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Problem Solving Posts

3 Ways of Looking at Profit (with 3 Harder Questions)

By posted August 31, 2011

As someone who is about to shell out hundreds of dollars in MBA application fees, you know that money makes the GMAT-world go round. Profit is an essential concept for any aspiring MBA admissions applicant. The GMAT tests this concept in both Problem Solving and Data Sufficiency questions in three main ways. Let’s examine the need-to-know formulas with three GMAT practice questions.

  1. A firm increases its revenues by 10% between 2008 and 2009. The firm’s costs increase by 8% during this same time. What is the firm’s percent increase in profits over this period, if profits are defined as revenues minus costs?(1) The firm’s initial profit is $200,000.(2) The firm’s initial revenues are 1.5 times its initial costs.

In this question from Grockit, we can start with our most basic Profit formula:

Profit = Revenue – Cost

Try this GMAT problem solving question for more GMAT practice!

Using Statement (1), we can say that 200,000 = R – C.

(1.1)r – (1.08)c = 200,000(1 + x), where x equals the amount of the increase. We still do not know R and C so we can’t find x. Insufficient.

Using Statement (2), 1.5c – c = p and (1.1)(1.5)c – (1.08)c = (1 + x)P. Here we can simplify.

.5c = p

.57c = (1 + x)p

Without continuing to solve, we can see that we can solve for x using substitution. .57c = (1 + x)(.5c), and dividing both sides by c will cancel out that variable and allow us to isolate x. Statement 2 is sufficient. Now to a more challenging question!

  1. A store purchased 20 coats that each cost an equal amount and then sold each of the 20 coats at an equal price. What was the stores gross profit on the 20 coats?

Statement 1. If the selling price per coat had been twice as much, the store’s gross profit on the 20 coats would have been $2400.
Statement 2. If the selling price per coat had been $2 more, the store’s gross profit on the 20 coats would have been $440.

This GMAT Prep question asks about gross profit.

Gross Profit = Selling price – Cost

For the value Data Sufficiency question, we need to know the price of each coat and the selling price of each coat. From the given information, we can use our known formula to set us the equation: P = 20 (s – c). So either we’ll need a value for s and a value for c, or we’ll need the value of (s – c).

Statement (1) tells us that $2400 = (20(2s – c)) or 2400 = 40s – 20c. We can divide both sides by 20 and simplify it to: 120 = 2s – c.  We still don’t know s and c. Insufficient.

Statement (2) tells us that 440 = 20(s + 2 – c). Let’s simplify: 440 = 20s + 40 – 20c.  400 = 20s – 20c.  400 = 20 (s – c). 20 = s – c. Sufficient. Even though we didn’t solve for s and c separately, we were able to find the value of (s – c). Sometimes DS will surprise you!

  1. If the cost price of 20 articles is equal to the selling price of 25 articles, what is the % profit or loss made by the merchant?A. 25% loss
    B. 25% profit
    C. 20% loss
    D. 20% profit
    E. 5% profit

Profit/Loss % = (Sales Price – Cost Price) / Cost Price x 100

The question asks about % profit or loss. It tells us that 20c = 25s, or 4c = 5s. So the ratio of the sales price to the cost price is 4/5.

Let’s simplify our Profit/Loss % formula by dividing each term by the cost price:

Profit/Loss % = (S/C – C/C) x 100

P/L% = (S/C – 1) x 100

We know that S/C = 4/5 for this problem. So we can plug in and solve:

P/L% = (4/5 – 1) x 100

P/L% = (-1/5) x 100

P/L% = -20%.   The answer is a 20% loss.

Find out how you can collaborate with your peers to reinforce your knowledge and theirs on Grockit today.

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GMAT Geometry Basics: Parallel Lines

By posted June 23, 2011

We can say that two lines are “parallel” if they never intersect and get neither closer nor farther away from one another. || is the symbol for parallel lines. In coordinate geometry, parallel lines have the same slope, but different x-intercepts.

Try this GMAT geometry practice question and test your skills today!

