GRE_Math_Bible_eBook

Word Problems 381

AGE PROBLEMS

Typically, in these problems, we start by letting x be a person's current age and then the person's age a years ago will be x – a and the person's age a years in future will be x + a. An example will illustrate.

Example: John is 20 years older than Steve. In 10 years, Steve's age will be half that of John's. What is Steve's age?

(A)2

(B)8

(C)10

(D)20

(E)25

Steve's age is the most unknown quantity. So we letx = Steve's age and then x + 20 is John's age. Ten years from now, Steve and John's ages will be x + 10 and x + 30, respectively. Summarizing this information in a table yields

Since "in 10 years, Steve's age will be half that of John's," we get 12 (x + 30) = x +10

x + 30 = 2(x + 10) x + 30 = 2x + 20 x = 10

Hence, Steve is 10 years old, and the answer is (C).

INTEREST PROBLEMS

These problems are based on the formula

INTEREST = AMOUNT TIME RATE

Often, the key to these problems is that the interest earned from one account plus the interest earned from another account equals the total interest earned:

Total Interest = (Interest from first account) + (Interest from second account) An example will illustrate.

Example: A total of $1200 is deposited in two savings accounts for one year, part at 5% and the remainder at 7%. If $72 was earned in interest, how much was deposited at 5%?

(A)410

(B)520

(C)600

(D)650

(E)760

Let x be the amount deposited at 5%. Then 1200 –x is the amount deposited at 7%. The interest on these investments is .05x and .07(1200 – x). Since the total interest is $72, we get

.05x + .07(1200 – x) = 72

.05x + 84 – .07x = 72 –.02x + 84 = 72 –.02x = –12

x = 600

The answer is (C).

Word Problems 383

Hard

7.Train X leaves New York at 10:00AM and travels East at a constant speed of x miles per hour. If another Train Y leaves New York at 11:30AM and travels East along the same tracks at speed 4x/3, then at what time will Train Y catch Train X?

(A)2 PM of the same day

(B)3 PM of the same day

(C)3:30 PM of the same day

(D)4 PM of the same day

(E)8 PM of the same day

8.An old man distributed all the gold coins he had to his two sons into two different numbers such that the difference between the squares of the two numbers is 36 times the difference between the two numbers. How many coins did the old man have?

(A)24

(B)26

(C)30

(D)36

(E)40

10.A man walks at a rate of 10 mph. After every ten miles, he rests for 6 minutes. How much time does he take to walk 50 miles?

(A)300

(B)318

(C)322

(D)324

(E)330

11.A project has three test cases. Three teams are formed to study the three different test cases. James is assigned to all three teams. Except for James, each researcher is assigned to exactly one team. If each team has exactly 6 members, then what is the exact number of researchers required?

(A)10

(B)12

(C)14

(D)15

(E)16

12.The combined salaries of three brothers is $90,000. Mr. Big earns twice what Mr. Small earns, and Mr. Middle earns 1 1/2 times what Mr. Small earns. What is the smallest salary of the three brothers?

(A)20,000

(B)22,000

(C)25,000

(D)30,000

(E)40,000

384GRE Math Bible

The next two questions refer to the discussion below:

Mike and Fritz ran a 30-mile Marathon. Mike ran 10 miles at 10 miles per hour and then ran at 5 miles per hour for the remaining 20 miles. Fritz ran the first one-third (by time) of the run at 10 miles per hour and the remaining two-thirds of the run at 5 miles per hour.

13.How much time in hours did Mike take to complete the Marathon?

(A)3

(B)3.5

(C)4

(D)4.5

(E)5

14.How much time in hours did Fritz take to complete the Marathon?

(A)3

(B)3.5

(C)4

(D)4.5

(E)5

15.A ship is sinking and 120 more tons of water would suffice to sink it. Water seeps in at a constant rate of 2 tons a minute while pumps remove it at a rate of 1.75 tons a minute. How much time in minutes has the ship to reach the shore before is sinks?

(A)480

(B)560

(C)620

(D)680

(E)720

16.When the price of oranges is lowered by 40%, 4 more oranges can be purchased for $12 than can be purchased for the original price. How many oranges can be purchased for 24 dollars at the original price?

(A)8

(B)12

(C)16

(D)20

(E)24

17.John has $42. He purchased fifty mangoes and thirty oranges with the whole amount. He then chose to return six mangoes for nine oranges as both quantities are equally priced. What is the price of each Mango?

(A)0.4

(B)0.45

(C)0.5

(D)0.55

(E)0.6

18.In a market, a dozen eggs cost as much as a pound of rice, and a half-liter of kerosene costs as much as 8 eggs. If the cost of each pound of rice is $0.33, then how many cents does a liter of kerosene cost? [One dollar has 100 cents.]

(A)0.33

(B)0.44

(C)0.55

(D)44

(E)55

386GRE Math Bible

Very Hard

25.The costs of equities of type A and type B (in dollars) are two different positive integers. If 4 equities of type A and 5 equities of type B together costs 27 dollars, what is the total cost of 2 equities of type A and 3 equities of type B in dollars?

