World Problems Flashcards
How do you translate this statement: “If Kelly received 1/3 times more votes than Mike”
K = M + 1/3 M
T or F. On data sufficiency, you can’t find solutions of variables with limited information if you know those variables must be integers.
False. You can
See page 18 and 19 of Manhattan Prep
A passenger train leaves the train depot 2 hours after a freight train left the same depot. If it overtakes the freight train in three hours, what is the same value in this problem?
Distance.
Note: Both trains traveled from the same spot and catch up at the same spot in the future.
Describe what condition is the same when both objects travel either:
1) towards each other and meet at one point
2) away from each other
3) in the same direction on same path (to catch up with each other) with one being further away than the other.
objects travel in the same time (t)
If two objects travel towards each other, what is their rate of travel?
The rate is the addition of object’s rates
Note: The objects reduce the distance themselves at this ADDITION rate
If two objects travel away from each other, what is their rate of travel?
The rate is the addition of object’s rates
Note: The objects increase the distance themselves at this ADDITION rate
In “working together” problem, what and when we can directly add value of each machine/ person?
when there is information about certain amount of work given in an hour, we can add all together the RATES
when we are not given directly the amount of work produced in an hour, what do we do in the “working together” problem?
We find the amount of work in an hour: 1/x , 1/y, 1/z and ADD them up.
On the GMAT, what is the range that weighted average will always be in?
between 20 and 30
in Weight Average problems, what is the weight?
It is the ratio
in Weight Average problems, what is the data point?
It is the specific value (i.g: 20 and 30)
When do we use Venn Diagram to solve set problems?
It should be used only for problems that involve three sets.
What do we use to solve two-set problems?
Double-set matrix
T or F. If there is actual quantities appear in the problem, you cannot assign a number to solve for solution
T.
p.g 88 Manhattan
Translate this into equation: If french fries cost twice as much as coleslaw.
F = 2C
Translate this into equation: 9 times as many paper back copies as hard copies were sold.
P= 9H
How do you calculate the ratio in weighted average problem, given the new average data?
x.(original ave - new ave) + y.(original ave - new ave) = 0
How do you translate this statement: From 2003 to 2006, # of employees decreased by a factor of 1/3
employees in 2006 = # of 2003- (1/3 x # of 2003)
or equal to 2/3 of # of 2003
How do you translate this statement: Twice as many paperpack fiction as paperpack nonfiction
PF = 2x PN
Tip: middle out, number in, last stay
What is best strategy to use for this problem?
Broad River Academy has 29 classrooms with an average of 17 students per classroom, and Lakeside School has 19 classrooms with an average of 18 students per classroom. Approximately, what percent of the town’s elementary school students attend Broad River Academy?
(A) 29% (B) 41% (C) 48% (D) 59% (E) 82%
- Lay out set up
- Estimation: (29).(17) / [29.17 + 18.19] = 29/ (29+19) = 29/48 ~ 30/50= 60%
60% is an overestimation but closer to answer choice (D) than any other choices
Considering the same travel distance
T or F: IF the speed increase by 20%, the time will reduced by 20%
False - For % increase/decrease, be careful even though the speed and time have reciprocal relationship:
- speed increases by 20% = 1+ 1/5 = 6 /5 of the actual speed
- hypothetical time will be 5/ 6 of the actual time or 1/6 time will be reduced (16.67%)
T or F: A fixed increase/decrease/difference matter much more percent-wise when the values you are working with are small than when they are large
T:
Think about Mr. Buffett analogy: $100mil fund will have higher % return than when you have $1B fund
Translate 25% Profit into the Cost: Rev Ratio?
Translate 20% Loss into the Cost: Rev Ratio?
25% Profit R- C 1.25 of the cost = 1+ 1/4 =5/4 —> flip C:R = 4/5
20% Loss = 0.8 of the cost = 1- 1/5 = 4/5 —-> flip C:R = 5/4
Translate/ Draw the inference:
- The difference between Mary’s & Jim’s salaries was twice the difference between Mary’s & Kate’s. If Mary has the highest salaries of the three.
