ACT Test Prep Math Concepts 104 Questions with Verified Answers
Undefined - CORRECT ANSWER division by zero; a/bc is ___ if b or c is equal to
... [Show More] zero
real number - CORRECT ANSWER number that has a location on the number line
imaginary number - CORRECT ANSWER on the ACT, ___ numbers involve the square root of a negative number; square root of -4
integer - CORRECT ANSWER this is a whole number.. includes negative whole numbers and zero
rational number - CORRECT ANSWER a number that can be expressed as a ratio of two integers
irrational number - CORRECT ANSWER a real number (with a location on the number line) that cannot be expressed precisely as a fraction or decimal; on the ACT, the most important ___ numbers are pi, the sqr. root of 2, and the sqr. root of 3
order of operations - CORRECT ANSWER PEMDAS; parentheses, exponents, multiplication and division (left to right), addition and subtraction (left to right)
how to handle absolute value bars - CORRECT ANSWER do what's inside them first and then take the absolute value of the result. treat ___ signs a lot like parentheses.
how to count consecutive integers - CORRECT ANSWER subtract the smallest from the largest, then add one
factors - CORRECT ANSWER the ___ of integer n are the positive integers that divide into n with no remainder. (six is a ___ of twelve)
multiples - CORRECT ANSWER the ___ of integer n are the integers that n divides into with no remainder. (twelve is a ___ of six)
prime number - CORRECT ANSWER a positive integer that has exactly two positive integer factors, one and the integer itself.
how to find the prime factorization of a number - CORRECT ANSWER keep breaking the integer up into factors until all factors are prime; 36 = 4 x 9 = 2 x 2 x 3 x 3
how to determine whether two integers are relative primes - CORRECT ANSWER break both down to their prime factorizations. if they have NO PRIME FACTORS IN COMMON, they are relative primes
how to find a common multiple of two numbers - CORRECT ANSWER multiply the two numbers together (this will not be the least common multiple unless the two are relative primes)
how to find the LCM, or the least common multiple - CORRECT ANSWER check out the multiples of the larger number until you find one that's also a multiple of the smaller; to find the LCM of 12 and 15, take the multiples of 15: 15 not divisible by 12, neither is 30 and neither is 45. but the next multiple of 15, 60, is divisible by 12 so its the LCM
multiples of 2 and 4 - CORRECT ANSWER an integer is divisible by 2 if the last digit is even and divisible by 4 if the last two digits form a multiple of 4
multiples of 3 and 9 - CORRECT ANSWER an integer is divisible by 3 if the sum of its digits is divisible by 3 and divisible by 9 if the sum of its digits is divisible by 9
multiples of 5 and 10 - CORRECT ANSWER an integer is divisible by 5 if the last digit is 5 or 0 and divisible by 10 if the last digit is 0
remainder - CORRECT ANSWER the ___ is the whole number left over after division. because 487 is 2 more than 485, the remainder of 487/5 will be 2
how to add/subtract fractions - CORRECT ANSWER find a common denominator then add or subtract the numerators
how to multiply fractions - CORRECT ANSWER multiply the denominators and multiply the numerators
how to divide fractions - CORRECT ANSWER invert the second fraction and multiply
how to convert a mixed number into an improper fraction - CORRECT ANSWER multiply the whole number part by the denominator, then add the product to the numerator. the result is the new numerator over the same denominator
how to convert an improper fraction into a mixed number - CORRECT ANSWER divide the denominator into the numerator to get a whole number quotient with a remainder. the quotient becomes the whole number part of the mixed number, and the remainder becomes the new numerator (same denominator).
reciprocal - CORRECT ANSWER switch the numerator and the denominator
how to compare fractions - CORRECT ANSWER convert to decimal, or re-express them with a common denominator
identifying the parts and the whole - CORRECT ANSWER the PART is usually associated with the verb is/are and the WHOLE is usually associated with the word of; half of the girls are brunettes (the whole is the girls - of the girls, and the part is the brunettes - are brunettes)
percent formula - CORRECT ANSWER part = percent x whole
percent increase - CORRECT ANSWER to increase a number by the percent, add the percent to 100%, convert to a decimal, multiply
finding the original whole after a percent increase - CORRECT ANSWER set up an equation. think of a 15% increase over x as 1.15x, a 47% increase over x as 1.47x and so on
combined percent increase and decrease - CORRECT ANSWER the price of a $100 item went up 10% one year and up 20% the next year. what is the combined percent increase? Year 1: 100 + (10% of 100) = 110. Year 2: 110 + (20% of 110) = 132. That is a combined 32% increase.
