The title of this post is: Flipping the sign to inequalities. The only difference to simplifying inequalities compared to normal equations is the times when we have to flip the sign. There are two consistent times when we must always flip the sign:

- When we multiple by a negative number.
- When we divide by a negative number.

While these situations may seem straight forward, the GMAT has found fun ways to increase the difficulty. Let’s look at an advanced Data Sufficiency question:

1. Is ?

1)

2)

As we evaluate the first statement, we see that the base of the exponents is the same . Since this inequality is just like an equation, we can drop like bases. However, do you know what happens when you square a fraction? If you square a number between 0 and 1, the number actually gets smaller. Thus, as we look at Statement 1, if we drop the fractional base, we have to flip the sign as well!

(To help clarify: . If we calculate the equation, we see the answer is – which is mathematically correct. If we drop the base without flipping the sign, the inequality reads , which isn’t mathematically accurate.)

Thus, Statement 1 simplifies to: . As we further simplify the inequality, we move our numbers around and achieve or or . Since , it is also greater than -4: Sufficient.

Let’s look at statement 2: Quadratics and inequalities are difficult when combined – the squared variable results in two possible solutions. For this example, a savvy test takers notices we can factor out from the equation. Translating this to . As we look at this situation, we know that either is negative (less than zero) or is negative (less than zero) – but not both. Here we probably have to test both situations.

Test 1: and – one positive and one negative (check out the blog post regarding binomials for further reading on the sign issue). If we solve for y in the above situation, we get the outcome of and . Is this possible? is both greater than 0 and less then -3? Nope. Thus, this scenario is not the right answer.

Test 2: and . If we solve for in this situation, we get the outcome of and . Is this possible? Yes! We can re-write this as . Or, in other words, Statement 2 is sufficient by itself!

The right answer to the above Data Sufficiency question? Either statement on its own is sufficient!

Let’s quickly review the times we need to flip the inequality:

- When we multiple by a negative number.
- When we divide by a negative number.
- When we remove exponential bases that are between 0 and 1
- When we are dealing with a quadratic equation (there will be two solutions: and ) – don’t forget that binomials usually have two solutions!

Good luck as you practice with inequalities. This was a complicated post! Don’t worry if it doesn’t set in immediately. Print out this screen and read it a couple times – you’ll get it. This is just one of the advanced math topics included among the Kaplan GMAT math material in the newly revised course. If you’ve gotten beyond the intermediate questions and algebra topics, you may be ready to work through advanced topics like this in preparation for the toughest questions on test day.

Brian Fruchey

Kaplan GMAT

What is the value of ?

(1)

(2)

We have two variables, and once we get both statements, we’ll have two equations, so we’ll be able to solve for . The answer is (C), or the third Data Sufficiency answer choice—together the statements are sufficient. If you’ve figured this out, that’s awesome. You’ve discovered how to save a lot of time on Test Day.

But I always tell students not to get trigger-happy. Before you pick (C), keep in mind that the GMAT often gives you situations in which we can get sufficiency with just one equation, or when two won’t be enough. Here are three of those situations:

**The Vanishing Variable**

What is the value of ?

(1)

(2)

Both equations have two variables, so how could one possibly be sufficient to solve for ? Let’s play with Statement (1) a bit so we can isolate . Distribute the right side of the equation to get . Then we can subtract from both sides, and poof! We have a single variable equation. We certain can solve for . The answer is (A), statement 1 alone is sufficient to answer the question. So before you settle for (C), ask yourself if you can eliminate a variable from one equation.

**Solving for a Relationship**

What is the value of ?

(1)

(2)

When the GMAT asks you to solve for a relationship between variables (a sum, difference, product, or quotient), ask yourself, Can I manipulate one of the statements to solve for that relationship? If you can do this, you’ll only need one equation for sufficiency. In this case, no amount of manipulating of Statement 1 can do the trick, but let’s play with Statement 2. Divide both sides by 3, and you get . We still don’t know what or is, but we *do* know what is. The answer is (B), statement 2 alone is sufficient.