Parallel lines can be tested in either Problem Solving or Data Sufficiency and though the concept itself is relatively simple, the test questions presented can appear somewhat complex. Don’t make assumptions about lines that look parallel but may in fact not be. Let’s look at an example of how the “eyes can deceive” from GMATPrep:

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GMAT Practice: A Process for Problem Solving

By posted March 15, 2011

Problem Solving is often overlooked strategy-wise, since it’s easy to place more emphasis on developing a process for the unfamiliar question-type, Data Sufficiency. However, it’s important to make sure you have a methodical step-by-step approach to these more traditional math questions. Even if you know how to do the math, reading too quickly, or jumping into the algebra without really understanding the question can lead you to the incorrect answer.

1.  What’s it really asking? Write down what the question is asking you to find. It may sound obvious, but the GMAT often requires an extra step. You may need to find 1/y, instead of y. Or you may be asked about the “ratio of girls in a class to boys in a class,” but have to solve for the two parts of the ratio first. If you don’t write it down, you will not see the end goal as clearly.

Need one-on-one GMAT help? Write to one of Grockit’s GMAT experts and find out how you can set up customized tutoring on Grockit.

2.  What’s the important info? Pull out any key numbers, variables, or phrases from the question and write them down on your scratch pad. This is the step most students skip. Don’t just scan the screen and start solving. Forcing yourself to slow down and process each piece of information will give your brain time to sort through it. This is especially important for word problems which can contain unnecessary extra information. This may lead you to find a faster way to solve!

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Carry It Over I (part 1)

By posted September 11, 2010

Carry It Over I (part 1)

One of the ways that the GMAT makes quantitative questions more challenging is by forcing you to carry over information from one type of environment to another.  This is most commonly seen in geometry questions, where the side of a triangle might also be a chord on a circle inscribed within a square, for example; because the test is going after your care and attention as much (or more) than it’s going after your math knowledge, carrying information over a key skill.  This “Carry It Over” series will challenge you to transfer information from one type of problem to another, within the same problem.  Let’s start with this one:

Raj is working on a set of Data Sufficiency problems for his December GMAT: a geometry problem, an algebra problem, and a data interpretation problem. He has determined that statement 1 of the geometry problem is insufficient on its own, that both statement 1 and 2 of the algebra problem are insufficient on their own, and that statement 2 of the data interpretation problem is insufficient on its own. By approximately what percent is the probability that all three answers are “C” greater if Raj figures out that statement 1 of the data interpretation problem is also insufficient on its own?

A) 2.3%

B) 2.8%

C) 3.3%

D) 5.6%

E) 8.3%

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Exponents

By posted September 2, 2010

An exponent refers to the number of times the base is a factor.  For example,  43 = 4 x 4 x 4 = 64. For a term with a coefficient in front of a variable raised to an exponent, it’s important to remember that the exponent only affects the variable. Knowing the order of operations is helpful to avoid simple exponent mistakes. For 2x2, first you would square x and then multiply the result by 2.

Any number to the 2 power is referred to as being “squared.” Any number to the 3 power is called being “cubed.”

When you multiply two terms with the same base, you can add the exponents.: 25 x 23 = 25+3 = 28

When you divide two terms with the same base, you can subtract the exponent of the numerator from the exponent of the denominator: 68 ÷ 62 = 66

If two exponents are separated by a parenthesis, you can multiply them: (82)5 = 810

On the GMAT look for ways to rewrite bases so they are the same.

97 x 3x = 317

(32)7 x 3x = 317

314 x 3x = 317

14  +  x = 17

x = 3

There is no quick way of combining exponents when the bases are added. Don’t be fooled if you see something like 32 + 36 = ?. The answer is NOT 38. To solve, you must multiple out each term and then find the sum.

Any nonzero number raised to a power of zero is equal to 1.

30 = 1

However 00 = undefined.

A negative exponent is another way of expressing a fraction: x-1 = 1 / x1

4-2 = 1/42 = 1/16

A fractional exponent is another way of expressing a root: x1/n = n√x1

72/3 = 3√72

81/3 = 3√8 = 2

When a fraction is raised to an exponent, you must distribute the power both to the numerator and the denominator:

(1/2)3 = 13 / 23 = 1/8

Notice how the fraction will actually decrease in number as the exponent increases.

