Tài liệu Master the Gre 2010 - Part 28 docx

Problems of Rate of Production and Rate of Travel A rate is a fraction that expresses a quantity per unit of time.. For example, the rate at which a machine produces a certain product is

34 Dr Kramer plans to invest $20,000 in an account paying 6% interest

annually How much more must she invest at the same time at 3% so that

her total annual income during the first year is 4% of her entire

investment?

(A) $32,000

(B) $36,000

(C) $40,000

(D) $47,000

(E) $49,000

The correct answer is (C) Letting x equal the amount invested at 3%, you can

express Dr Kramer’s total investment as 20,000 1 x The interest on $20,000

plus the interest on the additional investment equals the total interest from both

investments You can state this algebraically as follows:

0.06(20,000) 1 0.03x 5 0.04(20,000 1 x)

Multiply all terms by 100 to eliminate decimals, then solve for x:

6~20,000! 1 3x 5 4~20,000 1 x!

120,000 1 3x 5 80,000 1 4x

40,000 5 x

She must invest $40,000 at 3% for her total annual income to be 4% of her

total investment ($60,000)

Beware: In solving GRE investment problems, by all means size up the question to

make sure your calculated answer is in the ballpark But don’t rely on your intuition

to derive a precise solution Interest problems can be misleading For instance, in

attempting to solve the previous problem you might have guessed that Dr Kramer

would need to invest more than twice as much at 3% than at 6% to lower the overall

interest rate to 4%, which is not true

Problems of Rate of Production and Rate of Travel

A rate is a fraction that expresses a quantity per unit of time For example, the rate at

which a machine produces a certain product is expressed this way:

rate of production 5number of units produced

time

RATE OF PRODUCTION (WORK)

A simple GRE rate question might provide two of the three terms and then ask you for

the value of the third term To complicate matters, the question might also require you

to convert a number from one unit of measurement to another

35 If a printer can print pages at a rate of 15 pages per minute, how many

pages can it print in 21

2hours?

pages Enter a number in the box

The correct answer is (2250) Apply the following formula:

rate 5no of pages

time

The rate is given as 15 minutes, so convert the time (21

2 hours) to 150 minutes Determine the number of pages by applying the formula to these numbers:

15 5 no of pages

150

~15!~150! 5 no of pages 2,250 5 no of pages

A more challenging type of rate-of-production (work) problem involves two or more workers (people or machines) working together to accomplish a task or job In these scenarios, there’s an inverse relationship between the number of workers and the time that it takes to complete the job; in other words, the more workers, the quicker the job gets done

A GRE work problem might specify the rates at which certain workers work alone and ask you to determine the rate at which they work together, or vice versa Here’s the basic formula for solving a work problem:

A

x 1

A

y 5 1

In this formula:

• x and y represent the time needed for each of two workers, x and y, to complete

the job alone

• The letter A represents the time it takes for both x and y to complete the job working in the aggregate (together).

So each fraction represents the portion of the job completed by a worker The sum of the two fractions must be 1 if the job is completed

NOTE

In the real world, a team may

be more efficient than the

individuals working alone But

for GRE questions, assume that

no additional efficiency is

gained this way.

36 One printing press can print a daily newspaper in 12 hours, while another

press can print it in 18 hours How long will the job take if both presses

work simultaneously?

(A) 7 hours, 12 minutes

(B) 9 hours, 30 minutes

(C) 10 hours, 45 minutes

(D) 15 hours

(E) 30 hours

The correct answer is (A) Just plug the two numbers 12 and 18 into our work

formula, then solve for A:

A

121

A

185 1

3A

361

2A

365 1

5A

365 1

5A 5 36

A 536

5 or 7

1 5

Both presses working simultaneously can do the job in 71

5hours—or 7 hours, 12 minutes

RATE OF TRAVEL (SPEED)

GRE rate problems often involve rate of travel (speed) You can express a rate of travel

this way:

rate of travel 5distance

time

An easier speed problem will involve a single distance, rate, and time A tougher speed

problem might involve different rates:

• Two different times over the same distance

• Two different distances covered in the same time

In either type, apply the basic rate-of-travel formula to each of the two events Then

solve for the missing information by algebraic substitution Use essentially the same

approach for any of the following scenarios:

• One object making two separate “legs” of a trip—either in the same direction or as

a round trip

• Two objects moving in the same direction

TIP

In problems involving rate of work, use common sense to narrow down answer choices.

