Quantitative Section

A is what percent greater than B. What does this translate to?

A = [(x/100)B] + B

Strategy: When comparing fractions

Cross multiplication is usually the best way to go

4/250 = ?

Here we are multiplying by 4/4 not to cancel but to make the denominator easier to deal with... (4 * 4) / (250 * 4) = 16 / 1000

Rule: Circumference to Diameter ratio

Pi or 3.14

The greatest prime factor of 40,002 is x The greatest prime factor of 80,004 is y

Prime Factorization of a number*2 so all the prime factors are the same multiplied by 2

What is the value of 2 + 2^2 + 2^3 + 2^ 4 + 2^5?

Trick is to not go through this and solve... 1) set the equation equal to x 2) the equation 2x would add one 2 to each 2 already there: 2^2 + 2^3 + 2^ 4 + 2^5 + 2^6 = 2x But we want x, so subtract x from 2x so all the common terms cancel out you are left with 2x - x = x = 2^6 - 2

In a group of 40 people, 15 have visited Iceland and 23 have visited Norway. If 11 people have visited both Iceland and Norway, how many people have visited neither country?

Two ways to solve this: Double matrix and Total minus common For the matrix put as Iceland and Not Iceland and Norway and Not Norway *Notice that 15+23=38 that takes into account Norway, Iceland and Both. We need to cancel both (11) to get 27. That means 13 people visited neither.

1/(2 - .25) = ?

We know that .25 is 1/4 is there a way to convert 2 to where it has the same denominator? Of course...

*How to write this into an equation: A helicopter company charges $85 for the first kilometer of a trip and $5 for every kilometer after that. If the total cost of a trip was $365, how many kilometers were flown?*

y=85+5(x-1) since is x=1, y=85

a^2b^2

(ab)^2 so... (a^-n) (b^-n) = (ab)^-n

So, we know the one for addition and subtraction, here is the one for multiplication: x>0 xy = 2 yz = 5 xz = 10

(xy)(yz)(xz) = (2)(5)(10) (x^2)(y^2)(z^2) = 100 xyz = 10

*Corner cases, 0s and 1s* Column A p/q Column (p/q)^2

*1) Always check corner cases (0, 1); and multiple options 2) just because we have two different variables p & q, doesn't mean that the number they represent has to be be different* Here it can be seen that any fraction lower than 1 will be lower in column 2. But if both p and q are equal then the two columns will equal. And if the fraction were greater than 1, then squaring it will lead to a larger number.*

Column A The average (arithmetic mean) of 3^2, 3^4, 3^6 Column B 3^4

*A number with an exponent represents ONE number. So the three numbers are divided by 3 not 12.*

If x is a positive integer and x+2 is divisible by 10, what is the remainder when x2+4x+9 is divided by 10?

*A quadratic can be broken down to help you solve. We cant split the 9.* *Another thing is to pay attention to what x is and use that to plug in and solve.*

Cam is 20 percent taller than Bea, and Bea is 20 percent taller than Ann. Column A Cam's height minus Bea's height. Column B Bea's height minus Ann's height.

*An easy way to solve this is to set up with variable and compare the variable* A=x, B=1.2x, C=1.2(1.2x) column A = 1.44x - 1.2x = .24x column B = 1.2x-x = .2x *Also when order is available, the order is preserved. So you don't have to check multiple options. But if you solve this way, you don't have to worry about checking multiple option.*

To create paint with a certain shade of gray, one must combine 2.016 liters of black paint with every one liter of white paint. Approximately how many liters of white paint must be combined with 350 liters of black paint to create the certain shade of gray?

*Be careful about which direction to go in approximations.* We can round 2.016 down to 2 350/2=175 so... 350/2.016 must be less then 175... *To avoid confusion, first set up the problem... 350/2.016=x. Now it's easier to see that dividing by 2+ will gives us something less than 175*

If x > 21/40 , then x could be which of the following?