(1/2)4 = 14 / 24 = 1/16

A negative number raised to an even exponent will always be positive. The negative sign will cancel itself out.

(-2)2 = -2 x -2 = 4

However a negative numbers raised to an odd exponent will remain negative.

(-3)3 = -3 x -3 x -3 = -27

Large numbers and very small decimals are often expressed with exponents using scientific notation. Scientific notation involves writing the number as a product of a decimal and the number 10 raised to a certain power.

The number of the exponent indicates the number of places the decimal moves.

107 = 1 + 7 zeros = 10,000,000

.036 x 104 = 360 (the decimal moves four places to the right)

.0000000857 x 106 = .0857

5.6 x 10-4 = .00056 (Since it’s a negative exponent, the decimal will move to the left.)

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REPEAT AS NECESSARY – TIPS FOR ACING GMAT QUANT

By posted June 24, 2010

So you are starting the initial phases of your GMAT preparation and you are probably wondering how to ace that quantitative section score. You feel pretty well versed when it comes to verbal, but the math is what is nagging at you. You spend many sleepless nights tossing and turning about those combination/permutation questions, as well as rate and distance problems, not to mention multi-variable calculus questions that just might pop up (ok, that is a joke, there will be no calculus on the GMAT). But all kidding aside, what are the best ways to increase that math score to put you over that 700 milestone? Good question, and I hope to answer some of these concerns and calm some of the nerves so that you are confident going into the final weeks of your test preparation.

Back to the Basics

No matter how good you are at math, I would suggest buying a GMAT guide (Princeton Review has a good one, think it is called “Guide for Cracking the GMAT”, or something along those lines) and brushing up on all the different kinds of math questions that they might throw at you on test day. This might sound very rudimentary and obvious, but the fact is test-takers fail to realize and understand what types of math questions they will face come test day. If you have a solid working knowledge of the typical questions, this is the first step in boosting your GMAT quant score. As you are probably aware, there are some areas in quant that the last time you saw such math was when you were wearing tube socks and playing wall ball at recess. “What is the difference between a multiple and a factor?” you say? If you don’t know the answer to this, you definitely should get up to speed with these elementary, yet key fundamentals.

Make a Plan

Nothing big was ever accomplished without a plan. Organize your game plan, your plan of attack if you will, before you start studying. I’m not here to tell you how you should structure your plan, but lay out on paper how long you think you will need to review, do practice problems and practice tests, and then brush up on weaker areas. This will probably take at least two months, so have an idea of how you want to allocate your time and stick to it. If you don’t keep up with your plan or feel that you are pressed for time, perhaps you need to think about pushing your test date back. Regardless, just thinking about how you approach your studies will serve you well in the end.

Practice Practice Practice

As you go along in your studies, more specifically when doing hundreds of practice problems, you will invariably get a feel for and find your weaker quant areas. This is a good thing!! Don’t get discouraged when you come across these areas, it is an absolute blessing that you have identified your Achilles heel. You won’t be able to find these weaker areas if you aren’t practicing literally hundreds of questions (I suggest the Official Guide for GMAT Review for good practice problems). After hundreds of questions, you will be able to tell what your strengths and weaknesses are. As far as strengths go, it probably doesn’t make too much sense to keep practicing these types of problems that you are comfortable with. As you go along, start to hone in on the areas that are giving you trouble. Perhaps you can keep a spreadsheet or notes of what types of questions are holding you back.

Repeat as Necessary

Now that you have found your weak areas, I suggest doing multiple iterations of these types of questions. In your final few weeks, make it your goal to really focus on these areas and master these types of questions. This will give you confidence as your test day nears, as well as the ability to tackle those problems that you are missing. The key here is repetition; just do as many as you can. Soon you will discover a comfort level with once more difficult types of questions and you will be on your way to acing the exam. Most folks seem to struggle with time on the GMAT quant section, and increased exposure to hundreds of questions within your “bad” areas will really serve you well on test day. As you go from question to question during your exam, you will really thank yourself for your efforts. You will be amazed how many similar questions you see and will probably even tell yourself “Cool, I just did a question very similar to this last week” and the preparation will come in very handy. The Boy Scouts don’t have the motto “Be Prepared” for nothing. Put in your time and it will pay off. Good luck!!

