GRE Math
What is the length of arc ABC of circle O if the radius is 6 and the central angle AOC is 60 degrees?
2π C = 2πr → C = 2 x π x 6 = 12π Arc length = n/360(Circumference) → 60/360 x 12π = 1/6 x 12π = 2π
Allen, Betty, and Claire must wash the dishes. They decide to work in shifts of two people. How many shifts will it take before all possible combinations have been used?
3 (AB, AC, BC)
If one book costs $6.50, what's the cost of 12 copies of the same book?
$78 12(6.50) = 78 (multiply when you're given an amount for 1 item & asked for the total amount of many of these items)
June pays an average price of $14.50 for 6 articles of clothing. What is the total price of all 6 articles?
$87.00 Sum of terms= Average x Number of terms = 14.50 x 6
What is the range of the set (3, 8, 2, -6, 0, -2, 7)?
-14
What is the probability of tossing a coin 4 consecutive times and having the coin land heads up 0, 2, 3, or 4 times?
0.75 1. Find probability of coin landing heads up once → tossed 4 times & each toss has 2 outcomes, so the # of outcomes is 16 (2 x 2 x 2 x 2) & the total # of desired outcomes can be counted: HTTT, THTT, TTHT, or TTTH → 4 desired outcomes → prob of rolling 1 head = 4/16 = .25 2. To find probability of coin landing not exactly once subtract that probability from 1 → 1 - 0.25 = 0.75 (bc the total of all probabilities of all possible outcomes must equal 1)
What is the probability of rolling a 3 four consecutive times?
1 / 1,296 (1/6)(1/6)(1/6)(1/6) = 1 / 1,296 -To calc the probability of 2 or more independent events, multiply the probabilities of the individual events
Chord & Tangent (for a circle)
1. Chord: A line segment that joins 2 points on the circle. The longest chord of a circle is its diameter 2. Tangent: A line that touches only 1 point on the circumference of a circle. A line drawn tangent to a circle is perpendicular to the radius at the point of tangency
How do you calculate standard deviation?
1. Find the average/mean of the data points 2. Find the difference btw the average and each data point (subtract the mean from each #) 3. Square the differences 4. Find the average of the squared differences (add them & divide by the number of terms) 5. Take the square root of that average
A bag contains 10 marbles, 4 of which are blue & 6 of which are red. If 2 marbles are removed without replacement, what is the probability that both marbles removed are red?
1/3 1. Prob that the 1st marble removed will be red → 6/10 = 3/5 2. Prob that the 2nd marble removed will be red → 5/9 (fewer marbles so denominator is 1 less & 1 fewer red marble) 3. Prob that both marbles removed will be red → (3/5)(5/9) = 3/9 = 1/3 -Only focus on red marbles being chosen, don't do situations where blue is chosen 1st -To calc the prob of 2 or more dependent events occurring, multiply their individual properties
What is the length of the hypotenuse of a right triangle with legs of lengths 9 and 10?
181 Pythagorean theorem → a² + b² = c² 9² + 10² = c²
How many groups can be formed consisting of 2 people from room A and 3 people from room B if there are 5 people in room A and 6 people in room B?
200 Combination formula: n! / k!(n - k)! n! / k!(n - k)! n = # of whole group, k = # in subgroup 1. Insert the appropriate #s into the combination formula for each room & multiply the results 2. n! / k!(n - k)! → 5! / 2!(3!) = (5)(4)(3)(2)(1) / (2)(1)(3)(2)(1) → (5)(4) / 2 = 10 3. n! / k!(n - k)! → 6! / 3!(3!) = (6)(5)(4)(3)(2)(1) / (3)(2)(1)(3)(2)(1) → (6)(5)(4) / (3)(2) = 20 4. Multiply: 10 x 20 = 200
The exterior angles of any triangle add up to ___________
360
Pythagorean Theorem- Special Right Triangles (Isosceles right triangles- 30, 60, 90 & 45, 45, 90)
45, 45, 90- x : x : x√2 (x√2 = hypotenuse) 30, 60, 90- x : x√3 : 2x (2x = hypotenuse & x = side opposite the 30⁰)
How much time does it take to drive 300 miles at 60 mph?
5 hours (Time= Distance/ Speed → 300/ 60 = 5)
What is the area of the sector AOC in circle O if the radius is 6 and the central angle AOC is 60 degrees?
6π Area of sector = n/360(Area of circle) → 60/360 x πr² = 1/6(36π) = 6π
A menu has 3 choices for soup & 4 salad options; diners are permitted to select a soup or salad with their dinners. What is the total number of choices available?
7 (3 + 4 = 7) -Add in situations where choices are "or" rather than "and" (instead of multiply)
What is the value of 1/4 x 1/6 ÷ 3/7
77/22
Parallelogram
A quadrilateral with 2 pairs of parallel sides -Opposite sides are equal in length -Opposite angles are equal -Angles that are not opposite are supplementary to each other
Formula for Arc length
Arc Length = n/360(Circumference) → n/360 x 2πr (n = central angle measurement)
How do you find the area of a triangle?
