Quantitative - Word Problems

Question 1

The average score on a test taken by 10 students was x. If the average score for 5 of the students was 8, what was the average score in terms of x for the remaining students?

Accepted Answer

Is a list of n numbers has the sum S and average A, then S/n=A and S=n*A Therefore, becuase the average of hte 10 scores is x, the sum of the 10 scores is 10x. The sum of the 10 scores is also equal to S1*S2 S1=5*8 = 40 S2 = 10x - 40 The average of the remaining five scores is S2/5 = 10x-40/5 = 2x-8

Question 2

An experimental mathematics program was tried out of 2 classes in each of 32 elementary schools and involved 37 teachers. Each of the classes had 1 teacher and each of the teachers taught at least 1 but not more than 3 of the classes. If the number if teachers, who taught 3 classes is n, then the least and greatest possible values of n are

Accepted Answer

2*32 = 64 classes are taught by 37 teachers. x+ y+ z = 37 x+2y+3z = 64 x + 2z = 27 OR 2z = 27-m => 2z < 27 z < 13 SUBSTRACT BOTH TRY OUT ANSWER CHOICES

Question 3

The ratio of students to teacher sat a certain school is 30:1. If the student enrollment increases by 50 students and the number of teachers increased by 5, the ratio of students to teachers would be 25 to 1. What is the present number of teachers?

Accepted Answer

x= number of students present y = number of teachers present current ration : 30/1 = x/y future ratio: 25/1 = x+50 / y+5 SOLVE first equation for x and plug into second equation, solve for y

Question 4

Each point in the mileage table represents and entry indicating the distance between two cities. If the table were extended to represent the distances between all paris of 30 cities and each distance were to be represented by only one entry, how many entries would the table have then?

Accepted Answer

1. In a table with 30 cities, there are 30*30 = 900 boxes for entries. However, since a city does not neet to have any entry for a distance to and from itself, 30 entries are not needed on the diagonal through the table. 2. Thus, the necessary number of entries is reduced to 900-30 = 870 entries. 3. Each pair of city only needs one table entry, therefore the table needs to have only 870/2= 435 entries.

Question 5

If 3

Accepted Answer

possible values for x/3 are squares of 2,3,5,7 1 is not a prime number! possible values of x are then x = 3*Prime2 -> 12, 27, 75 => There are only 3 values

Question 6

Don estimated the labor cost of a job as 336$ and he was paid that amount. The job took 4 hours longer than he had estimates and consequently, he earned 2 $ less than his regular hourly rate. What was the time D had estimated for the job?

Accepted Answer

r = average hourly rate t = time r*t = 336 = estimated labor BUT (r-2)(r+4)= 336 EQUAL BOTH EQUATIONS (r-2)(r+4)= r*t SOLVE

Question 7

WORKING TOGETHER

Accepted Answer

1. ADD both hourly rates on the same denominator 2. TURN the fraction!

Question 8

In the first week of the year, Nancy saved 1$. IN each of the 51 weeks, she saves 1$ more than she had saved in the previous week. What was the total amount that Nancy saved during the 52 weeks?

Accepted Answer

SOLVE BY GROUPING! In dollars, the total amount saved is the sum of 1, (1+1), (1+1+1), and so on, up to and including the amount saved in the 52nd weeks, which was 52. The total amount saved is the Sum: 1+2+3+4+...50+51+52. GROUPING! => (1+52) + (2+51) + (3+50) + (4+39) + 5(38) + 6 ( 37) + 7 ( 36) .... + (26+27) This results in the number of 53 * 26 = 1378

Question 9

During her trip, F traveled x percent of the total distance at an average speed of 40 miles per hour and the rest of the distance at a speed of 60 miles per hour. In terms of x, what was F's speed for the entire trip?

Accepted Answer

1. ) PICK A SMART NUMBER : e.g. 100 Distance Rate Time = RATE / DISTANCE x 40 x/40 100-x 60 100-x/60 2. ) Total time for the trip: x/40 + 100-x/60 = x+200/120 3. ) The average speed d/t => 100*120 /x+200 = 12000/x+200

Question 10

A border of uniform width is placed around a rectangular photograph that measures 8 inches by 10 inches. If the area of the border is 144 square inches, what is the width of the border in inches?

Accepted Answer

Let x be the width, in inches of the border. The photograph with the border has dimensions (10+2x) inches and (8+2x) inches with an area of (10+2x)(8+2x) = (80+36x+4x2)square inches. SOLVE 4x2+36x-14=0 (x-3)(x+12) = 0 so x must be 3!

