Quant GRE Cards Flashcards
1
Q
What do we know about a square inscribed in a square?
A
It always creates 4 similar triangles at the corners of the larger square
2
Q
What is the product of the roots of the quadratic equation?
A
3
Q
How do you count the number of odd or even factors of large numbers?
A
4
Q
To determine the min or max possible value in a set for which you know the average, what should you do?
A
- To find the minimum possible value, maximize the other values.
- To find the maximum possible value, minimize the other values.
5
Q
What does this equal?
A
6
Q
How many paths of the shortest possible length from A to B are there? How do you get there?
A
Source: Magoosh Counting Practice Problems for the GMAT (problem 12)
Each one of them will consist of, in some order, two horizontal segments and two vertical segments. How many of these are there? Well, how many ways can we distribute two horizontal segments among four slots? 4C2 = 6. We put the horizontal segments in two slots, and then the two vertical segments must go in the remaining slots. There are six minimum slots.
7
Q
What do we know about squares of integers?
A
Only squares of integers have an odd number of positive divisors
8
Q
Immediate Reaction?
You have a triangle with some weird information and it’s clearly not one of the common triangle types (e.g. maybe the measure of a couple of its angles). See the attached pic for an example of this
A
One idea: Try to draw lines to divide the triangle into other shapes (e.g. 30-60-90 triangle) to discern more properties about it (e.g. side lengths)
9
Q
Describe the vertical stretches
A
10
Q
Describe the process for counting with identical terms
A
11
Q
Immediate reaction?
You have to square a number ending in 5
A
12
Q
What should you do if you have an even radical (square root, x^(1/4), etc.) in a function f(x) and you’re asked to find the roots of the function?
A
Check each of the roots to make sure there isn’t a negative output from a radical
13
Q
What’s the probability of A & B?
A
14
Q
If you list out the units digits for each number less than 10 raised to 1, 2, 3, 4, 5, etc., when would the units digit pattern start to repeat?
A
None of them have a pattern whose segment is longer than 4.
- Digits with pattern of 4: 2, 3, 7, 8
- Digits with pattern of 2: 4, 9
- Digits with pattern of 1: 1, 5, 6
15
Q
What should you do on every single percent increase question?
A
Make sure I didn’t forget to multiply by 100 in the percent increase equation!!!
16
Q
What should you be thinking when you look at this figure?
A
Angles t and 3t+8 are supplementary.
17
Q
Immediate reaction?
Exponent problem
A
- Write the number with the exponent by the simplest base possible
- If there’s a parentheses, try to factor numbers out of the parentheses if there’s a sum or difference inside. Do the simplified arithmetic in the parentheses.
18
Q
Immediate reaction?
You have to square a number ending in 10
A
19
Q
What do you do if you try to square an algebraic expression to remove the radical, then another square root appears?
A
Isolate the square root on one side, then square both sides.
20
Q
What can multiples be?
A
Positive, negative, or 0
21
Q
If the distances traveled at two different constant speeds are equal, which one is the average speed closer to?
A
If the distances are the same, average speed is always weighted towards the slower speed.
22
Q
What’s the divisibility rule for 11?
A
- Start with subtraction (between 1st and 2nd digit) regardless of whether there are an even or odd number of digits
23
Q
Describe the features of a box and whisker plot
A
Middle line is the median, 1st quartile line is median of data below the median, and 3rd quartile line is the median of data above the median. End lines are the minimum and maximum of the entire data set.
24
Q
One liquid solution of 7 liters contains 40 % water. A second liquid solution of 18 liters contain 20% water. If these two solutions are mixed, what percent of the mixture is water?
A
25.6
25
What do we know about remainders?
Always true: **divisor \> remainder**
26
Rules:
Dividing Evens and Odds
There are no rules
27
When are choosing numbers potentially a good strategy?
1. When there are variables in the question and variables in the answers
2. Percent increase questions
28
For the quadratic equation, what are the sum of the roots?
Vieta's Formula:
29
What are the function shifts
30
What are the strategies you should think through if you're stuck on a problem of the following types:
1. Geometry
2. QC with multiple inequalities
1. **Geometry:**
1. Did I discern angles from all sources possible?
* Sum of angles along lines = 180 (easy to miss when there are 3 or more angles along the same line)
* Parallel lines (mentally shift line onto other parallel line)
* Triangles
* Average angle from regular polygons
* Vertical angles
2. Have I marked all sides for which I know the length or ratio of lengths?
2. **QC with multiple inequalities**
1. ****if the left hand side is greater than the right hand side in both inequalities, then the sum of the left hand sides must be greater than the sum of the right hand sides. This simple algebraic move often simplifies matters considerably:
31
If you’re stuck on a “what must (or can’t) be true” question, what should you do?
