Tài liệu Master the Gre 2010 - Part 19 pptx

In multiple-choice questions, working backward from numerical answer choices works well when the numbers are easy and when few calculations are required, as in question 8.. TIP In multip

Know When—and When Not—to Work Backward

If a multiple-choice question asks for a number value, and if you draw a blank as far

as how to set up and solve the problem, don’t panic You might be able to work

backward by testing the answer choices in turn

8 A ball is dropped from 192 inches above level ground, and after the third

bounce, it rises to a height of 24 inches If the height to which the ball

rises after each bounce is always the same fraction of the height reached

on its previous bounce, what is this fraction?

(A) 1

8

(B) 1

4

(C) 1

3

(D) 1

2

(E) 2

3

The correct answer is (D) The fastest route to a solution is to plug in an

answer Try choice (C) and see what happens If the ball bounces up1

3as high as

it started, then after the first bounce it will rise up1

3as high as 192 inches, or 64 inches After a second bounce, it will rise1

3as high, or about 21 inches But the

problem states that the ball rises to 24 inches after the third bounce Obviously,

if the ball rises less than that after two bounces, it will be too low after three So

choice (C) cannot be the correct answer We can see that the ball must be

bouncing higher than one third of the way, so the correct answer must be a larger

fraction, meaning either choice (D) or choice (E) You’ve already narrowed your

odds to 50% Try plugging in choice (D), and you’ll see that it works:1

2of 192 is 96;

1

2of 96 is 48; and

1

2of 48 is 24.

Although it would be possible to develop a formula to answer question 8, it’s not

worthwhile, considering how quickly and easily you can work backward from the

answer choices

In multiple-choice questions, working backward from numerical answer choices works

well when the numbers are easy and when few calculations are required, as in

question 8 In other cases, applying algebra might be a better way

TIP

In multiple-choice Problem Solving questions, numerical answer choices are listed in order of value from least to greatest To work backward from the answer choices, start with choice (C) If choice (C) is too great, then you can assume choices (D) and (E) are incorrect and try choices (A) or (B) instead.

9 How many pounds of nuts selling for 70 cents per pound must be mixed

with 30 pounds of nuts selling at 90 cents per pound to make a mixture that sells for 85 cents per pound?

(A) 8.5 (B) 10 (C) 15 (D) 16.5 (E) 20 The correct answer is (B) Is the easiest route to the solution to test the

answer choices? Let’s see First of all, calculate the total cost of 30 pounds of nuts

at 90 cents per pound: 30 3 90 5 $27 Now, start with choice (C) 15 pounds of nuts at 70 cents per pound costs $10.50 The total cost of this mixture is $37.50, and the total weight is 45 pounds Now you’ll need to perform some long division The average price per pound of the mixture turns out to be between 83 and 84 cents—too low for the 85-cent average given in the question So you can at least eliminate choice (C)

You should realize by now that testing the answer choices might not be the most efficient way to tackle this question Besides, there are ample opportunities for calculation errors Instead, try solving this problem algebraically by writing and solving an equation Here’s how to do it The cost (in cents) of the nuts selling for

70 cents per pound can be expressed as 70x, letting x equal the number that

you’re asked to determine You then add this cost to the cost of the more expensive nuts (30 3 90 5 2,700) to obtain the total cost of the mixture, which

you can express as 85(x 1 30) You can state this algebraically and solve for x as

follows:

70 2 700 85 30

70 2 700 85 2 550

150 15 10

x x

=

=

At 70 cents per pound, 10 pounds of nuts must be added in order to make a mixture that sells for 85 cents per pound

Look for the Simplest Route to the Answer

In many Problem Solving questions, there’s a long way and a short way to arrive at the correct answer When it looks like you’re facing a long series of calculations or a complex system of equations, always ask yourself whether you can take an easier, more intuitive route to solving the problem

10 What is the difference between the sum of all positive odd integers less

than 32 and the sum of all positive even integers less than 32?

(A) 32

(B) 16

(C) 15

(D) 1

(E) 0

The correct answer is (B) To answer this question, should you add up two long

series of numbers on your scratch paper? In this case, it’s a waste of time, and you

risk committing calculation errors along the way A smart test-taker will notice a

pattern and use it as a shortcut Compare the initial terms of each sequence:

even integers: [2, 4, 6, ] ., 30

odd integers: [1, 3, 5, ] ., 29, 31

Notice that for each successive term, the odd integer is one less than the

corre-sponding even integer There are a total of 15 correcorre-sponding integers, so the

difference between the sums of all these corresponding integers is 15 But the

odd-integer sequence includes one additional integer: 31 So the difference is

16, which is 31 – 15

Start with What You Know

It’s easy to get lost in Problem Solving questions that are complex and involve

multiple steps to solve First, take a deep breath and start with information you know

Then ask yourself what you can deduce from it, leading yourself step-by-step to the

solution

11 In a group of 20 singers and 40 dancers, 20% of the singers are less than

25 years old, and 40% of the entire group are less than 25 years old What

portion of the dancers are younger than 25 years old?