Area = (1/2)(Base)(Height)
Formula for the circumference of a circle
C = πd or C = 2πr (bc d = 2r) (π = circumference / diameter)
What is the probability formula
Probability = Number of desired outcomes / Number of possible outcomes
Formula for volume and surface area of a rectangular solid
Volume = (Area of base)(Height) = (Length x Width)(Height) = lwh Surface area = Sum of area of faces = 2lw + 2lh + 2hw
Slope of a line
m = (∆ y / ∆ x) rise / run or change in y over change in x → (y2 - y1) / (x2 - x1)
(x + 5)(x - 2)
x² + 3x - 10
If x - 2z = 2(y - z), 2x - 6y + z = 1, and 3x + y - 2z = 4, what's the value of z?
z = 5 1. x = 2y - 2z + z → x = 2y 2. Plug 2y in for x in the other equations a. 2(2y) - 6y + z = 1 → z - 2y = 1 b. 3(2y) + y - 2z = 4 → 7y - 2z = 4 3. Multiply 1st equation by 2 → 2z - 4y = 2 4. Add the equations → 2z - 4y = 2 + (-2z + 7y) = 4 3y = 6 → y =2 5. Plug y into 2z - 4y = 2 → 2z - 4(2) = 2
What is the hypotenuse of an isosceles right triangle with legs length of 4?
4√2 Can use pythagorean theorem but is faster to use the special right triangle ratios Isosceles right triangle- Ratio of a leg to hypotenuse = x : x : x√2
If ♫ (x) = x² + 3 / x - 1 and 3r - 4 = 5, what is the value of ♫ (r)?
6 1. Find r → 3r - 4 = 5 → r = 3 2. Plug r in → 3² + 3 / 3-1 = 12 / 2 = 6
What is the area of a triangle that has a base of 4 and height of 3?
6 Area = 1/2bh → bh/2 → 4 x 3 / 2 = 6
Formula for Area of Sector
Area of Sector = n/360(Area of circle) → n/360 x πr² Sector: portion of a circle's area that's bounded by 2 radii & an arc (shaded section of circle) 1. Find the degree measure of the sector's central angle & figure out what fraction that degree measure is of 360 2. Multiply the area of the whole circle by that fraction
If x² - 16 / x² + 6x + 8 = y and x > -2, which of the following is an expression for x in terms of y? A. 1 + y / 2 - y B. 2y + 4 / 1 - y C. 4y - 4 / y + 1 D. 2y - 4 / 2 + y E. y + 4 / y + 1
B. 2y + 4 / 1 - y 1. Plug in a number greater than -2 → x=0; y= 0 - 16 / 0 + 0 + 8 = -16 / 8 = -2 2. y = -2 when x = 0 → Substitute -2 for y in the answer choices to find the choice that gives you 0 3. Only B gives you 0 (0/3= 0) (D gives you -8/0 = undefined)
★x is defined as the largest even number that is less than the negative square root of x. What is the value of ★81? A. -82 B. -80 C. -10 D. -8 E. 8
C. -10 (neg sq. root of 81 is -9 & the largest even # that's less than -9 is -10)
If a > 1, the ratio of 2a + 6 to a² + 2a - 3 is A. 2a B. a + 3 C. 2 / a-1 D. 2a / 3(3 - a) E. a-1 / 2
C. 2 / a-1 -Solve by picking a number & substituting it for the variable 1. a has to be greater than 1 so substitute a w/ 2 → 2(2) + 6 = 10 and 2² + 2(2) - 3 = 5 2. Question now reads, the ratio of 10 to 5 is what → 10:5 = 10/5 = 2 3. Find the answer choice that gives you 2 when you plug 2 in for a (check every answer choice!) (if more than 1 choice give you 2, pick another #)
If 5x/3 + 9 = x/6 + 18, x = A. 12 B. 8 C. 6 D. 5 E. 4
C. 6 -Solve normal or backsolve- plug in answer choices for x -Since x is in the numerator in the fractions & the denominators are 3 & 6, plug in a choice that's divisible by 3 & 6 (choices A & C)
What is the formula for calculating the probability of one or another event occurring? (Probability of A or B or both)
P(A or B) = P(A) + P(B) - P(A and B)
Events A and B are independent. P(A) is 0.60 and P(A or B) is 0.94. What is the probability that event B occurs?
P(B) = 0.85 Formula: P(A or B) = P(A) + P(B) - P(A and B) 1. Since the events are independent, P(A and B) = P(A) x P(B) 2. Plug in values given → 0.60 + P(B) - (0.60 x P(B)) = 0.94 0.60 + P(B) - 0.60P(B) = 0.94 0.60 + 0.40P(B) = 0.94 0.40P(B) = 0.34 P(B) = 0.85 (p.292)
Formula to find the perimeter of a rectangle
Perimeter = 2(Length + Width) bc the opposite sides are equal, the perimeter of a rectangle is twice the sum of the length & width (perimeter of a square = 4 (side)
If m = 4n - 10 & 3m + 2n = 26, find the values for m & n
m = 6 and n = 4 1. Substitute 4n - 10 for m into 3m + 2n to solve for n → n=4 2. Plug in 4 for n in either equation to solve for m → m=6
If 3(x - 6) + 8 - (2 - 4x) = 7 - 4(x + 2), what is the value of x?
x = 1 3x - 18 + 8 - 2 + 4x = 7 - 4x - 8 7x - 12 = 4x - 1 11 x = 11 → x = 1
If xy + 7 + 3x / 4 - 2y = 1, what is the value of x in terms of y?
x = 8y - 3 / 4y + 3 1. Get rid of the fraction by multiplying everything by 4 → 4(xy) + 4(7 + 3x / 4) - 4(2y) = 4(1) → 4xy + 3x = 8y - 3 2. Factor out x → x(4y + 3) = 8y - 3 3. Divide both sides by (4y + 3) → x = 8y - 3 / 4y + 3
Solve for x: 3 - x/4 ≥ 2
x ≤ 4 (have to flip the direction of the inequality when you multiply or divide by a negative #)
If the sales tax on a $12.00 lunch is $1.20, what is the total amount of the check?
$13.20 12.00 + 1.20 = 13.20 (add when you're given amounts of ind. quantities & asked to find the total)
If 350 widgets cost $20, how much will 1,400 widgets cost at the same rate?