*Break down fractions into components* 21/40 = 20/40 + 1/40 so... 1/2 + 1/40 19/36 would equal 18/36 + 1/36, well all we are comparing is 1/40 to 1/36 and we know that 1/36 is bigger so 19/36 would have to be bigger as well

2^n + 2^n + 2^n + 2^n = 4^(n + 3)

*Careful about how you group these together.* 4(2^n) = 4^(n+3) 2^2(2^n) = 4^(n+3) 2^(2+n) = 2^2(n+3) *You CANNOT add the 2^n's together because that would change the base.*

If k ≠ 0, k ≠ ±1, and (k^3 k k^4)^2/(k^n k) , then n =

*First combine all the like terms* (k^8)^2/(k^n+1)= k^14, now you can solve *Get the same base and use the fact the x^a = x^b, so a = b.*

Square ABCD has a side of 20. Circle O has a radius of r. If the circle has more area than does the square, then which of the following could be the value of r? Indicate all possible values of the radius. 8 9 10 11 12 13 14 15

*First solve what you know and write it down. Area of square is 400 and Area of a circle is 3.14(r^2). Start with C and start eliminating.*

The revenue generated by Company X is divided between Doug and Moira in a 6 to 5 ratio respectively. Column A Moira's share when the revenue generated by Company X is $15,700 Column B $7900

*Half and half* If it is split evenly then 7850, but Moria right now is receiving less than that so A has to be smaller so B is the right answer

In a group of 100 adults, 75 percent of the women are left-handed. If there are 12 right-handed women in the group, how many men are in the group?

*Here there are two people with two characteristics, left and right handed. We can use a double matrix method.*

Which of the following, when rounded to the nearest integer, are rounded to 3? 3.4 3.445 3.494 3.499 3.501 3.61

*Here we only look at the number directly to the right of the decimal point. Thus the first four are the right answers.*

A, B, and C are consecutive odd integers such that A < B < C. If A + B + C = 81, then A + C =

*I was right in that you have to set this up in terms of one variable. When the starting variable is not given in this case assume x is the odd and the next odd then would be x+2, and the next odd would be x+4.*

If k is an odd integer, which of the following must be an even integer?

*If you notice that logic is going to take some time. Use 1 for odd and 2 for even*

For positive numbers X,Y,Z we have Z/Y= 35/6 and Z/X=35/8. Column A X Column B Y

*It is a mistake to assume that Z = 35 and X = 8 and Y = 6* These are all ratios, so the best we can assume is that Z = 35k, X= 8k, and Y= 6k. And so we notice that 8k is always going to be bigger for any positive number than 6k.

84/400

*Notice that both of the numbers can be divided by 4* so.. 21/100 = 21%

If K is the least positive integer that is divisible by every integer from 1 to 8 inclusive, then K =

*Pay attention to the question. It is asking for the least positive integer. You can go through the multiplication and if all the ones work then the least is A.* Another way to solve is to figure out what the factors need to be to be divisible... (1)(2)(3)(2)(5)(7)(2) = 840. This is the lowest number that could be divisible by this number. Everything else is a multiple of it.

If xy ≠ 0 , and 75 percent of x equals 125 percent of y , which of the following is true?

*Realize that 75% is 3/4 and 125% is 5/4 and solve out that way.*

k is a positive number. If k is twice its reciprocal, and j is twice k, then jk =

*Solve for k immediately. Do not just leave it hanging.*

Column A 43 percent of 63 Column B 63 percent of 43

*Sometimes it is helpful to write out percents as 43/100 and 63/100.* So (43/100)*63 = (63/100)*43 through the associative property of multiplication.

The sum of 5 consecutive even integers is 0.

*The best way to solve this is to put it in terms of x* x+x+2+x+4+x+6+x+8=0 x=-4 so our numbers are -4, -2, 0, 2, 4

Column A 43 percent of 207 Column B 85

*To avoid calculator realize that 43% of 100 is 43, and so 43% of 200 must be 86. So 43% of 207 must be 86+. Thus A is the larger column.*

Nina has exactly enough money to purchase 6 widgets. If the cost of each widget were reduced by $1.25, then Nina would have exactly enough money to purchase 8 widgets. How much money does Nina have?

*To set this up you have to assign the right variables.* x=price of 1 widget n=amount of money nina has so... 6x = n is the price of each widget goes down by 1.25 and nina can still buy 8 then... 8(x-1.25) = n ... since both equations are equal to n 6x = 8(x-1.25)

If 3 apples and 4 bananas costs $1.37, and 5 apples and 7 bananas costs $2.36, what is the total cost of 1 apple and 1 banana?