Question 11

Of 200 students majoring in one or more of the sciences, 130 are majoring in chemistry and 150 are majoring in biology. If at least 30 of the students are not majoring in either chemistry or biology, then the number of students majoring in both chemistry and biology could be any number from

Accepted Answer

CREATE A VENN DIAGRAM If a least 30 students are not majoring in either chemisty or biology, then at most 200-30=170 students can be majoring in either or both. Since there are 130+150 = 280 biology AND chemisty majors (one and both) Then at least 280-170 = 110 major in both. The max number of students who will mahor in chemisty and biology is 130, since 130 is the number The max number of students who will mahor in chemisty and biology is 130, since 130 is the number

Question 12

A three digit code for a certain lock uses the digits 0-9 according to the following constraints: the first digit cannot be 0 or 1, the second digit must be 0 or 1 and the second and third digits cannot both be 0 in the same code. How many codes are possible?

Accepted Answer

1st digit = 3,4,5,6,7,8,9 = 8 possibilities 2nd digit = 0,1 = 2 possibilities 3rd digit = no restrictions = 10 digits 8*2*10 possibilities = 160 possibilities - special restriction that 8 codes are further eliminated (diagonal) 2-00, 3-00, 4-00, 5-00, 6-00, 7-00, 8-00, 9-00 160 - 8 = 152 possibilities

Question 13

A store reported total sales of 385 million for February of this year. It the total sales for the same month last year war 320 million, apron. what was the percent increase in sales?

Accepted Answer

APPROXIMATION 385-320 = 65/320 = 13/64 = approx. 13/65 = approx. 1/5 = 20%

Question 14

A certain experiment was tried out in 2 classes in each of 32 elementary schools and involved 37 teachers. Each of the classes had 1 teacher and each of the teachers taught at least 1, but no more than 3 of the classes. If the number of teacher who taught 3 classes is n, then the least and greatest possible values of n are 1) 0 and 13 2) 0 and 14 3) 1 and 10 4) 1 and 9 5) 2 and 8

Accepted Answer

2*32 = 64 classes are taught by 37 teachers. SET UP THE TWO EQUATIONS k= teachers who only taught 1 class m= teachers who taught 2 classes n= teachers who taught 3 classes k+2m+3n = 64 k+m+n = 37 m+2n = 27 CHECK WHICH ONE SATISFIES THE EQUATION! n= 14 => NO n= 13, k= 23 and m=1 to satisfy the condition !

Question 15

The present ratio of students/teachers of a school if 30:1. If the student enrollment increases by 50 students and the number of teacher increases by 5, the ratio of students to teachers would then be 35:1. What is the present number of teachers?

Accepted Answer

30/1 = s/t =>> s= 30t s+50/t+5 = 25/1 SOLVING FOR THE FIRST EQUATION FOR s gives s=30t. Substitute this value of s into the second equations and solve for t 30t + 50 / t+ = 25/1 30t + 50 = 25 t + 125 t=15

Question 16

Every year fr 4 years, a farmer increased the number of trees in a certain orchard by 1/4 of the number of trees in the orchard the preceding year. If all of the number of trees in the trees thrives and there were 6250 trees in the orchard at the end of the 4 year period, how many trees were in the orchard at the beginning?

Accepted Answer

At the end of the fourth year there will be 1,254 • n = 6250 n= 6250 / 1,254 1,25 = 5/4 = 54/44 625 * 10 = 54*10 = 2560

Question 17

During a trip, Francine traveled x percent of the total distane at an average speed of 40 miles per hour and the rest of the distance at an average speed of 60 miles per hour. In term of x, what was Francince’s average speed for the entire trip?

Accepted Answer

PICK A NUMBER: Assume for simplicity that the total distance is 100 miles. distance rate time x 40 x/40 100-x 60 100-x/60 1) SEARCH FOR THE TOTAL TIME: The total time for the trip is x/40 + 100-x/60 = 3x/120 + 200-2x/120 = 3x-2x+200 / 120 = x+200/120 2) CALCULATE THE AVERAGE SPEED: The average speed for the entire trip is = total distance / total time = 100/x+200/120 = 12000/x+200

Question 18

Working simultaneously at their respective constant rates, machine A and B produce 800 nails in x hours. Working alone at its constant rate, Machine A produces 800 nails in y hours. In terms of x and y, how many hours does it take Machine B, working alone at its constant rate, to product 800 nails?