**The Disguised Twin**

What is the value of ?

(1)

(2)

It seems like we have everything we need to pick (C) here. Two equations, two variables, we’re golden. Except dig a little deeper; Statement (2) should cause Déjà vu. Add 10y to both sides of the second equation and divide everything by 2, and you’ll discover that the two equations are identical—just dressed up a little differently. Since we really have just one equation with two variables, we have a recipe for insufficiency. The answer is (E), there is not enough information within these statements to answer the question, no matter how you use them or combine them.

**Going forward**

So you don’t necessarily have to solve these systems of equations when you see them in a DS question, but you will have to do some detective work before you pick (C). As we advise students in our newly revised GMAT courses, ask yourself the following questions when assessing this type of problem: Can you make a variable vanish? Can you solve for the desired relationship by manipulating an equation? Are these equations really different? When you know the anatomy of the test, you score higher.

Ben Leff

Kaplan GMAT

In short, it depends on the problem. One of the core competencies in Kaplan’s GMAT curriculum is Pattern Recognition, and our courses and materials aim to teach you to recognize which technique will get you to the answer *fastest*. There are certain “triggers” that let us know whether to combine or substitute; read below for the most fundamental triggers:

**Combination is ideal when you can easily eliminate a variable by adding or subtracting the equations.** For example, by adding the following equations, you can get rid the ’s.

Thus, , , and we are golden. You don’t want to substitute in this situation because it’s hard to get a “clean” value to substitute. Who wants to plug in for ? Fractions are no fun when you’re pressed for time.

However, **we should use substitution if we have a simple coefficient in front of one of the variables.**

This problem lends itself to substitution. It’s very easy to isolate x in the first equation (just subtract from each side to get ), so we can substitute for in the second equation. Doing so, we find that

Carrying out the algebra, we can find that .

Many combination aficionados would still argue that you could multiply the first equation by 3 and subtract the two equations to eliminate the ’s.

**There are many cases when choosing combination or substitution is merely a matter of personal preference.** If you feel more comfortable with one approach, that’s what you should use in these 50/50 situations on Test Day. Still, remember that your favored approach may change as you prep. For example, substitution may initially feels more familiar for most students, but many of them get more comfortable with combination as time goes on.

But, no matter what, you need to be ambidextrous! **Practice both approaches **so that you can choose the *right *one on test day.

Ben Leff

Kaplan GMAT

In working with GMAT students, I have seen that learning the basic quantitative content areas and practicing how and when to apply them to test questions frees test takers to tackle Data Sufficiency and Problem Solving questions with greater confidence and speed. The best way to practice foundational math content is through drills and practice, first working basic problems that get you back in the habit of mathematical operations such as using the distributive property, dealing with fractions properly, and applying the rules of exponents and radicals. Even if you feel that you have a good understanding of one of these math content areas, it still pays to practice with it until the use of operations and rules becomes second nature.

As I always tell my students, you should practice, practice, practice, and then practice some more! Once you have refreshed your basic skills, you will be better primed to approach the complicated verbiage and the multi-step calculations that can create obstacles on the GMAT Quantitative section. Moving forward to test questions, you can flip through your mental files more quickly and pull up the math knowledge that you have practiced extensively and etched into your brain. It’s amazing how much faster a Data Sufficiency question goes when you recognize properties of systems of linear equations and can determine, aha!, if you have two distinct linear equations, you have enough to solve for the value of the two variables presented in the question stem! If the very phrase ‘linear equation’ makes your eyes cross, you are definitely a candidate for a refresher on the math content areas that the GMAT focuses on – Arithmetic, Algebra, and Geometry.

Gina Allison

Kaplan GMAT

Every time I start to go over a question about two trains, two cars or, in one rather quirky problem, two earthworms, students roll their eyes and prepare for the worst.

