GRE Math
Raising an exponent to another exponent
multiply each exponent together
Even + odd =
odd
25x²-64y=
(5x+8y)(5x-8y)
f(x) = 1 - x +x² then, f(2) = `
1 - (2) + (2)² = 3 f(2) = 3
A $170 item is discounted 30%. What is the new price?
1 - .30 = .70 (.70)(170) = 119
Find how many odd factors 21,600 has.
12 odd factors
What are the prime numbers between 20 and 60?
23, 29, 31, 37, 41, 43, 47, 53, 59
Prime factorization of 9
3 x 3
What is 75% of 280?
3/4 x 280 = 210 4 goes into 280, 70 times.
If xy = 7 and x - y = 5, then x2 + y2 =
39 https://gre.magoosh.com/practices/57880654/q/130
For all numbers x and y, let (x💢y)=x²y. Then what is the value of 3💢4?
3=x 4=y 3²(4) = 36
Express |x-7|≤3 as an ordinary inequality.
4≤x≤10 7-3=4 7+3=10
If x ➤ y=y/x+y, what is the value of (3 ➤ 2) ➤ 1 ?
5/7 https://gre.magoosh.com/answers/463579952?review%5Bsubject_tag_ids%5D%5B%5D=5
What is 55% of 400?
50% of 400 = 200 5% if 400 = 20 200+20 = 220
(x²/7 - 10x/7 + 3)/(x/6 - 1/2)
6/7(x-7)
What is the prime factorization of 96?
96= 2 x 48 = 2 x 6 x 8 = 2 x (2x3)(2x2x2)= 2x2x2x2x2x3 = 2^(5) x 3 Break down each number that is not prime (6, 8) in terms of prime numbers
Proportions
An equation stating that two ratios are equal fraction = fraction
Don has x marbles. If y marbles are white, what percent of Don's marbles are NOT white? 100x/ x−y 100(x−y)/x (x−y)/100x x/100(x−y) 100y/x
B Don has x marbles. That's the total number of marbles, the whole. If y of the marbles are white, then the rest should be not white. number of not-white marbles = x - y That's the "part" and x is the "whole." Percent = part/whole×100 Percent = x−y/x ×100 = 100(x−y)/x Answer = (B) https://gre.magoosh.com/answers/465245751
E x O =
E
The roots of an algebraic expression are the x-values that make the expression equal to zero. What are the sum of the roots of the expression (x - 2)(x² + 7x + 12)?
First, we have to factor the quadratic: (x−2)(x^2+7x+12)=(x−2)(x+3)(x+4) Now, we have the three factors. The roots of these three factors are {+2, -3, -4}. The sum of those three numbers is = (+2) + (-3) + (-4) = -5
Solve. |2x+5|=x+1
No Solution. We cannot have an absolute value that is negative.
A negative squared =
a positive
If y=5+x and y=12-x, and if y²=x²+k, then k equals which of the following? a. 17 b. 25 c. 60 d. 119
c. 60 rearrange y=5+x and y=12-x y-x= 5; y+x=12 rearrange y²=x²+k k=y²-x² = (y-x)(y+x) = 5*12 = 60
Always do what before cross multiplying?
cancel before cross multiplying larger numbers
"Is" means
equal
E x E
even
even + even =
even
even - even =
even
odd + odd
even
odd - odd =
even
What is 80% of 200?
x = 80% x 200 x = .80 x 200 x=160
If (x − 3)² = x2 + bx + c, then b is equivalent to which of the following? −9 −6 3 6 9
-6 is the answer. expand the left side. https://gre.magoosh.com/answers/466536298?review%5Bpractice_session_id%5D=57909916
Basic consecutive facts
1. A set of n consecutive integers will always contain one number divisible by n. {20, 21, 22, 23, 24, 25, 26} Set of 7 --> 21 is divisible by 7. 2. if n is odd, then the sum of a set of n consecutive integers will be divisible by n. 20+21+22+23+24+25+26= 161 161/7 = 23
Prime Factorization
Every integer greater than 1, that's not prime, can be expressed as a product of primes. This product is called the prime factorization.
Divisibility Rule for 2
If a number ends in an even number: 2,4,6,8,or 0
7x+3y=5 2x-3y=13
Solution = x=2, y=-3
Percent decrease
percent of change where the new number is less than the original number P% decrease (multiplier for P% decrease) = 1 - (P% as a decimal) Ex: The multiplier for a 28% decrease is 1 - .28 = .72 New = (multiplier) (old)
Find the prime factorization of 144
144= 12*12=(3*4)(3*4)=(3*2*2)(3*2*2)= 3^(2)*2^(4) exponents are even (2) and (4)
For positive numbers a and b, let a🌀b= a/a+b What does p🌀p = ?
1st p = a 2nd p = b p/p+p = p/2p = 1/2
x and y are positive numbers such that 2x < y. Which of the following expressions must be greater than 1/2 ? Indicate all such expressions. x−1/y x/y 1−x/y y/x
1−x/y y/x plug in x=1 and y = 3 to each and solve. https://gre.magoosh.com/answers/466304149?review%5Bsubject_tag_ids%5D%5B%5D=5
N=135 is the lowest number in a set of 41 consecutive multiples of 5. What is the difference between the lowest and the highest number in the set?
200
Odd numbers
... -7, -5, -3, -1, 1, 3, 5, 7, ...
An office has 6 employees; there are 5 female employees and 1 male employee. In how many ways can a 3-person committee be created if the committee must include the male employee?
10 https://gre.magoosh.com/practices/57844110/q/855
Remainders Lesson
Let over number that does not divide evenly into number form of mixed numeral quotient: 17 ÷ 5 = 3 ²/₅ Integer quotient: 5 goes into 17 three times with a remainder of 2.
Divisibility Rules
Rules that tell if one whole number is divisible by another.
−2 < p < q < −1 Column A − ( p + q ) Column B 1
https://gre.magoosh.com/answers/461507737?review%5Bsubject_tag_ids%5D%5B%5D=5 The quantity in Column A is greater
Rebuilding the Dividend
if we multiply both sides of this equation by S (the divisor), we clear all fractions: D/S = Q + r/S D=SxQ+ r
Given the function f(x)=x²+kx+4, find the value of k if f(2)=18.
k=5
Finding the percent
multiplier = new price/old price *we have to remember to change to multiplier back to a percent.