(A) 20%

(B) 24%

(C) 40%

(D) 50%

(E) 60%

The correct answer is (D) To answer this question, you need to know (1) the

total number of dancers and (2) the number of dancers younger than 25 years old.

The question provides the first number: 40 To find the second number, start with

what the question provides and figure out what else you know Keep going, and

eventually you’ll arrive at your destination Of the whole group of 60, 24 are

younger than 25 years (40% of 60 is 24.) 20% of the 20 singers, or 4 singers, are

younger than 25 years Hence, the remaining 20 people younger than 25 must be

dancers That’s the second number you needed to answer the question 20 is 50%

of 40

Search Geometry Problem Figures for Clues

Some GRE geometry problems will be accompanied by figures They’re there for a reason: The pieces of information provided in a figure can lead you, step-by-step, to the answer

12 If O is the center of the circle in the figure above, what is the area of the

shaded region, expressed in square units?

(A) 3

2p

(B) 2p

(C) 5

2p

(D) 8

3p

(E) 3p The correct answer is (E) This question asks for the area of a portion of the

circle defined by a central angle To answer the question, you’ll need to determine the area of the entire circle as well as what percent (portion) of that area is shaded Mine the figure for a piece of information that might provide a starting point If you look at the 60° angle in the figure, you should recognize right away that both triangles are equilateral (all angles are 60°) and, extended out to their arcs, form two “pie slices,” each one1

6the size of the whole pie (the circle) What’s left are two big slices, each of which is twice the size of a small slice So the shaded area must account for1

3the circle’s area You’ve now reduced the problem

to the simple mechanics of calculating the circle’s area and then dividing it by 3

circumference) The area of any circle is pr2, where r is the circle’s radius Thus,

the area of the circle is 9p The shaded portion accounts for 1

3 the circle’s area,

or 3p

Sketch Your Own Geometry Figure

A geometry problem that doesn’t provide a diagram might be more easily solved if it

had one Use your scratch paper and draw one for yourself It will be easier than

trying to visualize it in your head

13 On the xy-coordinate plane, points R(7,23) and S(7,7) are the endpoints of

the longest possible chord of a certain circle What is the area of the

circle?

(A) 7p

(B) 16p

(C) 20p

(D) 25p

(E) 49p

The correct answer is (D) There are lots of sevens in this question, which

might throw you off track without at least a rough picture To keep your thinking

straight, scratch out your own rough xy-grid and plot the two points You’ll see

that R is located directly below S, so chord RS is vertical.

Accordingly, the length of RS is simply the vertical distance from 23 to 7,

which is 10 By definition, the longest possible chord of a circle is equal in

length to the circle’s diameter The circle’s diameter is 10, and thus its

radius is 5 The circle’s area is p(5)25 25p

Plug in the Numbers for “Defined Operation” Questions

One of your 14 Problem Solving questions might very well be a so-called defined

operation question These are odd-looking problems that might strike you as being

difficult But they’re really not In fact, the math usually turns out to be very easy

You’re being tested on your ability to understand what the problem requires you to do,

and then to cross your t’s and dot your i’s as you perform simple arithmetical

calculations

ALERT!

GRE geometry figures provide helpful information for solving the problem, but they’re not

intended to provide the

answer through visual measurement Make sure you solve the problem by applying

your knowledge, not by

looking at the figure.

14 Let be defined for all numbers a, b, c, and d by 5 ac 2 bd If

x 5 , what is the value of ?