$80
Factor the expression 9x² - 1
(3x + 1)(3x - 1) Difference of squares → 9x² = (3x)² & 1 = 1² → 9x² - 1 = (3x + 1)(3x - 1)
How to factor a difference of two squares? Ex: (a² - b²)
(a + b)(a - b)
If x² + 10x + 25 = 0, what is the value of x?
-5
Arrange in order from smallest to largest: 0.7, 0.77, 0.07, 0.707, & 0.077
0.07, 0.077, 0.7, 0.707, 0.77 (add 0s to the end of the #s until you reach the thousandth place & all have the same amount of #s & put into fractions)
Let x* be defined by the equation: x* = x² / 1 - x². Evaluate (1/2)*
1/3 (1/2)* = (1/2)² / 1-(1/2)² = 1/4 / 1-1/4 = 1/4 / 3/4 = 1/4 x 4/3 = 1/3
An operation is symbolized by ★ is defined by the equation x ★ y = x - 1/y. What is the value of 2 ★ 7?
13/7 Find the reciprocal of the 2nd quantity & subtract the reciprocal from the 1st quantity 1. Substitute 2 & 7 in the equation replacing x with 2 & y with 7 2. Reciprocal of 7 is 1/7, & subtracting from 2 gives you: 2- 1/7 = 14/7 - 1/7 = 13/7
There are 2 bowlers & you must find their average score per game. One has an average score per game of 100 & the other has an average score per game of 200. Supposed they bowled 4 games. Find their combined average.
150 1. Average formula= Sum of terms/ Number of terms 4(100) + 4(200) / 8 = 150 OR 2. Balanced Average: If both bowlers bowled the same # of games you can average their averages to find the combined average. So, their combined average will be the average of 100 & 200 (just average the 2 #s since they played the same # of games → 100 + 200/ 2)
1/3 + 3/5 + 1/2 - 13/15 = ?
17/30
17 is what percent of 85?
20% Set up proportion & cross multiply → 17/85 = x/100 → 1,700= 85x → x= 20
The ratio of domestic sales revenues to foreign sales revenues of a certain product is 3:5. What fraction of the total sales revenues comes from domestic sales?
3/8 1. The 3:5 ratio tells you that of every $8 in sales revenues, $3 comes from domestic sales and $5 from foreign sales. 2. So the ration of domestic sales revenues to total sales revenues is 3:8, or 3/8
The average annual rainfall in Boynton for 1976-1979 was 26 inches per year. Boynton received 24 inches of rain in 1976, 30 inches of rain in 1977, & 19 inches of rain in 1978. How many inches of rainfall did Boynton receive in 1979?
31 inches 1. Average formula: Sum of terms= Average x Number of terms 24 + 30 + 19 + x = 26 x 4 → 73 + x = 104→ x = 31 OR 2. Balanced Value: Sum of the differences btw each term & the mean must equal 0 (24 - 26) + (30 - 26) + (19 - 26) + (x - 26) = 0 → -5 (x - 26) = 0 → x = 31
2 years ago, 450 seniors graduated from the high school. Last year, 60 seniors graduated. By what percent did the number of graduating seniors increase?
33 1/3% 600-450= 150 → 150/450 (100%)= 33 1/3%
Joe is 16 & Mary is 12. Express the ratio of Joe's age to Mary's ago in lowest terms
4:3 (16/12 = 4/3, or 4:3)
If the average of 6, 3, -2, 5, 11, & x is 5, what is the value of x?
7 (Average= Sum of values / Number of values → or rewrite & use Sum of values = Number of values x Average) p. 233
The average of (-1, 3, 0, -2, 4, 2, x, y) is 3 and x - y = 2. What's the value of y?
8 Rearrange formula: Average= sum/ # of values → Sum = (average) x (# of values) 1. Sum = 3 x 8 = 24 2. Get x alone: x - y = 2 → x = 2 - y (plug in for x) 2. 24 = 4 + 2 + 2 + y + y → 16 = 2y → y = 8
How many groups of 30 students can be formed from a total of 240 students?
8 groups of 30 240 / 30 = 8 (divide when you're given the size of 1 group & the total size for many such identical groups & are asked how many of the small group fit into the larger one)
A certain teacher gave a grade of A to all students who scored in the 80th percentile or above on a recent test. The distribution of scores is shown below. What was the minimum score needed to receive an A on this test? Score Range: # of Students: < 70 2 70 - 74 7 75 - 79 15 80 - 84 8 85 - 89 3 90 - 94 2 > 94 3 Total Class 40
85 1. 80th percentile = 100% - 80% = 20% 2. 0.2 x 40 = 8 students (in 80th percentile)
If a ballet offers 3 candidate choices for Office A, 4 for Office B, and 2 for Office C, what is the total number of different ways a voter could fill out the ballot?
24 (3 x 4 x 2 = 24)
Jack drives a car that was manufactured in Europe, so the fuel economy readout on his dashboard is stated in L/km. If the value shown is 0.095, what is the fuel consumption rate stated in the American conversion of miles per gallon? (Note: 5 miles = 8 km & 1 gal = 3.8 L.)
25 mpg (Make conversions for each step) 1. Convert km to miles: 0.095 L/1 km x 8 km/5 mi = (0.095 x 8) L / 5 mi 2. Convert gallons/mile: (0.095 x 8) L/5 mi x 1 gal/3.8 L = 1 gal/5 mi x (0.095 x 8) / 3.8 3. Result= 0.4 gal/mi 4. Invert bc question asks for mi/gal → 1 / .04 = 25 (p. 235)
Find the perimeter of the triangle in terms of a if b = 2a and c = b/2
4a Perimeter = a + b + c → a + 2a + 2a/2
If 5 pounds of apples costs $6.75, what's the price of 1 pound of apples?