Even - odd =
odd
Divisibility Rule for 9
the sum of digits is divisible by 9. 1296, 1+2+9+6= 18, 18 is divisible by 9
"Of" means
to multiply
Solve for x. 3x/5 = 2/7
(3/5)x = 2/7 Multiply by 5/3 on both sides which cancels out 3/5 and leaves x = (5/3)(2/7) x=10/21
9x²-16=
(3x + 4)(3x - 4)
If x is a number such that x² + 2x - 24 = 0 and x² + 5x - 6 = 0, then x = -6 -4 -3 3 6
-6 We have x² + 2x - 24 = 0 and x² + 5x - 6 = 0 To solve for x, we set the equations to equal each other: x2 + 2x - 24 = x2 + 5x - 6 2x - 24 = 5x - 6 -24 = 3x - 6 -18 = 3x -6 = x
Convert to decimals 1/2, 1/4, 3/4
.50 .25 .75
Express 20<x<90 as an absolute value inequality.
1. find the midpoint (20+90/2) = 55 2. 90 and 20 are both a distance of 35 away from 55. 90-55 = 35 20 - 55 = 35 3. |x-55|<35
Express 5<x<17 as an absolute value inequality.
1. find the midpoint by taking the average of the two numbers. 5+17/2 = 11 2. 5 and 17 are a distance of 6 away from 11. 17-11 = 6 5-11 = 6 3. |x-11|<6
Solve. 4x+5y=1 5x+2y=14
1. multiply first equation by 2 ad second equation by 5. 2. add equations 3. plug x into either equation. x=4, y=-3
If 5x+2y=55 and 2x-y=19, then the value of x+y= ?
1. multiply second equation by -1 and the add them. 2. Divide by 3. X+Y = 12
In a large bucket of screws, the ratio of slot screws to Phillips screws is 11 to 4. There are no other varieties of screws in the bucket. If there are 320 Phillips screws in the bucket, what is the total number of screws in the bucket?
1200 I solved by Phillip screws/slot screws 4/11 = 320/x cross multiply to get 4x = 3520. divide by 4 x=880 (number of total slot screws) Add 880 + 320 = total number of screws=1200 https://gre.magoosh.com/answers/465245750
Appleton's population is 400 greater than Berryville's population. If Berryville's population were reduced by 900 people, then Appleton's population would be 3 times as large as Berryville's population. What is Berryville's current population? 1550 1650 1750 1850 1950
1550 Set A = Appleton's current population, and B = Berryville's current population. We know: A = B + 400 A = 3 × (B − 900) Substitute in for A: B + 400 = 3 × (B − 900) B + 400 = 3B − 2700 400 = 2B − 2700 3100 = 2B 1550 = B Answer: (A) https://gre.magoosh.com/practices/57853856/q/2
What is 50% of 128?
64
If D= dividend, S=divisor, Q=quotient, r=remainder, then
D/S = Q + r/S 17/5 = 3 + ²/₅
The exponents in the prime factors of a square all must be ______?
Even This means that if we see an unknown number in its prime factorization form, and all of the exponents are EVEN, we know the number MUST be a Perfect Square!
Which of the following is greater than √79? Indicate all possible values. 2√19 3√10 4√5 5√3 6√2 8 9 10
For positive numbers, squaring preserves the order of inequality. In other words,
An item originally cost $800. The price increased by 20%. What is the new price?
New = (multiplier)(old) New = (1.20) (800) = 960
A positive integer to the power of zero =
One
The first six terms of an infinite sequence are 2, 4, 4, 3, 7, 5 and these six terms repeat in the same order. (e.g., 2, 4, 4, 3, 7, 5, 2, 4, 4, 3, 7, 5 . . . ) Column A Term 49 Column B Term 50
Since these six terms repeat in order, every term that's a multiple of six will be 5. So term 48, which is a multiple of six, is 5. Term 49, then, must be 2, and term 50 will be 4. Answer: (B)
Solve. 2x+3y=15 x+2y=11
Solution x=-3, x=7 1. multiply the second equation by (-2) and then add equations. Y=7. 2. Plug in y=7 and solve for x. X=-3
Solve the system. 2x-y=5 2y-4x=-10
Solve with substitution. -10= -10 This is always true.
Odd x odd
odd always
changing from decimals to percents
we are multiplying by 100 so we move the decimal to places to the right.
If xy ≠ 0, and 75% of x equals 125% of y, which of the following is true? y is 25% of x y is 40% of x y is 60% of x y is 140% of x y is 1662/3% of x
y is 60% of x Part/whole = x/100 75/125 = x/100 --> reduced to 3/5 = x/100 --> cross multiplied and divided to get x = 60. another explanation in video: https://gre.magoosh.com/answers/465245744
The Difference of Two Squares
(a+b)(a-b)=a²-b²
The Square of a Difference
(a-b)²=a²-2ab+b²
If we have a system of 2 equations with 2 unknowns, we have 2 strategies to solve for x & y:
1. Substitution 2. Elimination
Elimination Method
1. We are always allowed to add two equations. 2. plug value after adding both together into other equation.
Simplify. (1/3 + 4/9)/(2/3+1/2)
LCM of all little denominators = 18 multiply by 18 on the numerator and denominator. 2/3
When Q is divided by W, the quotient is R and the remainder is E. Which of the following expressions is equal to E? RW + Q RW − Q Q − RW QW − R Q/RW
Q − RW Dividend = Divisor × Quotient + Remainder The relation above always works and you can check it by setting up such simple division problems in your mind. Coming back to the problem, it's given that "When Q is divided by W, the quotient is R and the remainder is E." Here, Q is equivalent to 9 in the example above, W is equivalent to 4, R is equivalent to 2 and E the remainder is equivalent to 1. Therefore we have Q = W ×× R + E Or, Q = WR + E As we need E, subtracting WR from both sides gives Q − WR = E Looking through the options, this is the same as E = Q − RW. Answer: (C) https://gre.magoosh.com/practices/57853856/q/310
Sector area
Sector area / total area = sector angle/360°
Solve the system. x-2y=5 3x-6y-8
Solve with substitution. 15≠8 - No solution. This means there is NO possible solution. The lines are parallels and will never intersect.
-4<5 -3x≤ 17
Start by subtracting 5 from each side. Then divide by -3 to get x by itself. 3>x≥-4
What happens when the divisor is LARGER than the dividend? 5÷12=??
The quotient = 0 with remainder of 5. 12 goes into 5 0 times. leaving remainder of 5.
changing from percents to fractions
We put the percent over 100. We may need to simplify after.