1. list possible values for the most constricting constraint (make sure you include all values in the range)
2. Start eliminating answers
32
33
What does |x| = -x mean?
34
What do we know about the products of prime numbers?
The product of any set of prime numbers (regardless of size) is unique. That is, no other set of prime numbers will have the same product.
35
How can you think about average speed?
Average speed can be thought of as an average of the speed he was traveling at every single moment during his journey—for instance, imagine that Davis wrote down the speed he was going during every second he was driving, then he averaged all the seconds. Since Davis spent more time going 50 mph than going 60 mph, the average speed will be closer to 50 than 60, and Quantity B is greater. If the distances are the same, average speed is always weighted towards the slower speed.
36
What does this equal?
37
What's the formula for two overlapping sets?
total = total(A) + total(B) - \<# in both\> + neither
38
What's the probability of A or B?
39
What's the fastest way to find the weighted average?
1. Find the step size:
* Find the difference between the averages of the two parts
* Divide this by the sum of the simplified ratio of the two groups (e.g. 3n+4n=7n so divide the difference between the averages of the two parts by 7)
2. Figure out which group the weighted average is closer to
3. Add or subtract the appropriate amount of steps (**smaller number in the simplified ratio**) from the one it’s closest to to find the weighted average
40
What's the equation of a circle?
(x-h)^2 + (y-k)^2 = r^2 where (h,k) is the center and r is the radius
41
What two equations can you set up for traveler/motion questions?
distance (d=rt) and time (t=d/r)
42
What's the area of a square in terms of it's perimeter?
p^2/16
43
What's the height of an equilateral triangle equal to?
h = sqrt(3)\*s/2 where s is the length of a side of the equilateral triangle
44
How do you find the number of shared factors between two numbers?
The shared factors are the factors of the GCF.
45
Immediate reaction:
Variables in the question and variables in the answers problem
Is it easy to derive the algebraic expression? If not, would choosing smart numbers be easier? Choose the strategy I think would be easiest for the problem
46
When we decide that we're going to try to solve a problem by backsolving (systematically testing the answer choices, what's our first thought?
Start with the median value of the answer choices (if the answer choices are numbers)
47
What should you be thinking for this problem:
A certain movie star's salary for each film she makes consists of a fixed amount, along with a percentage of the gross revenue the film generates. In her last two roles, the star made $32 million on a film that grossed $100 million, and $24 million on a film that grossed $60 million. If the star wants to make at least $40 million on her next film, what is the minimum amount of gross revenue the film must generate? Answer in terms of $ millions.
**Recognize that the salary difference comes completely from a percentage of the gross revenue! Thus, her percentage take, p, equals the difference in her salary over the difference in the overall revenue.**
One way to do this problem is to recognize that the star earned $8M more ($32M - $24M = $8M) when her film grossed $40M more ($100M - $60M = $40M). She wants to earn $40M on her next film, or $8M more than she earned on the more lucrative of her other two films. Thus, her next film would need to gross $40M more than $100M, or $140M.
48
What strategy should be used when you have parallel lines?
Mentally superimpose one of the lines on top of the other or connected to the other to make sure I've discerned all the angles possible from it.
49
What do we know about angles inscribed in a circle?
1. If two inscribed angles hold chords of the same length, the two inscribed angles are equal.
2. If the chord of an inscribed angle is the diameter of the circle, the inscribed angle is a right angle.
50
Problem Reaction:
You have two solutions being mixed.
**Set up your solution table and circle the value you're trying to find!**
## Footnote
Columns: % of each solution, volume of each solution
Rows: Solution A, Solution B, Final Solution
51
What do we know about the diameters of two circles?
The ratio of circumferences of two circles = the ratio of the diameters
52
List all numbers that equal their own square root.
0, 1
53
Immediate reaction?
You suspect two triangles are similar and want to confirm this
Any one of the following are sufficient to prove a pair of triangles are similar:
54
What's the divisibility rule for 7?
55
How do you reflect a point over the line y=-x?
Flip x and y and make each the opposite sign (e.g. (m,n) reflected over y=-x becomes (-n,-m)).
