GRE Quantitative: Arithmetic/Algebra
finding the least common denominator
1. Factor each denominator completely. Write any integral factor as a product of primes 2. find the product of the greatest power of each factor occurring in the denominators
Finding multiples of number
Adding the number to itself or multiplying by number. Can also find by dividing number by another ex) to find if 70 is a multiple of 14, divide 70 by 14, and you get whole number of 5, meaning it is a multiple
Example of graphing inequality: x>4
An inequality like this tells us that x can be any valye greater than 4. Can show this on a number line by putting an open circle on 4 and shading the numbers that are greater than 4.
Coordinates of a missing vertex example: You are graphing rectangle ABCD in the coordinate plane. The following three are vertices of the rectangle: A(2,1), B(5,1), and C(5,6). What are the coordinates of point D.
Point D will have to have the same coordinate as point A and the height of point C to make a rectangle. So coordinates (2,6)
greatest common divisor
The greatest number that divides into two or more numbers with no remainder.
In an arithmetic sequence whos common difference is negative...
The terms in the sequence are always decreasing
General recursive formula for arithmetic sequences
g(1)=A g(n)=g(n-1)+B
Equivalent ratio word problems: 5 pounds of avocados cost $9. Based on this, what is the cost for 1 pound of avocados?
- ratio is 5:9 - 5/5 to get 1 pound - 9/5 to get 9/5=1 4/5= $1.8
Multiplying 2 fractions example: 2/3 x 4/5
- (2x4)/(3x5) - 8/15 - Cant simplify, so this is it
Matching an input to a functions output example: The function f is defined as follows: f(t)=-2t+5=13 What is the input value for which f (t)=13
- -2t+5=13 - -2t=8 - t=-4
Solving systems of inequalities word problem example: Fleur wants to make tables and chairs. Each chair or table is made with the same number of wooden boards and nails. She has a total of 150 wooden boards and 330 nails. The following inequality represents the number of tables (t) and chairs (c) she can make with 150 wooden boards: 17t+6c </= 150 Additionally, the following inequality represents the number of tables and chairs she can make with 330 nails: 34t+ 27c </= 330 Does Fleur have enough boards and nails to make 3 tables and 9 chairs?
- 17x3+6x9 - 51+54 </= 150 YES enough - 34x3+27x9 - 102+243 </= 330 X Not true, not ennough - So she has enough boards, but not enough nails
Rewriting mixed number as improper fraction
- 3 and 4/5 - 3+ 4/5 - 1+1+1+4/5 - 5/5+ 5/5+ 5/5+ 4/5 - (5+5+5+4)/5 - = 19/5
How to find prime numbers
- 3 can be divided by 1 and 3 - 6 can be divided by 1, 2, 3, and 6
Inequality word problems example: John is making pasta dough. To make the dough, he mixed 3/2 cups wheat flour with 3/4 cups regular flour and some water. Write an inequality to express the relationship between wheat flour and regular flour
- 3/2 is the same as 1 and 1/2 if writing as mixed number so more than wheat flour - So 3/2 > 3/4 cups
Obtaining a function from an equation example: For a given input value b, the function f outputs a value a to satisfy the following equation 4a+7b=-52 Write a formula for f(b) in terms of b.
- 4a+7b=-52 - solve for a in terms of b! - subtract 7b from both sides -4a= -52-7b - a= -13- 7/4b - so if want formula for b, f(b)=-13- 7/4b
Contructing numerical expressions example: Alan found 4 marbles to add to his 5 marbles currently in his pocket. He then had a competition with his friends and tripled his marbles. Write a numerical expression to model this situation without performing any operations.
- 5 marbles+4 marbles - tripled marbles from this amount - 3(4+5)
Writing basic expressions as word problems example: The Running Aces card team won $548 playing in poker tournaments last year. The winnings were split evenly among the p players. How much money did each player recieve? Write your answer as an expression.