(A)15

(B)24

(C)35

(D)42

(E)55

26.How many coins of 0.5 dollars each and 0.7 dollars each together make exactly 4.6 dollars?

(A)1, 6

(B)2, 7

(C)3, 5

(D)4, 3

(E)5, 3

(A)20

(B)40

(C)60

(D)80

(E)100

29.Katrina has a wheat business. She purchases wheat from a local whol esaler at a particular cost per pound. The price of the wheat at her stores is $3 per pound. Her faulty spring balance reads 0.9 pounds for a pound. Also, in the festival season, she gives a 10% discount on the wheat. She found that she made neither a profit nor a loss in the festival season. At what price did Katrina purchase the wheat from the wholesaler?

(A)2.43

(B)2.5

(C)2.7

(D)3

(E)3.3

30.According to the stock policy of a company, each employee in the technical division is given 15 shares of the company and each employee in the recruitment division is given 10 shares. Employees belonging to both communities get 25 shares each. There are 20 employees in the company, and each one belongs to at least one division. The cost of each share is $10. If the technical division has 15 employees and the recruitment division has 10 employees, then what is the total cost of the shares given by the company?

(A)2,250

(B)2,650

(C)3,120

(D)3,180

(E)3,250

388GRE Math Bible

Answers and Solutions to Problem Set WEasy

1.Remember that Average Speed = Net Distance ÷ Time Taken. We are given that the time taken for the full trip is 30 minutes. Hence, we only need the distance traveled. We are given that the restaurant is 2 miles from home. Since Waugh jogs back along the same route, the net distance he traveled equals 2 + 2 = 4 miles. Hence, the Average Speed equals 4 miles ÷ 30 minutes = 4 miles ÷ 1/2 hour = 8 miles per hour. The answer is (E).

2.Since the answer is in minutes, we must convert the cyclist's speed (12 miles per hour) into miles per minute. Since there are 60 minutes in an hour, his speed is 12/60 = 1/5 miles per minute.

Remember that Distance = Rate Time. Hence,

24 = 15 t

Solving for t yields t = 5 24 = 120. The answer is (E). [If you forgot to convert hours to minutes, you may have mistakenly answered (B).]

3. Column B has 1,000 pounds of coal, and there are 16 ounces in 1 pound. So, Column B has 1,000 pounds = 1,000(16 ounces) = 16,000 ounces. Hence, each column weighs 16,000 ounces. The answer is

(C).

Medium

4. Column A:

First, place point A arbitrarily. Then locate point P 6 miles North of point A, and then locate a new point 8 miles East of P. Name the new point M. Now, Column A equals AM. The map drawn should look like this:

A S

Since the angle between the standard directions East and North is 90°, the three points A, P and M form a right triangle, with right angle at P. So, AM is a hypotenuse. By The Pythagorean Theorem, the hypotenuse equals the square root of the sum of the squares of the other two sides. Hence,

AM = AP2 + PM 2

=62 + 82

=36 + 64

=100

=10

Column B:

Similarly, place point B arbitrarily. Then locate point Q 8 miles South of it, and locate a new point 6 miles West of the point Q. Name the new point N. Now, Column B equals BN. The map should look like this:

Word Problems 389

Again, since the angle between standard directions is 90°, we have a right triangle BQN, with right angle at Q, and, by The Pythagorean Theorem mentioned above, the hypotenuse BN equals

BQ2 + QN 2

=82 + 62

=64 + 36

=100

=10

Since both columns equal 10, the answer is (C).

5.There are 16 ounces in a pound. Hence, each ounce equals 1/16 pounds. Now, 12 ounces equals

121/16 = 3/4 pounds. Hence, 5 pounds + 12 ounces equals 5 3/4 pounds. The answer is (C).

6.One ton has 2000 pounds. The capacity of the gunny bag is 13 tons. Hence, its capacity in pounds would equal 13 2000 pounds.

One pound has 16 ounces. We are given the capacity of each packet is 16 pounds and 4 ounces. Converting it into pounds yields 16 pounds + 4/16 ounces = 16 1/4 pounds = (16 4 + 1)/4 = 65/4 pounds.

Hence, the number of packets required to fill the gunny bag equals

(Capacity of the gunny bag) ÷ (Capacity of the each packet) = 13 2000 pounds ÷ (65/4) pounds =

132000 4/65 = 2000 4/5 =

1600

The answer is (A).

Hard

7. Train X started at 10:00AM. Let the time it has been traveling be t. Since Train Y started at 11:30AM, it has been traveling an hour and a half less. So, represent its time as t – 1 1/2 = t – 3/2.

Train X travels at speed x miles per hour, and Train Y travels at speed 4x/3 miles per hour. By the formula Distance = Speed Time, the respective distances they travel before meeting equals xt and (4x/3)(t – 3/2). Since the trains started from the same point and traveled in the same direction, they will have traveled the same distance when they meet. Hence, we have

Hence, Train Y will catch Train X at 4PM (10AM plus 6 hours is 4PM). The answer is (D).