Master gmat 2010 part 29 pdf

INVESTMENT PROBLEMSGMAT investment problems involve interest earned at a certain percentage rate on money over a certain time period usually a year.. GMAT investment questions usually in

The correct answer is (D) In the weighted-average formula, 315 annual gallons receives a

weight of 4, while the average annual number of gallons for the next six years (x) receives a

weight of 6:

378 51260 1 6x

10

3780 5 1260 1 6x

3780 2 1260 5 6x

2520 5 6x

420 5 x

This solution (420) is the average number of gallons needed per year, on average, during the

next 6 years

To guard against calculation errors, check your answer by sizing up the question Generally,

how great a number are you looking for? Notice that the stated goal is a bit greater than the

annual average production over the first four years So you’re looking for an answer that is

greater than the goal—a number somewhat greater than 378 gallons per year You can

eliminate choice (A) and (B) out of hand The number 420 fits the bill

CURRENCY PROBLEMS

Currency problems are similar to weighted-average problems in that each item (bill or coin) is

weighted according to its monetary value Unlike weighted average problems, however, the

“average” value of all the bills or coins is not at issue In solving currency problems, remember

the following:

• You must formulate algebraic expressions involving both number of items (bills or

coins) and value of items.

• You should convert the value of all moneys to a common currency unit before

formulating an equation If converting to cents, for example, you must multiply the

number of nickels by 5, dimes by 10, and so forth

29 Jim has $2.05 in dimes and quarters If he has four fewer dimes than quarters, how

much money does he have in dimes?

(A) 20 cents

(B) 30 cents

(C) 40 cents

(D) 50 cents

(E) 60 cents

The correct answer is (B) Letting x equal the number of dimes, x 1 4 represents the

number of quarters The total value of the dimes (in cents) is 10x, and the total value of the

quarters (in cents) is 25(x 1 4) or 25x 1 100 Given that Jim has $2.05, the following

equation emerges:

10x 1 25x 1 100 5 205

35x 5 105

x 5 3

Jim has three dimes, so he has 30 cents in dimes

You could also solve this problem without formal algebra, by plugging in each answer choice

in turn Let’s try this strategy for choices (A) and (B):

A 20 cents is 2 dimes, so Jim has 6 quarters 20 cents plus $1.50 adds up to $1.70 Wrong answer!

B 30 cents is 3 dimes, so Jim has 7 quarters 30 cents plus $1.75 adds up to $2.05 Correct answer!

MIXTURE PROBLEMS

In GMAT mixture problems, you combine substances with different characteristics, resulting

in a particular mixture or proportion, usually expressed as percentages Substances are measured and mixed by either volume or weight—rather than by number (quantity)

30 How many quarts of pure alcohol must you add to 15 quarts of a solution that is

40% alcohol to strengthen it to a solution that is 50% alcohol?

(A) 4.0

(B) 3.5

(C) 3.25

(D) 3.0 (E) 2.5

The correct answer is (D) You can solve this problem by working backward from the

answer choices—trying out each one in turn Or, you can solve the problem algebraically The

original amount of alcohol is 40% of 15 Letting x equal the number of quarts of alcohol that you must add to achieve a 50% alcohol solution, 0.4(15) 1 x equals the amount of alcohol in the solution after adding more alcohol You can express this amount as 50% of (15 1 x) Thus,

you can express the mixture algebraically as follows:

~0.4!~15! 1 x 5 ~0.5!~15 1 x!

6 1 x 5 7.5 1 0.5x 0.5x 5 1.5

x 5 3

You must add 3 quarts of alcohol to obtain a 50% alcohol solution

TIP

You can solve

most GMAT

currency

problems by

working

backward from

the answer

choices.

INVESTMENT PROBLEMS

GMAT investment problems involve interest earned (at a certain percentage rate) on money

over a certain time period (usually a year) To calculate interest earned, multiply the original

amount of money by the interest rate:

amount of money 3 interest rate 5 amount of interest on money

For example, if you deposit $1000 in a savings account that earns 5% interest annually, the

total amount in the account after one year will be $1000 1 0.05($1000) 5 $1000 1 $50 5

$1050

GMAT investment questions usually involve more than simply calculating interest earned on

a given principal amount at a given rate They usually call for you to set up and solve an

algebraic equation When handling these problems, it’s best to eliminate percent signs

31 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 GMAT 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, 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 OR WORK

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

A simple GMAT 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

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

print in 21

2hours?

(A) 1375

(B) 1500

(C) 1750

(D) 2250 (E) 2500

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

rate 5no of pages

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

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

15 5no of pages

150

~15!~150! 5 no of pages

2250 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 GMAT 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

• 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

33 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.

PROBLEMS OF RATE OF TRAVEL (SPEED)

GMAT 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, such as:

• 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

• Two objects moving in opposite directions

NOTE

In the real world,

a team may be more efficient than the individuals working alone But for GMAT questions, assume that no additional efficiency is gained this way.

TIP

In work problems, use your common sense to narrow down answer choices.

34 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 Richard’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

35 How far in kilometers can Scott drive into the country if he drives out at 40

kilome-ters 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

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 GMAT

overlapping set problems come in one of two varieties:

Single overlap (easier)

Double overlap (tougher)

36 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

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).

37 Adrian owns 60 neckties, each of which is either 100% silk or 100% polyester Forty

percent of each type of tie is 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

SUMMING IT UP

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

and prime numbers to better prepare yourself for the number theory and algebra

questions on the GMAT Quantitative section

• Use prime factorization to factor composite integers

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

radical sign to the outside of the sign

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

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

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 GMAT can be solved using what you’ve learned

about solving proportion and percentage questions Rate of production and travel

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