But these problems have received a bad rap. In order to solve them, students only need to be able to remember the basic rates formula, learn a two-step method and be able to differentiate between the two flavors in which this problem appears.

First up, the rate formula. A rate is just something per something. It can be dollars per jobs, people per team or any other ratio. However, the most common rate on the GMAT is speed. Speed is equal to distance divided by time. This is the only formula students need to know in order to solve a classic two trains problem.

Next, students need to remember a two-step process to solving these problems. Step 1 is to get the trains (or cars or earthworms) to start at the same time. Most GMAT problems of this ilk will have one of the trains starting earlier. Figure out how far this train has traveled by the time the second train starts and determine their distance apart at this time and you have completed step 1. Step 2 is to either add or subtract the train’s individual rates and, using the distance apart, calculate the missing piece (usually time) using the speed formula.

In order to implement this strategy effectively, the last piece of the puzzle is to know when to add the rates and when to subtract them. This is surprisingly straightforward. 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.

By remembering these three basic rules, you will be able to handle two train questions in under two minutes and save that time for truly time-consuming problems.

Bret Ruber

Kaplan GMAT

When most testtakers encounter problems such as the one above, they have a predictable, yet incorrect, reaction. Most people fret when faced with large numbers that would take more than two minutes to deal with. But this reaction is the exact opposite of the reaction you should have when faced with a large number.

Always remember: the GMAT is a test of your critical thinking abilities. While basic arithmetic skills are a prerequisite for achieving a high score on test day, the GMAT is not particularly concerned with your ability to divide 1,960 by 49.

And, not only is the GMAT not concerned with your ability to divide 1,960 by 49, but it also rewards those who know that the GMAT does not want them to do such laborious arithmetic.

The reaction you should have to problems such as the one above is not, “wow this problem is going to take a long time,” but, “the GMAT does not expect me to take the time to do all that division; there must be a shortcut here.”

That shortcut is prime factorization. Any number can be expressed as a series of primes multiplied together. If we can determine those primes, we can avoid time-consuming math.

Let’s return to the problem above, but this time we will express all the numbers in terms of their prime factors. In order to start the process of determining these factors, just think of any two numbers that multiply to equal 1,960:

1,960 = 196 x 10

Neither of these numbers is prime, so we repeat the process:

196* x 10 = 14 x 14 x 2 x 5

2 and 5 are prime, so they stay as they are, but 14 is not, again break it down, repeating the process on both sides of the equation until only primes remain. After breaking your original equation down into prime factors, you end up with:

7 x 2 x 7 x 2 x 2 x 5 = 7 x 7 x 2 x 2 x 2 x y

Then just cancel out any numbers appearing on both sides, and you are left with:

**y = 5**

**quick side note: to save time on test day, make sure you know your squares up to 15!*

*Bret Ruber*

Kaplan GMAT

Alongside the ability to recognize the different ways to write out a ratio, you must be able convert ratios quickly. The fastest way to do this is to pull out a trick you probably have not used since high school: cross multiplication.

Let’s say you are given the ratio of men to women at a party as 4:7, are told that there are 21 women at the party and asked for the total number of partygoers. To answer this question you must first see that the ratio of women to total partygoers is 7:11 – the eleven comes from adding four and seven together. Then you write your ratio as a fraction: . Next, set this fraction equal to a new fraction that puts the actual number of women on the same side of the fraction bar as they were in the initial ratio. You will end up with , where is the number of people at the party. Then multiply the numerator of each fraction with denominator of the other and set these equal to each other. This gives you 7(P) = 11(21). Once you are here, simply solve for . Remember, you can start by dividing both sides by seven in this case, which gives you P = 11(3) = 33.

*Bret Ruber*

Kaplan GMAT

Above all else, you should consider your relative strengths and weaknesses. Make this assessment based on recent practice. This means that your first step should be to take a diagnostic exam, which you can analyze to determine your strengths and weaknesses. Many students expect to be stronger in math than verbal, based on their experience in high school and are surprised when this is no longer the case. In order to ensure you accurately assess your trouble areas, you must take a diagnostic exam with GMAT style questions.