56
For a circle inscribed in a square, what's the ratio of the area of the circle to the area of the square?
circle area/square area = pi/4
57
What should you do if you’re asked to find the sum of a very difficult sequence?
Check if the first and last terms cancel each other out, or if many of the terms are positive/negative pairs.
58
What's the quadratic equation? What do we know about it?
y=a\*x^2+b\*x+c It opens upwards if a \> 0 and down if a \< 0.
59
What should you do when you have a pretty big factorial on either side of a QC question?
Divide the smaller factorial over to be the denominator underneath the larger factorial. This will allow a lot of factors to be cancelled.
60
What's the relationship between the area of a square and its diagonal?
A shortcut involves the relation between the area and the diagonal:
A = d^2 / 2
61
If you hear percent change when talking about figures of percentages, what should you think?
This is not a pure sum or difference! I still need to use the percent change formula and treat the percentages just like any other numbers. They're just asking for percent change on a set of percentages to throw me off.
62
For a problem where you're told the type of shape (e.g. square, triangle etc.) and given a couple points where the vertices are, what should you think first?
Don't draw the diagram just yet. First, check to see if the vertices share a common x coordinate or y coordinate and can be solved without drawing. This could save 30 seconds. This is helpful for problems like the one in the attached picture.
63
What happens to the standard deviation of a set when you do some common operations on it? (2)
64
What is a good percentage misconception to watch out for?
The difference/sums of percentages of two different values are not equal, even when the difference in the pure percentages is the same. Example:
* Reducing from 50% of 100 (50) to 40% of 10 (4) creates a difference of 46.
* Reducing from 30% of 100 (30) to 20% of 10 (2) creates a difference of 28.
* Even though these are both 10%, the differences are not equal
65
What is 2^5?
32
66
What does a sum of a set of numbers equal?
sum = average\*# of terms
67
How can this be interpreted?
The distance on the number line between x and 5
68
Immediate reaction:
Absolute value inequalities QC question
My strategy is **4 parts**, n order of what I should try first:
1. Can any of the variables be 0 based on the conditions? If so, does that simplify the problem?
2. Spend **no more than 15 seconds** trying to think about it logically. This could provide a quick solution.
3. Try to solve algebraically if and only if:
* \<=2 variables and \<=2 absolute values
* 3+ variables and only one absolute value
4. If the problem involves 3+ variables and multiple absolute values, plug in numbers!
69
What should you do if you need to find the average increase/decrease per unit time on a data analysis question?
Don't use inclusive counting!
70
What’s the fastest way to figure out which number in an ordered set is the median?
* For **odd** number of terms: Divide number of terms by 2 and round up. (e.g. 11/2=5.5, so the 6th term is the median)
* For **even** number of terms: Divide number of terms by 2 and take the average of that term and the next term up (e.g. 46/2=23 so take the mean of the 23rd and 24th terms)
71
What is the fastest way to figure out the new value after a certain percent increase from a known original value?
Just multiply by the multiplier (in this example, 1.3). That's faster than plugging into the percent increase formula.
72
How do you reflect a point over the line y=x?
Flip the x and y coordinates (e.g. the reflection of (m,n) is (n,m)).
73
Immediate reaction?
How many nonzero digits a decimal will have?
Try to factor out powers of 10 in the denominator or the numerator. These will have no effect on how many nonzero digits the number has.
74
What's the formula for percent change?
100\*(new - original)/original
75
Immediate reaction?
Find positive factors of a number
76
If you have the sum of a set of numbers and the number of data points in the set, what should you immediately think?
I can calculate the mean of the set.
77
If you are given a number and told that it's is the result after some other number was increased by p percent, how do you find the old number (i.e. the number before the percent change)?
**This works for percent decreases too, but** **you must make p negative (multiplier\<1)!!!**
**Example:** If $200 is a 20% markup, what was the original value?
200/1.2 = 100/(6/5) = 50\*5/3 = $83.33
78
What’s a good strategy if you’re stuck on a problem asking for the number of occurrences of a certain thing in numbers between x and y?
Use slots for each digit
79
If you're given the remainder when a variable, x, is divided by some number and the remainder when another variable, y, is divided by the same divisor, and asked to find the largest integer that must be a factor of the product, x\*y, how should you do this?
1. Write out rebuilding the dividend equations for x and y.
2. Multiply those equations together to find an equation for x\*y. Factor out everything you can from both equations. The integer that you factor out is the largest integer that must be a factor of the product x\*y.