For positive numbers p and q, let p θ q = p + 1/q. What does (1 θ 2) θ 3= ?
p = 1 q = 2 (1 + 1/2) = 3/2 (3/2) θ 3 = now 3 = q and (3/2) = p 3/2+ 1/3 = (find common denominator of 6) = 11/6
5x+7 < 2x-2
x < -3 ∅
(x³)² =
x⁶
(x³-4)(x³+4)
x⁶-16
If (8 - x)/(x + 1) = x, then x²+2x+3=
(8 - x)/(x + 1) = x 8 - x = x^2 + x 8 = x^2 + 2x 5 = x^2 + 2x - 3 5 https://gre.magoosh.com/practices/57880654/q/142
x²-49=
(x + 7)(x - 7)
x²y²-1=
(xy-1)(xy+1)
y²-25 =
(y-5)(y+5) the difference of two squares
Important Prime Facts
- 1 is not a prime number - 2 is the only EVEN prime number
Interchangeable ways to talk about factors Use 13 and 78
- 13 is a factor of 78 - 13 is a divisor of 78 - 78 is divisible by 13 - 78 is a multiple of 13 - 13 is part of the prime factorization of 78
How to find greatest common factor (GCF)
- Find the prime factorization of each number - Find the highest power that each of the same prime number has in common - multiply the highest power of each together to find the greatest common factor
Integer Strategies to Remember
- Know what an integer is - Know prime #'s below 60 - Know terminology for factors - Don't forget about zero and negative numbers
1/10
0.1
1/9
0.11111 repeat
1/8
0.125
1/6
0.16667
Machine A can make 350 widgets in 1 hour, and machine B can make 250 widgets in 1 hour. If both machines work together, how much time will it take them to make a total of 1000 widgets?
1 hour and 40 minutes (100 minutes) set up and cross multiply 600/60 mins =1000/x Solve for x x = 100 https://gre.magoosh.com/practices/57853856/q/73
Memorize perfect squares 1-10 11-15
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225
First 15 perfect squares
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225
Factor 1. 6x³-150x = 2. 2x²-22x+48= 3. 7x⁴-56x³-63x²= 4. -3x⁹+48xy⁴=
1. 6x(x²-25) = 6x(x+5)(x-5) 2. 2(x²-11+24) = 2(x-8)(x-3) 3. 7x²(x²-8x-9) = 7x²(x-9)(x+1) 4. 3x(4y²+x⁴)(2y+x²)(2y-x²) https://gre.magoosh.com/lessons/42-factoring-combined
Rules when there are more than one inequality in a problem:
1. Combine 2 inequalities (r<s and s<t, then r<s<t). The common term has to be greater than one term and less than the other term for this to work. 2. Adding inequalities. We can add inequalities with the same direction. 3. Subtracting inequalities. We can subtract inequalities with the opposite direction. 4. cannot multiply or divide inequalities together.
least common multiple (LCM)
1. Find prime factorization and the GCF 2. Write each number in the form of (GCF) x (another factor) 3. The LCM is the product of the 3 numbers
Counting factors of large numbers
1. Find the prime factorization (with exponents) 2. Make a list of exponents 3. add one to each exponent 4. Multiply all the numbers together (this product is the number of factors that n number has)
Procedure for quadratic equations
1. Get everything on one side of the equation, set equal to ZERO 2. DIVIDE off any GCF 3. Factor. 4. Use the Zero Product Property to create 2 linear equations and solve separately.
For questions involving remainders, two powerful strategies are:
1. Listing all dividends 2. Using the rebuilding formula (dividend)=(divisor)(quotient)+(remainder)
What is the LCM of 12 and 75?
1. Prime factorization and GCF: 12= 3x4 = 3x2x2 = 2²x3 75= 15x5 = 3x5x5 =3x5² Highest powers they have in common: 3 = GCF 2. Take the GCF and multiply it by whatever # gets you 12 and 75. 12 = 3x4 75 = 3x25 3. take each factor and find product: 3x4x25= 300 300 = LCM
Substitution Method
1. Solve one equation for one variable. We get one variable by itself on one side. 2. Plug in for value of x (or whichever you solved for first) to solve for y. 3. Now plug "y" value back into equation that was used to solve for "x"
For consecutive numbers, they are usually given in algebraic expression. (t² - 2t)(t-1) Find the consecutive number pattern
1. factor out t t(t-2)(t-1) = (t-2)(t-1)(t) Only if t is stated that it is an INTEGER you can do this.
Express the region -3≤x≤11 as an absolute value inequality.
1. find midpoint (-3)+11/2 = 4 2. 11-4 = 7 -3-4=-7 both are 7 away from 4. 3. |x-4-|≤7
Solve. 3x+2y=27 4x-4y=26
1. multiply first equation by 2. 2. add 3. plug in x=8 into equation to solve for y. y=3/2 x=8, y=3/2
To find the number of odd factors (prime factorization)
1. prime factorization 2. make list of odd exponents (ignore if there is a 2 in the exponent or any even number as exponent) 3. add one to each 4. multiply those numbers together = number of odd factors
Operations we can perform on both sides of inequalities
1. we can ADD and SUBTRACT the same thing to both sides and inequality stays the same. 2. we can MULTIPLY and DIVIDE both sides by a POSITIVE number and the inequality stays the same. 3. MULTIPLY OR DIVIDE by a NEGATIVE number REVERSES the order of the inequality!!! -x>3 → x<-3 (dividing by -1)
Counting factors in a Perfect Square
1. when making list of exponents, they will be even 2. when we add 1 to each on the list, the list will be all odd numbers 3. This means when you multiply the new list together, the product will be ODD! A perfect square always has an odd number of factors.
If the ratio of x to y is 4 times the ratio of y to x, then y/x could be? 1/4 1/2 1 2 4
1/2 https://gre.magoosh.com/answers/465245745 Translate the first part into words. xyyx = 4×yx4×xy Multiply both sides by xyyx to cancel the fraction on the right. (xy)(yx)2= 4 Take a square root of both sides. xyyx = ±2 Now, take the reciprocal of both sides. yxxy = ± 1221 We have only positive answers, so we have to choose +12+21. Answer = (B)
What is the average (arithmetic mean) of (8+2)(√8+√2)2and (8−2)(√8−√2)2?
10 https://gre.magoosh.com/practices/57844110/q/345
Prime factorization of 10, 12, 15, 24
10= 2 x 5 12 = 4 x 3 = 2 x 2 x3 (break down the 4 into prime number of 2) 15 = 3 x 5 24 = 8 x 3 = 2x2x2x3
Generating examples of possible dividends that when divided by a certain divisor, yield a specific remainder. Example: What numbers, when divided by 12, have a remainder of 5?