Once you have made this assessment you will know in which section to spend more time, however, you must also remember that this does not mean you should study your trouble areas exclusively. If you want to achieve the highest score possible, you need to study everything that is tested on the GMAT. Just because you are better in one area, does not mean you should ignore it. Even if you made no errors in a particular part of the exam, you need to reinforce that strength, to ensure you do not regress. This means that even the student with the most lopsided score, for example someone in the 95th percentile in verbal and the 30th percentile in math, should spend at least 20% of their time on their strength to make sure it remains a strength. Finally, it is a good idea to mix up your study sessions regularly, as opposed to going days or even weeks focused on just math or just verbal, so that you keep up your skills in both sections.

*Bret Ruber*

Kaplan GMAT

Remember how to find the volume of a cylinder? It’s okay most people don’t; grade school was a long time ago!

Knewton teacher Cole Entress helps you refresh your memory.

Also, in honor of the World Cup(!), we’re discounting our GMAT course by 10% through Monday night at 11:59pm EDT.

- Save
**$390**on a Knewton GMAT Course. Click here to get the coupon code - Click here to learn more about Knewton’s GMAT prep course
- Find all deals and discounts on Knewton

A few weeks back, I wrote a post that discussed Dan Meyer’s visionary talk about the future of math education. Halfway through the talk, Meyer mentions a classic problem in which students must determine how long it takes to fill a tank with water.

Usually, textbooks give students all (or most of) the necessary pieces and then ask them to construct the puzzle — that is, plug the numbers into a formula. Meyer, however, advocates doing away with all the information and simply posing the question: “How long will it take to fill the tank with water?” Students then have to figure out what they need in order to answer the question. This approach forces them to think patiently and creatively.

It should be no coincidence, then, that we at Knewton teach the exact same question in the Data Sufficiency portion of our GMAT course:

An empty rectangular tank has uniform depth. How long will it take to fill the tank with water?

- Water will be pumped at the rate of 480 gallons per hour (1 cubic foot = 7.5 gallons).
- The tank is 100 feet deep and 30 feet wide.

- (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
- (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
- (C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
- (D) EACH statement ALONE is sufficient.
- (E) Statements (1) and (2) TOGETHER are NOT sufficient.

Okay, so the question isn’t *exactly* the same. Meyer’s tank is an octagonal prism; ours is a rectangular solid. Details aside, both questions require the same level of critical reasoning.

Imagine that the prompt was pared down to simply, “How long will it take to fill the tank with water?” Give it a moment’s thought. What do you need to know to answer this question? This is, at its core, what the Data Sufficiency section tests. You never have to compute the final answer on a DS question; you just have to know what it would take to find the answer.

Here, you need to know how quickly the water is flowing into the tank — that is, you need a rate. Next, you need to know how big the tank is — you need a volume. The prompt tells you that the tank is rectangular, with uniform depth, so its volume will be a product of its length, its width, and its depth. In total, then, you need four quantities to answer the question: (1) rate, (2) length, (3) width, (4) depth.

Statement 1 gives you quantity (1), the rate, but nothing more; it can’t be sufficient. Statement 2 gives you quantities (3) and (4), the width and depth, but since it doesn’t give you the length or the rate, it can’t be sufficient, either. When you put the statements together, you’ve got quantities (1), (3), and (4), but you still don’t have quantity (2), the length of the tank. Even together, the statements are not sufficient.

If you approach this question critically, you can polish it off in a matter of seconds. “I need four quantities. I see only three quantities. Answer **choice E** is correct.” If you approach it passively, though, you get lost in the numbers and are far more likely to get it wrong. Nearly 25% of students do.

Surely, questions like this are rare. In general, standardized math tests and creative problem solving do not go well together. As we change the way we teach students by drawing them further into the conversation, we will also need to change the way we assess them. The sooner, the better.

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