$1.35 6.75 / 5 = 1/35 (divide when you're given a total amount for many items & asked for the amount for 1 item)
What is the probability of tossing a coin 4 consecutive times and having it land heads up exactly once?
0.25 -coin is tossed 4 times & each toss has 2 possible outcomes, so the # of outcomes is 16 (2 x 2 x 2 x 2 = 16) -the total # of desired outcomes can be counted: HTTT, THTT, TTHT, or TTTH → so 4 desired outcomes -probability of rolling 1 head is 4/16 = 1/4 = .25
Intersection of 2 sets & Union of 2 sets (what they are & how to determine them)
1. Intersection of 2 sets: A set that consists of all the elements that are contained in both sets (the overlap btw the sets - like in a venn diagram) -Written as A ∩ B 2. Union of 2 sets: The set of all the elements that are elements of either or both sets -Written as A ∪ B -Formula: |A ∪ B|= |A| + |B| - |A ∩ B| (adding the # of elements in A + B isn't correct bc that would count the elements in both sets twice) 3. Mutually Exclusive: If sets have no common elements. Their intersection is the empty set -Written as: |A ∩ C| = ∅ (∅ = empty set) → so |A ∪ C| = |A| + |C|
Quartiles & Interquartile Range
1. Quartiles: Arrange the terms in numerical order & subdivide the terms into 4 equal quarters. The largest # in the 1st group = 1st quartile (Q1), the largest in the 2nd group = Q2, etc. 2. Interquartile Range: Difference btw the values of the 3rd & 1st quartiles, Q3 - Q1. -Boxplot is a way to visually display data dispersion by quartiles→ L = least # in the data, G = greatest #, M = the median, Q1 = 1st quartile, Q3 = 3rd quartile
The ratio of red to blue to black pens in a box is 3:5:7. If all 75 pens in the box are one of these colors, how many are red?
15 1. Convert the part to part ratio to a part to whole ratio for red pens → 3 / 3 + 5 + 7 = 3/5 = 1/5 2. Set up a proportion & cross multiply → R/75 =1/5 → 5R=75→ R= 15
Ben spends $30 on his annual gardening budget on seed. If his total annual gardening budget is $150, what percent of his budget does he spend on seed
20% Percent= 30/150 (100%)= 1/5 (100%) = 20%
What is the value of 0.003 x 2.4 x 10⁴ ÷ 0.09 x 4 x 10⁻² ?
20,000 1. Multiply 0.003 x 10⁴ (move decimal 4 times to the right) → Numerator becomes 30 x 2.4 = 72 2. Multiply first terms in denominator → 0.36 x 10⁻² 3. Convert to 36 x 10⁻⁴ 4. Bring 10⁻⁴ to the numerator → 72 x 10⁴ / 36
A bag contains only red and blue plastic chips. There were 10 chips in the bag and 1 blue chip was removed. The probability of drawing a blue chip was then 1/3. How many red chips were in the bag?
6
Pythagorean Triples
3, 4, 5 (A right triangle w legs of lengths 3 & 4 & a hypotenuse of length 5) 5, 12, 13 (any multiple of these lengths makes another pythagorean triple- ex: 6² + 8² = 10²
In a bag of coins, 2/5 are pennies, 1/3 are nickels, 1/4 are dimes, & there are 5 quarters. If there are no other coins in the bag, what is the total number of coins?
300 1. C= 2/5C + 1/3C + 1/4C + 5 (C= total # of coins) 2. Multiply through by the LCM of 60 → 60C= 24C + 20C + 15C + 300 (p. 234)
The 4 finalists in a spelling contest win commemorative plaques. If there are 7 entrants in the spelling contest, how many possible groups of finalists are there?
35 Combination formula: n! / k!(n - k)! → n = # of whole group, k = # in subgroup 1. 7! / 4!(7 - 4)! → 7! / 4!(3!) 2. (7)(6)(5)(4!)/ (4!)(3!) → 7 factorial has all the factors of 4 factorial, so write 7! as (7)(6)(5)(4!) & cancel the 4! 3. Cancel the 4! in the numerator & denominator → (7)(6)(5) / (3)(2)(2) 4. Cross off the 6 in the numerator & the (3)(2) in the denominator → (7)(5) / 1 = 35
In a bag, the ratio of red marbles to blue marbles is 3:5. If there are only red and blue marbles in the bag, what's the ratio of the number of red marbles to the total number of marbles?
3:8 (can convert a part to part ration to a part to whole ratio bc you know there's only red & blue marbles- 3 + 5 = 8)
Factor the expression 3x³ + 12x² - 6x
3x(x² + 4x - 2)
What is the value of x³ - x² - 7(x - 1) if x = 3?
4 3³ - 3² - 7(3 - 1) = 27 - 9 - 14 = 4
Restaurant A has 5 appetizers, 20 main courses, and 4 desserts. If a meal consists of 1 appetizer, 1 main course, & 1 dessert, how many different meals can be ordered?
400 (5 x 20 x 4 = 400) -The # of possible outcomes from each set is the # of items in the set. So there are 5 possible appetizers, 20 possible main courses, & 4 possible desserts -The # of possible subgroups will always equal the number of items in the set (when finding the # of possible subgroups when choosing 1 item from a set)
Rectangle ABCD has an area of 72 and is composed of 8 equal squares. What is the area of the shaded region?
