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Math (ACT)

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Slope Formula
The steepness of a graph line; the ratio of the vertical change (the rise) to the horizontal change (the run).
What is the slope of the straight line passing through the points (-2,5) and (6,4)?
4-5/6- -2= -1/8, this graph will rise 1, and go left 8.
Slope-Intercept Formula
Ex: y= -4/5 - 7. The graph will go up 4, and to the left 5, and the y-intercept will be -7.
Line
A straight path of points that extends forever in two directions. A line doesn't have any thickness or width. Arrows sometimes show that the line goes on forever in either direction
Line segment
The set of points on a line between any two points on that line.
Midpoint
The point halfway between two endpoints on a line segment.
Midpoint Formula
(x₁+x₂)/2, (y₁+y₂)/2
Intersect
To cross. Two lines can intersect each other much like two streets cross each other at an intersection.
Vertical Line
A line that runs straight up and down,
Horizontal Line
A line that runs straight across from left to right. *Think horizon
Parallel Lines
Lines that run in the same direction and keep the same distance apart. Parallel lines never intersect one another.
Perpendicular Lines
Two lines that intersect to form a square corner. The intersection of two perpendicular lines for a right, or 90-degree, angle
Ray
A part of a line, with one endpoint, that continues without end in one direction.
What are the angle facts?
1. No negative angle exists
2. No zero angle exists
3. It's rare to see fractional angles
Ex: 45.50 degrees or 32 3/4 degrees
Acute Angles
In angle greater than 0°, but less than 90°
Right Angles
Angles that measure 90 degrees

Hint: If perpendicular line is show than it is considered a right angle
Obtuse Angles
An angle whose measure is greater than 90 degrees and is less than 180 degrees.
Straight Angles
An angle that measures exactly 180 degrees and forms a straight line.
Complementary Angles
Two angles whose sum is 90 degrees
Supplementary Angles
Two angles whose sum is 180 degrees. Also, there is another 180 degrees below the line, with a total of 360 degrees.
Vertical Angles
Opposite angles with be equal to each other.
Reflex Angles
Angles that have measures greater than 180 degrees and less than 360 degrees
Angle Note
Angles around a point total 360 degrees.
Angle Note
The exterior angles of any figure are supplementary to the interior angles and always total 360 degrees.
Transversal
A line that intersects two or more lines
Vertical Angles
Angles that are opposite of each other have equal .
Corresponding angles
Angles in the same position around two parallel lines and a transversal and have equal measures.
Equilateral Triangle
A triangle with three congruent sides and three equal angles. Because the three angles must add up to 180 degrees, all three angles of an equilateral triangle are always equal to 60 degrees.
Isosceles Triangle
A triangle with two congruent sides. The angles opposite those sides are also equal . If angle A is 50 degrees, then angle B is also 50 degrees.
Scalene Triangle
A triangle with no congruent sides
Right Triangle
A triangle that has one inside angle that is 90 degrees.
Hypotenuse
The longest side of a right triangle. The side opposite the right angle in a right triangle.
Sizing up Triangles
In any triangle, the sum of the lengths of two sides must be greater than the length of the third side
Exterior Angle of Triangle
The measure of an exterior angle of a triangle is equal to the sum of the two remote interior angles, so 1= 3 + 4
Similar Triangles
If the heights of two triangles are in a ratio 2:3, then the bases of those triangles are also in a ratio of 2:3
Similar Triangles
The ratio of the areas of similar triangles is equal to the square of the ratio of their sides

Ex: Two similar triangles have bases of 5 and 25, what is the ratio of the two triangles?
1. The ratio is 1:5 or 1/5
2. Square 1/5
3. 1/25
Tip
Don't assume that triangles are similar on the ACT just b/c they look similar to you.
Area of a Triangle
1/2bh (.5)(base)(height)
- Sometimes the height of a triangle may be outside the triangle itself.
Perimeter of a Triangle
The sum of the lengths of its sides.
Pythagorean Theorem
In this formula, a and b are the sides of the triangle and c is the hypotenuse. The hypotenuse is always opposite the 90-degree angle and is always the longest side of the triangle. This formula only works for right triangles.
Pythagorean Triples
Common ratios to find sides.
Ex: 3:4:5
Ex: 6:8:10 (2 times the first one)
Ex: 9:12:15 (3 times the first one)
Ex: 27:36:45 9 times the first one)
Ex: 5:12:13
Ex: 10:24:26 (2 times the first one)
Ex: 15:36:39 (3 times the first one)
Ex: 50:120:130 (10 times the first one)
Ratio for 45:45:90 Triangle/Isosceles Right Triangle
- x stands for the side of the figure
- This applies to an isosceles triangle

