Download TEN FOR TEN - Maine Prep

TEN FOR TEN®
MATH REFERENCE
Order of math operations (PEMDAS):
1. P is anything inside parentheses
2. Exponents
3. Multiplication and Division (in the order they occur left-to-right)
4. Addition and Subtraction (in the order they occur left-to-right)
NUMBER PROPERTIES/ARITHMETIC
Integers
-3
-2
-1
0
1
2
3
4
All of the above are integers. Integers can be positive or negative or even zero.
Absolute value
It’s the distance from zero. Forget every other definition you’ve learned for it, OK?
Prime numbers
•
A prime number has exactly two different factors, itself and 1.
•
So, is 1 prime? No, because 1 has only one factor, itself.
•
How about 9? Well, 9 has factors of 1, 3, and 9; check the definition—not prime.
•
How many even prime numbers are there? Hmmm, every even number but 2
seems to have at least 3 factors (itself, 1, and 2).
•
So, what are the first 10 prime numbers? Well, 2’s prime, and 3, and then 5, 7, 11,
13, 17, 19, 23, and 29. Why the gaps? Well, 15 has factors of 3 and 5; 21 has
factors of 3 and 7; 25 has a factor of 5; and 27 has factors of 3 and 9.
Zero
Zero is the SAT’s favorite integer. Multiplication by zero always equals zero so when a
product equals zero, zero must be a factor.
Even and odd
Adding two even numbers yields an even number. Adding two odd numbers yields an
even number. Adding an even number and an odd number yields an odd number.
2+6=8
3+5=8
2+7=9
Multiplication of any integer by an even number always results in an even number. Odd
products result when odd numbers are multiplied together.
2 x 12 = 24
2 x 13 = 26
3 x 13 = 39
MATH REFERENCE
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Negative numbers
Adding a negative number is equivalent to subtracting the absolute value of that
number (see Absolute value on page 1). 5 + (- 4) = 5 – 4 = 1
Subtracting a negative number is the same as adding the absolute value of the number.
5 – (- 4) = 5 + 4 = 9
The product/quotient of two negatives is positive. (-2)(-7) = 14 and (-14) ÷ (-2) = 7
The product/quotient of a negative and a positive is negative. (-5)(7) = -35; (-35) ÷ 7 =
(-5)
Just as multiplying two negative numbers results in a positive number, so raising a negative number to an even power also yields a positive result. (-2)4 = (-2)(-2)(-2)(-2) = 16
Raising a negative number to an odd power results in a negative number. (-3)3 =
(-3)(-3)(-3) = -27
Remainders
Here’s an easy way to find a remainder using your calculator: Let’s figure the remainder
if we divide 87 by 7. First, punch in 87/7. We get 12 and a decimal, right? OK, let’s now
subtract the whole number (here, 12), leaving just the decimal. Now, multiply the
remaining decimal by the number we originally divided by (7). Did you get a remainder
of 3? Try this one: 94/11.
Ratios and Direct Proportion
A ratio of 1:3 means that for every one that I have, you have three. A Direct Proportion is
a ratio. We usually have to figure out what it is.
Fraction/Decimal/Percent Conversion Practice
Convert these. Please put all fractions greater than 1 in the “improper” form that you
must use when solving grid-ins.
Fraction
1/2
Decimal
.5
.75
Percent
50%
3/8
.125
95%
3/20
1.25
225%
10/3
Average and Total
•
Median – the number in the middle
•
Average or mean – the sum of all the numbers in the group divided by how many
numbers we just added; so, 3 + 7 + 17 = 27; the average is 27/3, or 9. Consult the
AVERAGE AND TOTAL TEN FOR TEN for more details.
MATH REFERENCE
3
For any sequence of consecutive numbers, the median is always equal to the mean. For
example, in the consecutive sequence 6-12-18, both the median and the mean are 12.
Exponents and Roots
An exponent is shorthand that tells us how many times we need to multiply the base
number by itself. When in doubt, write the operation out longhand. For example:
•
Situation A: 25 x 23 = 2 x 2 x 2 x 2 x 2 times 2 x 2 x 2; when we multiply, we end up
with 8 factors of 2, which can be expressed as 28. Note that in this case you can
add the exponents.
•
Situation B: (25)3 = (2 x 2 x 2 x 2 x 2) (2 x 2 x 2 x 2 x 2) (2 x 2 x 2 x 2 x 2), right? We
multiply 2 by itself five times inside the parentheses, but then what’s in the
parentheses is cubed (which means it has to be multiplied by itself three times).
So, we end up with = 215. Note that in this case you can multiply the exponents.
•
Advanced stuff: (28)1/4 ; say it out loud: “Two to the eighth to the 1/4th power.”