80
How do you solve an absolute value inequality?
Create two equations:
1. Same equation without the absolute value sign
2. Flip sign, make the side without the absolute value sign negative, and remove the absolute value sign
81
Immediate reaction:
A number has an odd number of positive factors
It's the square of an integer. Only squares of integers have and odd number of positive divisors.
82
Immediate reaction?
**b \> 0**
The left side of the inequality is greater than 1, so it is positive. Because |a| is non-negative by definition, b would have to be positive.
83
What's a good strategy if you're stuck on a counting problem?
Can I count the ones that don't obey the restriction then subtract that from the total?
84
With counting questions that ask for how many arrangements of something you can make, what should you be thinking about?
Watch out for reduntant (duplicate) terms! e.g. how many ways can you arrange the letters in "illusion", need to divide by (2!\*2!) to account for the "i" and the "l" duplicates.
85
Immediate reaction?
How much would have to be added to x to make the fraction of x equal to another fraction?
**e.g.** Which of the following expressions approximates the number of women who would have to enlist in the Army to make the fraction of Army personnel who are women equal the fraction of Air Force personnel who are women?
Try to use the ratio of x to another part instead of x to the total. That could simplify the math considerably.
**e.g.** Make sure to use the ratio of men to women, not women to total.
86
What's the formula for three overlapping sets?
grand total = total(A) + total(B) + total(C) - \<# in 2 of the 3\> - 2\*\<# in all 3\> + \<# in none of the 3\>
87
If you try to apply the double matrix formula and realize you can’t fill in all the spots, what should you do?
1. First, double check to make sure you didn’t miss a constraint
2. If you didn’t miss a constraint, pivot and try to use the formula for two overlapping sets
88
Immediate reaction:
Percent increase question?
Would it be easiest to plug in numbers?
e.g. How much money is left over after a series of percent increases/decreases?
89
What are some special cases of absolute value equations you need to watch out for? (1)
When you have a variable in the absolute value and the same variable not in the absolute value, you need to check the answers that you get!
90
For a square inscribed in a circle inscribed in a larger square, what's the scale factor of between the small square and the large square and the ratio of their areas?
* side of large square / side of small square = sqrt(2)
* Area of large square / area of small square= 2
91
What do we know about the combinations formula?
nCr=nC(n-r)
92
What are "smart numbers" for the choosing smart numbers strategy?
93
Immediate reaction?
A problem that asks for percents and use no real numbers (maybe uses variables).
For problems that ask for percents and use no real numbers, it is almost always possible to use **100** as a starting number.
94
Rules:
Adding and subtracting evens and odds
95
What are the equations for work rate?
For one worker: w = r\*t
For multiple workers: Work = Individual Rate × Number of Workers × Time
96
Immediate reaction?
You try to solve for the roots of a function f(x) using the quadratic formula and find that it has a negative under the square root.
It has no real roots, so the function is either positive for all values x or negative for all values x.
97
If we know x^y \> x^z, how to y and z compare?
**If y and z are both positive:**
1. If **x \> 1**: y \> z
2. If **0 \< x \< 1**: y \< z
3. If **x \< 0**: Plug in to compare
98
What the formula for the area of a triangle with a vertex at the origin?
99
If you have an inequality that says an the absolute value of something is less than some variable, what should you think?
That variable must be positive, since the absolute value is less than it.
100
What's the process for determining which values of a variable make an expression an integer?
**Key idea:** The numerator must contain all of the factors in the denominator.
* Try to isolate the variable in a simple expression by pulling integers out of the expression.
* If variable is in the **numerator:**
1. Isolate the variable so that that term has its own denominator (i.e. there's no addition or subtraction in the numerator).
2. Compare the numerator with the denominator in that term. The variable in the numerator must have the factors in the denominator that are not present in the numerator
* If variable is in the **denominator**
1. ****Simplify the numerator as much as possible. Ideally get it into prime factors
2. The denominator must be a factor of the factors in the numerator
101
What should you do when you have one inequality with three unknowns on a Quantitative Comparison question?
Given one inequality with 3 unknowns, do not try to solve or simplify. Instead try plugging in values in an effort to prove D
102
For a circle inscribed in a square inscribed in a larger circle, what's the scale factor between the small circle and the large circle and the ratio of their areas?
* Scale factor: radius of large circle / radius of small circle = sqrt(2)
* Area of large circle / area of small circle = 2
103
For a circle inscribed in an equilateral triangle inscribed in a larger circle, what's the scale factor between the small circle and the large circle and the ratio of their areas?
