ask how much water you would need to add to make it 5% bleach. The key in this type of question is the part of the mixture that is constant – in this case the bleach. While we are adding water, the amount of bleach stays the same. First, determine how much bleach we have. 10% of 200 is 20 liters. Next, we know we want those 20 liters to equal 5% of our total. Since 20 is 5% of 400, our new total should be 400 liters. To go from 200 liters to 400 liters, you would need to add 200 liters of water, which would be the answer (For yet another way to solve this type of GMAT quantitative problem, check out this post).
The other type of mixture problem will ask you to combine two mixtures. For example, you could be told that mixture A is 20% bleach and 80% water, while mixture B is 5% bleach and 95% water. You could then be asked in what ratio these mixtures should be combined to achieve a mixture that is 10% bleach. You should solve problems such as this algebraically.
Both sides of your equation will represent the amount of bleach in the combined mixture. On one side you will represent the amount of bleach in terms of the individual mixtures. This will give you .2A + .05B. On the other side of the equation you will represent the amount of bleach overall, which is .1(A + B). Note that in these expressions A represents the total amount of mixture A and B represents the total amount of mixture B. Because these expressions both represent the total amount of bleach, we can set them equal to each other. This gives us .2A + .05B = .1(A + B). The ratio of A to B can be solved as follows:
.2A + .05B = .1(A + B)
.2A + .05B = .1A + .1B
.1A = .05B
A/B = .05/.1
A/B = 1/2
Now try the problem below to see how you do on your own.
Question:
Two brands of detergent are to be combined. Detergent X contains 20 percent bleach
and 80 percent soap, while Detergent Y contains 45 percent bleach and 55 percent
soap. If the combined mixture is to be 35 percent bleach, what percent of the final
mixture should be Detergent X?
(A) 10%
(B) 32_ 1_ 2 %
(C) 35%
(D) 40%
(E) 60%
Solution:
Step 1: Analyze the Question
This is a complex question, but there is a straightforward
solution. We are creating a new mixture from two others,
X and Y. X is 20% bleach, and Y is 45% bleach. The new
mixture is to be 35% bleach.
In other words, some amount of a 20% bleach mixture plus
some amount of a 45% bleach mixture will balance each
other out to a 35% bleach mixture.
Step 2: State the Task
Because this involves finding a particular balance between
Detergents X and Y, you can use the balance approach to
solve. We could use Algebra or Backsolving, but balance is
the most efficient. This will let us calculate the proportion
of Detergent X in the final mixture.
Step 3: Approach Strategically
The question does not state how many parts of Detergent
X are used, so call this x. And the question does not state
how many parts of Y are used, so call this y.
So 0.10y = 0.15x. To solve for a proportional amount, view
this as a ratio. Divide both sides by y and by 0.15 to get
the ratio:
0.10y = 0.15x
0.10 / 0.15 = x / y
10 / 15 = x / y
2 / 3 = x / y
So x:y is 2:3. Add the total to the ratio to determine how x
relates to the total: x:y:total = 2:3:5.
Thus x:total = 2:5. That’s 2 /5 , or 40%.
Answer (D) is correct.
GMAT Ratio Problems: Avoid the Algebra
July 14, 2012
Mastering ratio questions on the GMAT requires systematic organization of the individual pieces and a solid understanding of how ratios are typically presented and tested on test day. One of the most common presentations of ratios on test day is a question that presents a part:part or part:whole relationship and asks for the actual number of a part, the whole, or a difference between the parts.
The first thing to note about ratios is that they represent relationships between items. On the GMAT Quantitative Section, the ratio is usually in the simplest form; I call this multiple level 1 because it represents the smallest potential positive quantity for each aspect of the ratio. For instance, if a question tells you that the ratio of apples to oranges is 2:3, you know immediately that the minimum number of apples possible is 2 while the minimum number of oranges is 3 and the minimum total pieces of fruit is 5. Also, the actual number for those items must be a multiple of that minimum. Selecting the correct answer quickly on ratio questions commonly revolves around your ability to determine a multiple for one of the parts in a ratio relationship.
