Master gmat 2010 part 19 pot

Step 2: The five answer choices provide a couple of useful clues:• Notice that each answer choice includes all three letters p, q, and c.. These simple numbers make the question easy to

In the question, you started with six terms Let a through f equal those six terms:

19 5a 1 b 1 c 1 d 1 e 1 f

6

114 5 a 1 b 1 c 1 d 1 e 1 f

f 5 114 2 ~a 1 b 1 c 1 d 1 e!

Letting f 5 the number removed, here’s the arithmetic-mean formula, applied to the

remaining five numbers:

21 5a 1 b 1 c 1 d 1 e

5

105 5 a 1 b 1 c 1 d 1 e

Substitute 105 for (a 1 b 1 c 1 d 1 e) in the first equation:

f 5 114 2 105

f 5 9

Step 5: If you have time, check to make sure you got the formula right, and check your

calculations Also make sure you didn’t inadvertently switch the numbers 19 and 21 in your

equations (It’s remarkably easy to commit this careless error under time pressure!) If you’re

satisfied that your analysis is accurate, confirm your answer and move on to the next

question The correct answer is (C).

QUESTION 3

Question 3 is moderately difficult Approximately 50% of test takers respond correctly to

questions like it Here’s the question again:

3 If p pencils cost 2q dollars, how many pencils can you buy for c cents? [Note: 1 dollar

5 100 cents]

(A) pc

2q

200q

(C) 50pc

q

(D) 2pq

c

(E) 200pcq

Step 1: The first step is to recognize that this question involves a literal expression Although

it probably won’t be too time-consuming, it may be a bit confusing You should also recognize

that the key to this question is the concept of proportion It might be appropriate to set up an

equation to solve for x Along the way, expect to convert dollars into cents.

ALERT!

Careless errors, such as switching two numbers in a problem, is by far the leading cause of incorrect GMAT responses.

Step 2: The five answer choices provide a couple of useful clues:

• Notice that each answer choice includes all three letters (p, q, and c) So the solution

you’re shooting for must also include all three letters

• Notice that every answer choice but (E) is a fraction So anticipate building a fraction to solve the problem algebraically

Step 3: Is there any way to answer this question besides setting up an algebraic equation?

Yes In fact, there are two ways One is to use easy numbers for the three variables—for

example, p 5 2, q 5 1, and c 5 100 These simple numbers make the question easy to work

with: “If 2 pencils cost 2 dollars, how many pencils can you buy for 100 cents?” Obviously, the answer to this question is 1, so you can plug the numbers into each answer choice to see which choice provides an expression that equals 1 Only choice (B) fits the bill:

~2!~100!

~200!~1!5 1 Another way to shortcut the algebra is to apply some intuition to this question If you strip

away the pencils, p’s, q’s and c’s, in a very general sense the question is asking:

“If you can buy an item for a dollar, how many can you buy for one cent?”

Since one cent (a penny) is 1

100of a dollar, you can buy

1

100of one item for a cent So you’re probably looking for a fractional answer with a large number such as 100 in the denominator (as opposed to a number such as 2, 3, or 6) Answer choice (B) is the only choice that appears

to be in the correct ballpark Choice (B) is indeed the correct answer

Step 4: You can also answer the question in a conventional manner using algebra (This is

easier said than done.) Here’s how to approach it:

1 Express 2q dollars as 200q cents (1 dollar 5 100 cents).

2 Let x equal the number of pencils you can buy for c cents.

3 Think about the problem “verbally,” then set up an equation and solve for x:

“p pencils is to 200q cents as x pencils is to c cents”

“The ratio of p to 200q is the same as the ratio of x to c” (in other words, the two ratios are

proportionate)

p 200q5

x c pc 200q 5 x

Step 5: Our solution, pc

200q, is indeed among the answer choices If you arrived at this solution

using the conventional algebraic approach (step 4), you can verify your solution by substituting simple numbers for the three variables (as we did in step 3) Or if you arrived at your solution by plugging in numbers, you can check your work by plugging in a different set

of numbers or by thinking about the problem conceptually (as in step 3) Once you’re confident

you’ve chosen the correct expression among the five choices, confirm your choice, and then

move on to the next question The correct answer is (B).

