7. Word Problems Flashcards
1
Q
What are the four steps to breaking down any word problem?
A
(1) Desired. Identify the desired value (i.e. what the question is asking for)
(2) Unknowns. Identify unknown values and label them with variables (never forget units!)
(3) Relationships. Identify relationships and translate them into equations
(4) Solve. Use the equations to solve for the desired value
* NOTE: Generally, the most efficient way to find the desired value is to eliminate unwanted variables using substitution
* Steps do not need to be followed in strict order
2
Q
What rule applies to additive relationships?
A
The units of every term must be the same! Adding terms with the same units does not change the units
3
Q
What rule applies to multiplicative relationships?
A
Treat units like numerators and denominators. Units that are multiplied together DO change!
4
Q
What are some common word problem phrases for addition?
A
- Add, sum, total (of parts), more than: +
- The sum of x and y: x + y
- The sum of the three funds combined: a + b + c
- When fifty is added to his age: a + 50
- Six pounds heavier than Dave: d + 6
- A group of men and women: m + w
- The cost is marked up: c + m
5
Q
What are some common word problem phrases for subtraction?
A
- Minus, difference, less than: -
- x minus five: x – 5
- The difference between Quentin’s and Rachel’s heights (if Quentin is taller): q – r
- Four pounds less than expected: e – 4
- The profit is the revenue minus the cost: P = R – C
6
Q
What are some common word problem phrases for multiplication?
A
- The product of h and k: h * k
- The number of reds times the number of blues: r * b
- One fifth of y: (1/5) * y
- n persons have x beads each: total number of beads = nx
- Go z miles per hour for t hours: distance = zt miles
7
Q
What are some common word problem phrases for ratios and division?
A
- Quotient, per, ratio, proportion: /
- Five dollars every two weeks (5 dollars/2 weeks) -> $2.5 per week
- The ratio of x to y: x/y
- The proportion of girls to boys: g/b
8
Q
What are some common word problem phrases for average or mean?
A
- Sum of terms divided by the total number of terms
- Average of a and b: (a + b)/2
- Average salary of three doctors: (x + y + z)/3
9
Q
What is the profitability formula?
A
Profits = P*Q – VC – FC
10
Q
What is a conversion factor?
A
A fraction whose numerator and denominator have different units but the same value
11
Q
How do you handle units in word problems?
A
(1) Add or subtract quantities with units: ensure that the units are the same, converting first if necessary
(2) Multiply quantities with units: multiply the units, cancelling as appropriate
(3) Convert from one unit to another: multiply by a conversion factor and cancel
12
Q
What is the formula for rate or speed?
A
Rate = Distance / Time Speed = Distance / Time
Rate is measured in units of distance per unit of time (for example, miles per hour).
DERT: Distance Equals Rate Time. Notice, in addition, that the letters are in alphabetical order.
13
Q
What is the formula for distance?
A
Distance = Rate * Time
14
Q
What is the formula for Time?
A
Time = Distance / Rate
15
Q
What is the formula for work completed?
A
Work Completed = Rate * Time
Rate is measured in units of output per unit of time (for example, 5 widgets produced per minute).
*HIDDEN CONSTRAINT: All work rates must be positive
16
Q
How do you solve problems when people work together?
A
Add the rates – when two or more workers are performing the same task
Subtract the rates – when one worker is undoing the work of the other
17
Q
What is the formula for total personal earnings?
A
Total Personal Earnings = Wage Rate ($/hr) * Hours Worked (hrs)
18
Q
What is the formula for miles?
A
Miles = Miles per Hour * Hours Miles = Gallons per Hour * Gallons
19
Q
What are integer constraints?
A
- Some word problems, by their nature, restrict the possible values of the variables
- The most common restriction is that variables must be integers (e.g. cars, marbles, people, etc. must be integers)
- BEWARE: hidden constraints show up in data sufficiency because information that seems like it should be insufficient on its own actually does provide an answer
20
Q
How do you solve relative rate problems?