In order to manage all of the information in the question well, let’s pull out a tried-and-true tool, the basic chart. I know that most of us set up a proportion for these questions and solve algebraically. We absolutely can deal with ratios in a traditional math route through a proportion and algebraic equation; however, setting up the algebra traditionally leaves more room for error and can take a bit more time depending on the complexity of the question. The method is ultimately up to you, but in your GMAT prep you always want to learn to choose the most efficient and effective route for the particular question at hand. If you love the algebra, go for it (For those who want to avoid it more often, check out this additional GMAT strategy). In this post, I want to introduce and explain my favorite way to deal with this typical ratio question type – the ratio grid:
| Multiple Level | ||||
|
1 |
Step 1: Identify Ratio Parts
First, let’s take a look at the different pieces of data that can be presented in a ratio question. The first things to identify about the given ratio are the individual parts. In the example ratio of apples to oranges from above, we have a base ratio of 2:3. The stated parts of the ratio are apples and oranges, so we plug these into the chart with their respective minimums.
| Multiple Level | Part 1: Apples | Part 2: Oranges | ||
|
1 |
2 |
3 |
Step 2: Finish Level 1 Quantities
Next, finish out your base minimums, determining the minimum total by adding the minimum individual parts and determining the minimum difference by subtracting the smaller part from the larger part.
| Multiple Level | Part 1: Apples | Part 2: Oranges | Total Fruit | Difference |
|
1 |
2 |
3 |
5 |
1 |
Step 3: Strategically Eliminate
At this point, pause and eliminate answer choices strategically based on the minimums you have identified. The answer MUST be a multiple of the corresponding minimum that the question is asking for. Often this one step is the last thing that you will need to solve the question. For example, using our hypothetical apples : oranges scenario above, if a question asked for the total number of apples, eliminate anything that is not a multiple of 2 because the total number of apples must be some kind of multiple of 2. If the question is looking for the total number of oranges, eliminate all choices that are not a multiple of 3. If the question is asking for the total number of fruit in the basket, eliminate all choices that are not a multiple of 5 and so on. If that does not eliminate 4 choices, we move on to the next level to evaluate those choices that are left.
Step 4: Plug in the Actual Value From Question
In a typical ratio question, along with the base ratio, the test-makers will give you an actual total for one of the parts, the total, or the actual difference. Your task at this point is to plug the actual value in at the appropriate place in the chart and to determine which multiple level that actual value is sitting at. For instance, for our scenario of the apple to orange ratio of 2:3, if the question told you that there are 30 total pieces of fruit in the basket, we would plug 30 in below the original 5 and determine the multiple level by identifying what 5 must be multiplied by to get to 30. Since, 30 is 5 times 6, we are working on level 6.
| Multiple Level | Part 1: Apples | Part 2: Oranges | Total Fruit | Difference |
|
1 |
2 |
3 |
5 |
1 |
|
6 |
30 |
Step 5: Determine Your Answer
At this point you can solve for any of the remaining parts. For instance, if the question told you that in a basket the ratio of apples to oranges was 2:3 and the total pieces of fruit in the basket was 30 and asked for the actual number of apples in the basket, you would multiply 2 by 6 to get 12.
| Multiple Level | Part 1: Apples | Part 2: Oranges | Total Fruit | Difference |
|
1 |
2 |
3 |
5 |
1 |
|
6 |
12 |
30 |
Now, let’s walk through these steps using a realistic GMAT question:
The ratio of girls to boys in a class is 6:7. If there are 18 girls, how many total students are in the class?
A) 18
B) 21
C) 27
D) 28
E) 39
Step 1: Identify the Parts and Step 2: Fill in the Remaining Level 1 Quantities
First, look at just the base ratio to establish the minimums for level 1. The ratio of girls to boys in a class is 6:7.
| Multiple Level | Boys | Girls | Total Students | Difference |
|
1 |
6 |
7 |
13 |
1 |
Step 3: Strategically Eliminate
Because we are ultimately solving for the total number of students in the class, we need to eliminate anything that is not a multiple of 13, our minimum number of total students.
A) 18 – eliminate
B) 21 – eliminate
C) 27 – eliminate
D) 28 – eliminate
E) 39 – keep (MUST be the answer)
And we are done! Always check your multiples before you deal with the rest of the question. Sometimes, this is all that you need to correctly and strategically solve.
Let’s look at one more that is a bit more difficult.
Three investors, A, B, and C, divide the profits from a business enterprise in the ratio 5:7:8, respectively. If investor A earned $3,500, how much money did investors B and C earn in total?