*We can create new equations, by subtracting from other equations.* *Multiplication is the route that GRE wants you to take usually.*

Cylindrical tank A has radius x and height y. Cylindrical tank B has radius y and height x. x = 2y Column A Volume of Tank A Column B Volume of Tank B

*When given something like x=2y, set everything in terms of y because then you don't have to deal with FRACTIONS.* You don't need to plug in numbers simply set up the same equations and solve

Ben is three times as old as Ron. Ed is 8 years younger than Ben. Ron is 7 years older than Ken. If the sum of the ages of all four people is 161, how many years old is Ron?

*When there are multiple items, set the problem in terms of one variable. Here it makes sense to do it according to Ken's age, since he is the base and his age isn't related to anything else.* K=ken's age K+7=Ron 3(K+7)=Ben 3(K+7)-8=Ed

Strange Operators trick

*You can simplify the operation first, and you can make this go even faster* Do this if it is possible. Usually works with algebraic equations common on the standardized tests.

Seven years ago Bob was k times as old as Ann. If Ann is now 11 years old, what is Bob's present age in terms of k? *Do not keep re-writing formulas. Think logically about what is happening when you plug one thing into another.*

*You cannot treat the past as now. 7 years ago was 7 years ago. Treating the past as now will lead to double the right answer or lead you to subtract years instead of adding them.* You can organize age in a double matrix format with the people being of the rows and two time periods on the columns.

*When we see a sqrt, we the test only means the positive root*

-3 = sqrt(9) This says that -3 = +3 and that is not true. So that can't work.

*Tip with square roots*

-Besides squaring all of these, you can also write everything as roots, double the outside and put a square root over it and multiply it out. For example 2sqrt(13)=sqrt(4*13)=sqrt(52)

If four numbers are randomly selected without replacement from set {1, 2, 3, 4}, what is the probability that the four numbers are selected in ascending order?

-Best way to do this is to think about how many possible ways are there to organize 1,2,3,4? Well 24. And only one of those is in ascending order so 1/24 -Another way to thing about this is (1/4)(1/3)(1/2)(1/1)

*Roots*

-Careful about how the operator is applied! Work from inside to outside -With roots higher than 1/2 work in fractions

*Mathematical mistake to avoid.* (p+q)/p = 10/7. What is p-q/q solve by realizing that p/p + q/p and break down the second equation too. p/q - q/q

-DO NOT separate numerator and denominators. The fraction is one number it cannot be broken down further! -Instead trying breaking up the fraction into smaller components and solve out. Of course you can use can use reciprocals of a number so a/b can be b/a (to represent reciprocals)

Three friends are buying a gift for a friend. Declan contributes 4 dollars more than 1/4 the cost of the gift, Ed contributes 1 dollar less than 1/3 the cost of the gift, and Frank contributes the remaining 22 dollars. What is the cost of the gift?

-Good job on identifying what c was going to be *Notice that when you set this up: (c/4)+4 + (c/3)-1 + 22 = c *You do not have to combine the fractions with the whole numbers immediately. Think about making the process faster. Use the associative property of addition and add 1-3+22 to get 25 and then you are left with c/4+c/3+25. Much easier to deal with

Phil has three times as many DVDs as Sam as. Even after Phil gives Sam 6 DVDs, he still has 16 more DVDs than Sam has. What was the original number of DVDs that Phil had?

-Here make two equations but be careful about how to set up the second one... It is not like the age problem AND is a completely different equation than the first. 1) P = 3S 2) P - 6 = 16 + S + 6 S = 42

73^2 + 74^2 =

-One approach is just to do it the normal way -However based on the answer choices realize that the last digit of 73^3 is 9 and the last digit of 74^2 is 6 and so the last digit of the two numbers is 5... If the answer choice has the last digit as a 5 then that is the correct answer

*Stages and independency* Hal has 4 girl friends and 5 boy friends. In how many different ways can Hal invite 2 girls and 2 boys to his birthday party?

-Think about this in two stages, which are independent so it will be a multiplication of the two (stage 1 = girls) (stage 2 = boys)

Eighteen students attained an average score of 70 on a test, and 12 students on the same test scored and average of 90. What is the average score for all 30 students on the test? A) 78 B) 80 C) 82 D) 85 E) Cannot be determined

-This problem although not hard can become tedious if not approached with a GRE mindset -Do not multiple out the numbers, just write out the equation and see if things simplify before attempting to do arithmetic Ex: [(70*18) + (90*12)] / 30 = can be simplified first by removing the zeros since 10 can be factored from the top. Also, both the multiplication numbers are multiples of 3 and can be simplified as well