Accepted Answer

RA and RB are the constant rates in nails per hours at which Machine A and Machine B work, respectively. Then it follows from the given information, that RA+RB= 800/x and RA = 800/y Hence, 800/y + RB = 800/x or RB = 800/x-800/y = 800 (1/x - 1/y) = 800 (y-x / xy) Therefore, the time it would take for machine B to product 800 nails is fiven by 800/ 800 (y-x / xy) = xy/y-x

Question 19

In order to complete a reading assignment on time, Terry planned to read 90 pages per day. However, she read only 75 pages per day at first, leaving 690 pages to be read during the last 6 days before the assignment was to be completed. How many days in all did Terry have to complete the assignment on time?

Accepted Answer

LOOK FOR TWO EQUATIONS r days pages SLOW 75 n 75 * n FAST 6 690 TOTAL 90 n+6 75*n + 690 Then: 7n+690/ n+6 = 90 n = 10

Question 20

Bonny grows only pink parrots and green parrots. He grows three times as many green parrots as pink parrots. He feeds his parrots either soy beans or wheat, but no single parrot feeds on both kinds of food. Half of the parrots that feed on soy beans are green, and among the pink parrots, those that feed on soy beans are twice as many as those that feed on wheat. If 40 of Bonny's parrots feed on wheat, how many of the wheat feeding parrots are green?

Accepted Answer

Set the "Pink/Green" - "Soy/Wheat" table JEDEN EINZELNEN SATZ AUSEINANDERPFLÜCKEN ...among the pink parrots, those that feed on soy beans are twice as many as those that feed on wheat. Since this sentence involves the relation between subgroups and not totals, this would be a good place to start. If the pink parrots that feed on soy are twice as many as those that feed on wheat, the pink / wheat group is the smallest group in the question. Plugging in the variable into the smallest group is a great way to simplify the algebra, since it means you won't need to use fractions. Half of the parrots that feed on soy beans are green... If half of the parrots that feed on soy are green, the other half are pink. Since the two halves of the same whole are always equal, and the Pink / Soy box already contains 2x, put 2x in the Green / Soy box. If 40 of Bonny's parrots feed on wheat... Put 40 in the Wheat/Total box. If 40 of Bonny's parrots feed on wheat how many of them are green? - Place the ? in the Green / Wheat box. Soy Wheat Total Green 2x 7x 9x Pink 2x x 3x Total 4x 40 40+4x He grows three times as many green parrots as pink parrots. Since the total number of Pink parrots is 3x, and three times 3x is 9x, place 9x in the Total / Green box. Since the Green / Soy box and the Green / Wheat box add up to 9x, Green / Wheat = 9x-2x=7x. 7x+x = 40 8x=40 => x=35

Question 21

Two-Part Analysis: Work crews Alpha and Zeta are repaving a section of freeway in Los Angeles. Work crew Alpha started its work one week (40 working hours) earlier than work crew Zeta, and started on the north end of the freeway, working its way south at a rate of 12 meters per hour since starting the job. Now, work crew Zeta has started at the south end, working its way north at a rate of 10 meters per hour. The section of freeway that needs to be repaved is 1.5 kilometers long, including the section that has already been paved. Given that each crew will not necessarily work the same number of hours, which of the following answer choices represents the number of hours for each crew that will finish the project? Please make only one selection in each column.

Accepted Answer

To solve this problem, you must create a model that generates the potential solutions. speed hours distance Alpha 12 A+40 d Zeta 10 Z 1500-d Combine: 12(A+40) + 10Z = 1500 12 A + 10 Z = 1020 PLUG IN NUMBERS! 12A + 10Z = Distance To Be Paved 12(60) + 10Z = 1020 10Z = 300 Z = 30

Question 22

Hidden Plugging In for Fractions

Accepted Answer

Multiply the bottoms to find a good number to plug in! Calculate real numbers and plug into answer choices in the end!

Question 23

PI Percents: In Goshawk-Eurasian nature reserve 30 percent of the birds are Hawks, and 40 percent of the non-Hawks are Paddyfield-Warblers. If there are 25 percent as many Kingfishers as Paddyfield-Warblers in the reserve, then what percent of the birds in the nature reserve are not Hawks, Paddyfield-Warblers, or Kingfishers?

Accepted Answer

PERCENT PLUG IN: Plug in 100 as the number of birds in the nature reserve. Now work the problem: There are 30 Hawks, and 70 non-Hawks. Of the latter, 40 percent are Paddyfield Warblers (40% of 70 is 28). Now comes the interesting part:"...there are 25 percent as many Kingfishers as Paddyfield-Warblers in the reserve". If you keep reading this sentence and the only thing that comes to mind is "huh?", try substituting the convoluted words with simpler ones: " ...there are twice as many test-takers as test-writers in the GMAT..." = Test-Writers * 2= Test-Takers .....*2 "AS MANY = "Test-Takers In the same manner, [Paddyfield Warblers]·¼ =Kingfishers.