A
Create a third RT = D chart for the rate at which the distance between the bodies changes
(1) the bodies move towards each other
- e.g. two people decrease the distance between themselves at a rate of 5 + 6 = 11 mph
(2) the bodies move away from each other
- e.g. two cars increase the distance between the between themselves at a rate of 30 + 45 = 75 mph
(3) the bodies move in the same direction on the same path starting at the same point or different points
- e.g. person X and Y decrease the distance between themselves at a rate of 8 – 5 = 3 mph
21
Q
How do you determine the average rate?
A
- If an object moves the same distance twice, but at different rates, then the average rate will NEVER be the average of the two rates given for the two legs of the journey
- In fact, because the object spends more time traveling at the slower rate, the average rate will be closer to the slower of the two rates than to the faster
- In order to find the average rate, you must first find the total combined time for the trips and the total combined distance for the trips (NOTE: you can actually pick any smart number for the distance)
22
Q
How do you solve population growth or decay problems?
A
- Solve with a population chart – when some population increases by a common factor every time period
- Label the middle row “NOW” and work forward/backward, obeying any given conditions about the rate of growth or decay
- In some cases you might have to pick Smart Numbers for the starting point in your population
23
Q
What is an Average (arithmetic mean)?
A
Average = S / n, where S is the sum of all of the terms in the set, n is the number of terms in the set, and A is the average.
24
Q
What is a Weighted average?
A
In a weighted average, some data points contribute more than others to the overall average. This is in contrast to a regular average, in which each data point contributes equally to the overall average. A weighted average can be expressed with the formula A = [(D1)(W1) + (D2)(W2) + … + (Dn)(Wn)] / sum of weights, where each D represents a distinct data point, each W represents the weighting assigned to that data point, and A is the weighted average.
25
What is the Median?
Literally, the “middle” value in a set of numbers written in increasing (or decreasing) order. In a set with an odd number of terms, the median is the middle number. In the set 1, 3, 4, 6, 9, the median is 4. In a set with an even number of terms, the median is the average of the two middle numbers. In the set, 1, 3, 4, 6, the median is (3+4)/2 = 3.5.
26
What is the Standard Deviation (SD)?
A measurement used to describe the how far apart numbers in a set are. This is also called the “spread” or the “variation” of the set. Technically, SD is a measure of how far from that set’s average the data points typically fall. SD can be either positive or zero.
27
What do a small, large and zero SD indicate?
- A small SD indicates that the terms of the set are clustered closely around the average value of that set.
- A large SD indicates that the terms of the set are widely spread, with some terms very far from the average value of that set.
- An SD of zero indicates that all of the terms of that set are exactly equal to that set’s average.
28
How do you determine the weighted average with the ratio of weights?
- You do not necessarily need concrete values for the weights in a weighted average problem; -having just the ratios of the weights will allow you to find the weighted average
- Simply write the ratio as a fraction, and use the numerator and the denominator as weights; the sum of the two will be the denominator
29
How do you determine the weights with the data points and the average?
Look at the differentials between the data points and the average and make sure they cancel out; so you multiply both differentials by different numbers so that the positive will cancel out with the negative
e.g. x(+2) + y(-5) = 0
The weights are x = 5 and y = 2
30
How do you calculate the median?
(1) Sets with odd number of values: the median is the unique middle value when the data are arranged in increasing or decreasing order (i.e. median equals a value in the set)
(2) For even with even number of values: the median is the average of the two middle values when the data are arranged in increasing or decreasing order (i.e. median does not have to equal a value in the set)
31
How do you evaluate an entirely unknown set of numbers?
- Use alphabetical order to make the set a little more concrete
- If the problem is complex, it may be helpful to create a column chart; each column represents a number in the set; put the columns in order from shortest to tallest
32
What is the standard deviation?
Indicates how far from the average the data points typically fall
Small SD: indicates that the set is clustered closely around the average value
Large SD: indicates that the set is spread out widely, with some points appearing far from the mean
```
SD = Sqrt(Variance)
Variance = SUM(x – Mean)^2 / N
```
33
What happens to the SD when you add another number to the set equal to the mean?