Ex: If a question tells you that the side of a square is 5 and wants to know the diagonal of the square, you know immediately that is 5 square root of 2
Ratio for 30:60:90 Triangle
- x stands for the side of the figure
- Whatever the value of the short side of a 30-60-90 triangle, the hypotenuse is always twice as large. The medium side is always equal to the short side times the square root of 3.
Circle
A round shape that has no beginning or end.
Radius
The distance from the center of a circle to any point on the circle.
Diameter
The distance across a circle through its center.Twice the radius. Also the longest line in a circle.
Chord
A segment whose endpoints are on a circle
Arc
Any part of the circumference. The curved portion between the points of a circle.
Sector
The part of a circle that looks like a piece of pie. A sector is bounded by 2 radii and an arc of the circle.
Circumfrence
Distance around a circle.
The formula for the area of a circle is
πr²
The formula for the circumference is
2πr or πd
Four-sided figures /Quadrilateral
- All angles add up to 360 degrees.
-
Rectangle .
A parallelogram with four right angles

- Area is bh (base)(height)
- Perimeter is all sides added up
Square
A quadrilateral with 4 sides that are equal and has all 90 degree angles

- Area is bh (base)(height)
- Perimeter is all sides added up
Parallelogram
A quadrilateral with two pairs of parallel sides.
- Opposite angles are equal
- Opposite sides are parallel and equal
-
Trapezoid
A four-sided figure in which two sides are parallel.

- Area is 1/2(base 1)(base 2)(height)
- You can divide it by two triangles, or two triangles and a rectangle/square
Rhombus
A parallelogram with 4 congruent sides.