Question 24

c

Accepted Answer

A set is a group of objects. In the question we just saw there were 3 sets: Customers who dyed their hair blond, customers who dyed their eyebrows blond, and customers who straightened their hair. Definitions: Set = Group of objects. Subset = If all members of set B are also members of set A then B is a subset of A. Sample space = All the relevant objects from which it is possible to select a set. Complementary set - Is the set of all items in the sample space that are not included in set A. Mathematical phrasing: Not A. Everyday language phrasing: Everything but A. In special cases set B can be complementary to set A if everyday knowledge denotes that B is the only Not A that exists - boys/girls, smokers/non-smokers etc. Notice that opposites doesn't necessarily mean complementary: tall is not complementary to short, happy is not complementary to sad, etc. Intersection A ∩ B - All items that belong to set A and to set B, The intersection is included in both sets. Thus Set A of 12 fudge cake eaters is actually: 12 = 7 + 5[A] = [Only fudge] + [Both] And set B of funnel eaters is: 20 = 15 + 5[B] = [Only funnel] + [Both] Union [A,B] A U B - All items that belong to A or to B - The union is therefore comprised of: only fudge + both + only funnel

Question 25

A / Not A Table !

Accepted Answer

A/Not A Table: 1. Draw an empty 3×3 table.2. Place one set (Set A) as the the title of the first row.3. Title the second row as the complementary set (Not A) and the third row as the Total.4. Repeat the process in the columns for the other set in the question.5. Insert the data in the stem question to the proper cells.6. Mark the cell that denotes the requested value with a question mark.7. Calculate the value of the empty cells based on the rule that every row/column in the table fulfills the summation X+Y=Z.

Question 26

Sal's computer store sold 108 computers last week. The computers sold are either desktop computers or laptop computers, and each must come with one of two possible operating systems: the latest operating system "Quinta", or the previous version "ST". If 32 laptops were sold last week, and 52 of the computers were ordered with the "Quinta" operating system, while 36 desktop computers were sold with "ST", then how many laptops were sold with "Quinta"?

Accepted Answer

That's how the sentence "each must come with one of two possible operating systems: the latest operating system "Quinta", or the previous version 'ST'" Ensures that 3 essential terms are met: 1. There are only 2 sets.2. The 2 sets cover all of the sample space.3. There is no intersection between the sets. Table is then: Desktop Laptop Total Quinta ST Total

Question 27

A herd of goats consists of A males and B females. In that herd, half of the goats that have horns are females. If C goats of the herd do not have horns, then the total number of male goats in the herd that do NOT have horns, in terms of A, B and C is -

Accepted Answer

Note the word "half" in the question. It calls for plugging in even numbers. Such good numbers may be - A=16, B=14, C=10 Male Female TotalHorns x/2=10 x/2=10x=30-10=20no-Horns ?=6 10Total 161416+14=30

Question 28

c

Accepted Answer

c

Question 29

Ronnie and Bonnie are opposing participants in an online auction. Each has a constant bidding strategy: Ronnie always bids up by raising $40 over any previous bid. Bonnie always bids up by doubling any previous bid. The auction begins with an initial bid by a third player, and progresses as follows: Ronnie bids up first, and Bonnie follows. Ronnie bids up again, and Bonnie follows and raises the bid to a final $600. What was the dollar value of the initial price?

Accepted Answer

Backwards PLUG IN: 90 Assume the initial bid is $90. Ronnie raises it to $130, Bonnie bids up to $260, Ronnie bids up to $300 and Bonnie doubles the bid to the required $600.

Question 30

Tina and Rebecca had the same number of candies. Tina gave Rebecca 24 candies so that now Rebecca has five times as many candies as Tina has. Rebecca has now how many candies?

Accepted Answer

BACKWARD SOLVING If Rebecca has 36 candies now, she and Tina had 12 each to begin with, so how could Tina give her 24 candies?

Question 31

POE and Reverse PI Trick: Divisibility: Pugsley is three times as old as his sister Wednesday, who is 4 times as old as their pet Thing. If in 6 years Wednesday will be twice as old as Thing will be then, how old is Pugsley? (A) 36 (B) 45 (C) 48 (D) 56 (E) 60

Accepted Answer

ELIMINATE 56 RIGHT AWAY! Pugsley's age should be divisible by 12. Our previous work on answer choice C shows that Pugsley's age is 3 times Wednesday's age, and Wednesday's age is 4 times Thing's age. Wednesday's age must be divisible by 4, and a number 3 times greater than a number divisible by 4 must be divisible by 12. Since 56 is not divisible by 12, POE and move on.