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Properties of Numbers

By posted May 26, 2010

Math is all about numbers and you can imagine that quite a few concepts fall under properties of numbers.  Practice is the only way of familiarizing yourselves with this type of question but there a few things that you can keep in mind as you work your way through Grockit.

Even and Odd Numbers

Sometimes, all that a question tells you is that the product of two distinct numbers x and y is not even.  It then wants to know if x – y is always even.  How do you figure this out?

One way is to plug in numbers.  Suppose x = 5 and y = 3.  The product of x and y is 15 (odd) and x – y is 2 which is even.  Try this with a few other numbers and you can safely guess that x – y is always even if xy is odd.

Another way is to remember that

ODD * ODD = ODD

EVEN * EVEN = EVEN

ODD * EVEN = EVEN

In addition, for any number n, 2n is always even and 2n+1 is always odd.  Try it.  If n = 4, 2n = 8 which is even and 2n+1 = 9 which is odd.  If n = 5, 2n = 10 which is even and 2n+1 = 11 which is odd.

Knowing this, you can use it to solve the above problem.  If xy is odd, that means x and y must both be odd.  That means that if x = 2n + 1 and y = 2m + 1, then x – y = 2n – 2m = 2(n – m).  2 times of anything is always even, and thus, you have just proved algebraically that x – y is always even.

Other good things to remember is that

EVEN + EVEN = EVEN

ODD + ODD = EVEN

ODD + EVEN = ODD

Estimating quickly

Being about to round up and down and estimate quickly is always a good skill.  Knowing how to estimate fractions, in particular, can save you a fair bit of time on the GMAT.

Given this question: which of the following fractions is greater than ¼?

  1. 12/50
  2. 3/11
  3. 2/9
  4. 4/17
  5. 6/24

You should be able to recognize quickly that 6/24 = ¼ so choice E is out.  Next look at choices C and D.  9 is close to 8 and ¼ of 8 is 2.  That means 2/9 is less than 2/8 i.e. 2/9 is less than ¼.  The same can be said of 4/17.  17 is close to 16 and ¼ of 16 is 4.  That means that 4/17 < 4/16.

Looking at choice A, any fraction over 25 or 50 can easily be converted into its decimal form.  12/50 = 24/100 = .24 < .25  So choice A is out too.  That leaves us with Choice B as the answer

Prime Numbers

By definition, prime numbers are only divisible by 1 and themselves.  This is a very useful property to know.  For example, if x and y are distinct prime numbers, then you know that x/y is not an integer because y is not a factor of x, or x would not be prime.

Prime factors

Knowing how to prime factorize will help you tackle almost any factorization problem.  Take the number 70 for example.  In order to find the prime factors of 70, divide 70 by the smallest prime, 2.  If that’s not possible, move on to the next prime.  70 is divisible by 2, which gives you 35.

Try dividing 35 by 2 again.  If that’s not possible, more on to the next prime.  35 is not divisible by 3, so try 5.  35/5 = 7 and since you are left with a prime number itself, you can stop there.

Through this constant dividing process, we can determine that 70 = 2*3*5

Try and see if you can find the prime factors of 244.

(You should get 168 = 2*2*2*3*7)

Let’s try applying it to this problem: The product of three positive integers is 70. If all of the integers are greater than 1, what is the sum of the greatest two integers?

  1. 2
  2. 7
  3. 12
  4. 14
  5. 35

We worked out just not that 70 = 2*5*7, meaning that the two largest integers are 5 and 7.  That means that the sum is 12.

Squares

The last important thing to know about numbers is what perfect squares are.  A perfect square is an integer that is the square of another integer.  For example, 4, 9, 16, 25, 36 are perfect squares since they are integers and they are the squares of 2, 3, 4, 5 and 6 respectively.