(A) 1 (B) 2 (C) 4 (D) 5 (E) 10

The correct answer is (B) In defining the diamond-shaped figure as “ac 2 bd,”

the question is telling you that whenever you see four numbers in a diamond like this, you should plug them into the mathematical expression shown in the order given The question itself then requires you to perform this simple task twice

First, let’s figure out the value of x If x is the diamond labeled as x, then a 5 5,

b 5 4, c 5 2, and d 5 1 Now, we plug those numbers into the equation given,

then do the simple math:

x x x

=

(5 2) (4 1)

10 4 6

Now we tackle the second step Having figured out the value of x, we can plug it into our second diamond, where a 5 6, b 5 10, c 5 2, and d 5 1 Again, plug in the

numbers and do the math:

(6 3 2) 2 (10 3 1) 5 12 2 10 5 2

As you can see, the math is very easy; the trick is understanding what the question is asking: which is to have you “define” a new math operation and then carefully plug in the numbers and work out the solution With a little practice, you’ll never get a defined operation question wrong

THE DATA INTERPRETATION FORMAT

Data interpretation is a special Problem Solving format designed to gauge your ability

to read and analyze data presented in statistical charts, graphs, and tables, and to calculate figures such as percentages, ratios, fractions, and averages based on the numbers you glean from graphical data

Expect to find three to five data interpretation questions (possibly in sets of two or three) mixed in with your other Quantitative Reasoning questions Each question in a data interpretation set pertains to the same graphical data Each question (and each

set) involves either one or two distinct graphical displays Four types appear on the

GRE most frequently: tables, pie charts, bar graphs, and line charts

Note the following key features of GRE data interpretation questions:

NOTE

Expect 3 to 5 data

interpretation questions on the

exam Some may appear in

sets pertaining to the same

graphical data.

• Important assumptions will be provided Any additional information that

you might need to know to interpret the figures will be indicated above and below

the figures Be sure to read this information

• Some questions might ask for an approximation This is because the test

makers are trying to gauge your ability to interpret graphical data, not your

ability to crunch numbers to the nth decimal place.

• Many of the numbers used are almost round This feature relates to the

previous one The GRE rewards test takers who recognize that rounding off

numbers (to an appropriate extent) will suffice to reach the correct answer

• Some questions may be long and wordy Solving a data interpretation

problem may call for multiple steps involving various graphical data, so the

questions can be lengthy In fact, you may have more trouble interpreting the

questions than the graphical data

• Bar graphs and line charts are drawn to scale That’s because visual

esti-mation is part of what’s required to analyze a bar graph or line chart’s graphical

data However, pie charts will not necessarily be drawn to scale (You’ll interpret

them strictly by the numbers provided), and visual scale is irrelevant when it

comes to analyzing tables

• Figures are not drawn to deceive you or to test your eyesight In bar

graphs and line charts, you won’t be asked to split hairs to determine precise

values These graphs and charts are designed for a comfortable margin for error

in visual acuity Just don’t round up or down too far

• You may need to scroll vertically to see the entire display Some vertical

scrolling might be required to view the entire display, especially the information

above and below the chart, graph, or table If so, don’t forget to scroll up and down

as you analyze each question

The 5-Step Plan

Here’s the 5-step approach that will help you handle any data interpretation question

(or set of questions) Just ahead, we’ll apply this approach to a two-question set

STEP 1: LOOK AT THE “BIG PICTURE” FIRST

Before plunging into the question(s), read all the information above and below the

figure(s) Look particularly for the following:

• Totals (dollar figures or other numbers)

• Whether the numbers are expressed in terms of hundreds, thousands, or millions

• How two or more figures are labeled

• Whether graphical data is expressed in numbers or percentages

STEP 2: READ THE ENTIRE QUESTION CAREFULLY

Read the entire question very carefully As you do so, divide the question into parts, each

of which involves a distinct step in getting to the answer Pay particular attention to

what the question asks for For example:

NOTE

Data interpretation questions are most often based on tables, pie charts, bar graphs, and line charts However, you might encounter some other type of graphic, so don’t be surprised to see something else.

• Does the question ask for an approximation or a precise value?

• Does the question ask for a percentage or a raw number?

• Does the question ask for a comparison?

• Does the question ask for an increase or a decrease?

In breaking down the question into tasks, look for a shortcut to save yourself pencil work

STEP 3: PERFORM THE STEPS NEEDED TO REACH THE ANSWER

Look for a shortcut to the answer For questions calling for approximations, round numbers up or down (but not too far) as you go

STEP 4: CHECK THE CHOICES FOR YOUR ANSWER

If an “approximation” question asks for a number, find the choice closest to your answer Look for other answer choices that are too close for comfort If you see any, or if your solution is nowhere near any of the choices, go to step 5

STEP 5: CHECK YOUR CALCULATIONS

Check all of your calculations and make sure the size and form (number, percentage, total, etc.) of your solution conforms with what the question asks Check your rounding technique Did you round off in the wrong direction? Did you round off too far?