- Area is 1/2(base 1)(base 2)(height)
- You can divide it by two triangles, or two triangles and a rectangle/square
Angle Facts
- The longest side is opposite the largest angle
- The shortest side is opposite the smallest angle
- The third side of any triangle is always less than the sum and greater the difference of the other two sides.
Multiples of 10
The first four positive multiples of 10 are 10, 20, 30, and 40.
Factors
Factors of a number are always equivalent to that number of smaller
Multiples
Multiples of a number are always equivalent to that number or larger
What is the reciprocal of -1/4
-4
What is the opposite reciprocal of 2?
-1/2
When a number and its reciprocal are multiplied, what is the product?
1
How many times does 2 go into 15 evenly? How much is left over?
7, 1
Which terms describes the number 6?
Number, Integer , Positive, Even
Which terms describe the number -1/5?
Number, Negative
Which terms describe the number 0?
Number, Integer, Even
When 14 is dived by 3, it has a _______ of 2.
Remainder
The numbers 2, 4, 6 are listed in ______ order.
Consecutive
The numbers 3, 1, 14 are not listed in ______ order.
Consecutive
List the factors of 12.
1, 2, 3, 4, 6, 12.
List the four multiples of 12.
12, 24, 36, 48.
List the factors of 30.
1, 2, 3, 5, 6, 10, 15, 30.
List the four multiples of 30.
30, 60, 90, 120.
Is 12 a multiple or factor of 24?
Factor
Is 8 one of the factors of 64?
Yes
What is the greatest common factor of 27 and 45?
9
What is the greatest common factor of 9 and 36
9
What is the least common multiple of 9 and 12?
36
What is the least common multiple of 24 and 48?
48
Prime number
A whole number that has exactly two factors, 1 and itself.
What are the single digit prime numbers?
2, 3, 5, 7
What is the only even prime number?
2
Is 1 a prime number?
No, it does not have indistinct factors.
What real number could you square to get a product of -1?
None, it's imaginary. (i)
Write 0.5 as a fraction?
1/2
Write 3 as a fraction
3/1
In the number 0.16666, what digit is coming next?
6
In the number 0.191919, what digit is coming next?
1
When you multiply two numbers with common bases,
add the exponents.
When you divide two numbers with common bases,
subtract the exponents.
When you raise an exponential number to a power,
multiply the exponents.
You can add or subtract square roots only when the numbers under the square root
are the same.
Consecutive
In increasing order.
Collinear
Points that lie on the same line
Difference
Answer to a subtraction problem
Even
Divisible by two
Fraction
A way of expressing the division of numbers by stacking one over the other.
Greatest Common Factor
The largest factor that two or more numbers have in common.
Imaginary
The square root (or any even root) of a negative number.
Integers
All real numbers other than decimals or fractions.
Irrational
A number than can be expressed as a decimal but not a fraction.
Least Common Multiple
The smallest multiple (other than zero) that two or more numbers have in common.
Negative
Less than 0.
Odd
Not divisible by two.
PEMDAS
Parentheses, Exponents, Multiplication, Division, Addition, Subtraction (order of operations)
Positive
Greater than 0.
Prime
A number that has itself and 1 as its only factors.
Product
The result of multiplication.
Real
Zero, all positive and negative integers, fractions, decimals, and roots.
Radical
Another word for root.
Reciprocal
The inverse of a number--flip the numerator and the denominator.
Remainder
The number left over when a number is not divisible by another number.
Rational Number
A number that can be written as a fraction
Sum
The result of addition.
John has x red pencils, and three times as many red pencils as blue pencils. If he has four more yellow pencils than blue pencils, then in terms of x, how many yellow pencils does John have?
#21 DRILL
For all x = 3 (line through equality), which of the following is equivalent to the expression? 3x^2 - 7x - 6/x - 3
#17 DRILL
In my dear Aunt Sally's math class, there are four exams. The first three exam scores are averaged, and the resulting score is averaged with the final exam score. If a, b, and c are the first three exam scores, and f is the final exam score, which of the following examples gives student't final score in the class?
#36 DRILL
If x - z = 6 and y = 3x - 2 - 3z, then y =
#25 DRILL
If $600 were deposited in a bank account for one year and earned interest of $42, what was the interest rate?
42/600 x 100= 7%
In a piggy bank, there are pennies, nickels, dimes, and quarters that total $2.17 in value. If there were
#49 drill
Polygons
- Sum of angles in a n-sided polygon: (n-2)180
- Angle measure of each angle in a regular n-sided polygon: (n-2)180/n
Triangle
A polygon with three sides. All inside angles add up to 180 degrees. The largest angle of a triangle is always opposite to its larger side.
Pentagon
5 sides; This can be split into 3 Triangles, so 180 x 3 = 540
Know your rules for 0
- 0 is an even number
- 0 is neither positive nor negative
- Anything multiplies by 0 is 0
- 0 raised to any power is 0
- Anything raised to the 0 power is 1
Hexagon
6 sides, This can be split into 4 triangles, so 180 x 4 = 720
Heptagon
7 sides, This can be split into 5 triangles, so 180 x 5 = 900
Octagon
8 sides, This can be split into 6 triangles, so 180 x 6 = 1080
Nonagon
9 sides, This can be split into 7 triangles, so 180 x 7 = 1260
Decagon
10 sides, This can be split into 8 triangles, so 180 x 8 = 1440
Secant Line
straight line joining two points on a function
Tangent Line
A line that intersects a curve once and only once
Graphing < or > on a coordinate plane
Dotted line
Graphing ≥ or ≤ on a coordinate plane
Solid line
SOHCAHTOA
Trig relationships
SOH
Sin=Opposite/Hypotenuse
CAH
Cos=Adjacent/Hypotenuse
TOA
Tan=Opposite/Adjacent
Csc
1/sin
Sec
1/cos
Cot
1/tan
Regular Polygon
A polygon with all sides and all angles equal
A circle with center O has a radius of r. What is the area of a circle with a radius three times larger?
9πr², you must square the 3 following the area formula.
How to find percentage?
Part/Whole x 100%
You are thrown 100 pitches and you hit 20 of them. What percent of pitches did you hit? What's the ratio of the pitches you hit to the pitches you missed?