The average length of 10 objects is 25 inches. If the average length of 2 of these objects is 20 inches, what is the average length of the remaining 8 objects? A) 22 1/2 B) 24 C) 26 1/4 D) 28 inches E) Cannot be determined

-This question is not hard but can be hard depending on the choices that the test makers can put down if you forget what you are looking for -We want the average of the 8 objects left so... [(20*2)+(8x)]/10 = 25 If you just get x and forget to to add the 8 then you will just get the sum of the remaining 8 scores not their averages

Strategy for Multiple Correct Answers

-Usually you have to go through and check each one to see which one is correct -Key is to identify an equation that you can quickly go through and plug each one into it to see if the answers work or not

Number Property Rule:

-With fractions less than 1, let's say you have 3 of them, then there is absolutely no way to get a number greater than 3, since then the fractions would have to be greater than 1 -But it is possible to get a whole number just not greater than the number of fractions available - sum of value of fractions less than one < the number of fractions available

In a certain town in Connecticut, the ratio of NY Yankees fans to NY Mets fans is 3:2, and the ratio of NY Mets fans to Boston Red Sox fans is 4:5. If there are 300 baseball fans in the town, each of whom is a fan of exactly one of those three teams, how many NY Mets fans are there in this town?

1) Always just like the given information as is, so you don't confound the information. Then proceed to answer the question 2) For ratios always write down in terms of n. 6n:4n:5n

Strategy: Rule for exponents

1) x^2 (or any other even number): We don't know if x is negative or positive 2) x^3 (or any other odd number): We know for a fact based on the outcome that x can only be positive or negative

*Division by the decimal or fraction*

1/decimal OR 1/fraction is going to be a BIGGER number than 1/whole number

*Look out for problems in which the equations are equal to the same thing! Then you can set the part equal to each other.* If 100,000 microns = 1 decimeter, and 1,000,000,000 angstroms = 1 decimeter, how many angstroms equal 1 micron?

100,000m=1,000,000,000a

*Tips for Roots*

2^n+2^n=4^n 2^3 = (2)(2)(2) = basically the root shows us that the base is 2 and we have 3 of them

*It is better to first just assign variables and the determine the ratio. Be careful about how you read these.* Sue planted 4 times as many apple seeds as she planted orange seeds. 15 percent of the apple seeds grew into trees, and 10 percent of the orange seeds grew into trees. If a total of 420 apple trees and orange trees grew from the seeds, how many orange seeds did Sue plant?

A=apple seeds O=orange seeds 4O=A

*Strategy: Geometry*

Always label everything you want to find with your own variables and then try to make an equation with what you know

*Key: Units Comparisons* Machine A produces 11 nails in 14 secs Machine B produces 7 nails in 9 secs Quantity A: Time is takes machine A to produce a nail Quantity B: Time is takes machine B to produce a nail

Careful about how you set this up. We don't want nails by second but "seconds per nail"... so.. 14secs/11 nails ?? 9secs/7nails Now we can cross multiply to get the right answer "B" Otherwise is we had done 11/14 ?? 7/9 we would've have gotten A since machine A can produce more nails per second since it is faster. But this questions is asking about time and B takes more time!

A man rode a bicycle a straight distance at a speed of 10mph and came back the same distance at 20mph. What was the man's total number of miles for the trip back and forth, if his traveling time was 1hr?

D1 = D2 D1 = 10mph * T1 D2 = 20mph * T2 T1+T2 = 1 So be pretty easy to solve now. When you are given information regarding time see if it equals a certain number.

If y+2q=15, q+2p=5, and p+2y=7, then find the value of p+q+y

Do not approach this by solving for each variable, they want what p+q+y is so solve that way y + 2q + _ = 15 _ + q + 2p = 5 2y+ _ + p = 7 So, 27 = 3y+3q+3p The answer is 9

If 0 < x < 1, then which of the following must be true? I) 2x < 2 II) x-1 < 0 III) x^2 < x

Do this by splitting the original into two parts

Given that a/b is less than 1, a>0, b>0. Which of the following must be greater than 1?

Do this by taking the statement a/b < 1 and manipulating is around

a>1 Quantity A: (a-3) (2a+1) Quantity B: (a-3) (a+2)

Here is one of the fundamental rules of comparisons: The equations must hold in EVERY SINGLE SCENARIO -So if A is bigger with one set of numbers, but is smaller with another set of numbers then the relationship is indeterminable

sqrt(28) + sqrt(175) = a*sqrt(b) What is a+b?