If one worker’s rate is faster than another’s, the combined rate is twice as fast as a rate

somewhere between the two

individual rates.

37 Janice left her home at 11 a.m., traveling along Route 1 at 30 mph At

1 p.m., her brother Richard left home and started after her on the same road at 45 mph At what time did Richard catch up to Janice?

(A) 2:45 p.m.

(B) 3:00 p.m.

(C) 3:30 p.m.

(D) 4:15 p.m.

(E) 5:00 p.m.

The correct answer is (E) Notice that the distance Janice covered is equal to

that of Richard—that is, distance is constant Letting x equal Janice’s time, you can express Richard’s time as x 2 2 Substitute these values for time and the

values for rate given in the problem into the speed formula for Richard and Janice:

Formula: rate 3 time 5 distance

Janice: (30)(x) 5 30x Richard: (45)(x 2 2) 5 45x 2 90

Because the distance is constant, you can equate Janice’s distance to

Rich-ard’s, then solve for x:

30x 5 45x 2 90 15x 5 90

x 5 6

Janice had traveled 6 hours when Richard caught up with her Because Janice left at 11:00 a.m., Richard caught up with her at 5:00 p.m

38 How far in kilometers can Scott drive into the country if he drives out at

40 kilometers per hour (kph), returns over the same road at 30 kph, and spends 8 hours away from home, including a 1-hour stop for lunch?

(A) 105 (B) 120 (C) 145 (D) 180 (E) 210 The correct answer is (B) Scott’s actual driving time is 7 hours, which you

must divide into two parts: his time spent driving into the country and his time

spent returning Letting the first part equal x, the return time is what remains of the 7 hours, or 7 2 x Substitute these expressions into the motion formula for

each of the two parts of Scott’s journey:

Formula: rate 3 time 5 distance

Going: (40)(x) 5 40x Returning: (30)(7 2 x) 5 210 2 30x

TIP

Regardless of the type of

speed problem, start by setting

up two distinct equations

patterned after the simple

rate-of-travel formula

(r 3 t 5 d ).

Because the journey is round trip, the distance going equals the distance

returning Simply equate the two algebraic expressions, then solve for x:

40x 5 210 2 30x

70x 5 210

x 5 3

Scott traveled 40 kph for 3 hours, so he traveled 120 kilometers

Problems Involving Overlapping Sets

Overlapping set problems involve distinct sets that share some number of members

GRE overlapping set problems come in one of two varieties: single overlap and double

overlap As you might guess, the latter variety is generally more difficult

39 Each of the 24 people auditioning for a community-theater production is

an actor, a musician, or both If 10 of the people auditioning are actors and

19 of the people auditioning are musicians, how many of the people

auditioning are musicians but not actors?

(A) 10

(B) 14

(C) 19

(D) 21

(E) 24

The correct answer is (B) You can approach this relatively simple problem

without formal algebra: The number of actors plus the number of musicians

equals 29 (10 1 19 5 29) However, only 24 people are auditioning Thus, 5 of the

24 are actor-musicians, so 14 of the 19 musicians must not be actors

You can also solve this problem algebraically The question describes three

mutually exclusive sets: (1) actors who are not musicians, (2) musicians who are

not actors, and (3) actors who are also musicians The total number of people

among these three sets is 24 You can represent this scenario with the following

algebraic equation (n 5 number of actors/musicians), solving for 19 2 n to answer

the question:

~10 2 n! 1 n 1 ~19 2 n! 5 24

29 2 n 5 24

n 5 5

19 2 5 5 14

40 Adrian owns 60 neckties, each of which is either 100% silk or 100%

polyester Forty percent of each type of tie are striped, and 25 of the ties are silk How many of the ties are polyester but not striped?

(A) 18 (B) 21 (C) 24 (D) 35 (E) 40 The correct answer is (B) This double-overlap problem involves four distinct

sets: striped silk ties, striped polyester ties, non-striped silk ties, and non-striped polyester ties Set up a table representing the four sets, filling in the information given in the problem, as shown in the next figure:

striped non-striped

silk polyester

40%

60%

?