You hit 20% of the pitches. The ratio of hits to misses is 1:4 or 1/4.
At a restaurant, diners enjoy an "early bird" discount of 10% off their bills. If a diner orders a meal regularly priced at $18 and leaves a tip of 15% of the discounted meal (no tax), how much does he pay in total?
First, calculate the 10% discount. Next, add on the 15% tip. You should get $18.63.
What would be a 10% discount of $18?
10/100=discount/$18, Cross multiply and dived by 100. $1.80 should be the discount.
To find 10% of any number, move the decimal point of that number over one place to the left.
10% of 500 is 50, 10% of 50 is 5, and 10% of 5 is .5.
To find 1% of a number, move the decimal point of that number over two places to the left.
1% of 500 is 5, 1% of 50 is .5, and 1% of 5 is .05.
When 15% of 40 is added to 5% of 260, the resulting number is:
First 15/100=?/40 should equal 6. Then, 5/100=?/260 equals 13. 6 + 13= 19.
Zach's practice bag contains 4 blue racquetballs, one red racquetballs, and 6 racquetballs. If he chooses a ball at random, which of the following is closest to the probability that the ball will NOT be green?
First there are a total of 11 balls. 5/11 are not green while 6/11 are green. So, 5/11 x 100 (part/whole) x 100 is .45% or 45%. A shortcut for this problem would be to look at the probability. 5/11 is roughly around 50%, but we know it is just under it. So, many of the choices could be eliminated right off the bat.
If the ratio of 2x to 5y is 1/20, what is the ration of x and y?
So, 2x/5=1/20. Begin by isolating x and y by cross multiplying. You should have x/y=5/40. So, this should reduce to 1/8.
If 3.7 inches of rain fall in Chicago during the first 4 days of November, and the rain continues to fall at this pace for the rest of the month, approximately how many feet of rain will fall during December? (The month of December has 31 days)
Begin by calculating how many inches there are per day, that would be .925. Multiply this number by the 31 days and you should get 28.675 inches for the month of December. But, the question asks how many feet. Since there are 12 inches in a foot, divide 28.675 by 12 , you should get 2.4 ft if you round to the nearest tenth. Or you could use the part over whole formula. So, 3.7/4=?/31, then cross multiply and divide by 12 inches.
Mean (Average)
The sum of the values in a set of numbers, divided by the number of values in the set.
Median
The one in the middle.
Mode
The element that appears the most.
What is the average of 9, 12, and 6?
All three numbers add up to 27. Since there are 3 items, 27/3= 9
Over 9 games, a baseball team had an average of 8 runs per game. If the average number of runs for the first 7 games was 6 runs per game, and the same number of runs was scored in each of the last 2 games, how many runs did the team score during the last game?
First, find the total number of runs in the first 9 games, which should equal 72 (9 x 8= 72). Then find the total number for the first 7 games, which should equal 42 (7 x 6= 42). Now find the difference from the totals and you should get 30. Since there are 2 games left, divide 30 by 2, and you should get 15.
The starting team of a baseball club has 9 members who have an average of 12 home runs apiece for the season. The second-string team for the baseball club has 7 members who have an average of 9 home runs apiece for the season. What is the average number of home runs for the starting team and the second-string team combined?
First, we want to find the total number of runs for each team. First the first team it should equal 108 (9 x 12= 108). The second team should it equal 56 (7 x 8= 56). Add both of these totals, and you should get 164. Since there are a total of 16 members (9 + 7= 16), dived 164 by 16 and you should get 10.25.
At the school cafeteria, students can choose from 3 different salads, 5 different main dishes, and 2 different desserts. If Isabel chooses one salad, one main dish, and one desert for lunch, how many different lunches could she choose?
3 x 5 x 2= 30 (Multiply all number since there is only one of each). Use slot method. _ _ _=
At the school cafeteria, 2 boys and 4 girls are forming a lunch line. If the boys must stand in the first and last places in in, how many different lines can be formed?
Use the slot method. Since we know there are 6 people, _ _ _ _ _ _. Since the boys must be at the end of the lines, 1 _ _ _ _ 2. Then, the girls can go in any order. So, 1 4 3 2 1 2= 48.
Elias has to select one shirt, one pair of pants, and one pair of shoes. If he selects at random from his 8 shirts, 4 pairs of pants, and 3 pairs of shoes, and all his shirts, pants, and shoes are different colors, what is the likelihood that he will select his red shirt, black pants, and brown shoes?
8 x 4 x 3= 96. Since there is one of each, 1/96.
< or >
Open circle for inequalities.
Quadrants
Equation for Circle
(h,k) is center, r is radius.
Equation for Ellipse
(h,k) is center, 2a is horizontal axis (width), 2b is vertical axis (height).
Equation for Parabola
y= x^2
There are 600 school children in the Lisle district. If 54 of them are high school seniors, what is the percentage of high school seniors in the Lisle district?
54/600 x 100% = 9%
Passes to the Renaissance Fair cost $9 when purchased online and $12 when purchased in person. The group sponsoring the fair would like to make at least $4,000 from sales of passes. If 240 passes were sold online, what is the minimum number of tickets that must be sold in person in order for the group to meet its goal?
The group has sold 240 passes online at $9 each. This means that they have already made (240)(9) = $2,160. In order to reach their goal, they will need $4,000 - $2,160 = $1,840. Since the question asks how many in-person tickets they will need to sell, you can approximate how many tickets will get them $1,840 by dividing: $1,840 / $12 = 153.333. Since there is no such thing as 0.333 tickets, you have to round up to get 154 tickets.
#10
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What is the least common multiple of 25, 16, and 40?