- 548/p
Equivalent ratio word problem: You are throwing a party and need 5 liters of Yoda Soda for every 12 guests. If you have 36 guests, how many liters of soda do you need?
- 5:12 - 36 is three times as many as 12 - three times 5 is 15 - So for 36 guests, you would need 15 liters of soda
Solving quadratic equations by factoring example: x^2-3x-10=0
- All we need to do is factor and solve like before - So (x+2)(x-5)=0 -x+2=0 - x=-2 -x-5=0 -x=5
Equivalent ratio word problems: Burger Barn makes dipping sauce by mixing 2 spoonfuls of honey with 1/2 spoonful of mustard. Sandwich Town makes dipping sauce by mixing 4 spoonfuls of honey with 1 spoonful of mustard. Which dipping sauce has a stronger mustard flavor?
- BB=2:1/2 - ST= 4:1 - To find this out, need to mutiply by 2 to find equivalent ratio. So 4:1 - They have the same concentration of mustard and same ratio of honey to mustard
Finding equivalent fractions using multiplication example: What number could replace a below? 2/3 = a/12
- First, figure out what to multiply 3 to by to get 12. 2/3 x ?/4= 12 - Next, we multiply the numerator by the same number as the denominator. 2/3 x 4/4= 8/12 - 2/3= 8/12, so we can replace a with an 8.
Interpreting two-variable inequalities word problem example: Goku wants to get over 9000 comments on his youtube videos. He gets the same number of comments for each cat video he uploads, and he gets the same number of comments for each dog video that he uploads. In the following inequality, C represents the number of cat videos and D represents the number of dog videos that Goku should upload to achieve his goal. 750C+450D>9000 According to the inequality, how many comments does each cat video get, and how many comments does each dog video get? Can Goku achieve his goal by uploading 8 cat videos and 9 dog videos?
- In this, assume C is the number of cat videos and 750 is the number comments he gets regularly. Same goes with D. - So each cat video gets 750comments -Each dog video gets 450 comments - if we have 750 times eight plus 450 times 7, is this greater than 9000? 750(8)+450(7)>9000? - 6000+3150 - 9150>9000, so yes, he can achieve his goal
Equation with variable in denominator example: 7- (10/x) = 2+ (15/x)
- need to multiply ENTIRE side by x, not just number, as well as right hand side - so x(7- 10/x) = (2+ 15/x )x - 7x-10 = 2x+15 - 5x-10=15 - 5x= 25 - x=5
Two step equation word problem: Marcia has jsut opened her new computer store. She makes $27 on every computer she sells and her monthly expenses are $10,000. What is the minimum number of computers she needs to sell in a month to make a profit?
- Let x= # of computers sold - Profit=27x-10,000 - We want profit to be greater than 0, so think about number of computers that wold get us to 0. See what will have her break even - So 0=27x-10,000 - 10,000=27x - x= 370.3 -She can't sell .3 of a computer, and she can't sell 370 bc then it's less than the amount to break even (370.3), so she needs to sell 371 computers
Percent change word problem example: Mobile users in India have gone up 20% in a year. There are 540 million mobile users today. How many mobile users were there in India last year?
- Mobile users have increased by 20% to 540 million.Need to ask, which number when increased by 20% becomes 540? - If we increase a number by 20%, it becomes 100%+20%=120% of itself - to find last years mobile users, letsask ourselves: 120% of what equals 540? - 120/100 x users= 540 - users=540x 100/120 users= 450 - There were 450 million mobile users in India last year
Equivalent ratio word problems: An ice cream shop uses the following ingredients to make 1 sundae. Ice cream- 2 scoops Sprinkles- 4 spoonfuls whipped cream- 2 tablespoons How many sundaes did the shop make if they used 32 spoonfuls of sprinkles?