What do we do? If you don’t know, take a look at Situation B, where we
multiplied the exponent inside the parentheses by the exponent outside the
parentheses, as in 2(8 x ¼) = 22?
Roots multiply and divide the same way rational numbers do:
5 x 7 = 35
5 x 7 = 35
35 ÷ 5 = 7
35 ÷ 7 = 5
(Advanced) Fractional powers: A fractional exponent contains both a power
(numerator) and a root (denominator). So, 82/3 means that we need to raise 8 to the 2nd
power (or square it) and then take the cube (3rd) root or we need to take the cube root
of 8 and then raise that to the 2nd power.
ALGEBRA
Solve algebraic equations by isolating the variable on one side (using SADMEP, the
reverse of PEMDAS—see page 1).
In the following equation, solve for x:
2 4
x + 27 = 81
3
Step 1: Subtract 27 from both sides of the equation:
2 4
x = 54
3
Step 2: Multiply both sides by
3
(the reciprocal of the coefficient):
2
3 2
3
( )( )x4 = 54( )
2 3
2
x4 = 81
Step 3: Eliminate the Exponent by taking the fourth root of both sides:
MATH REFERENCE
4
4
x4 =
4
81
Step 4: Solve:
x=3
Cross multiplication is a quick and easy way to get rid of two denominators.
If
a
c
=
then ad = bc
b d
GEOMETRY
Perimeter/Area/Volume: In a room with shag wall-to wall carpeting, the perimeter is the
total measurement of the edges of the carpet (2 lengths plus 2 widths), the area is the
square measure of the carpet, and the volume is the cubic measure of the carpet,
which we can calculate by multiplying the area by the height of the shag!
Since a cube has six sides, its surface area will be six times each side’s area.
a
b
c
d
e
f
g
h
Properties of Lines/Degree Measurements of Angles
Parallel lines cut by a transversal (any third line that isn’t parallel) form angles as shown
above:
Supplementary: The sum of the angles on one side of a line is 180°. So, angles that add
up to form a straight line, such as a and b, or b and d, have a sum of 180°.
Vertical: Angles opposite each other, such as a and d, or g and f, have equal degree
measures.
Corresponding angles, such as a and e, or b and f, have equal measures.
Alternate interior angles, such as c and f, or d and e, have equal measures.
Supplementary Interior angles on the same side of a transversal, such as c and e, add up
to 180°.
To find out the sum of the interior angles of any polygon, break the figure up into the
smallest possible number of triangles. Multiply the number of triangles times 180° to
determine the total interior degree measure. For example, draw a pentagon—into how
many triangles can you break it? (For those who like formulas, (n – 2)180°, where n
represents the number of sides of the polygon.)
MATH REFERENCE
5
Triangles
Important things about triangles:
•
The length of the longest side of a triangle is less than the sum of the two shorter
sides. So, if we have sides of 4 and 7, our third side has to be less than 11.
•
The three angles of a triangle add up to 180°.
•
The hypotenuse is the longest side of a right triangle.
•
The longest side of any triangle is opposite its largest angle, and its smallest side is
opposite its smallest angle. Equal angles are opposite sides of equal length.
Popular SAT Triangles:
•
Similar triangles have equal angle measures and thus sides in proportion.
•
Equilateral: three equal angles of 60° and three sides of equal length.
•
Isosceles: two equal angles and two equal sides.
•
Right triangles have one right angle. Pythagoras’ theorem says: a2 + b2 = c2,
where a and b are the legs of a right triangle and c is the hypotenuse. Knowing
any two side lengths of a right triangle allows us to compute the third.
Circles
You will always be given a value for the circumference, diameter, radius, or area; to
solve, you’ll need to compute one or more of the others.
•
To compute the Circumference, multiply the Diameter by pi (π). Do not ever
convert π to a decimal number unless the problem tells you to.
•
To compute the Area, square the Radius and multiply it by pi (π).
Since a circle surrounds a point, and every point on a plane is surrounded by 360°, every
circle contains 360°. So, ¼ of a circle contains 90°, ½ of a circle contains 180°, etc. Circle
problems often test your ability to combine your knowledge about angle (“pizza slice”)
measurements within a circle with the circle’s overall CdrA measurements.
For example: For a circle of radius 4, how long an arc on that circle is created by a 45°
angle?
Answer: First, an arc is a portion of the circumference, so we have to move up from r to
C. If r is 4, then d is 8, which makes C 8π. Since 45° is 1/8 of the angle measurement of
the circle, the arc it describes is 1/8 of the circle’s circumference. Its measurement is π.
Finally:
Q: What can we do to both sides of an equation?
A: Anything we want!