Given that 25 ties are silk (see the left column), 35 ties must be polyester (see the right column) Also, given that 40% of the ties are striped (see the top row), 60% must be non-striped (see the bottom row) Thus, 60% of 35 ties, or 21 ties, are polyester and non-striped

TIP

Use a Venn diagram

(overlapping circles) or table

to organize the information in

an overlapping-set problem

and to help you visualize the

solution.

SUMMING IT UP

• Make sure you’re up to speed on the definitions of absolute value, integers,

factors, and prime numbers to better prepare yourself for the number theory and

algebra questions on the GRE Quantitative Reasoning section

• Use prime factorization to factor composite integers

• GRE questions involving exponents usually require that you combine two or more

terms that contain exponents, so review the basic rules for adding, subtracting,

multiplying, and dividing them

• On the GRE, always look for a way to simplify radicals by moving what’s under

the radical sign to the outside of the sign Also, make sure you know the rules for

combining radical terms using the four basic operations

• Most algebraic equations you’ll see on the GRE exam are linear Remember the

operations for isolating the unknown on one side of the equation Solving

alge-braic inequalities is similar to solving equations: Isolate the variable on one side

of the inequality symbol first

• Weighted-average problems and currency problems can be solved in a similar

manner by using the arithmetic mean formula

• Mixture and investment problems on the GRE can be solved using what you’ve

learned about proportion and percentage questions Rate of production and travel

questions can be solved using the strategies you’ve learned about fraction

problems

OVERVIEW

• Lines and angles

• Triangles

• Quadrilaterals

• Circles

• Polygons

• Cubes and other rectangular solids

• Cylinders

• Coordinate geometry—the xy-plane

• Summing it up

In this chapter, you’ll first review the following concepts—all of which involve

plane geometry:

• relationships among angles formed by intersecting lines

• properties of any triangle

• the Pythagorean theorem

• characteristics of special right triangles

• properties of isosceles and equilateral triangles

• properties of and differences among squares, rectangles, parallelograms,

and trapezoids

• properties of circles

• arcs and their relationships to other features of a circle

• relationships between circles and tangent lines

• relationships created by combining circles with other geometric figures

• properties of polygons (including those with more than four sides)

• properties of solids (cubes and other rectangular solids, and cylinders)

Later in the chapter, you’ll review the basics of coordinate geometry These

basics include the following:

• features and characteristics of the xy-plane

.

• applying the midpoint and distance formulas to problems involving lines and lines segments

• plotting and defining two-dimensional figures (e.g., triangles and rectangles) on

the xy-plane

As in Chapters 9 and 10, the review in this chapter focuses strictly on what you need

to know for the GRE and provides one or two GRE-style questions for each reviewed concept

LINES AND ANGLES

Lines and line segments are the basic building blocks for most GRE geometry problems A GRE geometry question might involve nothing more than intersecting lines and the angles they form To handle the question, just remember four basic rules about angles formed by intersecting lines:

Vertical angles (angles across the vertex from each other and formed by the same two lines) are equal in degree measure, or congruent (≅) In other words, they’re the same size

If adjacent angles combine to form a straight line, their degree measures total

180 In fact, a straight line is actually a 180° angle

If two intersecting lines are perpendicular (⊥) to each other, they form right (90°) angles

The sum of the measures of all angles where two or more lines intersect at the same point is 360° (regardless of how many angles are involved)

Note that the symbol (≅) indicates that two geometric features are congruent, meaning

that they are identical (the same size, length, degree measure, etc.) The equation

AB≅CD means that line segment AB is congruent to line segment CD The two equations∠A≅ ∠B and m∠A 5 m∠B are two different ways of symbolizing the same relationship: that the angle whose vertex is at point A is congruent (equal in degree

measure, or size) to the angle whose vertex is at point B (The letter m symbolizes

degree measure.)

Angles Formed by Intersecting Lines

When two or more lines intersect at the same point, they form a “wheel-spoke” pattern with a “hub.” On the GRE, “wheel-spoke” questions require you to apply one or more

of the preceding four rules