Mary, Jim & Kate’s salaries belong to a group of consecutive space integer with Kate’s salaries in the middle:
- J, K, M (i.e: 2,4,6 -> 6-2 = 2 (6-4)
Define the mode
the number that appears the most in a set of data
How do you find the rate at the moment of a compound period (i:e: compound annually) ?
(annual rate/ 12 months) x numbers of months at compound period
If an account has annual rate of 8%, compounds semiannually, what is the rate at the moment of compound period?
If an account has annual rate of 8%, compounds quarterly, what is the rate at the moment of compound period?
(8%/12 ) x 6 = 4% first 6 months & 4% the rest of the year
(8%/12 ) x 3 = 2% each quarter
Note: the total of each month rate needs to sum up to 8%
What does “each was paid in proportion to the number of hours he or she worked” mean?
That total amount paid x (the number of hours per person/ total number of hours for everyone) or
Each is paid at the same rate $/hour
How does the increase (or decrease) in constant rate differ from a fix amount?
Increase/decrease in constant rate = increases (or decreases) by a fixed percentage in a given time interval.
- a(2) = k a (1)….a(3) = k a(2)….a(n) = a(1) x k^(n-1)
- its the compounding interest graph
Increase/Decrease by a fix amount = Consecutive Sequence: a(n) = a(1) + k (n-1)
- its the linear equation
How do we know if the 9 times hard copies include its original number?
There were 36.000 hardback copies of a certain novel sold before the paperback version was issued. From the time 1st paperback copy was sold until the last copy of the novel was sold, 9 times as many paperpack copies has hardback copies were sold.
The problem quite explitcitly indicates that during the specific period (2nd sentence), paperback is 9 times of the only number of hardback copies for the period that paperback copies were sold.
What is the hidden theme in this DS problem?
A certain dealership has a number of cars to be sold by its salespeople. How many cars are to be sold?
(1) If each of the salespeople sells 4 of the cars, 23 cars will remain unsold.
(2) If each of the salespeople sells 6 of the cars, 5 cars will remain unsold.
Divisibility & Unit conversion:
- 4x + 23 = 6x + 5 -> number of salesperson
- Note the assumption: same # of hired salespeople & same # of cars sold.
Of the 230 single-family homes built in City X last year, how many were occupied at the end of the year?
(1) Of all single-family homes in City X, 90 percent were occupied at the end of last year.
(2) A total of 7,200 single-family homes in City X were occupied at the end of last year.
1) Practice Quant reading just as how you do on Verbal section
2) No information relating to single-family homes built in either statements. Therefore, anwer is E
Intuitively draw the conclusion about the rate relationship
It takes Machine X twice as long to produce the lot of cans as it takes Machines X and Y running simultaneously to produce the lot.
-taking twice as long then rate of Machine X is 1/2 the rate of both machine X & Y working together
- machine Y has the same constant rate as X: two identical machines cut down the time by half
If a tank is leaking water at a constant rate of 3gal/hr, and became empty less than 12hrs. Does the tank contain more than 30 gallons of water?
Yes & No:
1) No: if the tank emptied at one hour, w/ constant rate of 3gal/hr, the tank contain 3 gal, which is less than 30
2) Yes: If the tank emptied at 11 hours, the tank would contain 33 gal, which is more than 30
Tim and Élan are 90 miles away from one another. They are starting to move towards each other simultaneously, Tim at a speed of 10 Mph and Élan at a speed of 5 Mph. If every hour they double their speeds, what is the distance that Tim will pass until he meets Élan?
The fact that Tim & Elan both double their speeds every hours doesn’t change the original ratio of speed 2:1. Therefore, their distance must follow the ratio of 2:1 (as their travel time is the same)
when you see double matrix…
In a certain group of 50 people, how many are doctors who have a law degree?
(1) In the group, 36 people are doctors.
(2) In the group, 18 people have a law degree.