Adding a number equal to the mean would decrease the SD of the set because it is the same as the mean of the set
34
How do you find the sum using an average? (e.g. sum of all GMAT scores, using the average of 720 given 300 test takers)
```
Sum = average * number of terms
Sum = 720 * 300
```
35
If a number is added or deleted from an average, how do you determine the value of that number?
Number Added = New Sum – Original Sum
| Number Deleted = Original Sum – New Sum
36
What is the average of a sequence of evenly spaced numbers?
Average of the first and last terms
37
What are evenly spaced sets?
Sequences of numbers whose values go up or down by the same amount (the increment) from one item in the sequence to the next
E.g. {4, 7, 10, 13, 16}
38
What parameters must be known for an evenly spaced set to be fully defined?
(1) The smallest (first) or largest (last) number in the set
(2) The increment (always 1 for consecutive integers)
(3) The number of items in the set
39
What are consecutive multiples?
Special cases of evenly spaced sets: all of the values in the set are multiples of the increment
E.g. {12, 16, 20, 24}
*All sets of consecutive multiples are evenly spaced
40
What are sets of consecutive integers?
Special cases of consecutive multiples: all of the values in the set increase by 1, and all intgers are multiples of 1.
E.g. {12, 13, 14, 15, 16}
* All sets of consecutive integers are sets of consecutive multiples
* NOTE: Consecutive integers alternate between EVEN and ODD numbers
41
What are the properties of all evenly spaced sets?
(1) Median of Terms = Average of Terms
(2) Average = (First + Last) / 2
(3) Sum of Terms = Average * Number of Terms
(4) Number of Terms = Last – First + 1
42
What are consecutive integers?
Integers that follow one another from a given starting point, without skipping any integers. 3, 4, 5, and 6 are consecutive integers but 3, 5, 6, and 14 are not.
Consecutive Even = 2, 4, 6, 8, etc
Consecutive Odd = 1, 3, 5, 7, etc
Consecutive Multiples = 4, 8, 12, 16, etc (multiples of 4)
43
What is an evenly spaced set?
A set of numbers in which each number is a set distance, or increment, from the next. 1, 4, 7, 10 is an evenly spaced set with an increment of 3.
44
What are the rules of an evenly spaced set?
(1) The arithmetic mean (average) is equal to the median. The set comprising 1, 2, 3, 4, and 5 has a mean of 3 and a median of 3.
(2) The mean and the median of the set are both equal to the average of the first and last terms in the set. For the set {1, 2, 3, 4, 5}, the first term is 1 and the last term is 5. The average of these two terms is 3, which also equals both the median and the mean of the entire set.
(3) The sum of the elements in a set equals the mean multiplied by the number of items in the set. For the set {1, 2, 3, 4, 5}, the mean is the middle term, 3. There are five terms in the set, so the sum is 3*5 = 15
45
What does an inclusive set mean?
Given "a set of integers between15 and 19 inclusive," we include the numbers 15 and 19 in the set: {15,16,17,18,19}. Given "a set of integers between 15 and 19," we do not include 15 and 19 in the set: {16,17,18}.
46
What is the rule for the product of n consecutive integers and divisibility?
The product of n consecutive integers is always divisible by n! Given 6*7*8*9, we have n = 4 consecutive integers. The product of 6*7*8*9, therefore, is divisible by 4! = 4*3*2*1 = 24.
47
What is the rule for the sum of n consecutive integers?
There are two cases, depending upon whether n (number of integers in the set) is odd or even:
(1) If n is odd, the sum of the integers is always divisible by n. Given 6+7+8, we have n = 3 consecutive integers. The sum of 6+7+8, therefore, is divisible by 3.
(2) If n is even, the sum of the integers is never divisible by n. Given 6+7+8+9, we have n = 4 consecutive integers. The sum of 6+7+8+9, therefore, is not divisible by 4.
48
What is the rule of the product of three consecutive integers?