12+5=17 - dividing 17 by 12 goes in once with remainder 5. 24+5=29 36+5=41 48+5=53 multiples of 12 + 5 each have remainder of 5
If (1/x + x²) =16 then, (1/x²+x²)= ? 4 8 14 16 18
14. You use FOIL to find the answer. https://gre.magoosh.com/practices/57844110/q/121
If xy = 5 and x² + y² = 12, then x/y+y/x =?
2 and 2/5 https://gre.magoosh.com/practices/57931328/q/124 Let's first combine these two fractions into one. We can obtain a common denominator of xy if we multiply the left fraction by x/x and the right fraction by y/y...
in a set of 3 consecutive integers, you could have
2 evens and 1 odd or 2 odds and 1 even
All prime numbers below 60 & facts about prime numbers
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 39, 41, 43, 47, 53, 59 - Prime numbers only have 2 factors: 1 and itself. - Lowest prime number is 2 - Only even prime number is 2 - Prime numbers are POSITIVE
(3×10²⁰)×(8×10³⁰)= 2.4 x 10^50 2.4 x 10^51 2.4 x 10^60 2.4 x 10^61 2.4 x 10^301
2.4 x 10^51 https://gre.magoosh.com/practices/57880654/q/2606
Find the integer quotient and remainder 20 ÷ 6 95 ÷ 7
20 ÷ 6 = yields 3 with remainder 2 95 ÷ 7 = yields 13 with remainder 4 (13 x 7 = (7*10) + (7*3)= 91 →95-91 = 4)
What is the GCF of 720 and 1200?
240
What is 60% of 60?
36
Find the GFC for 360 and 800?
360 = 6*6*10 = (2*3)(2*3)(5*2) = 2^(3) * 3^(2) * 5 800 = 8*10*10 = (4*2)(5*2)(5*2) = (2*2*2)(5*2)(5*2) = 2^(5)*5^(2) What is the highest power of 2 each have in common? 3 What is the highest power of 3 each have in common? 0 What is the highest power of 5 each have in common? 1 Thus, the GCF = 2^(3) * 5 = 40
18 is what percent of 45?
40%
A machine is making thermometers at a rate of 135 every 18 minutes. How many thermometers will this machine make in an hour?
450. This is a rate so we do 135/18 = x/60 Thermometer/ minutes 60 minutes are in 1 hour. https://gre.magoosh.com/answers/465550007?review%5Bpractice_session_id%5D=57789675
If 4P+3Q-4R/P-R = 19, then Q/P-R=?
5 Start with rearranging order of denominator and separating them.
Joan has 100 candies to distribute among 10 children. If each child receives at least 1 candy and no two children receive the same number of candies, what is the maximum number of candies that a child can receive?
55 https://gre.magoosh.com/practices/57844110/q/46
Find how many factors 8400 has.
60 factors 1. Find the prime factorization (with exponents) - 8400 = 84 x 100 = (10 x 10) (3 x 28) = (5 x2)(5x2)(3)(3x8)= (5 x2)(5x2)(3)(3)(4x4 = 2x2x2x2) = (5x2)(5x2)(3)(3)(2x2x2x2)= 5²x3²x2⁶ 2. Make a list of exponents 3. add one to each exponent 4. Multiply all the numbers together (this product is the number of factors that n number has)
change to percent .68 0.075 2.3
68% 7.5% 230%
Simplify this complex fraction. (x/2 + 5/4) / (x/3 + 3/2) =
6x+15/4x+18 LCM of the "little" denominators = 12 Little denominators are (2, 4, 3, 2)
2/x ≥ 1/3
6≥x because they are all positive, you can cross multiply or multiply both sides by 3 to get 6.
56 is what percent of 800?
7% 56 = x(800) divide by 800 on both sides →56/800 → 7/100→0.07→move decimal two places to the right = 7% OR cross multiply to get 7%.
Find how many factors 21,600 has.
72 factors
Find how many even factors 21,600 has.
72-12=60 even factors
If n is an integer greater than 50, then the expression (n^2 - 2n)(n+1)(n-1) must be divisible by which of the following? 8 12 18
8 and 12 factor out an n in the first one - n(n-2)(n+1)(n-1) Rearrange - (n-2) (n-2) (n) (n+1) 4 consecutive integers
A box contains 4 red chips and 2 blue chips. If two chips are selected at random without replacement, what is the probability that the chips are different colors?
8/15 https://gre.magoosh.com/practices/57844110/q/837
Compare the fractions: 8/33 ? 24/100
8/33>24/100 we can multiply 8/33 times 3 on the top and bottom to get 24/99. 24/99> 24/100. Slightly larger that 24%. When we have the same numerator (24) but a smaller denominator, the fraction is bigger.
Which of the following numbers are divisible by 6 but not by 18? Indicate all possible values. 1296 2744 3072 4356 5832 6000 7290 8112
8112 6000 3072
Which of the following numbers is divisible by 36? Indicate all possible values. 1296 2160 3438 4608 5346 6144 7000 8244
8244 2160 4608 8244 https://gre.magoosh.com/answers/465550006?review%5Bpractice_session_id%5D=57789675 36 = 4x9. So each has to be divisible by 4 and 9. Divisibility rule for 4 - last 2 digits have to be divisible by 4. Divisibility rule for 9 - sum of digits divisible by 9. Because 4 and 9 are a factor pair for 36, if a number is divisible by both 4 and 9, that means that number is also divisible by 36, so our work would be done.
5 inequality symbols
< > ≠ ≤ ≥
To find the even number of factors (prime factorization)
Calculate the total number of factors - the odd number of factors = even number of factors
How to test if a number less than 100 is prime:
Check whether it is divisible by one of the prime numbers less than 10 (2, 3, 5, or 7). If a number less than 100 is NOT divisible by any prime divisor less than 10, then the number has to be prime! (if it is divisible by at least one of those numbers it CAN'T be prime)
W is a two-digit positive integer divisible by 4. Column A The number of different integers that could be the units digit of W Column B 3
Column A is greater than Column B. ∅ The correct answer is (A): the quantity in Column A is greater. It is possible for a two-digit multiple of 4 to end in any even digit: 0, 2, 4, 6, or 8. For example, the numbers 0, 4, 8, 12, and 16 are all divisible by 4, but each has a different units digit. Thus, "the number of different integers that could be the units digit of W" is 5. Since Quantity A equals 5 and Quantity B equals 3, Quantity A is greater. = 5 https://gre.magoosh.com/answers/465550008?review%5Bpractice_session_id%5D=57789675
Changing from percents to decimals
Dividing by 100, so we move the decimal point 2 places to the left.