1) Be wary of the assumption that neither categories is ZERO
2) Only If the statement indicate that a member must be either or, you can make the assumption for 1)
A clothing manufacturer makes jackets that are wool or cotton or a combination of wool and cotton. The manufacturer has 3,000 pounds of wool and 2,000 pounds of cotton on hand. Is this enough wool and cotton to make at least 1,000 jackets?
(1) Each wool jacket requires 4 pounds of wool, and no cotton
(2) Each cotton jacket requires 6 pounds of cotton, and no wool
- Notice the language asking: how many jackets in general, and not a specific type
- Always think why the information in the stimulus is there. DON’T TAKE IT for granted.
For data sufficiency
What technique can we use to test whether two linear equations of X & Y (rev, units, costs..) has a unique solution or they are just the same equation in disguise?
Calculate the slopes of two equation - if they are equal then they are parallel, and hence they are just the same equation
How come there are more than one possible solutions for A &C?
the regular admission fees were ¥5,500 for each adult and ¥4,800 for each child. Because there were at least 10 people in the group, each paid an admission fee that was 10% less that the regular admission fee. How many children were in the group?
(1) The total of the admission fees paid for the children in the group was ¥4,860 more than the total of the admission fees paid for the adults in the group.
Because there is no upper limit of total A +C, which is given by the problem that is greater than 10
Under what condition that this following paradox can occur?
At a certain clothing store, customers who buy 2 shirts pay the regular price for the first shirt and a discounted price for the second shirt. The store makes the same profit from the sale of 2 shirts that it makes from the sale of 1 shirt at the regular price. For a customer who buys 2 shirts, what is the discounted price of the second shirt?
Since the store makes the same profit selling 2 shirts as 1 shirt, the shirt at a discounted price must bring zero profit to the store. In result, the discounted price must be equal to the cost of the shirt to store (sale price - cost = 0)
Express the discounted price in term of original price & saving
If Mel saved more than $10 by purchasing a sweater at a 15 percent discount, what is the smallest amount the original price of the sweater could be, to the nearest dollar?
Percentage discount (15%) of the Original Price >= the amount of Saving ($10)
- 0.15 X > 10 -> X = 1000/15
Why this statement is sufficient?
For a certain city’s library, the average cost of purchasing each new book is $28. The library receives $15,000 from the city each year; the library also receives a bonus of $2,000 if the total number of items checked out over the course of the year exceeds 5,000. Did the library receive the bonus last year?
1. The library purchased an average of 50 new books each month last year and received enough money from the city to cover this cost.
1) Extract all the given info: 50 books/month x 12 x28 = $16,800.
2) Since it is said that the library received enough money to cover this cost, it must have received the bonus
Key lession for this problem
A car dealer sold x used cars and y new cars during May. If the number of used cars sold was 10 greater than the number of new cars sold. Which of the following expresses this relationship ?
(A) x > 10y
(B) x > y + 10
(C) x > y – 10
(D) x = y + 10
(E) x = y – 10
Read carefully about the relationship: was 10 greater than (Ans D) # was greater than (Ans B)
Use the efficient conversion unit to find distance difference/ minute
While on a straight road, Car X and Car Y are traveling at different constant rates. If Car X is now 1 mile ahead of Car Y, how many minutes from now will Car X be 2 miles ahead of Car Y ?
(1) Car X is traveling at 50 miles per hour and Car Y is traveling at 40 miles per hour.
Car X is traveling at 5 miles/6 mins. (50/60)
Car Y is traveling at 4 miles/6 mins. (40/60)
Every 6 mins, Car X moves away from car Y at a rate 1miles. Therefore, it would take 12 mins for 2 miles away between car X and Y
What is the key lesson from this?
At Perry High School, the ratio of students who participate in either the band program or the choral program to students who participate in neither program is 3 to 8. If 220 students attend Perry High School, how many of them participate in neither program?
It is not double matrix problem - Not every question with either/neither requires double matrix solution
Use limited algebra & test case
If a motorist had driven 1 hour longer on a certain day and at an average rate of 5 miles per hour faster, he would have covered 70 more miles than he actually did. How many more miles would he have covered than he actually did if he had driven 2 hours longer and at an average rate of 10 miles per hour faster on that day?