- Ratio of spoonfuls of sprinkles to sundaes is 4:1 (4 spoonfuls for 1 sundae_ - 32/4= 8 so 8 times as many sundaes - So 8 sundaes for 32 spoonfuls of sprinkles
Adding mixed numbers with like denominator example: 2 5/13+ 7 6/13
- Rewrite 2 and 5/3 as 2+5/13+ 7 + 6/13 (broke up mixed numbers into whole number parts) - add 2 +7+5/13+6/13 - 9 11/13
Solving factored quadratic equations example: (x-1)(x+3)=0
- This is a product of two expressions that is equal to 0. Note any x value that makes either (x-1) or (x+3) 0, will make their product 0. - (x-1)(x+3)=0 - x-1=0 - x=0 -x+3=0 -x=-3 -Substituting either x=1 or x=-3 into the equation will result in the true statement of 0=0, so they are both solutions to the equation
Linear equations with unknown coefficients example: a(5-x)=bx-8
- Want to expand everything out, distribute a and get all x on one side - 5a-ax=bx-8 - Subtract bx from both sides. Since want all x terms on left and non x terms on rights, you also subtract 5a from both sides: -ax-bx=-8-5a - Factor out an x next. Multiply both sides by -1 first: ax+bx=8+5a - x(a+b)=8+5a - divide both sides by a+b - x= (8+5a)/(a+b)
Linear equations with unknown coefficients example: ax+3x=bx+5
- Want to solve for x, answer will be in terms of a,b, and other numbers - try to get all x on one side, so subtract bx from both sides - ax+3x-bc= 5 - Can factor x out: x(a+3-b)= 5 -to solve for x, divide both sides by thing x being multiplied: x= 5/ a+3-b
Evaluating expressions w variables example: cubes: The surface area of a cube is equal to the sum of the areas of its six sides. The surface area of a cube with side length x is given by the expression 6x^2. Jolene has two cube-shaped containers that she wants to paint. One cube has a side length 2. The other sube has a side length 1.5. What is the total surface area she has to paint?
- We know that the surface area of each cube is going to be 6x^2, where x is dimensions of the cube. So surface area of 1st cube will be 6(2^2). Second cube will be 6(1.5^2) - Sum of two cubes, so equation is 6(2^2)+6(1.5^2) - 24+13.5 - 37.5
Functional notation word problem exmple: Let M(t) denote the account balance M (measured in dollars), t days since it was opened. What does the statement M(30)-M(0)=100 mean?
- When input 30 into function, youte doing to get M(30). - This account balance was 30 days after opened - M(0) is the balance after 0 days, or the initial balance - So theyre taking balance after 30 days and subtracting from initial balance, and saying it is equal to 100 - So arjun made a profit of $100 over the first 30 days since the account was opened
Sums of consecutive integers example: The sum of 4 consecutive odd integers is 136. What are the 4 integers?
- consecutive=one right after other. So consec odd integers would be 3,5,7,9 as example - Let x=smallest of the 4 integers - To get to 5, would be x+2, bc if x+1, yu'd just get an even number. Then x+4, x+6, etc. - In general if x is smallest of integers, we can define other three as x+2, x+4, and x+6. Take sum of them and set equal to 136 - x+x+2+x+4+x+6= 136 -4x+12=136 - 4x=124 - x= 31 - integers= 31, 33, 35, 37
Evaluating functions from equation example: The function f(x) is defined as f(x)=49-x^2. Find the value of f(5)
- f(5)=49 -(5)^2 - 49-25 -24
Adding fractions with unlike denominators example: 9/10+ 1/6
- find common denominator to convert both these into (least common multiple of 6 and 10) - usually to find, easier to start by finding lcm of bigger number. In this case, 10, and say, is 10 divisible by 6? No. Is 20 divisible by 6? No. Is 30 divisible by 6? Yes. - so rewrite both of these as ?/30. - to get from 10 to 30, multiply by 3, so have to multiply numerator by 3 as well- 27/30 - same thing with the next one. To get, multiply both by 5= 5/30 - 27/30 + 5/30 - 27+5= 32 - 32/30. Can reduce, as common factor between these of 2. so divide by 2= 16/15. - To write as mixed number, 15 goes into 16 1 time, with a remainder of 1, so this is the same thing as 1 and 1/15.