45 2 approaches: 1.) Break the area into smaller pieces whose separate areas you can find & add those together OR 2.) Find the area of the whole figure & find the area of the region(s) you're not looking for & subtract it from the whole 1. Area of each square is 9 → bc 8 equal squares form a total area of 72 (72 / 8 = 9) 2. Each side of a square is 3 → bc the area of a sq. equals the square of the length of a side (√9 = 3)
If the coordinates of point A are (3, 4) and the coordinates of point B are (6, 8), what is the distance between points A and B?
5 1. Draw a diagram to help visualize - Plot points A & B and in line segment AB. Then draw a right triangle w AB as its hypotenuse & label the point of intersection C 2. Point C has the same x-coordinate as B & same y as A (bc perpendicular) → C (6, 4) 3. Distance btw points A & C is the difference btw their x-coordinates & distance btw B & C is the difference btw their y-coordinates → AC = 3 & BC = 4 4. Use pythagorean theorem to find the hypotenuse (or realize this was a 3, 4, 5)
Which is greater: 5/6 or 7/9?
5/6 (cross multiply or find a common denominator)
What is the sum of the interior angles of a pentagon?
540 1. Draw a pentagon & divide it into triangles 2. Multiply the number of triangles by 180 to get the sum of interior angles 3. 3 x 180 = 540
The ratio of x to y is 5:4. The ratio of y to z is 1:2. What's the ratio of x to z?
5:8 1. Need to y's in the 2 ratios to equal each other bc then can combine the ratios (x:y:z) 2. Multiply 2nd ratio by 4 to make the y's equal → y to z= 4:8 3. Combine w first ratio to make x to y to z ratio → 5:4:8 4. So x to z= 5:8
Nelson can mow 200 sq. meters of lawn per hour. John can now 100 sq. meters of lawn per hour. Working simultaneously but independently, how many hours will it take Nelson & John to mow 1,800 meters of lawn?
6 hours 1. Add Nelson's rate to John's rate to get a combined rate (200 + 100 = 300 m per hour) 2. Divide total number of lawn area by the combined rate (1,800 / 300= 6)
John takes 3 hours to weed the garden, & Mark takes 2 hours to weed the same garden. How long will it take them to weed the garden together?
6.5 (6/5) hours T= ab/ a + b (or 1/T = 1/a + 1/b) 1. 3 x 2 / 3 + 2 = 6/5
Our gardeners have decided they're tired of weeding & have purchased 3 robots. Unfortunately, the robots malfunction a lot. Robot A takes 4 hours, robot B takes 6 hours, & robots C takes 10 hours to weed the garden. How long will it take these robots to weed the garden together?
60/31 hours Adding the rates: 1/T= 1/a + 1/b + 1/c 1/T= 1/4 + 1/6 + 1/10 → 30/120 + 20/120 + 12/120 = 62/120 = 31/60 → T= 60/31 Using the formula: T= a x b x c / ab + ac + bc T= 4 x 6 x 10 / 4x6 + 4x10 + 6x10 = 240/ 24 + 40 + 60 = 240/124 = 60/31
In a spelling contest, the winner will receive a gold metal, the 2nd place finisher will receive a silver metal, the 3rd place finisher will receive a bronze metal, and the 4th place finisher will receive a blue ribbon. If there are 7 entrants in the contest, how many different arrangements of award winners are there?
840 (7 x 6 x 5 x 4 = 840) -Gold can be won by any of the 7 ppl -Silver can be won by any of the remaining 6 ppl -Bronze can be won by any of the remaining 5 ppl -Blue ribbon can be won by any of the remaining 4 ppl (think of each place that needs to be filled as a blank space. The 1st place can be filled w any of the items in the group, the 2nd can be filled w any of the items except for the one used to fill the 1st place, etc.)
Isosceles Triangle
A triangle that has 2 equal sides (are opposite of 2 equal angles)
Formula for the area of a circle
A=πr²
Quantity A: Area of right triangle ABC, where AB = 5 and BC = 4 Quantity B: 6 A. Quantity A is larger B. Quantity B is larger C. They are equal D. Their relationship cannot be determined from the information given
D. Their relationship cannot be determined from the information given May think that ABC is a 3:4:5 right triangle but don't know which sides are given → if AB is the hypotenuse then it's a 3:4:5 & the area is 1/2(3 x 4) = 6 → If AC (the missing side) is the hypotenuse the area is 1/2(4 x 5) = 10
Pythagorean Theorem
The square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides. a² + b² = c² (c² = hypotenuse)
Formula for volume and surface area of a cylinder
Volume = (Area of base)(Height) = πr²h Lateral surface area = (Circumference of base)(Height) = 2πrh (lateral surface: the "pipe" surface as opposed to the circular "ends") Total surface area = Area of circular ends + Lateral surface area = 2πr² + 2πrh (need the height of the cylinder & the radius of the base to find volume & surface area)
Formula for volume and surface area of a cube
Volume = lwh = edge x edge x edge = e³ (since l = w = h, the formula for volume can be stated in terms of any edge) Surface area = Sum of areas of faces = 6e²
What is the combination formula
n! / k!(n - k)! n = number of items in the group as a whole k = number of items in each subgroup formed ! = factorial (ex: 5! = 5 x 4 x 3 x 2 x 1 = 120)
3x + 3y = 18 and x - y = 10, find the values of x and y
x = 8 and y = -2
Equations of lines (linear equation)
y = mx + b m = slope (∆ y / ∆ x) b = point where the line intercepts the y-axis (value of y when x = 0) -Lines that are parallel to the x-axis have a slope of 0 & equation y = b -Lines that are parallel to the y-axis have the equation x = a (a is the x-intercept of that line)
If -1 < x < 1, but x is not 0, which has the greater value |x⁴| or |x⁵|?