1) S (speed) x T (time) = D (distance)
(S+5)(T+1) = D + 70 -> ST +5T+S+5 = 70+D,
then 5T + S = 65
2) Test case w/ T=1, S = 60 then D= 60 & the extra will have 2x65 = 130 (exactly 70 extra miles), this means: 3 (2hrs longer) x70 (10mph faster) = 210 or 150 more miles than the original S & T
When you see profit, unit & cost difference, what strategy can you use?
A grocer purchased a quantity of bananas at 3 pounds for $0.50 and sold the entire quantity at 4 pounds for $1.00. How many pounds did the grocer purchase if the profit from selling the bananas was $10.00?
profit amount difference/ Unit
Since we know the quantity must be divisible by 3 & 4, we can pick # of units as 12 -> Cost $2/12 lbs while Rev is $3/12 lbs —> $1 profit/ 12lbs. Therefore, $10 profit will have 120lbs
Translate all possible relationships & Find assumption
If an organization were to sell n tickets for a theater production, the total revenue from ticket sales would be 20 percent greater than the total costs of the production. If the organization actually sold all but 5 percent of the n tickets, the total revenue from ticket sales was what percent greater than the total costs of the production?
1) Sell n tickets will have 20% profit, so 5% unsold ticket n will have less than 20% profit
2) The sale price for each ticket is 20% greater than the cost production of each ticket
3) The sale price for each ticket is implied constant
- Actual revenue = 0.95 Projected Rev
- Projec Rev = 1.2 Cost
Can you see the analogy w/ other question?
In 1980 John’s salary was $15,000 a year and Don’s salary was $20,000 a year. If every year thereafter. John receives a raise of $2,450 and Don receives a raise of $2,000, the first year in which John’s salary will be more than Don’s salary is
This question is a variation on the “Chase Down” prompts (e.g. A car chasing another car or a runner chasing another runner) that are in the same rare category of questions as Combined Rates. The method for solving is also the same:
- $5000 (Distance Difference)/ Relative Speed (2450-2000) = 12 years
Excluding $80, determine the value of lbs for break-even saving
Whizzo Chocolate Company uses only the following shipping methods: Western Food Sender
a) WFS non-refrigerated service: $50 plus $10 times each pound
b) WFS refrigerated service: $80 plus $15 times each pound
W/o $80 cost, we save $5/lb. However, because of the $50 base cost, we won’t save as much for certain number of lbs (i.e: In fact, with different method of shipping, 1lb will lead us to pay extra $45 instead of saving). Therefore, the break-even - saving = cost - 5X = 50 or X= 10lbs
- Hence, we have to ship more than 10lbs in order to make the non-refrigerated service cheaper
Translate the bold part into algebraic expression
Regulators are likely to end what are, in effect, long-standing exemptions permitting pilots of small turboprop aircraft at small carriers to fly as many as 20 percent more hours per month than pilots at larger airlines do, and consequently some carriers could be forced to hire additional pilots.
Pilots at small carriers are allowed to fly more hours than those at larger airlines (X). How much more?
- 1%….20% -> If X is the # of hours then: X < Y < 1.2X
Why is this statement alone is sufficient?
MachinesXandYproducedidenticalbottlesat different constant rates. Machine X, operating alone for 4 hours, filled part of a production lot; then Machine Y, operating alone for 3 hours, filled the rest of this lot. How many hours would it have taken Machine X operating alone to fill the entire production lot?
(1) Machine X produced twice as many bottles in 4 hours as Machine Y produced in 3 hours.
Machine Y produce A bottles in 3 hrs then Machine X will produce 2A in 4hrs. Therefore, the total number of bottles for THE JOB is 3A bottles.
- Using proportional technique, we have (3A x 4)/ 2A = 6 hours
What is the key lesson in this problem?
On a certain road, 10 percent of the motorists exceed the posted speed limit and receive speeding tickets, but 20 percent of the motorists who exceed the posted speed limit do not receive speeding tickets. What percent of the motorists on that road exceed the posted speed limit ?