INTEGER PROPERTIES SECTION!!!
Divisibility, Multiples, Prime Numbers
Multiplying even and odds
E*O=E E*E=E O*O=O
The Greatest Common Factor (GCF) of 48 and 72 is 4 6 12 24 48
Find the prime factorizations of both numbers, and mark the factors they have in common: They both have at least three factors of 2 and one factor of 3, and 2*2*2*3 = 24, so this is the GCF. Answer = D (24) 48= 6x8 = (2x3)(2x2x2x2) 72= 8x9 = (2x2x2x2)(3x3) What is the highest power of 2 they have in common? 3 What is the highest power of 3 they have in common? 1 Multiply those together: 2³x3 = 24 2x2x2x3=24
Quadratic Equations
Has to be set equal to zero. Most often these equations have 2 different solutions. The coefficient x², will equal 1 (unless numerical GCF can be factored out from all 3 terms).
Absolute Value
How far a number is from zero. Absolute value is always positive.
M is a positive two-digit number. When the digits are reversed, the number is N. If K = M + N, which of the following is true? K must be even K cannot be square K cannot be divisible by 13 K must be divisible by 11 If M is even then K must be even
K must be divisible by 11 https://gre.magoosh.com/answers/465558844?review%5Bpractice_session_id%5D=57790740
When positive integer N is divided by positive integer P, the quotient is 18, remainder 7. When N is divided by (P+2), the quotient is 15, remainder 1. What is the value of N?
N=151 If D= dividend, S=divisor, Q=quotient, r=remainder N = 18 x P + 7 N = 15 x (P + 2) +1 18P +7 = 15P +31 Solve for P. P = 8 Plug P=8 into N = 15(P+2)+1 N=151
x ≠ 0 Column A 1/(x⁻⁸) Column B √x¹⁶
The two quantities are equal When you take a square root of an exponent, you are raising it to the (1/2). So (x¹⁶)^½ = x⁸ When you move a negative exponent to the numerator, it becomes a positive exponent = x⁸ https://gre.magoosh.com/practices/57859312/q/16498
Column A The number of prime numbers divisible by 13 Column B The number of prime numbers divisible by 2 The quantity in Column A is greater The quantity in Column B is greater The two quantities are equal The relationship cannot be determined from the information given
The two quantities are equal rime numbers are only divisible by themselves and 1. Both 13 and 2 are prime numbers. So there is exactly 1 prime number divisible by 13 (13), and exactly 1 prime number divisible by 2 (2). So both columns are equal to 1.
f(x) = (x+2)², g(x) = 2x − 3 Column A f(g(−1)) Column B f(g(2)) The quantity in Column A is greater. The quantity in Column B is greater. The two quantities are equal. The relationship cannot be determined from the information given.
The two quantities are equal. f(g(-1)) = 9 f(g(2)) = 9 https://gre.magoosh.com/answers/462275223?review%5Bsubject_tag_ids%5D%5B%5D=5
Four consecutive positive integers have a sum of 802. Column A the least of the four integers Column B 199
The two quantities are equal. We know that they have to be consecutive. 199+200+201+202 = 802 199 is the least of the four integers. https://gre.magoosh.com/practices/57859312/q/12582
For positive numbers a and b, let a❤️b=2a²+b. What does (1❤️2)❤️3=?
a = 1, b = 2 plug in 2(1)²+2 = 4 Now, 4=a and 3=b. plug in 2(4)²+3 = 32+3 = 35
Divisibility
a number is divisible by anther if, there is no remainder
Multiplying a power times another power
add both powers together
If a number larger than 100 appears on any question,
chances are you can find the PRIME FACTORIZATION to help answer the question.
Adding and subtracting even and odds
even + even = Even number Even - even = Even number Odd + odd = even number Odd - odd = even number Even + odd = odd number Even - odd = odd number
Describe what f(x) means.
f = function name (x) = the input, the x is a placeholder. It is there to show us where the input goes and what happens.
Given the function f(x)=x²+4x-21 evaluate the following: f(0) = f(3) = f(-1) =
f(0) = -21 f(3) = 0 f(-1) = -24
f(x)=x²+4 Then, f(3) =
f(3)= (3)²+4=13 f(3) = 13
If f(x)=3x-7, then f(4x)= f(2x-5)= f(x²-3x+4)=
f(4x)= 12x-7 f(2x-5)= 6x-22 f(x²-3x+4)= 3x²-9x+5₆
Given f(x) = x²-2x-1, find f(x²+3) =
f(x²+3) = x⁴+4x²+2 plug in (x²+3) each time there is an x.
If the number of equations is ______ than the number of variables, you will ______ be able to solve for the individual values of variables.
fewer than; not
A popular website requires users to create a password consisting of digits only. If no digit may be repeated and each password must be at least 9 digits long, how many passwords are possible? 9! + 10! 2 x 10! 9! x 10! 19! 20!
https://gre.magoosh.com/answers/465245743 2 x 10!
If 1/x = 0.4, then 1/x+2 = 1/8 1/5 2/9 1/4 2/7
https://gre.magoosh.com/answers/465263407 2/9
If x < 0 and (2x²)² = 64, then what is the value of x?
https://gre.magoosh.com/answers/466983039?review%5Bpractice_session_id%5D=57957178 -2 (2x2²)²=64 ⇒2x² × 2x²=64 ⇒4x⁴=64 ⇒x⁴=64/4=16 Notice that 2⁴=16 and as 4 in an even number, (−2)⁴ is also equal to 16. As x < 0, x must be equal to −2. So, −2 is our final answer.
There are three sections of Algebra at Fairview High School. Section A has 20 students, section B has 27 students, and section C has 30 students. Last Tuesday, each section took the same exam. The average scores on the exams were 90 in section A, 80 in section B, and 70 in section C. Column A The average (arithmetic mean) score on the exam, for students in all three sections, taken together. Column B 80
https://gre.magoosh.com/practices/57859312/q/16314 Column B is greater. In section A, the average score of the 20 students was 90. Thus, SumA = 20 × 90 = 1800 In section B, the average score of the 27 students was 80: SumB = 27 × 80 = 2160 Finally, in section C, the average score of the 30 students was 70: SumC = 30 × 70 = 2100 Therefore, the sum of scores of all Algebra students at Fairview High School was 1800 + 2160 + 2100 = 6060 And the total number of Algebra students is 20 + 27 + 30 = 77 Thus, the average score on the exam, for students in all three sections, taken together was 606077≈78776060≈78 which is Quantity A. Since Quantity B = 80, Quantity B is greater.
consecutive
in a row, one following another
The remainder is ALWAYS _____ than the divisor.
less than the divisor 0≤remainder<divisor
The price of an item decreases from $250 to $200. What was the percent decrease?
multiplier = new price/old price multiplier = 200/250 →4/5=.80 multiplier = .80 1-.80 = .20 →20% (subtract to get percent)
If 2x - y = 10 and x/y = 3, then x = -10 2 4 6 12
solve with substitution to get x=6. start by multiplying both sides of x/y=3 by y - x=3y. then plug in x into other equation.