|x⁴| (since x is btw -1 & 1 it will be a fraction & as positive fractions are raised to greater powers they become smaller)
Factor the polynomial x² + 6x + 9
(a + 3)² -x² & 9 are both perfect squares & 6x is 2(3x) -so this polynomial is in the form of a² + 2ab + b² -a = x & b = 3
Factor the expression ac - 2bc - 2bd + ad
(a - 2b)(c + d) 1. Factor c out of the first half → c(a - 2b) 2. Factor d out of the second half → d(a - 2b) 3. c(a - 2b) + d(a - 2b) → (a - 2b) is a common term so factor it out 4. (a-2b) (c + d)
Factor the polynomial x² - 3x + 2
(x - 1)(x - 2) 1. The last term (2) is the product of the 2 missing terms 2. The coefficient of the 2nd term (-3) is the sum of the 2 missing terms 3. Find the factors of 2 that add up to -3 (-2 & -3)
If the price of a $120 dress is increased by 25%, what is the new selling price?
$150 (% increase= increase (100% / original price) 1. Increase/120 (100%) = 25% → = Increase/120 = 1/4 2. Increase= 120/4 3. Increase= 30 4. 120 + 30 = 150
If an investor invests $100 at 20% simple annual interest, how much does she have at the end of 3 years?
$160 1. 20% of $100 is $20 2. After 3 yrs he will have 3 yrs of interest, or $60 (20 x 3) 3. 60 + 100 = $160
If an investor invests $100 at 20% compounded annually, how much does he have at the end of 3 years?
$172.80 1. First yr: Earns 20% of 100= $20 → $120 2. Second yr: Earns 20% of 120= $24 → $144 3. Third yr: Earns 20% of 144= $28.80 → $172.80
A certain brand & style of shoe was priced at $120. The store owner was concerned it wasn't selling well & decided to mark the list price down by 15%. Sales of the shoe only increased slightly, so the owner offered an additional 10% discount from the sales price at checkout. Sales tax of 8% is added to all purchases. What would a customer have to pay for the shoes with tax added?
$99.14 1. 120 x 0.15= 18 → 102 - 18 = 102 2. 102 x 0.10= 10.2 → 102 - 10.2 = 91.8 3. 91.8 x 0.08 = 7.34 → 91.8 + 7.34 = 99.14
If a represents the median value of a group of numbers, b is the mode, and c is the range, what is the value of a + b - c for the set (3, 7, -4, 2, -5, 0, -2, 7)?
-4 a = 1, b = 7, c = 12
If list A consists of the numbers 1, 3, 3, 3, 3, 3, 3, 5 and list B consists of the numbers 2, 3, 3, 3, 3, 3, 3, 4, how much greater is the standard deviation of list A than that of list B?
0.5 1. Find the mean for each list → List A mean: 3; List B mean: 2 2. Subtract the mean from each # → List A: -2, 0, 0, 0, 0, 0, 0, 2; List B: -1, 0, 0, 0, 0, 0, 0, 1 3. Square each difference & add them → List A: (-2)² + (2)² = 8 List B: (-1)² + (1)² = 2 4. Find the average → List A: 8/8 = 1 & List B: 2/8 = .25 5. Take the square root of the average → List A: √1 & List B √.25 = .50 Difference btw the SD = 1 - .5 = .5
What is the range of (-3, 0 , 2, x, 6, y, 3) if x² + 8x = -16 and y² - y = 6?
10 1. Find value of x: x² + 8x + 16 = 0 → (x + 4)(x + 4) = 0 → x = -4 2. Find value of y: y² - y - 6 = 0 → (y - 3)(y + 2) = 0 → y = 3, y = -2
The average of 63, 64, 85, & x is 80. What is the value of x?
108 1. Average formula: Sum of terms= Average x Number of terms 63 + 64 + 85 + x = 80 (4) → x=108 OR 2. Balanced Value: Think of each term in terms of its position relative to the ave. 80 -63 is 17 less than 80, 64 is 16 less than 80, & 85 is 5 greater than 80 -The 3 terms (17 + 16 - 5) total 28 less than the average -X must be 28 greater than the average to restore the balance at 80 → x= 28 + 80= 108
Working together, Kendra, Latasha, & Melanie can complete a certain task in 4 hours. If Kendra alone could complete the task in 8 hours & Latasha could complete the task in half the time it would take Melanie, how long would it take Latasha to complete the task by herself?
12 hours 1/T= 1/K + 1/L + 1/M 2. 1/4 = 1/8 + 1/L + 1/2L (2L bc Latasha can do it in half the time as Melanie so M=2L) 3. Multiply each by the LCM of the denominators (8L) to clear the fractions → 8L(1/4) = 8L(1/8) + 8L(1/L) + 8L(1/2L) → 2L= L + 8 + 4 → L= 12
Below shows the inventory of light bulbs at a hardware store. If the total number of bulbs in stock was 400, how many were less than 60 watts? 100 watts: 15% 25 watts: 10% 40 watts: 20% 60 watts: 30% 75 watts: 25%
120 (less than 60 watts = 30% → 400 x .30 = 120)
In a certain sequence, if x > 2, then s(x) (subscript x) = s subscript (x - 1) + s subscript (x - 2) + 3. If s subscript 1 = 0 and s subscript 2 = 4, what is the value of s subscript 4 (s4)
14 -To find s4 the values of s3 & s2 must first be determined (bc a series of #s beginning w the 3rd # in the series is composed of the previous # plus the 2nd prior # plus 3) -Need the # of the preceding #s to calculate the value of any # 1. S3= S2 + S1 + 3 p. 258 → 4 + 0 + 3 = 7 2. S4 = 7 + 2 + 3 - 14 (p. 258)
The total weight of licorice sticks in a jar is 30 ounces. If the average weight of each licorice stick is 2 ounces, how many licorice sticks are there in the jar?