* Scale factor: radius of large circle / radius of small circle = 2/1
* Ratio of areas: 4/1
104
What do we know about symmetrical distributions?
Mean = median = 50th Percentile
105
What do we know about the angles of a triangle?
Two angles of a triangle must be acute, the other could be acute, right, or obtuse
106
What if you need to add or subtract inequalities with signs facing the same way but one has "or equal to" and the other doesn't?
The resulting sign does not have the "or equal to" sign
107
For an equilateral triangle inscribed in a circle inscribed in a larger equilateral triangle, what's the scale factor of between the small triangle and the large triangle?
side of large triangle / side of small triangle = 2/1
108
What do we know about ratios?
**Ratios can be negative!!!**
109
What do you know if x is even and sqrt(x) is a prime number?
x=4
110
What do we know about the GCF and LCM?
For any two integers P and Q: P\*Q = LCM \* GCF
111
If you have a QC question where you're given an equation that relates x and y, and you're asked to compare the values of x and y, what should you do? (see example problem in attached picture)
Isolate either x or y in the equation, and plug it into the comparison so that the comparison only involves one variable.
112
Problem Reaction:
You're given the ratio of the perimeters of two similar shapes.
Immediately recognize you know the scale factor!!! This is likely key to the problem
113
When you have the square of some number x, how do you get the square of:
1. x + 1
2. x - 1
1. Add 2x + 1
2. Subtract (2x-1)
114
Rules:
Multiplying Evens and Odds
If there's any evens numbers being multiplied, the product is even. Otherwise, its odd.
115
What can an equilateral triangle be split into?
1. 6 congruent 30-60-90 triangles.
2. 4 smaller equilateral triangles
116
For a square inscribed in a circle, what's the ratio of the area of the square to the area of the circle?
square area/circle area=2/pi
117
Describe the approach if you're given a mixture question and told (e.g. [this question](https://greprepclub.com/forum/topic12442.html))
1. The percentages of the constituent mixtures A and B
2. The percentage of the final mixture
and you're asked to find the amount of B that needs to be added to a certain amount of B.
Immediately
1. Set up your table!!!
2. Write down this proportion: **Volume A /Volume B = Distance from B / Distance from A**
3. Plug in all numbers to the proportion and add a variable where necessary
4. Solve for the variable.
5. Quickly double check that the mixture (A or B) that has more volume in the final mixture is the mixture whose percentage is closer to the final mixture's percentage.
118
List 4 pythagorean triplets
119
How do you find the possible values for the total size of a set when you have the ratio of the individual components of that set?
1. Combine all components into one ratio
2. Simplify (divide) the ratio until the smallest number in the ratio is a prime number
3. Add the components of this simplified ratio together
4. **The total must be a multiple of the sum of the components of the simplified ratio**
120
To find the roots of a function f(x), what do you do?
1. Get all instances of x on one side of the equation
2. Remove radicals
3. Set that equal to 0
121
Immediate reaction?
Percent change
It could be a percent decrease or increase! Don't assume it's a percent increase.
122
Immediate reaction?
x \< |x|
x \< 0
123
What do we know about squaring a number?
Squaring a number **does not change whether it’s odd or even**
124
What are the rules for whether a negative value of a square root should be used on the GRE?
From Magoosh (source: [Positive and Negative Square Roots on the GRE](https://magoosh.com/gre/2016/positive-and-negative-square-roots-on-the-gre/)):
1. **If the actual square root symbol (officially known as the principal square root) does appear in the question, you should only consider the positive number** because the square root symbol, by definition, only has positive outputs. In all cases in which this symbol appears as part of the question itself, you NEVER consider the negative square root, and ONLY take the positive square root.
2. **If the square root symbol does not appear in the actual question, always consider the negative value.** What does appear is, for instance, a variable squared, or some other combination of algebra that leads to a variable squared, and you yourself, in your process of solving the problem, have to take the square root of something in order to solve it. The act of “square rooting” is not initiated by the test maker in the act of writing the question; rather, it is you who initiated the square-rooting. In this case, 100% of the time, you ALWAYS have to consider both the positive and negative square roots.
125
What are the properties of reflections in the x-y plane?
126
Problem reaction:
Absolute value QC question with multiple variables.
First thought: Can these variables be 0? If so, does that simplify the problem considerably?