Any three consecutive integers will always include one multiple of three. So, if (x), (x – k) and (x – 1) are consecutive integers, then their product must be divisible by three. Note that (x) and (x – 1) are consecutive, so the three terms would be consecutive if (x – k) is either the lowest of the three, or the greatest of the three:
(x – k), (x – 1), and (x) are consecutive when (x – k) = (x – 2), or k = 2
(x – 1), (x), and (x – k) are consecutive when (x – k) = (x + 1), or k = -1
Note that the difference between k = -1 and k = 2 is 3. Every third consecutive integer would serve the same purpose in the product x(x – 1)(x – k): periodically serving as the multiple of three in the list of consecutive integers. Thus, k = -4 and k = 5 would also give us a product that is always divisible by three.
49
What do we know about x = y + 1?
We are dealing with two consecutive integers. In any pair of consecutive integers, one of the integers must be even and one must be odd
50
How many integers are there from 6 to 10?
For consecutive integers, the Number of Terms = Last – First + 1
Number of integers = 10 – 6 + 1 = 5
*DO NOT forget to be inclusive of the extremes
51
How many multiples of 7 are there between 100 and 150?
For consecutive multiples, the number of multiples = (Last – First)/Increment + 1
Number of multiples = (147 – 105)/7 + 1 = 7
*The bigger the increment, the smaller the result, because there is a larger gap between the numbers that you are counting
52
What is the sum of all integers from 20 to 100, inclusive?
Average = (100 + 20)/2 = 60
Number of Terms = 100 – 20 + 1 = 81
Sum of all Integers = 60 * 81 = 4,860
53
What is the average of an odd number of consecutive integers (e.g. 1, 2, 3, 4, 5)?
Will ALWAYS be an integers because the middle number will be a single integer
54
What is the average of an even number of consecutive integers (e.g. 1, 2, 3, 4)?
Will NEVER be an integer because there is no true middle number; therefore, the middle number for an even number of consecutive integers is the average of two consecutive integers
55
Is k^2 odd?
(1) k – 1 is divisible by 2
(2) The sum of k consecutive integers is divisible by k.
Answer D.
(1) SUFFICIENT. k – 1 is even, thus k is odd. An odd number times itself will result in an odd number
(2) SUFFICIENT. Sum of k consecutive integers is divisible by k. Therefore, this sum divided by k is an integer. Moreover, the sum of k consecutive integers divided by k is the average of that set of k integers. As a result, statement 2 is telling you that the average of the k consecutive integers is itself an integer. If the average of this set of consecutive integers is an integer, then k must be odd
Takeaway:
-For any odd number of consecutive integers, the sum of those integers is divisible by the number of integers
56
What is an overlapping set?
Translation problem that involves two or more given sets of data that partially intersect with each other
57
What is the double-set matrix and when is it used?
- A table whose rows correspond to options for one decision, and whose columns correspond to the options for the other decision
- The last row and the last column contain totals, such that the bottom right hand corner contains the total number of everything or everyone in the problem
- Use on GMAT overlapping set problems involving only two categorizations or decisions
58
How do you approach a problem with 2 sets of data with three choices?
-As long as the choices do not overlap (e.g. no respondent can give more than one response), use a double-set matrix and extend the chart (e.g. if respondents can answer “Yes”, “No” or “Maybe”
59
What is a Venn Diagram and when is it used?
- A Venn Diagram is a chart showing three overlapping circles
- Use for problems involving three overlapping sets
- RULE: work from the inside out
- In order to find the total, just add all seven sections together (3 one’s, 3 two’s, 1 three)
60
What are the formula for solving overlapping set problems with three data sets?
Memorize the overlapping set formulas; given 3 groups (x, y, z)
(1) Act Tot = Only(X) + Only(Y) + Only(Z) + Only(XY) + Only(YZ) + Only(XZ) + XYZ
(2) Act Tot = Tot(X) + Tot(Y) + Tot(Z) – (XY + YZ + XZ) – 2(XYZ)
61
WP Strategy Guide, Ch 5, Q 6. Of 60 children, 30 are happy, 10 are sad, and 20 are neither happy nor sad. There are 20 boys and 40 girls. If there are 6 happy boys and 4 sad girls, how many boys are neither happy nor sad?