15 sticks Number of terms= Sum of terms / Average (30/2)
If a website calls for a 3-letter password but no two letters can be the same, what is the total number of possibilities of letter choices?
15,600 (26 x 25 x 24 = 15,600) -no 2 letters can be the same so reduce the # of choices for the 2nd & 3rd letters
There are 2 bowlers & you must find their average score per game. One has an average score of 100 and has bowled 4 games and the other has an average of 200 and has bowled 16 games. What is their weighted average?
180 1. Average formula: 4(100) + 16(200) / 20 2. Balanced Average: The bowler bowling an average of 200 bowled 4/5 of the games, so the combined average will be 4/5 of the distance along the # line btw 100 & 200, which is 180
A:B is 3:7, C:D is 15:11, & B:C is 14:5, what is A:D?
18:11 A/B= 3/7, C/D= 15/11, B/C= 14/5 1. Manipulate ratios so they share a numerator or denom. 2. Multiply both parts of A:B by 2→ A/B = 6/14 3. Flip B:C→ B/C = 5/14 4. Now A:C can be compared bc have same denom → A/C = 6/5 5. Multiply both parts of A:C by 3 (to match CD)→ A/C = 18/15 6. Flip C:D→ D/C = 11/15 7. A/D= 18/11 (since demons are equal) OR "Chain the ratios" 1. Look at just variables. Note that A/B x B/C x C/D = A/D bc the Bs & Cs cancel out 2. Plug the values into this equation→ 3/7 x 14/5 x 15/11 → 3/1 x 2/1 x 3/11 = 18/11
The mean of a group of 6 numbers is 15. If the values of one-third of the numbers are increased by 12 each, what is the new mean?
19 1. Rearrange formula→ Average = sum / # of values → Sum = (# of values) x (average) → 6 x 5 = 90 2. 1/3 of 6 = 2 → Increase of 2 #s by 12 = Sum increased by 24 3. Divide new sum by 6 → 90 + 24 = 114 → 114 / 6 = 19
If 3x - 7 ≥ -1, what is the minimum value of x?
2 (test numbers greater than -1 until you get an answer that is greater than or equal to -1)
A store is open for 10 hours each day. Below shows the number of customers each hour for the past week. What was the relative frequency of the number of customers observed btw noon & 2 pm? Time: Customers: 10 - 11 am 22 11 - 12 pm 38 12 - 1 60 1 - 2 44 2 - 3 33 3 - 4 27 4 - 5 31 5 - 6 38 6 - 7 49 7 - 8 58
26% 1. Find total number of customers → 400 2. Number of customers btw 12 & 2 → 60 + 44 = 104 3. Percent of total → 104 / 400 = .26 = 26%
The ratio of x to y is 1:4. If the value of x were increased by 1 without changing the value of y, the ratio of x to y would become 1:3. If z= x + 2y, what is the value of z?
27 1. Restate the ratios as fractions: x/y = 1/4 → y= 4x 2. Ratio if x increased by 1: x + 1 / y = 1/3 3. Cross multiply: y= 3x + 3 4. Set the 2 values equal to each other: 4x = 3x + 3 → x= 3 5. y= 4x → y= 12 6. z: 3 + 2(12) = 27
Factor the expression 2a + 6ac
2a(1 + 3c) (greatest common factor= 2a, so factor out 2a)
Express a - b² / b - a in terms of x if a= 2x & b= 3
2x-9 / 3-2x (can't simplify or evaluate any further w/out more info)
(3x² + 5x + 7) - (x² + 12)
2x² + 5x - 5 To add or subtract polynomials, combine like terms → (3x² - x²) + 5x + (7 - 12) = 2x² + 5x - 5
If @@(x) = x + 3 / x - 1 and ##(y) = y² - 2y + 3, what's the value of @@(##(2))?
3 1. ##(2) → 2² - 2(2) + 3 = 3 (now plug 3 in for @@(x)) 2. @@(3) → 3 + 3 / 3 - 1 = 3 (plug number in for ##, then plug that value in
A teacher surprised her students with a 5 question pop quiz. 10% of the students answered no questions correctly, 20% had 2 right answers. 30% answered 3 questions correctly, 30% had 4 right answers, & 10% had a perfect score. What was the weighted average of the number of correct answers.
3 Weighted Avg= .1(0) + .2(2) + .3(3) + .3(4) + .1(5) / .1 + .2 + .3 + .3 + .1 = 0 + .4 + .9 + 1.2 + .5 / 1 = 3
Jen & Boris are graduate assistances helping a professor grade student tests. Working together, they're able to complete the task in 1 hour & 12 minutes. If Jen could have graded the entire batch of tests by herself in 2 hours, how long would it have taken Boris to complete that task by himself?
3 hours Combined Work Formula: T = AB / A + B 1. 72 = 120B / 120 + B (converted to minutes) 2. Cross multiply → 72(120 + B) = 120B 3. Distribute the multiplication → 72(120) + 72B = 120B 4. Group the terms w the variable → 72(120) = 48B 5. Divide both sides by 24 → 3(120) = 2B → B= 3(60) = 3 hrs
Abdul recently made a 200 mile trip. For the first 30 miles, he traveled at an average speed of 45 mph. His average speed for the next 50 miles was 60 mph. Adbul averaged 50 mph for the final portion of the trip. How long did it take Adbul to compete his journey?