Here to add radicals you have to get a common radical and we get 2sqrt(7) + 5sqrt(7) = 7sqrt(7) Now... Do not be confused a is obviously 7 b is also 7 NOT sqrt(7). sqrt is just an operator of sorts, the number inside is b. *Variables can only be numbers not the operators on them. They can be positive or negative though.*

*Combinations in stages* Car X can come with any of these 5 additional features: sunroof, stereo, tinted windows, leather seats and cruise control.

Here we have five stages where we can decide to either have an option or not so the combinations then become (2)(2)(2)(2)(2)=32 Can this be solved using combinations? Yes, though it's a little more complicated. We have 5 options, so we need to choose 0, 1, 2, 3, 4, and 5 options and add them together for our total possibilities. That gives us: 5C0+5C1+5C2+5C3+5C4+5C5 1+5+10+10+5+1 32

Strategy: Dash Method for Counting

How many number are there below 20,000 which have prime numbers as each of their digits? Answer is 340; figure it out

If (p+1)/(r+1) = 1 then...

I just want to highlight here that you can keep going when a variable is equal to another.. p+1 = r+1 equals... p = r ... You can keep going p/r = r/r = 1

*Triangle inside a Square*

If the triangle's base is the entire side, then the ratio of area of triangle to area of rectangle will always be 1:2

Geometry Rule:

If two opposite sides of a quadrilateral are 90-degrees then that means that the other two angles must form 180-degrees and can be bisected so that two sides of that shape will be equal and the other two will be equal (try to draw a 90-140-90-40; it's kind of like a weird rhombus-like figure)

*When things are equal to each other, write out in terms of one variable.*

In 2003, the number of girls attending Jefferson High School was equal to the number of boys. In 2004, the population of girls and the population of boys both increased by 20 percent. Which of the following could be the total student population at Jefferson High School in 2004? G=B so G=G so total equals 2G. In 2004 1.2(2G) = total this is equal to 12(x/5) so we need a number divisible by 3 and 4

*Mean and Median are equal if the list is spread out evenly OR if the deviation from the mean is equal.*

List S = {16, 9, 23, X, 13, 16} In list S, the mean and median and mode are all equal to one another. What is the value of X? the deviation that would make this work is +3 so 19 has to be the answer.

*Use prime factorization to your advantage. If you must check answers, start with E first.* If k is an integer, what is the smallest possible value of k such that 1040k is the square of an integer?

My first intuition was that squares of a number have an odd amount of total factors because all their individual prime factors must be symmetrical and even. This approach leads to (10)(104) = 2,5,2,2,2,13,k we need to group these evenly so all factors are symmetrical (2,2,5,13)(2,2,k) so... k must be 65

Strategy: When equation/inequality is set up

Once an equation or inequality is set up or given then first try to figure out how to manipulate (add or subtract or multiple or divide both sides by a variable or number) to see if you can get it to equal what you need. This should be especially obvious if the question is asking for a formula like answer rather than a straightforward number

*If you are comparing things and something pops up every time in the comparison. This thing can be left out to only focus on the remaining part.*

Population density is the population of a region divided by its geographic area. Of the seven continents, Asia has by far the largest population density. Which continent has the second largest? Africa North America South America Europe Australia -Here we have %of population/%of land. We can leave out the population/land and just focus on the percents.

*Strategy: Set up correctly, and then start plugging and chugging if nothing else makes sense. Something will make itself apparent.* a, b and c are positive integers. If b equals the square root of a, and if c equals the sum of a and b, which of the following could be the value of c? A) 21 B) 30 C) 45 D) 72 E) 100 F) 331

Setup: b^2 = a c = a + b so... c = b^2 + b -> *b(b+1)* (DO NOT THINK that b^2+b=b^3) -Now notice that we are dealing with consecutive integers and c is the result of two consecutive integers. Notice further that if that is the case then b(b+1) must be an even number so we can narrow down to B, D, and E but only B and D can be results of two consecutive integers *Also, think about it this way. c = b^2 + b. So go down the list starting from 1. Notice that once you get to 5 you notice that 5*5=25 + 5 = 30 so B could work... Also notice that 72 can work (b = 8) *You are asked what C could be. That doesn't mean you have to find what a and b have to be. Set up the problem. Solve as far as you can. And then start plugging and chugging if nothing else makes sense.*

If 6(x^5) = y^2 and x = 1/y, then y = ?