Question 32

Mrs. Gonzales divided $600 evenly among her children. Later, when Mrs. Gonzales was away, Jose, her eldest, took $10 from each of his brothers and sisters giving him a total of $160. If Mrs. Gonzales has fewer than 12 children, how many children does she have?

Accepted Answer

The right answer is 5. Note that he tkaes money from n-1 persons!

Question 33

A plate with oatmeal cookies and chocolate chip cookies was eaten up by the cookie monster and by the omelet monster. Of the cookies eaten by the cookie monster, one third were oatmeal cookies. The omelet monster did not eat any of the oatmeal cookies. If the number of chocolate chip cookies eaten by the cookie monster is equal to the number of cookies the omelet monster ate, then which of the following is true?

Accepted Answer

CHOOSE NUMBERS and PLUG them IN 3-1=2.The Omelet monster did not eat any oatmeal cookies - write 0 in the Omelet / Oatmeal box. The table looks as follows: Cookie Omelet Oatmeal 1 0 1Chocolate 2 2 4Total 3 2 5

Question 34

v

Accepted Answer

DRAW a framework with arrows: W <-- 6* T ---> +9 R R -> +2 W ---> +2 T = 5

Question 35

Jerome anticipated that the WebWeb.com stock price would fall and sold all his WebWeb.com stocks for $5 per stock. He paid $10,000 tax on the revenue. A week later, Jerome became convinced that the WebWeb.com stock price would rise, and he used the money that he had gotten from selling the WebWeb.com stocks to purchase them again, this time at $6 per stock. If Jerome ended up having 4000 WebWeb.com stocks fewer than the number he had before selling them, how many WebWeb.com stocks did he have originally?

Accepted Answer

Start by plugging in answer choice C: 14,000 Assume Jerome had sold 12,000 stocks for 12,000×$5=$60,000. After the tax deduction he had $50,000 left, with which he bought $50,000/$6 stocks. This does not fit the information given in the problem because clearly $50,000/$6 is not different from 12,000 by exactly 4000 It may be difficult to determine whether the correct answer is smaller or bigger than 12,000, so just go on plugging in answers and POEing.

Question 36

Basic Stop Signs for time to Reverse PI, regardless of its level of difficulty

Accepted Answer

An invisible variable Numbers in the answer choices A specific question Time to Reverse PI, regardless of its level of difficulty

Question 37

Stop Signs on Reverse PI

Accepted Answer

Basic Stop Signs: Invisible variable in the problem. Specific question. Numbers in the answer choices. Advanced Stop Signs: Feeling an algebraic urge to set up one big equation. The problem presents a long convoluted story. You feel stuck, not sure how to approach the problem. The advanced Stop Signs are more apparent. However, the basic Stop Signs which are sometimes overlooked, are essential to defining a Reverse PI situation.

Question 38

When Plugging In AGAIN: DOZEN F numbers

Accepted Answer

DOZEN F numbers; These numbers, in turn, cause the answers to behave differently, exposing the sometimes answers and leaving only the one always true answer. DOZEN F stands for: Different (e.g. if your good number was even, try odd; If the original was a multiple of x, try a prime; small (e.g. x=3) Vs. huge numbers (e.g. x=10,000), fractions vs. integers, positive vs. negative, etc.) One - although one is a bad number which can present multiple matches to your goal, plugging in "1" is sometimes a special case often overlooked. Zero - ditto. Equal numbers for different variables - another often overlooked special case. If the problem uses X and Y Negatives - The average test taker automatically thinks of variables X and Y as positive integers Fractions - Plug in a Dozen F AFTER Plugging good numbers, in order to "break" the question and make the answer choices behave differently.

Question 39

Gerry decided to spend $450 on fireworks. If he buys regular crackers rather than deluxe crackers, he pays $3 less per cracker and gets 40 crackers more. How much does a deluxe cracker cost? 4.50 5 7.50 9 10

Accepted Answer

This problem has the two major identifiers of Reverse PI questions: There are numbers in the answer choices, and the question is specific ("How much does a deluxe cracker cost?”.) Answer choice C contains an ugly, "not worth my time" number. Avoid it for the time being, and start by plugging in B or D (but never start with the extremes, A or E.) The overall strategy is to realize that the answer should be greater than B, but smaller than D, and thus has to be C. Reach this conclusion by plugging in B and D, which are much easier numbers to work with. The question asks for the price of a deluxe cracker. If we take that to be $5 (answer choice B), and work the problem with this cost in mind, we find the following:

GMAT (Fach) / Quantitative - Word Problems (Lektion)