3 hours & 54 minutes Time= Distance / Speed (use this formula to determine the time for each leg of his trip) 1. T= 30/45 = 2/3 2. T= 50/60 - 5/6 3. T= 120/50 = 2 2/5 (distance found by subtracting the 2 known distances from the total) 4. Convert to minutes rather than finding a common denominator → 2/3 (60) + 5/6 (60) + 2 2/5 (60) = 40 + 50 + 144 = 234 5. Divide 234 minutes by 60→ 3 hours & 54 min (p. 233)
The lengths of boards cut at a sawmill are normally distributed with a mean of 96 inches and a standard deviation of 0.05 inch.What is the approximate probability that a randomly selected board will be longer than 96.10 inches?
3% -A boarder longer than 96.10 inches would be .10 inches longer than the mean value -The SD of the boards is 0.05, so a 96.10 inch board would be 2 SDs above the mean -The boards are normally distributed so 50% are below the mean, 34% are w/in the area btw the mean & 1 SD above the mean , & 13% are btw 1 & 2 SDs above the mean -Probability that a random board would be less than or equal to 96.10 inches → 50% + 34% + 13% = 97% -Probability that a board would be longer → 100% - 97% = 3%
Rafael has three more bus tokens than Taz. Chan has twice as many as Rafael. If Chan had 1 more token, he would have 3 times as many as Taz. How many tokens does the group have?
37 R = 3 + T; C = 2R ; C = 3T - 1 1. Substitute 2R for C in the 3rd equation → 2R = 3T - 1 2. Double the 1st equation → 2R = 2T + 6 3. Set both R equations equal to each other → 2T + 6 = 3T - 1 → T = 7 4. Plug T in the 1st equation to find R → 3 + 7 = 10; R = 10 5. C = 2R → 2(10); C = 20 6. 7 + 10 + 20 = 37
The 7th & 8th grade students in a particular school were given the same math test. A total of 110 7th graders and 100 8th graders completed this test. The overall average score for the 2 grades was 37.4. If the 7th graders averaged 36.4, what was the average score of the 8th grade students?
38.5 1. Rearrange formula: Average= sum/# of values → Sum = (average) x (# of values) 2. Sum of 7th graders= 36.4 x 110 = 4,004 3. Sum of 8th graders= 37.4 = 4,004 + x / 210 → x = 3,850 4. Average of 8th grade = 3,850 / 100 = 38.5
How many hours will it take to fill a 500 liter tank at a rate of 2 liters per minute?
4 1/6 hours (Time= Quantity/ Rate → 500/ 2= 250 minutes)
If ! and ∫ are digits and (!!)(∫∫) = 60∫, what is the value of ∫
5 1. Each symbol represents a digit from 0-9, so the product of the multiplication equals a value from 600 to 609 2. The 2 quantities multiplied each consist of a 2 digit integer in which both digits are the same 3. List the relevant 2 digit integers → 00, 11, 22, 33, 44, 55, 66, 77, 88, 99 4. Test which can be multiplied evenly into the 600 to 609 range → only (11)(55) fits the requirement → ∫ = 5
The price of a share of company A's stock fell by 20% two weeks ago & by another 25% last week to its current price. By what percent of the current price does the share price need to rise in order to return to its original price A. 45% B. 55% C. 66 2/3% D. 75% E. 82%
C. 66 2/3% 1. Pick a value for the original price ($100) → 100- 20 (20%) = $80 2. 25% = 1/4 and 1/4 of 80= $20 3. 80 - 20 = $60 4. Price must rise by $40 to return to its original price → So 40 is what percent of the current price, 60? 5. 40/60 (100%) = 2/3 (100%) = 66 2/3%
If 3 - 5y = 10 & 6y - 2x = 20, what is the value of x + y?
x + y = 30 1. Align the 2 equations so the terms w the same variables are in the same order bc simply adding the 2 equations gives the answer 3x - 5y = 10 -2x + 6y = 20 =x + y = 30
If 5x + 5y = 20, what is the value of x + y?
x + y = 4 (just divide both sides by 5)
What are the possible values of x if 9x² - 36 = 0
x = -2 and x = 2 (3x + 6)= 0 and (3x - 6)= 0
x² - 3x + 2 = 0. Solve for x.
x = 2; x = 1 1. Factor → (x - 2)(x - 1) 2. Set both factors equal to 0 → x - 2 = 0 and x - 1 = 0
What are the possible values of x if 6x² - 10 = 11x?
x = 5/2 and x= = -2/3 6x² - 10 = 11x → 6x² - 11x + 10 = 0 (2x - 5)(3x + 2) = 0
If y = 2x, and 3(y + 6x) - 7(2y + 3) = 11, what is the value of y?
y = -16 1. Plug in 2x for y in the 2nd equation → 3(2x + 6x) - 7(4x + 3) = 11 → x = -8 2. Plug x in → y = 2 (-8) = -16
If x - 15 = 2y & 6y + 2x = -10, what's the value of y?
y = -4 1. Solve first equation for x by adding 15 to both sides → x = 2y + 15 2. Substitute 2y + 15 for x in the second equation → 6y + 2(2y + 15) = -10
If x + 4y = 19 and 2x - y = 11, what is the value of y
y = 3 1. x = 18 - 4y 2. Plug x in for y → 2(19 - 4y) - y = 11
In the formula: z = xy / a + yb, solve for y in terms of x, z, a, & b
y = az / x - bz 1. Multiply both sides by a + yb → z(a + yb) = xy 2. Distribute the z → az + ybz = xy 3. Put all terms containing a y on 1 side → az= xy - ybz 4. Factor out the common factor (y) → az= y(x - bz) 5. Divide to get y alone → az/x - bz = y