a. 27
b. 32
c. 320
d. 400
e. 16,000
Use the answer choices and work backwards. Find the smallest number that divides evenly by 16, 25, and 40. This is 400: 400/16 = 25; 400/25 = 16; and 400/40 = 10. I
#14
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In triangle XYZ, Angle Y is a right angle and Angle Z measures less than 52 degrees. Which of the following phrases best describes the measure of angle X
The sum of all measures in a triangle are 180 degrees. Angle Y is described as a right angle, which means it is 9- degrees. The other two triangles, therefore, will have to add up to 90 degrees to complete the 180 degrees in the triangle. If angle Z measures less than 52 degrees., then Angle X must be greater than 90-52 = 38 degrees. it cannot be equal to 38 degrees because Angle Z is less than 52 degrees.
An integer, x, is subtracted from 6. That difference is then multiplied by 3. This product is 15 more than half the original integer. What is the equation of this relationship?
Translate the English in this problem into the math of the answer choices. The problem reads, "An integer, x, is subtracted from 6. That difference is then multiplied by 3." This translates into the following (6-x) x 3 or 3(6-x). The second part of the problem reads, "This product is 15 more than half the original integer." Therefore 3(6-x) will equal "15 more than half of the original integer": 15 + 1/2x or 1/2x + 15. Therefore, 3(6-x)= 1/2x + 15.
The employees of two factories, X and Y, are comparing their respective production records. Factory X has already produced 18,000 units and can produce 120 units per day. Factory Y has produced only 14,500 units but can produce 155 units per day. If d represents the number of days, which of the following equations could be solved to determine the number of days until X's total production equals Y's total production.
Find the total production of each company after d days. Factory X starts with 18,000 units and can produce 120 units per day, so its production after d days will be 18,000 + 120d. Factory Y starts with only 14,500 units, but can produce 155 units per day, so its production after d days will be 14,500 + 155d. Since you are looking for the number of days after which the total production of each factory will be equal, set the two equations equal to one another: 18,000 + 120d = 14,500 + 155d
If a + 3b = 27 and a -3b = 9, then b =
Set up a system of equations to find a:
( a + 3b = 27)
(a - 3b = 9)
______________________
2a = 36
a = 18 <------ Use this value to find b
Plug 18 into to any of the two formulas and you should get b = 3.
#25
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When asked the price, in dollars, of his fancy calculator, Albert responded, " If you take the square root of the price, then add 3/8 the price, the result is 66." What is the price, in dollars, of Albert's calculator?
√x + (3/8)x = 66 . Simply find a square root number. 11 will work once trying all numbers. So, √144 + (3/8)(144) = 66
For the imaginary number i, which of the following is a possible value of i^n if n is an integer less than 5.
Raise i to the 5th power: -1^5 = -1
#45
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#44
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All of the following statements about rational and/or irrational numbers must be true EXCEPT:
the sum of two rational numbers is rational
the product of any two rational numbers is rational
the sum of any two irrational numbers is irrational
the product of a rational number and an irrational maybe be rational or irrational
the product of any two irrational numbers is irrational (false) If you multiply √2, an irrational number, by itself, you get a rational number.
Amplitude
...

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