A) $4,000
B) $4,900
C) $5,600
D) $9,500
E) 10,500
Step 1 & 2: Identify Parts and Finish out Grid
First we look at the base ratio to establish some minimums. Three investors, A, B, and C, divide the profits from a business enterprise in the ratio 5:7:8, respectively. In this chart, I modified the total column slightly because the goal of the question is to deal with the sum of B and C.
| Multiple Level | A | B | C | Sum of B and C |
|
1 |
5 |
7 |
8 |
15 = (7+8) |
Step 3: Strategically Eliminate
The goal is to solve for the sum of investors B and C, so the answer MUST be a multiple of 15. Eliminate every choice that isn’t a multiple of 15.
A) $4,000 – eliminate
B) $4,900 – eliminate
C) $5,600 – eliminate
D) $9,500 – eliminate
E) $10,500 – Keep – MUST be the answer
Let’s look at one last variation of this pattern.
At a certain zoo, the ratio of sea lions to penguins is 4 to 11. If there are 84 more penguins than sea lions at the zoo, how many sea lions are there?
A) 24
B) 36
C) 48
D) 72
E) 132
Steps 1 & 2: Set up the base ratio
At a certain zoo, the ratio of sea lions to penguins is 4 to 11.
| Multiple Level | Sea lions | Penguins | Total together | Difference |
|
1 |
4 |
11 |
15 |
7 |
Step 3: Strategically Eliminate
Since we are solving for the number of sea lions, eliminate any choice that is not a multiple of 4.
A) 24 – Keep
B) 36 – Keep
C) 48 – Keep
D) 72 – Keep
E) 132 – Keep
Step 4: Plug in Actual Value and Determine Multiple Level
“If there are 84 more penguins than sea lions at the zoo” is the given difference between the pieces. To get to 84 you must multiply 7 by 12, so we are at multiple level 12.
| Multiple Level | Sea Lions | Penguins | Total Together | Difference |
|
1 |
4 |
11 |
15 |
7 |
|
12 |
84 |
Step 5: Determine your answer
Since we are looking for the number of sea lions, multiply the base 4 by the multiplier 12. Therefore, “C” is our final answer.
| Multiple Level | Sea Lions | Penguins | Total Together | Difference |
|
1 |
4 |
11 |
15 |
7 |
|
12 |
48 |
84 |
The big takeaway for ratio questions on the GMAT is that most are about multiples. Make sure that you check multiples before you waste too much time walking through the problem completely. Also, organize your information systematically on test day to efficiently and effectively walk through every question on the test.
Do you have a favorite way to approach these questions? Post it in the notes so that we can all check it out.
GMAT Fences
June 4, 2012
In my years of teaching, I’ve seen all kinds of clever solutions to GMAT math problems. I’ve also seen all kinds of errors. Some are utterly bizarre—and fortunately, seldom repeated, because the students who make those mistakes usually face-palm when they review their tests and go on to learn from their missteps. But some errors are so common and so often repeated that they earned their own names. One such example is the “fencepost error.”
Here’s a simple example: Say we are setting up a straight fence that’s exactly 100 ft long, with posts every 10 feet. How many posts do we need?
Did you say 10? Tempting, but that’s the right answer to the wrong question. There are 10 sections of fence, each 10 feet long. But there are actually 11 fenceposts, because you start with a fencepost, at 0 feet!
See?
This error can trap the unwary GMAT student in a few different ways. The most common is in finding sums of series consecutive integers. The formula for the total of an evenly spaced list is to multiply the average value of that list by the number of items. So, you need to know exactly how many numbers you’re adding. So if a question asks for the sum of numbers between, say, 37 and 59, some students might just say there are 59 – 37 = 22 numbers on that list—but that doesn’t count 37, the first number. To ensure you solve for the correct value, you need to add the first “fencepost” and count all 23 numbers (write them out and count if you want to confirm!)
Also, rather oddly, this error shows up on the verbal section in sentence correction too. Clauses are the “fenceposts” in sentences, in a sense, and connecting words (therefore, however, so) are the fence sections. You can tell a sentence is improperly constructed if you don’t have exactly one more clause than you have connecting words.
Keep this error in mind as you solve your practice problems—and today’s question of the day. It’s a common mistake, but also an easy one to avoid once you’re aware of it. Practice will make sure you’re not fooled on Test Day.