The following is a simple application of the above concept.  The only thing that might confuse you at first is the phrasing of the question.  Which of the following is NOT equal to an integer squared?

  1. Root 16
  2. Root 9
  3. 27/3
  4. 37-12
  5. 49

The question is essentially asking, which of the 5 choices is not a perfect square.  After you simplify the choices, pick the one that cannot be square rooted.  (Did you get choice B?)

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Interest and Compound Interest Problems

By posted March 10, 2010

Interest and Cmoneyompound Interest

There are two types of interest problems on the GMAT, and they include simple interest and compound interest. Simple interest is the most basic and is a function of P, the principle amount of money invested, the interest rate earned on the principle, i, and the amount of time the money is invested, t (this is usually stated in periods, such as years or months). The resulting equation is:       I = iPt

In basic terms, the above equation tells us the amount of interest that would be earned on a principle amount invested (P), for a given time (t) at a given interest rate (i).

Example:

If you invested $1,000 (P = your principle) for one year (t = one year) at 6% simple interest (i = given interest rate), you would get $60 in interest at the end of the year and would have a total of $1,060.

For compound interest, you would earn slightly more.

Let’s look at similar type problem, though this one involves compound interest.

Mr. Riley deposits $500 into an account that pays 10% interest, compounded semiannually. How much money will be in Mr. Riley’s account at the end of one year?

For compound interest, first you need to divide the interest rate by how many compound periods there are. So for in the above question, because we are compounding semiannually, we need to divide 10% by 2 (because of 2 compounding periods), and if we were compounding quarterly, we would need to divide 10% by 4.

In the above question, Mr. Riley deposited $500 into his account at a rate of 10% compounded semiannually and the bank will divide his interest into two equal parts. They will pay 5% interest (10%/2) at the end of six months, and then will pay another 5% at the end of the year. Compound interest can essentially be translated into “interest paid on interest”, meaning that after one period, you are paid interest on the interest that was paid in prior periods, hence the phrase “compounding”.

So at the end of the six months, Mr. Riley has $525 because the bank paid $25 in interest ($500*5%) into his account. For the second half of the year, Mr. Riley is then paid 5% on the $525 balance that was in his account at the end of the first six months. This interest is equal to $525*5% = $26.25. Therefore, at the end of the year, Mr. Riley has $551.25, which is equal to his balance of $500, plus the $25 interest paid at the end of 6 months, plus $26.25 paid at the end of the year. Mr. Riley earns $1.25 more with this compound interest than he would have been paid if he were paid only 10% simple interest (would have been only $550). The lesson? Compound interest always pays more!

Let’s look at another similar type of problem that involves interest.

Money invested at x%, compounded annually, triples in value in approximately every 112/x years. If $2500 is invested at a rate of 8%, compounded annually, what will be its approximate worth in 28 years?

A. $3,750

B. $5,600

C. $8,100

D. $15,000

E. $22,500

At first glance, this one seems pretty tricky because you are given x% as the interest rate and it asks you about compounding and it might seem difficult where to find a starting point for this. For this one, it might be a bit easier to think about this without the use of compound interest, which might unnecessarily confuse you. Here, we are given x% as 8%, so all we need to do is take 112/8 = 14. Thus, we know that the money triples in value every 14 years. Further, we know that the money will triple exactly twice in 28 years, once in 14 years and one more time at the 28th year. So first we need to multiply the original $2500 invested by 3 to get the balance at the end of year 14 (because it triples), to get $7,500 (or $2,500*3). Now, we know that this balance of $7,500 will triple again, so the final balance at the end of the next 14 year period will be $22,500 (or $7,500*3). The correct answer choice is E.

Overall, the three types of interest problems you will most likely encounter come test day will be simple interest, compound interest, and word problems involving the mention of interest, but that can be solved without the application of interest or compound interest methods. The key to deciphering between compound interest and simple interest is to see how many periods the interest is paid….interest paid in one period is simple interest and interest “paid on interest” in multiple periods is compound interest. Finally, remember that some questions can be solved intuitively.