how to set up a ratio - CORRECT ANSWER put the number associated with the word OF on top and the quantity associated with the word TO on the bottom, the reduce the fraction; ratio of 20 oranges to 12 apples = 20/12 = 5/3
how to solve a proportion - CORRECT ANSWER cross multiply
average rate - CORRECT ANSWER to find the average speed for 120 miles at 40 mph and 120 miles at 60 mph, DO NOT just average the two speeds. first find the total distance (240 miles) and the total time (3 hours for the first leg and 2 hours for the second leg = 5 hours). average speed = 240/5 = 48 mph
average of evenly spaced numbers - CORRECT ANSWER just average the smallest and the largest; avg. of all integers 13 through 77 is the same as the avg. of 13 and 77.
using the average to find the sum - CORRECT ANSWER sum = average x number of terms
how to find a missing number when you're given the average - CORRECT ANSWER USE THE SUM. if the avg. of four numbers is 7, then the sum of those four numbers is (4 x 7) 28 because of the formula "sum = average x number of terms." if three of the numbers are 3, 5, and 8, add these together. this value comes out to be 16 of that sum of 28, which leaves 12 for the (missing) fourth number
probability: counting the possibilities - CORRECT ANSWER if there are m ways one event can happen and n ways a second event can happen then there are M x N ways for the two events to happen; 5 shirts and 7 pairs of pants to choose from means you can put together 5 x 7 = 35 different outfits
probability - CORRECT ANSWER probability = favorable outcomes / total possible outcomes
how to multiply powers (with the same base) - CORRECT ANSWER add the exponents
how to divide powers (with the same base) - CORRECT ANSWER subtract the exponents
raising powers to powers - CORRECT ANSWER multiply the exponents
simplifying square roots - CORRECT ANSWER factor out the perfect squares under the radical, "unsquare" them, and put the result in front; square root of twelve = square root of 4 x 3 = square root of 4 x square root of 3 = 2 square roots of 3
adding and subtracting roots - CORRECT ANSWER this can only be done if the part under the radicals is the same; 2 square roots of 3 + 5 square roots of 3 = 7 square roots of 3
multiplying square roots - CORRECT ANSWER product of square roots is equal to the square root of the product; sqr. root of 3 x sqr. root of 5 = sqr. root of 15
dividing square roots - CORRECT ANSWER quotient of square roots is equal to the square root of the quotient; sqr. root of 6 / sqr. root of 3 = sqr. root of 2
factoring the difference of squares - CORRECT ANSWER a^2 - b^2 = (a - b) (a + b)
factoring the square of a binomial - CORRECT ANSWER a^2 + 2ab + b^2 = (a + b)^2 and
a^2 - 2ab + b^2 = (a - b)^2
simplifying an algebraic function - CORRECT ANSWER find factors common to the numerator and denominator and cancel them out
what does "solve for x in terms of y" mean? - CORRECT ANSWER solve for x; isolate the x variable
how to solve a system of equations - CORRECT ANSWER combine the equations in such a way that one of the variables cancels out
solving an equation that includes absolute value signs - CORRECT ANSWER think about the two different cases (two different equations so that means you'll get two different answers); I x - 12 I = 3 : x - 12 = 3 AND x - 12 = -3
solving an inequality - CORRECT ANSWER do whatever is necessary to isolate the variable. remember that when you multiply or divide both sides by a negative number, you must REVERSE THE SIGN
finding the distance between two points - CORRECT ANSWER use the Pythagorean theorem, special right triangles, or the distance formula
distance formula - CORRECT ANSWER square root of (all of the following)
(x sub 2 - x sub 1)^2 + (y sub 2 - y sub 1)^2
slope formula - CORRECT ANSWER m = change in y / change in x = rise / run
slope intercept form - CORRECT ANSWER you need to put the equation into this form to find the slope of the line; y = mx + b
using an equation to find an intercept - CORRECT ANSWER to find the y-intercept, plug in zero for x. to find the x-intercept, plug in zero for y.