Question of the Day:
In a new housing development, trees are to be planted along the sidewalk of a certain street. Each tree takes up one square foot of sidewalk space, and there are to be 14 feet between each tree. How many trees can be planted if the road is 166 feet long?
A) 8
(B) 9
(C) 10
(D) 11
(E) 12
Solution:
Step 1: Analyze the Question
Though this is not a Geometry question, a quick sketch of the situation will help illustrate how to solve it. 
So we know that the unit of one tree and one space is 1 foot + 14 feet = 15 feet.
Step 2: State the Task
To find how many trees can be planted, determine the feet required for a tree and the space between trees. Divide the total length of the street by the unit of one tree and the space between trees.
Step 3: Approach Strategically
Each tree takes up 1 foot, and each space takes up 14 feet. Together they take up 15 feet. Now find how many times 15 goes into the total number of feet on one side of the street:
166 / 15 = 11, with a remainder of 1 foot.
We can plant one last tree in the remaining foot, bringing the total number of trees to 12. This means along the street, we can plant 12 trees with 11 spaces between them, as long as we start and end with a tree. (E) is correct.
Step 4: Confirm Your Answer
Make sure your answer makes sense in the context of the question. Did you take into account the remainder of the division? Will an entire tree fit in the remaining space? You can use these questions to confirm your work.
GMAT Average Speed Problems
May 30, 2012
Imagine you are driving from Chicago to Los Angeles, and you want to know what your average speed needs to be to reach Los Angeles in a certain number of hours. You would probably start by determining the speed you will be able travel during certain parts of your journey. Since most of the distance will be covered by highway, you might plan to travel most of the distance at 70 miles per hour. However, you will also want to plan for some traffic when you are still in or near Chicago and when you get close to Los Angeles. During these parts of your journey let’s say you can plan to travel at 30 miles per hour.
When calculating the average speed at which you will be traveling, you need to avoid the trap of just averaging these speeds together and planning on an average speed of 50 miles per hour. Because the vast majority of your journey will take place at 70 miles per hour and only a relatively small portion will take place at 30 miles per hour, simply averaging the speeds is not sufficient. You need to account for the difference in the amount of time you will be driving at each speed. Once you do so, your average speed will be much closer to 70 miles per hour than 30 miles per hour.
The same principle will apply when you see average speed questions on the GMAT. Average speed is defined as total distance divided by total time, rather than the average of the speeds. Additionally, the average of the speeds will almost always be offered as an answer choice, so be sure to avoid it.
This can be especially tricky when a problem gives little information other than the two speeds. On test day, you should think strategically and pick a number for the distance, calculate the times using this number, and then plug into the average speed formula described above. Give it a try on the following problem.
Problem:
A canoeist paddled upstream at 10 meters per minute, turned around, and drifted downstream at 15 meters per minute. If the distance traveled in each direction was the same, and the time spent turning the canoe around was negligible, what was the canoeist’s average speed over the course of the journey, in meters per minute?
(A) 11.5
(B) 12
(C) 12.5
(D) 13
(E) 13.5
Answer
Step 1: Analyze the Question
A canoeist goes one rate in one direction, turns around, and goes back at a different rate. Whenever you deal with one entity that has different rates at different times, set up a chart to track the data. Otherwise, you’ll find yourself in a six-variable, six-equation system that will take a long time to work through. Also, notice that although the distance and time are never mentioned, there are no variables in the answer choices. Whenever variables will cancel out, consider Picking Numbers.
Step 2: State the Task
Our task is to calculate her average speed for the whole journey.
Step 3: Approach Strategically
The formula is this:
Average Speed = Total Distance / Total Time
But we’re seemingly told nothing about time, and the only thing we know about distance is that it is the same in both directions. So what to do? As with almost every problem involving a multistage journey, set up this chart:
| Rate | Time | Distance | |
| part 1 of trip | |||
| part 2 of trip | |||
| entire trip |
Now plug in the data we’re given:
| Rate | Time | Distance | |
| upstream |
10 |
||
| downstream |
15 |
||
| entire trip |
Now we see clearly that we’ll be able to know the time if we know something about the distance. Since we know whatever variable we put in place will cancel out by the time we get to the answer choices, let’s just pick a number for distance—one that will fit neatly with a rate of 10 and a rate of 15. A distance of 30 should work well:
| Rate | Time | Distance | |
| upstream |
10 |
30 |
|
| downstream |
15 |
30 |
|
| entire trip |
At this point, we can fill in the rest of the chart very straightforwardly. The entire distance is 60. The time taken upstream must be 3, and the time taken downstream must be 2. That makes the entire time 5.
| Rate | Time | Distance | |
| upstream |
10 |
3 |
30 |
| downstream |
15 |
2 |
30 |
| entire trip |
5 |
60 |
The speed for the entire trip, then, is 60 / 5 = 12. Answer (B).