Check out Grockit for more GMAT quantitative practice!

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Weighted Averages On The GMAT

By posted February 26, 2010

This post will introduce weighted average questions you’ll see on the GMAT.  There is one main formula you need to solve simple GMAT Average questions:6

  • Average = SUM / # of observations

Note that this can be rearranged to read:

  • SUM = Average x (# of obs)
  • # obs = SUM / Average

So, if you are given ANY 2 of the 3 values, you should be able to find the 3rd. For example:

John drinks an average of 1.5 cups of water/day. After how many days has he drank 3 gallons of water? (1 gallon = 16 cups.)

In this case, we are looking for the number of days (or observations) such that we total 48 cups (3 gallons) of water.

# = SUM / Average

  • # days = 48 cups / 1.5 cups/day
  • # days = 32 days

NEVER AVERAGE AVERAGES!

Class A  has 15 students and an average height of 60”. Class B has 20 students. What is class B’s average height if the average height of both classes is 65”.

One might say:  (A + B) / 2 = 65”; A = 60”; so B must be 70”. However, keep in mind:

TOTAL AVERAGE = TOTAL SUM / TOTAL OBS

CLASS A + CLASS B = BOTH

  • 15 students + 20 students = 35 students
  • 60” average + 68.75” average = 65” average
  • 900” total in A + 1375” total in B = 2275” total in Both

The given information is in black. The necessary intermediate steps are in blue, and the red is your answer. Note that the average of the averages ≠ total average. We must calculate each average separately, and to do this we need the SUM and # of observations for each category. This brings us to the idea of WEIGHTED AVERAGES.

A WEIGHTED AVERAGE is needed when you are taking average of a large group in which there are subgroups with a different number of observations in each. Take a look at this generalized formula, assuming there are 3 groups A, B and C.

(Average of A x Obs in A) + (Average of B x Obs in B) + (Average of C x Obs in C)

(Obs in A) + (Obs in B) + (Obs in C)

Think of weighted averages like a tug of war between numbers. The “stronger” one side (dog) is, the more that weighted average (tennis ball) will be “pulled” in that direction.

In the previous question, we had:

CLASS A + CLASS B = BOTH

  • 15 students + 20 students = 35 students
  • 60” average + 68.75” average = 65” weighted average

Note that the weighted average is CLOSER to B’s average than it is to A’s. This is because there are 20 students in Class B compared to only 15 students in Class A.

Two More Examples

At a certain restaurant, the average (arithmetic mean) number of customers served for the past x days was 75. If the restaurant serves 120 customers today, raising the average to 90 customers per day, what is the value of x?

A. 2

B. 5

C. 9

D. 15

E. 30

WITHOUT using the formula, we can see that today the restaurant served 30 customers above the average. The total amount ABOVE the average must equal total amount BELOW the average. This additional 30 customers must offset the “deficit” below the average of 90 created on the x days the restaurant served only 75 customers per day.

30/15 = 2 days. Choice (A).

WITH the formula, we can set up the following:

  • 90 = (75x + 120)/(x + 1)
  • 90x + 90 = 75x + 120
  • 15x = 30

x = 2  Answer Choice (A)

Use whichever makes more sense to you!

Anita spent a total of $780 on 52 bottles of wine for her wedding. She then decided to buy 8 bottles of sparkling wine for the toasts, as well. Was the average (arithmetic mean) price per bottle of wine less than $20?

(1) Each bottle of sparkling wine cost more than $15.

(2) Each bottle of sparkling wine cost less than $40.

Take another look at what exactly the question is calling for: the TOTAL average price of all the wine at the wedding. We should look at the suggested average ($20) and use that as our threshold amount.

  • 60 bottles * $20/bottle = $1200 total
  • $1200 total – $780 (given) = $420 (left for sparkling wine)
  • $420 / 8 bottles = $52.50/bottle of sparkling wine (for the total average to equal $20)

Which of the answer choices are conclusively above or below $52.50/bottle of sparkling wine? Only (2). With (1), we can be below OR above the threshold, so (1) is not sufficient.