SOME ADVANCED TECHNIQUES

Now let’s take a look at some more advanced methods of Problem Solving In this next section,

you’ll:

• Apply the success keys you learned earlier to more challenging Problem Solving

questions

• Learn additional success keys that apply to certain types of Problem Solving

questions and apply these keys to example questions

The first thing you’ll want to do is scan the answer choices to see what all or most of them

have in common, such as radical signs, exponents, factorable expressions, or fractions Then

try to formulate a solution that looks like the answer choices

4 If a Þ 0 or 1, then

1

a

2 22

a

5

2a 2 2

a 2 2

a 2 2

(D) 1

a

2a 2 1

The correct answer is (A) Notice what all the answer choices have in common: Each one is

a fraction in which the denominator contains the variable a, and no fractions appear in the

numerator or denominator That’s a clue that your job is to manipulate the expression given

in the question so that the result includes these features First, place the denominator’s two

terms over the common denominator a Then, divide a from the denominators of both the

numerator fraction and the denominator fraction (this is a shortcut to multiplying the

numerator fraction by the reciprocal of the denominator fraction):

1

a

2 22

a

5

1

a 2a 2 2

a

2a 2 2

USE COMMONSENSE “GUESSTIMATES” TO NARROW THE FIELD

If the question asks for a numerical value, you can probably narrow the answer choices by estimating the value and type of number you’re looking for Use your common sense and real-world experience to formulate a rough estimate for word problems However, don’t expect

to eliminate all answer choices except the correct one by using common sense alone

5 A spinner containing seven equal regions numbered 1 through 7 is spun two times

in a row What is the probability that the first spin yields an odd number and the second spin yields an even number?

(A) 2

7

(B) 12

49

14

(D) 1

2

(E) 4

7

The correct answer is (B) This problem involves the concept of probability Common sense

about basic probability should tell you that, with odds of close to 50% of spinning the desired

type of number on each of the two spins, the odds of spinning such a number twice in a row should be less than 50% So you can eliminate choices (D) and (E) Your odds of answering the question correctly are now 1 in 3 But notice that the remaining choices—(A), (B), and (C)—are closely grouped in value Also notice that, in each of these remaining choices, the denominator contains the sort of number you could end up with when you apply a mathematical operation to the numbers given in the question

Conclusion: You’ve probably reached the limits of applying common sense, and you’ll need to solve the problem mathematically to find the correct choice Here’s how to do it There are four odd numbers (1, 3, 5, and 7) and three even numbers (2, 4, and 6) on the spinner So the chances of yielding an odd number with the first spin are 4 in 7, or4

7 The chances of yielding

an even number with the second spin are 3 in 7, or3

7 To determine the probability of both events occurring, combine the two individual probabilities by multiplication:

4

73

3

75

12 49 Notice the “sucker” answer choice in this question: Answer choice (D) provides the simple average of the two individual probabilities:4

7and

3

7 Aside from the fact that

1

2, or 50%, is too high a probability from a commonsense viewpoint, (D) should strike you as too easy a solution

to what appears to be a complex problem

WHEN TO PLUG IN NUMBERS FOR VARIABLES

If the answer choices contain variables (like x and y), the question might be a good candidate

for the “plug-in” strategy Pick simple numbers (so the math is easy) and substitute them for

the variables You’ll definitely need your pencil for this strategy

6 If a train travels r 1 2 miles in h hours, which of the following represents the

number of miles the train travels in 1 hour and 30 minutes?

(A) 3r 1 6

2h

h 1 2

(C) r 1 2

h 1 3

h 1 6

(E) 3

2~r 1 2!

The correct answer is (A) This is an algebraic word problem involving rate of motion

(speed) You can solve this problem either conventionally or by using the plug-in strategy

The conventional way: Notice that all of the answer choices contain fractions This is a clue

that you should try to create a fraction as you solve the problem Here’s how to do it Given

that the train travels r 1 2 miles in h hours, you can express its rate in miles per hour as

r 1 2

h In

3

2 hours, the train would travel this distance:

S3

2DSr 1 2

h D53r 1 6

2h The plug-in strategy: Let r 5 8 and h 5 1 Given these values, the train travels 10 miles

(8 1 2) in 1 hour Obviously, in 11

2 hours the train will travel 15 miles Start plugging these

r and h values into the answer choices You won’t need to go any further than choice (A):

3r 1 6

2h 5

3~8! 1 6 2~1! 5

30

2, or 15 Plugging the values into choice (E) also gives an answer of 15, but you should eliminate choice

(E) because it omits h Common sense should tell you that the correct answer must include

both r and h.

WHEN—AND WHEN NOT—TO WORK BACKWARD FROM NUMERICAL ANSWER CHOICES

If a Problem Solving 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, each one in turn

7 A ball is dropped 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 rise 1

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’ll be way 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 greater fraction, 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; and1

2of 48 is 24.

Although it would be possible to develop a formula to answer the question, doing so would be senseless, considering how quickly and easily you can work backward from the answer choices Working backward from numerical answer choices works well when the numbers are easy and when few calculations are required, as in the preceding question In other cases, however, applying algebra might be a better way

8 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) 10

(B) 12

(C) 15

(D) 20

(E) 24

The correct answer is (A) 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 0.90 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 long division The average weight of the mixture turns out to be between

83 and 84 cents—too small valued for the 85 cent average given in the question At least you

can 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 2700) 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:

70x 1 2700 5 85~x 1 30!