Finding least common denominator example: rewrite the following two fractions 2/8 and 5/6
- find multiples of 8 and 6 first. 6:6, 12, 18, 24, 30... 8: 8, 16, 24, 32.. -LCM=24 - Rewrite these fractions with 24 as denominator: 2/8 = /24. - what times 8 is 24? 3. So multiply numerator and denominator by 3. So 6/24 - 5/6= ? 24. - What times 6 is 24? 8. So multiply num and denom by 4. So 20/24
Example of whole ratio word problem: In language class, the girl to boy ratio is 5 to 8. If there are a total of 65 students, how many girls are there?
- girls/total students= 5/13 (8+5=13) - 65/13=5, so multiply by 5 - 5x5=25 -25/65
Negative exponent example: 2 ^-4
- how many times are we going to divide by 2, is what the negative exponent means - 1 x 1/2 x 1/2 x 1/2 x 1/2 (exact same thing as dividing by 2 four times) -
Equation with variables on both sides fractions example: 3/4x +2= 3/8x -4
- mutiply both sides by some number to get rid of fractions. Best number to do it by is smallest that goes into both - in this case, it is 8, bc 8 is the LCM of 4 and 8. so multiply both sides by 8 - 8(3/4 x +2)= 8(3/8x -4) - (8x3)/4 = 6 - 6x+16= 3x-32 - 3x+16= -32 - 3x= -48 - x=-16
Modeling with systems of inequality example: Luis recieves a gift card worth $25 to an online retailer that sells digital music and games. Each song costs $0.89, and each game costs $1.99. He wants to buy at least 15 items with this card. Set up a system of inequalities that represents this scenario
- s=songs, g=ggames - s+g>/=15 - 0.89s+1.99g</=25
Writing basic expressions as word problems example: Susie ran a race. She ran 5 miles an hour, and the race took her t hours to complete. How long was the race? Write your answer as an expression
- taker her speed times number of hours - So 5t
Percent word problem: 78 is 15% of what number?
- x is number - if we take 15% of x, multiply by 15%, we get 78. - Now just have to solve for x - 0.15x=78 - divide both sides by 0.15 - 78/0.15= 520
Writing basic expressions as word problems example: Phil recieved a prize of x dollars from a poker tournament. The tournament cost him 100 dollars to enter. What were Phils net winnings from the tournament? Write your answer as an expression.
- x-100
Testing a solution to a system of equations example: Is (-1,7) a solution for the system of linear equations below? x+2y=13 3x-y=-11
- x=-1 and y=7 - -1+2(7)=13? -1+14=13 - 13=13, so yes does satisfy first equation - 3(-1)-(7)=-11? -3-7=-11? =10=-11 X, so does not satisfy second equation - Not a solution for the system equations bc it doesnt satisfy both
Percent change word problem example: 30% off on guavas (only today). 6 guavas at sale price=$12.60. How much are 2 guavas at full price?