Answer Choice (B)

Now, you’re score will be above average! Please visit the Grockit forum or leave a comment here if you have more questions on weighted averages.

Good luck!

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Estimation Strategies For GMAT Problem Solving

By posted January 19, 2010

The GMAT is not testing who is the fastest at long division. It is a test that seeks to measure problem solving skills that are not necessarily the “textbook” ways to discover solutions. Let’s discuss some estimation strategies, which are not used as often as they should be.

1. Round Up AND Round Down When Multiplying

Be aware of the direction in which you are altering the result. If you want to estimate a product of two “ugly” numbers, you can move one up and one down, which is an attempt to minimize the error in your estimation. For example:

658*436 = 286,888

If we round 658 UP to 700 and 436 DOWN to 400, we can approximate using:

700*400 = 280,000

2. Round In The Same Direction When Dividing

When you want to approximate a fraction, you can either adjust only the numerator (or denominator) or move both in the same direction. For example:

8/19 = .4210526…

8/20 = 0.4 (Note that increasing the denominator, will decrease the fraction.)

9/20 = 0.45 (Note that increasing both top and bottom will increase the fraction.)

Your estimate is somewhere between .40 and .45.

3. Remember These Other Helpful Tips

  • Peek at your answer choices: If your answer choices are relatively far apart, this could be hint that approximation is helpful. If the answers are very tight together, you may still estimate, but you have to be more careful and do due diligence.
  • Geometry shortcut 1: √2 =~ 1.4 and √3 =~ 1.7. Try to commit these to memory, as they are very common.
  • Geometry shortcut 2: Be careful when using pi = 3. Recognize that you are using a smaller number, so your result will be smaller too. Test makers love to give tempting answer choices that assume pi = 3. It’s not.
  • Geometry shortcut 3: Even though you cannot assume charts are drawn to scale, they can still be a resource. Obtuse/acute angles are typically shown as much, and angles can be approximated in many circumstances. That’s not to say “if it looks like a right angle, it must be 90.” But you can use the drawing as a guide to your estimation.
  • Use the extremes: If you are given a range, it helps to plug in those extremes to see between which values your answer falls. This will focus your attention on the cases that are above (or below) those endpoints.

Two Examples

If a square has a perimeter of 80 inches, what is the approximate length of its diagonal, in inches?

A. 20

B. 28

C. 40

D. 56

E. 112

This question uses the word “approximate,” so that should be a very big hint that you will need to find a number “close enough.” If P = 80, then s = 20. The diagonal is essentially a hypotenuse of a 45-45-90 triangle, so d = 20√2.

Two strategies:

1) 20√1 = 20 and 20√4 = 40. Therefore 20 < 20√2 < 40. (B) 28 is the only option.

2) Since we remember that √2 =~1.4, we can simply multiply 20*1.4 = 28. (B).

Addison High School’s senior class has 160 boys and 200 girls. If 75% of the boys and 84% of the girls plan to attend college, what percentage of the total class plan to attend college?

A. 75

B. 79.5

C. 80

D. 83.5

E. 84

84 is an obscure number. When you see obscure numbers, that is another sign that you may want to look for an approximating shortcut.

Firstly, we should eliminate the overtly incorrect choices. This will be (A) 75 (since that’s the low extreme) and (E) 84 and (D) 83.5 (since they are both essentially equal to the high extreme).

Secondly, find the average of the given percents. Since there are more girls than boys, we know that the weighted average will be closer to the girls’ percent than the boys’ percent. By finding 79.5% as the mean of 75% and 84%, we are given the low extreme. Again, we recognize the weight placed on 84%, making the answer higher than 79.5. (C) 80 it is!

(For similar questions in the future where we actually need to calculate, we could drop the extra “0” from 160 and 200. The ratio of 16:20 is the same (4:5), and the calculation is much easier.)

Any other estimation tricks? Just post in the comment field or check out Grockit’s forums for more strategies on GMAT math.

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