70x 1 2700 5 85x 1 2550

150 5 15x

10 5 x

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

85 cents per pound

FIND THE EASIEST ROUTE TO THE ANSWER

If the question asks for an approximation, then you know that precise calculations won’t be

necessary and you can safely “round off” the numbers as you go But even in other questions,

you can sometimes eliminate all but the correct answer without resorting to precise

calculations

9 What is the difference between the sum of all positive odd integers less than 102

and the sum of all positive even integers less than 102?

(A) 0

(B) 1

(C) 50

(D) 51

(E) 101

The correct answer is (D) To see the pattern, compare the initial terms of each sequence:

odd integers: {1,3,5 99,101}

even integers: {2,4,6 100}

Notice that, for each successive term, the even integer is one more than the corresponding odd integer There are a total of 50 corresponding integers, so the difference between the sums of all these corresponding integers is 250 But the odd-integer sequence includes one additional integer: 101 So the difference is (250 1 101), or 51

SEARCH GEOMETRY FIGURES FOR CLUES

Most geometry problems are accompanied by figures The pieces of information a figure provides can lead you, step-by-step, to the answer

10.

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 This multi-step question is as complex as any you might encounter on the GMAT But there’s no need to panic; just start with what you know, then move step-by-step toward the answer Mine the figure for a piece of

information that might provide a starting point DOCD is your first “stepping stone.” Here are

the steps to the answer:

You know that OC and OD are congruent (equal in length) because each one is the

circle’s radius In any triangle, angles opposite congruent sides are also congruent

(the same size, or degree measure) Thus, ∠ODC must measure 60°—just like

∠OCD.

For any triangle, the sum of the measures of all three interior angles is 180° Thus,

∠COD measures 60°, just like the other two angles.

Vertical angles created by two intersecting lines are congruent Thus, ∠AOB also

measures 60°

By the same reasoning as in steps 1 and 2, each angle of DABO measures 60° Notice that the

length of AB is given as 3 Accordingly, the length of each and every side of both triangles is 3.

Since this length (3) is also the circle’s radius (the distance from its center to its,

circumference), you can determine the circle’s area The area of any circle is pr2, where r is

the circle’s radius Thus, the area of the circle is 9p

Now determine what portion of the circle’s area is shaded The four angles formed at the

circle’s center (O) total 360° You know that two of these angles account for 120°, or1

3of those 360° ∠AOC is supplementary to ∠DOC ; that is, the two angles combine to form a straight

line, and so their measures total 180° Therefore, ∠AOC measures 120°.

120° is1

3of 360° Thus, the shaded portion accounts for

1

3the circle’s area, or 3p.

If you look at the 60° angle in the figure, you might recognize right away that both triangles

are equilateral 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 With this intuition, the problem

is reduced to the simple mechanics of calculating the circle’s area, then dividing it by 3

SKETCH A GEOMETRY FIGURE TO SOLVE A PROBLEM

A geometry problem that does not provide a diagram might cry out for one That’s your cue to

take pencil to scratch paper and draw one yourself

11 A rancher uses 64 feet of fencing to create a rectangular horse corral If the ratio of

the corral’s length to width is 3:1, which of the following most closely approximates

the minimum length of additional fencing needed to divide the rectangular corral

into three triangular corrals, one of which is exactly twice the area of the other two?

(A) 24 feet

(B) 29 feet

(C) 36 feet

(D) 41 feet

(E) 48 feet

The correct answer is (B) Your first step is to determine the dimensions of the rectangular

corral Given a 3:1 length-to-width ratio, you can solve for the width (w) of the field using the

perimeter formula:

2~3w! 1 2~w! 5 64

8w 5 64

w 58

Accordingly, the length of the rectangular corral is 24 feet Next, determine how the rancher must configure the additional fencing to meet the stated criteria This calls for a bit of sketching to help you visualize the dimensions Only two possible configurations create three triangular corrals with the desired ratios:

The top figure requires less fencing You can determine this fact by calculating each length (using the Pythagorean theorem) Or you can use logic and visualization Here’s how As a rectangle becomes flatter (“less square”), the shorter length approaches zero (0), at which point the minimum amount of fencing needed in the top configuration would decrease, approaching the length of the longer side However, in the bottom design, the amount of fencing needed would increase, approaching twice the length of the longer side

Your final step is to calculate the amount of fencing required by the top design, applying the

theorem (let x 5 either length of cross-fencing):

821 1225 x2

64 1 144 5 x2

208 5 x2

x 5=208 ' 14.4 Thus, a minimum of approximately 28.8 feet of fencing is needed Answer choice (B) approximates this solution

Since the question asks for an approximation, it’s a safe bet that estimating =208 to the nearest integer will suffice If you learned your “times table,” you know that 14 3 14 5 196, and 15 3 15 5 225 So=208 must be between 14 and 15 That’s close enough to zero in on choice (B), which provides twice that estimate