APPLYING THE 5-STEP PLAN

Let’s apply the 5-step approach to a pair of questions, based on the following two related pie charts

We’ll perform step 1 before we even look at the first question based on this data

TIP

Use rounding and estimation

to answer a data

interpretation question—but

only if the question calls for an

approximation.

totals are provided, but that the pie segments are all expressed only as percentages.

That’s a clue that your tasks for this set of data might be to calculate dollar amounts

for various pie segments

Now that you’ve familiarized yourself with the graphical display of data, proceed to

step 2 for the first question

Step 2: Carefully read the question, but not the five lettered answer choices yet (This

question is near the lowest level of difficulty; about 85% of test takers would answer it

correctly.)

15 Which of the following pairs of divisions, each pair considered separately,

accounted for more than 50 percent of Company XYZ’s income during the

year?

I Division A and Division B

II Division C and Division D

III Division A and Division C

(A) I only

(B) III only

(C) I and II only

(D) I and III only

(E) I, II, and III

Just by reading the question, you know that it involves only the chart on the left

(“Income”) and that you won’t need to convert percentages into dollar figures So it’s a

straightforward question that you should not consider skipping Nevertheless, be sure

to read it very carefully: You’re looking for any pair that accounted for “more than 50

percent” of XYZ’s income, not 50 percent or more (This distinction makes the

dif-ference between the correct answer choice and an incorrect one, as you’ll soon see.)

Step 3: You need to combine three different percentage pairs (indicated by Roman

numerals I, II, and III) Be careful to combine the correct percentages from the chart

on the left Here are the results:

I Division A and Division B (38% + 12% = 50%)

II Division C and Division D (20% + 30% = 50%)

III Division A and Division C (38% + 20% = 58%)

Step 4: The only pair among the three that accounted for more than 50% of XYZ’s

income is pair III Because only pair III meets that criterion, the correct answer must

be (B): “III only.” If you have extra time, go to step 5

Step 5: Check the pie chart on the left and the three division pairs (I, II, and III)

again to make sure the percentages you used and your addition are correct Once

you’ve confirmed they are, go on to the next question The correct answer is (B).

Let’s apply the same five-step approach to a moderately difficult question (About 50%

of test takers would respond correctly to questions like it.)

Step 1: You’ve already performed this step, so move to step 2.

NOTE

The Roman numeral format illustrated in Question 15 is not used very frequently on the GRE Expect no more than one

or two Problem Solving questions in this format.

Step 2: Carefully read the question, but not the five answer choices yet:

16 During year X, by approximately what amount did Division C’s income

exceed Division B’s expenses?

(A) $125,000 (B) $127,000 (C) $140,000 (D) $180,000 (E) $312,000

This question involves three tasks: (1) calculate Division C’s income; (2) calculate Division B’s expenses; and (3) compute their difference

There’s no shortcut to these three tasks, although you know from the question stem that calculating approximate values will probably suffice Go on to step 3

Step 3: Division B’s expenses accounted for 26% of XYZ’s total expenses, given as

$495,000 Rounding off these figures to 25% and $500,000, Division B’s expenses totaled approximately $125,000 Income from Division C sales was 20% of total XYZ income, given as $1,560,000 Rounding this total down to $1,500,000, income from Division C sales was approximately $300,000 Income from Division C sales exceeded Division B’s expenses by approximately $175,000

Step 4: Scan the answer choices The only one that’s close to this solution is choice

(D) If you have extra time, go to step 5

Step 5: Make sure that you started with the right numbers Did you compare C’s

income with B’s expense (and not some other combination)? If you’re satisfied that the numbers you used were the right ones and that your calculations are okay, confirm

your response and move on to the next question The correct answer is (D).

DATA INTERPRETATION STRATEGIES

Applying the five-step approach to the two sample questions in the previous section highlighted certain tips, techniques, and strategies for data interpretation Here’s a list

of them (The last one applies to certain other types of graphical data.)

Don’t Confuse Percentages with Raw Numbers

Most data interpretation questions involve raw data as well as proportion—in terms of

either percent, fraction, or ratio Always ask yourself: “Is the solution to this problem a

raw number or a proportional number?” (You can be sure that the test designers will bait

you with appropriate incorrect answer choices.)

Be Sure to Go to the Appropriate Chart (or Part of a Chart) for Your Numbers

When it comes to GRE data interpretation, carelessly referring to the wrong data is the

ALERT!

Be sure to consult the

appropriate portion of

the chart, graph, or table for

the information you need to

answer a data interpretation

question Your ability to find

the right data is a big part of

what’s being tested.