For questions about "even" and "odd", remember to
substitute 1= odd , 2= even
Percent increase
the percent of change when the new amount is greater than the original Multiplier for P% increase = 1 + (P% as a decimal) New = (multiplier)(old) Ex: The multiplier for 46% increase is 1 + .46 = 1.46.
x/2 + 5/3 = (4x/3) + 1
x = 4/5 LCM = 6 Multiply each number by 6. 3x+10 = 8x+6 Get x on one side.
Which of the following inequalities is equivalent to 12 - 3x < -18 x > 10 x < 10 x > -10 x < -10 x > 2
x > 10 https://gre.magoosh.com/answers/463019777?review%5Bsubject_tag_ids%5D%5B%5D=5
x⁹-x=
x(x⁴+1)(x²+1)(x+1)(x-1) Start by factoring out an x. x(x⁸-1) = x(x⁴+1)(x⁴-1) You can factor again because of difference of squares in (x⁴-1). x(x⁴+1)(x⁴-1) = x(x⁴+1)(x²+1)(x²-1) You can factor again because of difference of squares in (x²-1) x(x⁴+1)(x²+1)(x²-1)= x(x⁴+1)(x²+1)(x+1)(x-1)
Solve. x+2y=11 2x+3y=15
x+2y=11 x=11-2y Plug in x=11-2y into 2x+3y=15. 2(11-2y)+3y=15 y=7 Now plug in y=7 into x=11-2y x=11-2(7) x=-3 Solution = (x=-3, y=7)
x²+10x-24=0
x=-12 or x=2
x²-3x-43=11
x=-6 or x=9
Given the function f(x)=x²+4x-21, find the value(s) of x that would satisfy f(x) = 24
x=-9 or x=5, therefore, f(5) = f(-9) = 24
Solve. |x=4|=3x+2
x=1 Only one solution.
Solve. |1+2x|=4-x
x=1 OR x=-5 plug in and test both answers.
Solve. |3x+2|+1=5
x=2/3 or x=-2
If x²+12x-45/x+15 = 22, then x=?
x=25
If √2x²+2xy+13y²=x+3y then x =
x=2y https://gre.magoosh.com/practices/57853856/q/143
2x²+4x+25=x²+10x+16
x=3
x/2+ 5/4 = x/3 +3/2
x=3/2 LCM = 12 Multiply each side by 12. 12(x/2) +12(5/4) = 12(x/3)+12(3/2)
w-2x+3y=13 2w+x-4y=-14 3w-x+2y=8
y= 3, x=-2, w=0 https://gre.magoosh.com/lessons/1007-three-equations-with-three-unknowns
In a set of 4 consecutive integers,
you must have 2 evens and 2 odds.
Zero to raised to a power = 0² =
zero 0² = 0
Consecutive multiples of a number example
{25, 30, 35, 40} are consecutive multiples of 5
After a 30% increase, the price of something is 78$. What was the original price?
$60 1 + .30 = 1.3 multipler 78= (1.3)x divide by 1.3 on both sides x = 60
25s²-t⁴=
(5s-t²)(5s+t²) the difference of two squares
The Square of a Sum
(a+b)²=a²+2ab+b²
Even numbers
... -8, -6, -4, -2, 0, 2, 4, 6, 8, ...
4/5
.80
1/1000
0.001
1/100
0.01
Find the factors of 36
1, 2, 3, 4, 6, 9, 12, 18, 36 9 factors total 36= 36x1 36=2x18 36=3x12 36=4x9 36=6x6 (the 6 is only counted once)
To find GCF or LCM
LCM = P x Q / GCF ALWAYS CANCEL BEFORE MULTIPLYING. (P and Q are just integers that the question will give us.)
Change to decimal. 42.5% 4% 0.25%
Move to places to the left. 0.425 0.04 0.0025
Simplify a complex fraction
We multiply the numerator and denominator of the big fraction by the LCM of all the denominators of the little fraction.
The price of an item increases from $200 to $800. What was the percent increase?
m = 800/200 = 4 This multiplier was created by adding 1 + (p% increase). To find just p%, subtract 1. 4-1 = 3 →300% increase
Cross multiplication
multiplying the numerator of one ratio by the denominator of the other ratio in a proportion. 5/7 = 3/x --> 5x=21 --> x=21/5
1/5
0.2
Convert to decimals 1/3
0.33333333 repeat
3/8
0.375
2/5
0.4
3/5
0.6
5/8
0.625
2/6
0.666 repeat
7/9
0.77777 repeat
if p+1/2p + p-1/3p = 2 What is the value of p ?
1/7 = p https://gre.magoosh.com/practices/57957178/q/16246 LCM = 6. Multiply each fraction so that they have a common denominator of 6.
In a jar, there are 24 marbles, each of which is either red or blue. Which of the following CANNOT be the ratio of red marbles to blue marbles? 1:2 3:5 1:1 4:3 7:5
4:3 If the ratio of red to blue is a:b, then (a + b) would represent the size of the whole. The total number of marbles, 24, must be divisible by (a + b). For example, if the ratio R:B = 1:2, then the reds are 1331 and the blues are 2332, and the whole must be divisible by 3. As it happens, 24 is divisible by 3, so (A) is a possible ratio. (B) 3 + 5 = 8 and 24 is divisible by 8—this is a possible ratio (C) 1 + 1 = 2, and 24 is divisible by 2—this is a possible ratio (D) 4 + 3 = 7, and 24 is NOT divisible by 7—this is NOT a possible ratio (E) 7 + 5 = 12, and 24 is divisible by 12—this is a possible ratio Answer = (D) https://gre.magoosh.com/answers/465263405
Factor and divisor relationship
Every factor is a divisor, every divisor is a factor.
Divisor
If C/A = B, then we say A is a divisor of C, because it divides evenly into C. One number divides evenly into another when the quotient is an integer. We can say that C is divisible by A. There is not difference between factor and divisor (they mean the same thing).