Set up an extended double-set matrix
Boys: Happy 6, Unhappy ?, Neither X, Total 20
Girls: Happy ?, Unhappy ?, Neither ?, Total 40
Total: Happy 30, Unhappy 10, Neither 20, Total 60
Boys: Happy 6, Unhappy 6, Neither 8, Total 20
Girls: Happy ?, Unhappy ?, Neither ?, Total 40
Total: Happy 30, Unhappy 10, Neither 20, Total 60
X = 8
62
WP Strategy Guide, Ch 5, Q 8. The 38 movies in the video store fall into the following three categories: 10 action, 20 drama, and 18 comedy. However, some movies are classified under more than one category, 5 are both action and drama, 3 are both action and comedy, and 4 are both drama and comedy. How many action-drama-comedy movies are there?
Set up a Venn Diagram
Three Categories: X
Two Categories: 5 – X, 3 – X, 4 – X
One Category: 2 + X, 11 + X, 11 + X
Three Categories: X
Two Categories: 12 – 3X
One Category: 24 + 3X
Total = 38
X = 2
63
WP Strategy Guide, Ch 4, Q 3. In the sequence of eight consecutive integers, how much greater is the sum of the last four integers than the sum of the first four integers?
```
ALTERNATIVE # 1
*Pick numbers; find the average of the last half of the set and work backwards to find the sum of the first half of the set
1, 2, 3, 4, 5, 6, 7, 8
Avg Last Four = (8 + 5)/2 = 6.5
Sum Last Four = 6.5 * 4 = 26
Avg First Four = (4 + 1)/2 = 2.5
Sum First Four = 2.5 * 4 = 10
Difference = 16
```
ALTERNATIVE # 2
*Represent each unknown with a line
_, _, _, _, +4, +4, +4, +4
Difference = 4 * 4 = 16
ALTERNATIVE # 3
Last Four = (n + 4) + (n + 5) + (n + 6) + (n + 7) = 4n + 22
First Four = n + (n + 1) + (n + 2) + (n + 3) = 4n + 6
Difference = (4n + 22) – (4n + 6) = 16
64
WP Strategy Guide, Ch 4, Q 7. If the sum of the last three integers in a set of seven consecutive integers is 258, what is the sum of the first four integers?
Average Last Three = 258/3 = 86 (6th term)
Set {81, 82, 83, 84, 85, 86, 87}
Average First Four = (84 + 81)/2 = 82.5
Sum of First Four = 82.5 * 4 = 330
65
WP Strategy Guide, Ch 4, Q 8. What is the sum of 100 to 150 minus the sum of 125 to 150?
Sum 100 to 150 – Sum 125 to 150 = Sum 100 to 124
Average = (124 + 100)/2 = 112
Number of Terms = 124 – 100 + 1 = 25
Sum 100 to 124 = 112 * 25 = 2,800
66
Ch 9, Q 4. Norman is 12 years older than Michael. In 6 years, he will be twice as old as Michael. How old is Norman now?
(1) Translate the first sentence into an equation
N = M + 12
(2) Translate the second sentence into an equation
N + 6 = 2(M + 6)
(3) Rewrite the first equation to be in terms of M
M = N – 12
(4) Substitute N – 12 for M in the second equation
N + 6 = 2(N – 12 + 6)
N + 6 = 2N – 12
N = 18
67
Ch 9, Q 8. Louise is three times as old as Mary. In 5 years, Louise will be twice as old as Mary. How old is Mary now?
(1) Translate the first sentence into an equation
L = 3M
(2) Translate the second sentence into an equation
L + 5 = 2(M + 5)
(3) Substitute 3M for L in the second equation
3M + 5 = 2M + 10
M = 5
68
Ch 9, Q 15. Arnaldo earns $11 for each ticket that he sells, and a bonus of $2 per ticker for each ticket he sells over 100. If Arnaldo was paid $2,400, how many tickets did he sell?
11x + 2(x – 100) = 2,400
13x – 200 = 2,400
13x = 2,600
x = 200
69
Ch 9, Q 20. Every week, Renee is paid $40 per hour for the first 40 hours she works, and $80 per hour for each hour she works after the first 40 hours. If she earned $2,000 last week, how many hours did she work?