Since we are given an equation we can multiply the original by BOTH side to the one we need so x[6(x^5) ] = [y^2](1/y) = 6(x^6) = y

If x+y=7 and xy=4, then find the value of x^2 + y^2

Strategy: -Whenever you see the x^2 + y^2 of any format think of the following three rules: 1) (x+y)^2 2) (x-y)^2 3) x^2 - y^2 In this case approach via solving for (x+y)^2 = 49 and you are given xy=4 so 2xy=8 Hence x^2 + y^2 = 49 - 8 = 41

*Read very carefully and consider what it means. Especially if a number is negative* If [x] is defined as the greatest integer less than or equal to x. [-3.4] + [21] =?

The answer is NOT 18 and is a trick answer -The key is to notice "less than or equal to. If the number were 3.4, then 3 would make sense. But the number is -3.4, so the integer less than this would be -4 not -3 which would be larger! -4 + 21 = 17

*Dealing with average questions*

The equation can be simplified to average*number of items = sum of total items -Now the sum of total items can be dispersed any which way. It can be evenly distributed, it can be all towards one side or all towards the other side. There is no way to compare. *Thus the answer is usually D*

*Remember that when multiplying by a negative you can choose what part of the term you want to apply it to!* If 3xm + 2ym - 2yn - 3xn = 0 and m ≠ n , then what is the value of y in terms of x?

The equation simplifies to 2y(m-n)=3mx(n-m) since we want to cancel the m-n, apply -1 to 2y and apply -1 to (m-n) on the right hand side

*Assumptions on the math part*

The only assumption you can make is that if you see a line it is straight and not bent (can't make assumptions about parallel, etc.)

*When asking to check rules that you know, spell out the rule and do not doubt it and just check each one against it.* *For triangles this has to hold up for all sides! a > b+c but you also have to check if b > a+c. If not you have a problem.*

The sides of a triangle are 1, x, and x^2. What are possible values of x? We know that the sum of two sides must be greater than the third side. Plug in the values given for x and solve these out.

What are all the values of x such that (x-7)(x+3) is positive?

There is only one way multiplication of two things is positive: -When both the values are positive (x-7) > 0 and (x+3) > 0 OR -When both the values are negative Use that to figure this out

Strategy: Plugging in 1s and 0s

These work really well for easy problems [(-3x)^3]/[-3(x^3)]

sqrt(12 * 32 * 54)

This can be broken down into: sqrt(12) * sqrt(32) * sqrt(54)

*Remainder question* When positive integer N is divided by 167, the remainder is 35, and when positive integer K is divided by 167, the remainder is 17. What is the remainder when 2N+K is divided by 167?

To get the original number that is divided we can *add the remainder + divisor* -This is not the smallest number that can be divided. The smallest that can be divided is the remainder itself!

John is now m years old and Sally is 4 years older than John. Which represents Sally's sage 6 years ago?

Well right now J = m and S = 4+m Six years ago: S - 6 = 4 + (J - 6) So our answer is to the right of the equation m-2. Do not try to keep going and move 6 from Sally's side! That is not what the question is asking!

*We can get averages from percentages. It just depends what kind of information is given to us.* Percent is out of 100 so 0 households have 0 televisions. 8 households have 1*8 televisions. 24 households have 2 each so a total of 48, etc... so it would be total television divided by 100 households. BUT we can't solve this because of the "5 or more". We need the exact number.

What is the average (arithmetic mean) number of televisions per household? 0=16% 1=8% 2=24% 3=20% 4=20% 5 or more=12%

*Tip for quicker calculation*

When two fractions are set equal to each other. We can cross multiple and then multiply by something to cancel the denominators. n (15+ (10/n)) = can be simplified to 15n+10

*How to write out the equation* Whenever Art Dealer sells a sculpture, he earns a 20 percent commission on the first $12,000 of the sale price plus 15 percent of the sale price in excess of $12,000. If Art earned a $3,900 commission on the sale of a certain sculpture, what was the sale price?

commission = .20(12000) + .15(x - 12000)

If x^2 - y^2 = 12 and x - y = 4, then x =

recognize the that (x+y)(x-y)=12 so (x+y)=3 since we want x just subtract x+y from x-y to eliminate y and get 2x=7 so x=3.5

Recognize that if we are multiplying by the same percent and we are looking for the difference then...

x%(a-b)= x%a - x%b