Step 4: Confirm Your Answer
Reread the question stem, making sure that you didn’t
miss anything about the problem.
To all of you strategic thinkers out there, can you spot a way to quickly eliminate 3 of the answer choices without doing any calculations?
Avoid a GMAT Train Wreck
May 17, 2012
Ever since I started teaching GMAT classes, I have taken note of any references to standardized tests I come across in television shows and movies. In the six years of doing so, I have found that these references almost always follow the same pattern. One of the characters needs to take a standardized test that they find difficult or boring. In order to illustrate this to the other characters, they will read an example of one of the questions on the exam. Invariably, the question they read involves two trains leaving two different stations at two different times and traveling towards each other.
Because of this, rate problems that feature two trains (or cars or people or anything else) have a bit of a bum rap. These questions are seen, unjustly, as difficult, time consuming and complicated. However, by learning only a few basic rules, you can handle these questions quickly and correctly.
The first step is to make sure the trains leave at the same time. If one train leaves earlier than the other one, calculate the distance the earlier train will have travelled by the time the later train leaves. Subtract that distance from the distance originally separating the trains, and use the that new distance as the total distance.
The second step will depend on the exact type of problem. If the trains are coming towards each other or going away from each other, add their speeds. If one train is catching up to the other, subtract their speeds. Use this result as the total speed.
Once you have calculated the total distance and total speed, you can solve for the time as you would on any other rate question. You just plug these numbers into the same formula you used back in step one to find the earlier trains distance, which is distance = rate x time.
The problem below is a perfect example of this type of question. While it looks complicated at first, draw a diagram and then follow the steps outlined above to reach the correct answer.
Problem:
Train A left Centerville Station, heading towards Dale City Station, at 3:00 p.m. Train B left Dale City Station, heading toward Centerville Station, at 3:20 p.m. on the same day. The trains rode on straight tracks that were parallel to each other. If train A traveled at a constant speed of 30 miles per hour and Train B traveled at a constant speed of 10 miles per hour, and the distance between the Centerville Station and Dale City Station is 90 miles, when did the trains pass each other?
(A) 4:45 p.m.
(B) 5:00 p.m.
(C) 5:20 p.m.
(D) 5:35 p.m.
(E) 6:00 p.m.
Solution:
Begin as you do for any word problem, by understanding the basic situation. Two trains, 90 miles apart, start moving toward each other at different times. One train moves at 30 mph, the other at 10 mph. Our task is to determine the times at which the trains pass each other, which is to say when they will be at the same point on these 90-mile tracks.
Our story begins at 3:00 pm when train A leaves. It is going 30 mph. The next event happens at 3:20 pm when Train B leaves its station going 10 mph. In the 20 minutes before train B leaves, train A has travelled 10 miles. This leaves 80 miles of track between them when train B starts at 3:20. This is now our total distance.
The question is then, “How fast will the two trains close that distance?” Here we add the speeds to get total speed: 30 + 10 = 40.
So, at a combined rate of 40 mph, how long will it take them to close an 80-mile gap? Time = distance/speed. Thus, 80/40 = 2 hours to close the gap. It will be 5:20 at that point. Answer (C) is correct.
On the GMAT, Use Primes to Crack Big Numbers
May 12, 2012
Everyone studying for the GMAT wants to identify the skills that will lead directly to the greatest point increases. While this can be difficult to do, given the adaptive nature of the exam, some skills definitely do come into play more often than others.
One of the most important skills to master for the GMAT is prime factorization. Finding prime factors can be useful on many different types of questions. On test day, if you are stuck on a question and unsure of how to solve, remember the big number rule. The big number rule is simply this: if you see a big number, one that is so large it is unreasonable to work with, find its prime factors. Once you have those factors, you should be able to simplify.