40(40) + 80(x – 40) = 2,000
1,600 + 80x – 3,200 = 2,000
80x = 3,600
x = 45
70
Ch 9, Q 47. Bingwa the African elephant can lift 6% of his body weigh using his trunk alone. If Bingwa weighs 1,000 times as much as a white handed gibbon, how many gibbons can Bingwa lift at once with his trunk?
We do not know how much Bingwa weighs but we know that he weighs 1,000 gibbons.
6% of 1,000 gibbons = 60 gibbons
71
Ch 9, Q 51. Sal is looking for a clean, unwrinkled shirt to wear to work. 2/3 of his shirts are dirty, and of the remainder, 1/3 are wrinkled. If Sal has a total of 36 shirts, how many shirts does he have to choose from for work?
2/3 Dirty
1/3 Clean
Of the 1/3 Clean shirts, 1/3 are wrinkled and 2/3 are not wrinkled.
(2/3) * (1/3) * 36 = 72 / 9 = 8 shirts are clean and unwrinkled
72
Ch 9, 54. Of all the homes on Gotham Street, 1/3 are termite-ridden, and 3/5 of these are collapsing. What fraction of the homes are termite-ridden, but NOT collapsing?
1/3 termites
(1/3) * (3/5) = 3/15 termites and collapsing
(1/3) * (2/5) = 2 /15 termites and not collapsing
73
WP Strategy Guide, Ch 2, Q 4. Nicky and Cristina are running a race. Since Cristina is faster than Nicky, she gives him a 36 meter head start. If Cristina runs at a pace of 5 meters per second and Nicky runs at a pace of only 3 meters per second, how many seconds will Nicky have to run before Cristina catches up to him?
Nicky is originally 36 meters ahead of Cristina. If Nicky runs at a rate of 3 meters per second and Cristina runs at a rate of 5 meters per second, then the distance between the two runners is shrinking at a rate of 5 – 3 = 2 meters per second
Rate * Time = Distance
2 * t = 36
t = 18 seconds
74
WP Strategy Guide, Ch 2, Q 5. Did it take a certain ship less than 3 hours to travel 9km? (1km = 1,000m)
(1) The ship’s average speed over the 9km was greater than 55m per minute
(2) The ship’s average speed over the 9km was less than 60m per minute
Answer A. Rephrase the information to be meters and minutes and solve for the rate. The question asks whether the ship traveled 9km in less than 3 hours. Therefore, the question is really asking us if r > 50
Rate * Time = Distance
Rate * 180 = 9,000
180r = 9000
r = 50
(1) SUFFICIENT. r > 55. Thus, r is definitely greater than 50
(2) INSUFFICIENT. r 50
75
WP Strategy Guide, Ch 2, Q 6. Twelve identical machines, running continuously at the same constant rate, take 8 days to complete a shipment. How many additional machines, each running at the same constant rate, would be needed to reduce the time required to complete a shipment by 2 days?
Let the work of one machine be r. Then the work of 12 machines is 12r
Rate * Time = Work
12r * 8 = 96r
To figure out how many machines are needed to complete this work in 8 – 2 = 6 days, set up another row and solve for the unknown rate:
Original: 12r * 8 = 96r
New: Rate * 6 = 96r
We need four additional machines
76
WP Strategy Guide, Ch 2, Q 7. Al and Bob shared the driving on a certain trip. What fraction of the total distance did Al drive?
(1) Al drove for 3/4 as much time as Bob did
(2) Al’s average driving speed for the entire trip was 4/5 of Barb’s average driving speed for the trip
Answer C. Rephrase the question to say what is the ratio of Al’s driving distance to the entire distance driven? Alternatively, since the entire distance is the sum of only Al’s distance and Bob’s distance, you can simply find the ratio of Al’s distance to Bob’s distance
(1) INSUFFICIENT. You have no rate information so you do not have definitive distance relationships
(2) INSUFFICIENT. As with statement 1, you have no definitive distance relationships
(C) SUFFICIENT.