- x=cost of 6 guavas at full price - x-.30 (taking 30% off of full price to figure out sale price - 1x-0.30x=12.60 - 0.7x=12.60 - x=12.60/0.7 - x= $18.00 (full price of 6) - $3 per guava at full price (18/6) - $6
Writing expressions with variables examples: - take the quantity -3 times x, and then add 1
-3x+1
Writing expressions with variables examples: - 6 plus the products of -1 and x
-6+-1x or -6-x
Evaluating expressions with variables temperature example: Express 25 degree celcius (C) as a temperature in degrees farenheit (F) using the formula F=(9/5)C+32
-F= 9/5 x 25/1+32 - Can divide the num and denom of our product by 5. So, 9/1 x 5/1 - F=9x5+32 -F=45+32 - F=77
Writing explicit formulas example: 5,8,11
-First term of sequence is 5 and common diff is 3 - Can get nth term by taking first term 5 and adding common difference 3 repeatedly for n-1 times - this can be written algebraically as 5+3(n-1)
Explicit formulas example: a(n)=3+2(n-1)
-In this, n is any term number and a(n) is the nth term. - In order to find fifth term, plug n=5 into explicit formula: - a(5)=3+2(5-1) -=3+2x4 = 3+8 =11
Writing recursive formulas example: 5,8,11
-Two parts of formula should give following info: first term (5) and the rule to get any term from its previous term (which is add 3). So.. - c(1)=5 - c(n)=c(n-1)+3
Recursive formulas give us two pieces of info
-first term of sequence and pattern rule to get any term from the term that comes before it
Writing expressions with variables examples: - The sum of -7 and the quantity of 8 times x
-sum=addition of - -7+8x
Divisibility tests for 2,3,4,5,6,9,10
2,799,588 - can see if divisible by two by looking at last digit. 2 does go into 8, so it is - for three, add all numbers up 2+7+9...= 48. Add these three=12, which is clearly divisible by 3 - look at last two digits of four and see if divisible by 4. - for 5, if your final digit is 5 or 0, it is divisible by 5 - for 6, you have to be able to be divisible by both 2 and 3 to be divisible by 6 -for 9, sum all digits, and if sum is divisible by 9. Must be divisible by 3! - for 10, just need to have a 0 in the ones place.
Example of equivalent fractions: 6/15
3 is lcm of 6 and 15, so divide by that. New fraction is 2/5
Ratio
A comparison of two quantities. ex) There are 4 monkeys and 5 bananas= 4:5
composite number
A whole number greater than 1 that has more than two factors.
Standard explicit formula of an arithmetic sequence whos first term is A and common difference is B
A+B(n-1)
Terms
Each number in a sequence
Finding factors of number
Find whole numbers that a number is divisible by. ex) all factors of 120 are 1,2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120
equivalent fractions
Fractions that have the same value
When evaluating functions from graph, for ex when finding f(-1) for f(x)
Look at graph, and go to -1 on x axis. On the graph in this case, the y axis for -1 is equal to 6. So f(-1)=6
When factoring an equation...
Make sure to arrange the equation so one of the sides equals 0!!
Multiplying Fractions
Multiply the numerators and put the product over the product of the denominators
Adding fractions with like denominators example: 3/15+7/15
Since the denominators is the same, you don't add them together, you just add the top two. So =3+7/15= 10/15. Which you divide by 5 (lcm), and get 2/3
Whole ratio
To simplify ratio, divide by their greatest common divisor. ex) ratio of 8:20 can be simplified to 2:5, as gcd is 4
Equivalent Ratios
Two ratios that have the same value ex) Three ratios that are equivalent to 7:6 - 14:12 (7x2=14,6x2=12) - 21:18 (7x3=21, 6x3=18) - 42:36(7x6=42, 6x6=36) - 63:54(7x9=63, 6x9=54)
Prime numbers
a number which can only be divided by itself and 1
arithmetic sequence
a sequence whos pattern involved adding or subtracting a number to each term to get the next number. In this, the difference between consecutive terms is always the same
explicit formulas for arithmetic sequences
allow us to plug in the number of the term interested in and we will get value of that term
improper fraction
fraction where numerator is greater than or equal to denominator. Ex) 9/4, 5/5, 7/3
Mixed number
number consisting of a whole number and a proper fractions. ex) 4 1/2, 1 3/8, 12 5/6
Sequences
ordered list of numbers
Divisibility Rules
patterns used to determine whether a number divides evenly into another number without actually completing the division
Quadrants
the four sections of the coordinate plane