127
If you need to figure out which large number is divisible by a different number that's pretty big, what do you do? e.g. Which 5-digit number is divisible by 24
Take the prime factorization of the divisor (in this case, 24), and start checking the answer choices for the factors from smallest to largest (e.g. 2, then 4, then 8) while eliminating choices. The answer is the choice with all the factors.
128
Immediate reaction?
Find the number of multiples of a certain number in a range
1. Use a calculator to find the indexes of the highest and lowest terms:
* **In****dex of highest multiple =** Upper bound of range/the number (round down)
* **In****dex of highest multiple** = Lower bound of range/the number (round up)
2. Use inclusive counting rules between the indexes of the lowest and highest multiples in the range
129
What can we do if we have the root of an equation?
Plug it into x to solve for any one coefficient in the equation
130
What are the equations for a parabola?
131
If you’re trying to derive an algebraic equation for something and it’s taking a while, what should you do?
Choose smart numbers and plug into the answer choices!!!
132
What are the divisibility rules for 3, 4, 5, 6, 8, and 9?
3 : sum of digits divisible by 3
4 : the last two digits of number are divisible by 4
5 : the last digit is either a 5 or zero
6 : even number and sum of digits is divisible by 3
8 : if the last three digits are divisible by 8
9: sum of digits is divisible by 9
133
When are the volume of a cube and the surface area of a cube equal to each other?
when side length = 6
134
Describe the situation in which you’d use the three criteria venn diagram and the situation in which you’d use the double matrix method
* **When to use the three-criteria venn diagram**: A Venn diagram is best when we have three categories and all three can overlap, or 2 of 3 can overlap.
* **When to use the double matrix method**: When we can put everything into either A or B and also into C or D.
135
What are the rules for operations on inequalities?
1. Squaring
* If both sides are positive, you can square without changing the sign
* If both sides are negative, you need to flip the sign when you square it
* Cant square it if signs are different
2. Can always raise both sides to an odd power
3. Addition
* Can only add inequalities with the same signs
4. Substraction
* Can only subtract inequalities of opposite signs
5. Multiplication
* Multiply both sides by positive, don't have to change sign
* Multiply both sides by negative, do have to flip the sign
136
What should you be thinking when you see this equation?
Just because there are two variables doesn't mean I can't solve it.
137
What should you think if you have factors of 2 and 5 each raised to some power next to each other?
I can make 10s with those. That could be key to solving the problem.
138
What are question types you should always double check because you fuck it up?
* Geometry questions asking for the proportion of the figure that is shaded
* Questions asking for an algebraic expression as the answer (variable in the answers problems)
* Data analysis questions
139
What is 2^10?
1024
140
Describe the approach for this question ([source](https://www.varsitytutors.com/gmat_math-help/word-problems/problem-solving-questions/mixture-problems))
A chemist wants to make one liter of 25% hydrochloric acid solution. He only has two concentrations on hand - 10% and 50%. How much of the 10% solution will the chemist use?
We know the percentages of the constituent solutions and the final solution, and we know the volume of the final solution, so:
1. Find the ratio of the volumes and add these to the table using the multiplier, n.
2. Solve for n by setting up the equation sum of the volumes of the constituent mixtures (using n) = the final volume.
3. Plug in back into the volume of the appropriate equation
141
What do we know about evenly spaced sets?
Mean = median
142
What are you immediately thinking?
In the given inequality, if y is moved to the right hand side, we have:
| x| \< y
**Because y is greater than the absolute value of x, y must be positive.**
143
What do we know about parallelograms?
1. Equal opposite sides
2. Equal opposite angles
144
What’s a root of a function?
* A root of a function f(x) is an a value of x such that 𝑓(x)=0 (need to move everything in the equation to one side and have the other equal 0)
* The x intercepts
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Immediate reaction?
A number has three positive divisors
Only squares of primes have three positive divisors
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Immediate reaction?
All the values given in a problem and its answers are percents, ratios, or fractions
If all the values given in a problem and its answers are percents, ratios, or fractions of some unknown, then the problem will probably be easiest to solve by stipulating values for the unknowns
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What should you do if you have a problem giving or asking for discounted/marked up prices and original prices?
Write down the meaning of the variables you use explicitly (e.g. whether the variable represents discounted or original)
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What do you do if you have the total in one group and the number who are in both groups and you want to find the number who are only in one of the groups?
Subtract the number in both groups from the total number of the subgroup for which you want to know how many are exclusive to that group.