How do we find all the positive factors of 36?
List the factor pairs: the pairs of numbers that have a product of 36. 1, 36 2, 18 3, 12 4, 9 6, 6 36 has 9 positive factors, including 1 and itself. For small numbers less than 100, we can count the positive factors simply by making a list of factor pairs.
5/3+3/2x = 2+2/3x What is the value of x? Give your answer as a fraction.
Multiply by LCM = 6 to get ride of fractions then solve for x. 2/5 https://gre.magoosh.com/practices/57957178/q/6148
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?
Set A = number of apple seeds, and O = number of orange seeds. We know: A = 4*O number of apple trees = 0.15*A number of orange trees = 0.1*O 0.15*A + 0.1*O = 420 --> 15*A + 10*O = 42000 Substitute A = 4*O: 15*(4*O) + 10*O = 42000 60*O + 10*O = 42000 70*O = 42000 O = 600 https://gre.magoosh.com/practices/57844110/q/13
changing from fractions to percents
Turn into decimal and then slide decimal point 2 places to the right
Rules for canceling proportions
We cannot cancel diagonally!! We can to vertical cancellation in the same fraction. We can cancel horizontally across 2 numerators. We can cancel horizontally across 2 denominators.
Multiple Relationships
- if I add 2 multiples of a number together, the sum will be a multiple - if I subtract 2 multiples of a number, the difference itself will be a multiple - if I multiply 2 multiples together, the product itself will be a multiple
5/6
0.8333 repeat
7/8
0.875
The numbers {a,b,c} are three positive integers. If a×b×c/14 equals an integer and b×c/4 equals an integer, what is the smallest possible integer value of a? 1 2 4 7 14
1 https://gre.magoosh.com/answers/465245754 Since this is a multiple choice problem, one way to solve this is to start by plugging in the lowest integer value they give us. The first answer choice happens to be a = 1. If we let a = 1, then the first fraction given to us becomes: 1×b×c/14 We're told that this fraction must equal an integer. Since a = 1, this means that b × c must be some multiple of 14, such as 14, 28, or 56. That's the only way that the denominator can evenly divide into the numerator to yield an integer. If b × c is not a multiple of 14, then we'd end up with a decimal value. The second fraction given to us tells us there's another condition we have to satisfy. If b × c divided by 4 equals an integer, then that means that b × c must also be a multiple of 4. So, if b × c is both a multiple of 14 and 4 such as 14 × 4 = 56, then we will have satisfied all the conditions of this problem. Since b and c are allowed to be any positive integer, then they could very well be b = 4 and c = 14. In that case, both fractions will result in integers and thus the lowest value for a can indeed be 1. This makes (A) the correct answer. Now, we could have also discovered this same information without plugging in (but plugging in just makes it a little easier to spot). If we regroup the first fraction, we could express it as: a×b×c/14 In other words, this is a way to see that a could be any positive integer we want as long as b × c is a multiple of 14. Likewise, the 2nd fraction shows us that b × c must also be a multiple of 4. So, as long as b × c meets both those conditions, then a could take on any positive integer value. Since 1 is the lowest answer choice given to us, that's the answer.
What are the prime numbers less than 20?
2, 3, 5, 7, 11, 13, 17, 19
Change to fraction 20% 92% 0.02%
20/100→1/5 92/100→23/25 .02/100→move decimal over until you get a whole number (move decimal over 2 places in numerator and denominator→2/10000→1/5000
240 is 30% of what number?
240 = (.30)(x) 240/.30 → make denominator a whole number by moving decimal over 2 places. we move decimal over 2 places on top too. → 2400/30 = 800 x = 800
Change fraction to percent 3/8 2/3 59/100 17/1000
3/8→0.375→slide decimal 2 places to the right→37.5% 2/3 →0.6667→slide decimal 2 places to the right→66.67% 59/100 →59% 17/1000→0.017→1.7%
What is a prime number?
A number that is divisible by 1 and itself
Positive integers
All numbers greater than zero. {1, 2, 3, 4, 5, 6, 7, ...}
In Blattodea Hotel, the ratio of single-bed rooms to double-bed rooms is 4 to 11. In Fremont Hotel, the ratio of single-bed rooms to double-bed rooms is 3 to 8. Column A The number of single-bed rooms in Blattodea Hotel Column B The number of single-bed rooms in Fremont Hotel
Relationship cannot be determined It's very important to remember what a ratio is and isn't. We are given a statement "In Blattodea Hotel, the ratio of single-bed rooms to double-bed rooms is 4 to 11." This means the number SBR could be 4 and DBR could be 11, but it also could be any multiples of those numbers: 8 SBR, 22 DBR 12 SBR, 33 DBR 16 SBR, 44 DBR 20 SBR, 55 DBR etc. A ratio is only a statement about the simplified fraction, and we cannot draw conclusions about individual numbers given in the ratio. Similarly, we have the statement "In Fremont Hotel, the ratio of single-bed rooms to double-bed rooms is 3 to 8." This hotel could have 3 SBR and 8 DBR, or it could have: 6 SBR, 16 DBR 9 SBR, 24 DBR 12 SBR, 32 DBR 15 SBR, 40 DBR etc. The number of SBR in Blattodea Hotel could be 4 or 20 or any other multiple of 4. The number of SBR in Fremont Hotel could be 3 or 15 or any other multiple of 3. We see 4 < 15 but 20 > 3. The inequality could go either way. Answer = (D)
Ashley's score was 20% higher than Bert's score. Bert's score was 20% lower than Charles' score. Column A Ashley's score Column B Charles' score
Start with Charles score. Plug in 100 for his score. take away 20% to get Bert's score. Once you find Bert's score, add 20% of that. Charles score is greater than Ashley's store. Charles - 100 20% of 100 = 80. 80 x 20% = 16. Add 16 to 80 = 96 = Ashley's score.
In a certain lemonade recipe, the ratio of lemon juice to water is 2 to 7. Column A Amount of lemon juice in 63 ounces of lemonade Column B 16 ounces
The correct answer is (B): the quantity in Column B is greater. We are given that the ratio of lemon juice to water is 2 to 7, which means that in every 2 + 7 = 9 ounces of lemonade, there are 2 ounces of lemon juice. Column A: The amount of lemon juice in 9 ×× 7 = 63 ounces of lemonade = 2 ×× 7 = 14 ounces. We can also express this algebraically. Let x be the number of ounces of lemon juice in 63 ounces of lemonade. Then 29=x6392=63x x=29×63=14x=92×63=14 Therefore, the quantity in Column B is greater.