Every positive integer that is not prime, with the exception of 1, can be broken down into a series of prime numbers multiplied together. Additionally, each series of primes is unique and will only result in a single integer.
For example, let’s say you want to find the prime factors of 20. Start by identifying any two numbers that multiply to equal 20. We will choose 4 x 5. 5 is prime, so we cannot break it down further, but 4 is not, so we repeat the process. If you try to think of two numbers that multiply to equal 4, you will only come up with 2 x 2, which are both prime. Thus, the prime factorization of 20 is 2 x 2 x 5.
Now let’s say that instead of choosing 4 x 5, you choose 2 x 10. 2 is prime, so we leave it alone, but we need to break down 10. 10 equals 2 x 5, thus we again find the prime factorization of 20 is 2 x 2 x 5. Notice that no matter what numbers we choose, as long as the math is correct, we reach the prime factors.
Try the question below and see if you can figure out how prime factorization can help you solve.
Problem:
Is q a multiple of 48?
(1) q is a multiple of 6.
(2) q is a multiple of 8
Solution:
The stem asks whether q is a multiple of 48. It’s a yes/no question so you should see if you can answer the question with both a yes and a no. The prime factorization of 48 is 2 x 2 x 2 x 2 x 3, so we need four 2’s and one 3 to know that q is a multiple of 48.
Statement 1: This tells you that q is a multiple of 6. The prime factorization of 6 is 2 x 3, so we know q has one 2 and one 3, but we do not know if it has the additional three 2’s we need, so this statement is not sufficient.
Statement 2: If q is a multiple of 8, we know that it must include the prime factors 2 x 2 x 2, which gives us three 2’s. Since we do not know if it will have the additional 2 and one 3, it is insufficient.
Statements 1 & 2: If q is a multiple of 6 and 8, it must have one 2 and one 3 in its prime factors, so it will divide by 6, and three 3’s in its prime factors, so it will divide by 8. This means that q must have three 2’s and one 3. Note that one of those 2’s is being used to divide by 6 and by 8, since we need to be able to divide by both, but not at the same time. However, to reach 48 the prime factors would need to include one additional 2. Since we do not know if q would include a fourth 2 or not, the statements are still not sufficient. Therefore, the answer is (E).
Translation on the GMAT
April 21, 2012
One of the big GMAT skills that is often overlooked by students is translation. Any time you decide approach a word problem using algebra, you will need to translate the English in the question stem into an algebraic equation. While this seems as if it would usually be fairly straightforward, the GMAT will often find ways to make it more difficult. A translation error will often lead to a trap answer, so it is essential that you learn how to translate difficult statements before test day.
To understand why translation can be more difficult than it seems, think about translating a foreign language. If you only need to translate one word, you can usually just find the equivalent word in English. Similarly, if a GMAT problem uses the phrase “more than” you know that it must translate to addition.
However, when you try to translate an entire sentence from a foreign language to English, the process becomes much more complicated. You do not need to only determine the matching English words, but also need to consider the structure and meaning of the sentence as whole, since the foreign language might construct sentences differently than English. Along the same lines, when you translate a word problem to math, you must consider the relationships between the different parts of the sentence and check to see if any words are used in unusual ways. You might get lucky and have a problem that uses the word “equals”, but you are just as likely to see the words “costs”, “weighs”, and “sells” used. Identifying when these other words translate to “equals” will be key to writing a correct equation.c
Below, you will find a word problem that can be solved algebraically. Be careful on your translation and then check out the solution to see how you did.
Problem:
Jacob is now 12 years younger than Michael. If 9 years from now Michael will be twice as old as Jacob, how old will Jacob be in 4 years?
(A) 3
(B) 7
(C) 15
(D) 21
(E) 25
Step 1: Analyze the Question
We must first translate the question into algebraic equations, then apply the techniques of combination or substitution to solve for Jacob’s age.
Step 2: State the Task
Translate the question stem, then solve for Jacob’s age using substitution.
Step 3: Approach Strategically
Equation for Michael’s and Jacob’s current ages: M = J + 12.
Equation for Michael’s and Jacob’s ages 9 years from now: M + 9 = 2(J + 9)
We now have two distinct equations with two unknowns that we can solve. We can substitute the first equation into the second for M, yielding the equation:
( J + 12) + 9 = 2 (J + 9)
J + 12 + 9 = 2J + 18
J + 21 = 2J + 18
Thus, J = 3. So if Jacob is 3 years old now, then in 4 years, he will be 7 years old. Don’t forget to add these 4 years at the end.