equation for a circle - CORRECT ANSWER (x - h)^2 + (y - k)^2 = r^2
h = x-coordinate of origin
k = y-coordinate of origin
equation for a parabola - CORRECT ANSWER y = ax^2 + bx + c
equation for an ellipse - CORRECT ANSWER (x^2 / a^2) + (y^2 / b^2) = 1
a, b are the radius on the x and y axes respectively
the horizontal axis is the major axis while the vertical axis is the minor axis
when two lines intersect... - CORRECT ANSWER adjacent angles are supplementary and vertical angles are equal
a transversal across two parallel lines forms... - CORRECT ANSWER four equal acute angles and four equal obtuse angles, and any of the acute angles is supplementary to any of the obtuse angles
the three interior angles of any triangle add up to ___ - CORRECT ANSWER 180 degrees
an exterior angle of a triangle is equal to... - CORRECT ANSWER the sum of the remote interior angles
the three exterior angles of any triangle add up to ___ - CORRECT ANSWER 360 degrees
similar triangles - CORRECT ANSWER triangles that have the same shape; this means that corresponding angles are equal and corresponding sides are proportional
area of a triangle - CORRECT ANSWER 1/2 x base x height
height = perpendicular distance between the side that's chosen as the base and the opposite vertex
Pythagorean theorem - CORRECT ANSWER for all RIGHT triangles:
(first leg)^2 + (second leg)^2 = (hypotenuse)^2
3-4-5 triangle - CORRECT ANSWER if a right triangle's leg-to-leg ratio is 3:4 or if the leg-to-hypotenuse ratio is 3:5 or 4:5, then it's a 3-4-5 triangle and you don't need to use the Pythagorean theorem to find the third side. just figure out what MULTIPLE of 3-4-5 it is
5-12-13 triangles - CORRECT ANSWER if a right triangle's leg-to-leg ratio is 5:12 or if the leg-to-hypotenuse ratio is 5:13 or 12:13, then it's a 5-12-13 triangle, you don't need to use the Pythagorean theorem to find the third side. just figure out what MULTIPLE of 5-12-13 it is
30-60-90 (degrees) - CORRECT ANSWER the sides of a 30-60-90 triangle (only a right triangle) are in a ratio of 1:the square root of 3:2. Pythagorean theorem is not necessary; if the hypotenuse is 6, then the shorter leg is half that (3), so then the longer leg is equal to the shorter leg (3) times the square root of 3 = 3 sqr. roots of 3
45-45-90 (degrees) - CORRECT ANSWER the sides of a 45-45-90 right triangle are in a ratio of 1:1:square root of 2. if one leg is 3, then the other leg must also be 3, and the hypotenuse is equal to a leg (3) times the sqr. root of 2 = 3 sqr. roots of 2
rectangle - CORRECT ANSWER four-sided figure with four right angles. opposite sides are equal and diagonals are also equal. the perimeter is equal to the sum of the lengths of the four sides AKA 2(length + width)
parallelogram - CORRECT ANSWER two pairs of parallel sides. opposite sides are equal, and opposite angles are equal. consecutive angles add up to 180 degrees
trapezoid - CORRECT ANSWER a quadrilateral with one pair of parallel sides and one pair of nonparallel sides
area of a rectangle - CORRECT ANSWER length x width
area of a parallelogram - CORRECT ANSWER base x height
area of a square - CORRECT ANSWER (side)^2
area of a trapezoid - CORRECT ANSWER (base one + base two / 2) x height
in other words, the average of the bases (the two parallel sides) times the height (the length of the perpendicular altitude)
the sum of the measures of the interior angles of a polygon is... - CORRECT ANSWER (n - 2) x 180, where n is the number of sides
to find one angle of a regular polygon... - CORRECT ANSWER divide the sum of the angles by the number of angles (which is the same as the number of sides)
FORMULA: ((n - 2) x 180) / n
circumference of a circle - CORRECT ANSWER 2 x pi x r
r = radius
length of an arc - CORRECT ANSWER if n is the measure of the arc's central angle, then the FORMULA is: (n / 360)(2)(pi)(r)
area of a circle - CORRECT ANSWER (pi)(r^2)
area of a sector - CORRECT ANSWER a sector is a piece of the area of a circle. if n is the measure of the sector's central angle, then the FORMULA is: (n / 360)(pi)(r^2)
surface area of a rectangular solid - CORRECT ANSWER SA = 2lw + 2wh + 2lh
l = length, w = width, h = height
volume of a rectangular solid - CORRECT ANSWER V = lwh
volume of a cube - CORRECT ANSWER V = e^3
if e is the length of an edge of the cube
volume of a cylinder - CORRECT ANSWER V = (pi)(r^2)(h)
volume of a cone - CORRECT ANSWER V = (1/3)(pi)(r^2)(h)
volume of a sphere - CORRECT ANSWER V = (4/3)(pi)(r^3)
sine of acute angles - CORRECT ANSWER sine = opposite / hypotenuse
cosine of acute angles - CORRECT ANSWER cosine = adjacent / hypotenuse
tangent of acute angles - CORRECT ANSWER tangent = opposite / adjacent
cotangent of acute angles - CORRECT ANSWER 1 / tangent = adjacent / opposite
secant of acute angles - CORRECT ANSWER 1 / cosine = hypotenuse / adjacent
cosecant of acute angles - CORRECT ANSWER 1 / sine = hypotenuse / opposite
simplifying trigonometric expressions - CORRECT ANSWER use the inverse function definitions along with the fundamental trigonometric identity
the fundamental trig. identity - CORRECT ANSWER sin^2x + cos^2x = 1
graphing trigonometric functions - CORRECT ANSWER use the x-axis for the angle and the y-axis for the value of the trig. function. use special angles (0, 30, 45, 60, 90, 120, 135, 150, 180, etc.) to plot key points
sine graph - CORRECT ANSWER starts at 0, goes up to 1 (at pi over 2) then comes back down to 0 (at pi)
cosine graph - CORRECT ANSWER starts at 1, goes down to 0 (at pi over 2), goes to -1 (at pi) [Show Less]