Percents as multipliers
The decimal form of a percent is called a multiplier for that percent. this is because we multiply by this form to take a percent of the number.
Divisibility Rule for 4
The last 2 digits are divisible by 4
Divisibility Rule for 5
The last digit is 0 or 5
Divisibility Rule for 6
The number is divisible by both 2 and 3 (even; and add up digits)
Anne pays 150 percent more for a wholesale widget than Bart pays. Anne's retail price per widget is 15 percent greater than the wholesale price she paid. Bart's retail price per widget is 185 percent greater than the wholesale price he paid. Column A Anne's retail price per widget Column B Bart's retail price per widget.
The quantity in Column A is greater The best way to approach this question is to plug in numbers. Let's say that Bart's wholesale price per widget is $100. The question tells us that Bart's retail price per widget is 185 percent greater than the wholesale price he paid. So to find Bart's retail price, we need to do the following calculation: Bart's retail price = $100 + (185% of $100) Bart's retail price = $100 + (1.85 ×× $100) Bart's retail price = $100 + ($185) Bart's retail price = $285 So column B is $285. Now let's think about Anne. The question tells us that Anne pays 150 percent more for a wholesale widget than Bart pays. Now, keep in mind that we decided to use $100 for Bart's wholesale price. This means that to find out what Anne pays (wholesale), we have to find 150% of $100, and then add that number to $100. So: 150% of $100 = 1.5 ×× $100 = $150 $150 + $100 = $250 Thus, Anne's wholesale price will be $250. The question also tells us that Anne's retail price per widget is 15 percent greater than the wholesale price she paid. So to find Anne's retail price, we need to do the following calculation: Anne's retail price = $250 + (15% of $250) Anne's retail price = $250 + (0.15 ×× $250) Anne's retail price = $250 + (15% of $250) Anne's retail price = $250 + ($37.50) Anne's retail price = $287.50 So column A is $287.50. Thus, column A is greater than column B, and the answer is A. https://gre.magoosh.com/answers/465245740
For positive numbers p and q, p+qp−q=32 Column A p + q Column B 5
The relationship cannot be determined from the information given. There are a couple tricky things about this problem. The first is one of the GRE's favorite traps, the word "number". Of course, positive number could be a whole number, such as 7 or 6000, or a fraction, such as 113131, or a decimal such as pi or 13√13 --- all of which are positive. Don't fall for the trap that "number" must be a whole number, a positive integer --- that kind of thinking is severely penalized on the GRE. Next, we cannot not equate the numerators and denominators separately: that's totally invalid. We can NEVER consider "across the numerators" and "across the denominators" as two separate equations. That's another gigantic trap. We have to consider the fraction as a whole. We have no guarantee that the fraction on the left is written in simplest form. When a fraction equals 2332, the numerator and denominator could be both very big or both very small. For example: 23=20003000=0.00020.000332=30002000=0.00030.0002 The algebraic fraction given in the prompt has a numerator of (p - q) and a denominator of (p + q). Notice that Column A is the denominator. Well, as we see above, the denominator (p + q) could equal 3000, or it could equal 0.0003, so for different choices, it could be much greater than 5 or much less than 5. We can't determine. Answer = D. https://gre.magoosh.com/answers/465245741
Integers
The set of whole numbers and their opposites. {...-3,-2,-1,0,1,2,3,...}
Divisibility Rule for 3
The sum of the digits is divisible by 3
The numbers p and q are both positive integers. Column A p/q Column B (p/q)^2
https://gre.magoosh.com/answers/465263401 When we square a fraction smaller than one, it gets smaller. If, for example, p = 1 and q = 4, then then this fraction gets smaller when we square it. When we square any number bigger than one, it gets bigger. If, for example, p = 7 and q = 1, then then this number gets bigger when we square it. Lastly, if p and q are equivalent (for example, p = 5, q = 5), that means column A will be equal to 1 (55=155=1), as will column B ((5555)2 = 12 = 1). Depending on choices, the square could be greater than, less than, or equal to the original, so the answer cannot be determined. Answer = D.
Country 1 has an average temperature of 12 degrees Celsius. If the ratio of average temperature of Country 2 to Country 1 is 2 to 3, in which temperature category is Country 2? A B C D E
https://gre.magoosh.com/practices/57846385/q/16511 C2: C1 2 : 3 x : 12°c 2/x = 3/12 cross multiply x=8 B because 8 is in the range of b
Factor
if A*B=C then we say A & B are factors of C. Ex: 3 is a factor of 6. Ex: 25 is a factor of 100. Notice that 1 is a factor of every positive integer. Notice that every integer is a factor of itself. Thus, every positive integer greater than one has at least two factors: 1 and itself.
The price of an item increases from $60 to $102. What was the percent increase?
m = 102/60 = 17/10 = 1.7 1.7 - 1.0 = .70 → 70%
Multiples - think of the tree picture
multiples mean you multiply so the number gets bigger. Above the line is the multiples, below the line is the factors. Factors of 12: 1, 2, 3, 6, 12 Multiples of 12: 24, 36, 48, etc...
Compare the fractions: 11/14 ?? 77/100
multiply first fraction by 7 on top and bottom. 77/99 is greater than 77/100. Same numerator but smaller denominator = bigger fraction,
An orchard contains only cherry trees, apple trees and peach trees. The ratio of apple trees to peach trees is 2:3, and the ratio of cherry trees to peach trees is 2:1. There are 33 trees altogether. Column A. Column B Number of peach trees 8
peach trees = 9 so A is bigger than b. In order to be able to relate the three types of trees, we need to use the type of tree present in both ratios: the peach tree. To do so, the number of peach trees in one ratio needs to equal the number of peach trees in the other. Currently, we have ratios with 3 peach trees and 1 peach tree. To make those values equal, we multiply the second ratio by 3:C : P 2 : 1 --> 6 : 3The ratio is still the same (2112 = 6336). However, now we are able to compare apples, peaches, and cherries: A : P : C - 2 : 3 : 6 For every 3 peach trees, there are 2 apple trees and 6 cherry trees. If we sum these values, we get 3 + 2 + 6 = 11 trees. (2/11; 3/11; 6/11) That means that in every set of 11 trees, there will be 3 peach trees, 2 apple trees, and 6 cherry trees. We want to know how many peach trees there are in the orchard, which contains 33 trees total. That means that there are 3 sets of 11 trees in the orchard, and in each set there are 3 peach trees. 3 × 3 = 9 peach trees total.