Step 4: Confirm Your Answer
We can plug in J = 3 into the original question stem to confirm our answer. Be careful with your calculations, since (A) is a trap for those who don’t add 4 years to his current age.
Absolute Value on the GMAT
April 19, 2012
Most students learn that absolute value is the positive version of a number. Thus, the absolute value of 7 is 7 and the absolute value of -7 is also 7. While these absolute values are correct, many GMAT problems will be more straightforward if you learn the true definition of absolute value, which is the distance a number is from zero on a number line. Thus, the absolute values of 7 and -7 are 7 because both numbers are 7 away from zero on a number line.
To understand how absolute value works, imagine you live in a house right in the middle of a block. The street has 5 houses to the left of your house and 5 houses to the right of your house. Whether you walk two houses to the left or two houses to the right you will be 2 houses away from your home. Now, instead of picturing a block, consider the same concept as a number line in which your house represents zero. If you walk 2 houses to the left, you are at -2 and if you walk 2 houses to the right, you are at 2. Normally, we use positive and negative numbers to distinguish these as different spots. However, if we are concerned with the distance from us rather than the location relative to us, we no longer need to distinguish between positive and negative numbers. Since that distance cannot be negative – it is either 2 to the right or 2 to the left – we must define the distance in positive terms. Thus, absolute value will always be positive.
Keeping the concept of absolute value in mind, try the problem below and then check out the solution to see how you did.
Problem:
Which of the following could be the value of x, if |4x – 2| = 10?
(A) -3
(B) -2
(C) 1
(D) 2
(E) 4
Analyze: We see an equation with an absolute value sign in the Q-stem. Also, what important GMAT wording do we see? “could be the value”—that means that there are at least two possible solutions for x in the equation and the absolute value sign means the expression could equal 10 or –10.
Task: Pick the answer that makes the equation true. There may be another solution, too, but only one of the answer choices will work.
Approach Strategically: To solve a normal algebraic equation, we have to do the same thing to both sides in order to isolate the variable.
We’ll do the same here, but we’ll set up two equations because of the 2 possible answers that the expression can equal, like so: 4x – 2 =10 and 4x – 2 = –10.
Solving for 4x – 2 = 10 gives us x = 3.
That’s not one of our answer choices.
Solving for 4x – 2 = – 10 gives us x = -2 which is answer choice B.
You could also backsolve here but it’s important to know the rules of absolute value on Test Day.
Multiple Variables on the GMAT
April 7, 2012
While preparing for the GMAT, you have probably heard that if you have multiple variables for which to solve, you need as many equations as you have variables to do so. However, as is the case with many GMAT topics, just knowing the rule will not be enough. You will also need to know three exceptions to this rule that regularly appear.
First, if you have two variables, but only one equation, you can solve for one of those variables if the other variable cancels out. Note that it is still not possible to solve for the variable that cancels – in fact, that variable would have an infinite number of solutions. Second, if you are asked to solve for an expression, such as x + y, rather than an individual variable, you may be able to do so with fewer equations than variables. In these cases make sure to do the algebra, since you cannot always solve for the expression, and remember that you still will not know the values of the two variables on their own. Third, if the two equations are algebraically identical, even though they appear different at first, they only count as one equation for purposes of our rule. For example, x + y = 2 and 2x + 2y = 4 are the same equation, since the former equation is the latter equation divided by two.
These exceptions show up on data sufficiency more than problem solving, since they all involve knowing if you are able to solve for a variable or expression. The sample problem below is one such question, so make sure to keep the exceptions in mind as you try it.
Problem:
What is the value of (3x – 3y)/2?
(1) x + y = 22/3
(2) x – y = 10/3
Solution:
Step 1: Analyze the Question Stem
We can simplify the expression in the stem to 3(x-y)/2. To determine the value of this expression, we will need to know the value of x-y.
Step 2: Evaluate the Statements
Statement (1) provides two variables in one equation. We cannot re-express this statement to provide the value of x-y. Nor can we determine the values of x and y, which would allow us to calculate x – y. Statement (1) is not sufficient. Eliminate (A) and (D).
Statement (2) directly provides the value of x – y and is sufficient, thus (B) is the answer.
Answer: B