Careful reading of the key words/modifiers & Double Matrix map outline:
- 10% of the motorists exceed the posted speed limit & receiving speeding tickets = 0.1 of the total motorists on that road (X) or 0.1X
- 20% of the motorists who exceeded the posted speed limits (Y) do not receive ticket or 0.2Y
- Find Y/X relation via: 0.1 X + 0.2Y = Y -> 0.1 X = 0.8Y so Y/X = 12.5%
Why this statement is not sufficient?
A jewelry dealer initially offered a bracelet for sale at an asking price that would give a profit to the dealer of 40 percent of the original cost. What was the original cost of the bracelet?
(2) The jewelry dealer sold the bracelet for $1,953.
Its the trap - the key word is “initially offered”, and hence we can infer that the dealer sold the bracelet at a profit could higher or lower than 40% of the original cost
Express the constant rate relationship here
The net value of a certain stock increased at a constant rate during the ten-year period between 1990 and 2000. What was the value of the stock in the year 1998?
(1)In 1991, the value of the stock was 130 U.S dollars.
(2)In 1992, the value of the stock was 149.5 U.S dollars.
Constant rate = a(3)/ a(2) = 149.5/130 = 1.15 - or 15% increase by each year
- a(1998) = 9th term of sequence
- a (1990) = 1st term of sequence
hence : a (1998) = a (1990) x 1.15^(9-1)
What do you need to pay attention to?
The cost of deluxe hamburger and a shake is $7.00. If prices do not vary with quantity purchased, how much does one shake cost?
(1) If the price of the shake were 1/2 of its present price, a shake would cost 1/5 of what the deluxe hamburger costs.
The wording of hypothetical scenario:
- Price of hypothetical (H) = 1/2 P
- P (H) = 1/5 Price of hamburger (H)
-> 1/2 P = 1/5H
-> H + P = $7
so this statement is sufficient
Why the answer is not 460/15 or 30? What is the hidden lesson?
A farmer decides to plant a row of trees along one side of a road. He decides to plant one tree every 15 metres. If the road is 460 metres long what is the maximum number of trees the farmer could plant?
C. 30
D. 31
If we divide 460 by 15 then the 1st tree will be on the 15 meter mark, leaving the potential starting point empty, of which we can plant the 1st tree instead
- This is the sequence test: A = {0, 15, 30, 45, 60 … 450}
- 450 = 0 + 15 (n-1) with n = number of trees
T or F
You can use weighted average for the average speed, time, or rate
F- There are reciprocals in Work, Rate & Distance relationship
Interprete this passage
A certain clock marks every hour by striking a number of times equal to the hour, and the time required for a stroke is exactly equal to the time interval between strokes. At 6:00 the time lapse between the beginning of the first stroke and the end of the last stroke is 22 seconds.
At 6:00, 22 seconds include = time of 6 strikes + time of the 5 intevals (between each strike)
- each strike time = interval time –> 22 seconds/ 11parts or 2sec/ part
T or F: Distance Ratio X: Y = Ratio of Rate X:Y when both travel at the same amount of time
Train X, traveling at a constant rate, completed the 100-mile trip in 5 hours; Train Y, traveling at a constant rate, completed the 100-mile trip in 3 hours. How many miles had Train X traveled when it met Train Y ?
True - Since they travel the same mount of time when they meet, the distance is only dependent on the ratio of rate
- Train X will travel in ratio 5:3 to train y: 5x 100/ (3+5) = 62.5 miles
Use CR thinking mental math model w/o using equation
Company X has 50 employees and company Y has 60 employees. Both companies have same number of full time employees, but company Y has 3 more than twice the number of part-time employees that Company X has. How many part-time employees does Company Y have?
1) Since # of full-time between two companies are the same, the 10 extra employees difference (60-50) is accounted by the difference between part-time employees
2) 2 times P(X) + 3 - P(X) = 10 -> Part-time at Comp X = 7, then Comp Y has 2*7+3= 17