Al: (4/5)r * (3/4)t = (3/5)rt
Bob: r * t = rt
% of Total Distance = (3/5)rt / (8/5)rt = 3/8
77
WP Strategy Guide, Ch 2, Q 8. Mary and Nancy can each perform a certain task in m and n hours, respectively. Is m
Answer D. First, set up an RTW chart
Rate * Time = Work
Mary: 1/m * m = 1
Nancy: 1/n * n = 1
(1) SUFFICIENT. Find out how much time it would take for the task to be performed with both Mary and Nancy working. Then, set up the inequality describing the statement
t(1/m + 1/n) = 1
t(m+n)/mn = 1
t = mn / (m+n)
2(mn / (m+n)) > m
2mn > m^2 + mn
mn > m^2
n > m
(2) SUFFICIENT. You can reuse the computation of t.
2(mn / (m+n))
78
WP Strategy Guide, Ch 3, Q 7. On a particular exam, the boys in a history class averaged 86 points and the girls in the class averaged 80 points. If the overall class average was 82 points, what was the ratio of boys to girls in the class?
b(4) + g(-2) = 0
b = 1
g = 2
The ratio of boys to girls is 1:2
79
WP Strategy Guide, Ch 3, Q 8. S = {1, 2, 5, 7, x} If x is a positive integer, is the mean of set S greater than 4?
(1) The median of set S is greater than 2
(2) The median of set S is equal to the mean of set S
```
Answer B. Rephrase the question:
(15 + x)/5 > 4
X > 5 ?
OR is 3 + x/5 > 5
(1) INSUFFICIENT. For the median of the set to be greater than 2, x must also be greater than 2. If 2
```
80
WP Strategy Guide, Ch 3, Q 9. {9, 12, 15, 18, 21} Which of the following pairs of numbers, when added to the set above, will increase the standard deviation of the set?
I. 14, 16
II. 9, 21
III. 15, 100
II and III ONLY. The mean of the set is 15
I. The numbers 14 and 16 are both very close to the mean. Additionally, they are closer to the mean than four of the numbers in the set, and will reduce the spread around the mean. This pair will REDUCE the SD
II. The numbers 9 and 21 are relatively far from the mean. Adding them to the body-text will increase the spread of the set and INCREASE the standard deviation
III. Adding the number 15 to the set would actually decrease the standard deviation; adding the number 100 would greatly increase the standard deviation of the set. This pair of numbers will INCREASE the standard deviation
81
WP Strategy Guide, Ch 6, Q 2. Two racecar drivers, Abernathy and Berdoff, are driving around a circular track. If Abernathy is 200 meters behind Berdoff and both drivers drive at their respective constant rates, how long, in seconds, will it take for Abernathy to catch up to Berdoff?
(1) The circumference of the racetrack is 1,400 meters
(2) Abernathy is driving 25 meters per minute faster than Berdoff
Answer B. If one driver is 200 meters behind another driver, all you need to know in order to determine the time it takes for the rear driver to catch up is the rate at which the rear driver is gaining on the front driver. Rephrase the question: “What is the difference in the two driver’s rates
(1) INSUFFICIENT. Knowing the length of the track does not tell us anything useful.
(2) SUFFICIENT. If Abernathy is gaining at 25 meters per minute, then it will take him 8 minutes to catch up (time = distance/rate = 200/25, which is equal to 8)
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WP Strategy Guide, Ch 6, Q 6. Boys and girls in a class are writing letters. There are twice as many girls as boys in the class, and each girl writes 3 more letters than each boy. If boys write 24 of the 90 total letters written by the class, how many letters does each boy write?
Girls: # of children (2b), letters/child (x + 3), # of letters written (2bx + 6b)
Boys: # of children (b), letters/child (x), # of letters written (bx)
```
bx = 24
2bx + 6b = 66
2(24) + 6b = 66
6b = 18
b = 3
```
```
x(3) = 24
x = 8
```
Each boy wrote 8 letters.
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Recognition: Quick Sell Outlet sold a total of 40 TVs, each of which was either a model P TV or a model Q TV. Each model P sold for $p and each model Q sold for $q. The average selling price of the 40 televisions was $141. How many of the 40 TVs were model P TVs?
*Statistical average: when given total quantity, express as x and given quantity minus X
write expression as pX + q(40 - X) = 40 * 141
