- Draw Pictures
- Apply Logic
- Answer the Question That They Ask You
- Check the Choices
- Pick Numbers for the Variables
- Skip Around
- Read the Questions Carefully
- Look for "New Operations"
- Arithmetic
- Algebra
- Geometry
- Multiple Choice Versus Student-Produced Response
- Practice Questions
- Answer Explanations
This chapter is from the book
This chapter is from the book
This chapter is from the book
Arithmetic
This book assumes a basic understanding of arithmetic. Our focus will be on reviewing some general concepts and applying those concepts to questions that might appear on the PSAT math sections.
Understanding Operations Using Whole Numbers, Decimals, and Fractions
Following are some simple rules to keep in mind regarding whole numbers, fractions, and decimals:
Ordering is the process of arranging numbers from smallest to greatest or from greatest to smallest. The symbol > is used to represent "greater than," and the symbol < is used to represent "less than." To represent "greater than and equal to," use the symbol ≥; to represent "less than and equal to," use the symbol ≤.
The Commutative Property of Multiplication can be expressed as a x b = b x a, or ab = ba. When two numbers are multiplied together, the order they are in doesn’t matter. For example: 2 x 3 = 6, and 3 x 2 = 6.
The Distributive Property of Multiplication can be expressed as a(b + c) = ab + ac. Multiply each element within the parentheses by the element outside of the parentheses. For example: 2(2x + 4) = 4x + 8.
The Associative Property of Multiplication can be expressed as (a x b) x c = a x (b x c). It doesn’t matter in which order the numbers are multiplied. For example: (2 x 3) x 4 = 6 x 4, or 24; 2 x (3 x 4) = 2 x 12, or 24.
The Order of Operations for whole numbers can be remembered by using the acronym PEMDAS:
When a number is expressed as the product of two or more numbers, it is in factored form. Factors are all the numbers that will divide evenly into one number. For example, 1, 3, and 9 are factors of 9: 9 ÷ 1 = 9; 9 ÷ 3 = 3; 9 ÷ 9 = 1.
A number is called a multiple of another number if it can be expressed as the product of that number and a second number. For example, the multiples of 4 are 4, 8, 12, 16, and so on, because 4 x 1 = 4, 4 x 2 = 8, 4 x 3 = 12, 4 x 4 = 16, and so on.
The Greatest Common Factor (GCF) is the largest number that will divide evenly into any two or more numbers. The Least Common Multiple (LCM) is the smallest number that any two or more numbers will divide evenly into. For example, the GCF of 24 and 36 is 12 because 12 is the largest number that will divide evenly into both 24 and 36. The LCM of 24 and 36 is 72 because 72 is the smallest number that both 24 and 36 will divide evenly into.
Multiplying and dividing both the numerator and the denominator of a fraction by the same non-zero number will result in an equivalent fraction. For example,
,
which
can
be
reduced
to
.
This works because any non-zero number divided by itself is always equal to 1.When multiplying fractions, multiply the numerators to get the numerator of the product, and multiply the denominators to get the denominator of the product. For example,
.-
To divide fractions, multiply the first fraction by the reciprocal of the second fraction. In mathematics, the reciprocal of any number is the number that, when multiplied by the first number, will yield a product of 1. For example: The reciprocal
of
is
because
.
So, 
, which equals 
. -
When adding and subtracting like fractions (fractions with the same denominator), add or subtract the numerators and write the sum or difference over the denominator. So,
,
and
. -
When adding or subtracting unlike fractions, first find the Lowest (or Least) Common Denominator (LCD). The LCD is the smallest number that all the denominators will divide evenly into. For example, to add
and
,
find
the
smallest
number
that
both
4
and
6
will
divide
evenly
into.
That
number
is
12,
so
the
LCD
of
4
and
6
is
12.
Multiply by 
to
get
,
and
multiply
by
to
get
.
Now
you
can
add
the
fractions,
as
follows:
,
which
can
be
simplified
to
. -
To convert a mixed number into an improper fraction, first multiply the whole number by the denominator, add the result to the numerator, and place that quantity over the denominator. For example,
,
or 
.
To
convert
an
improper
fraction
into
a
mixed
number,
first
determine
how
many
times
the
denominator
will
divide
evenly
into
the
numerator.
Then,
place
the
remainder
over
the
denominator,
as
follows:
In
the
improper
fraction 
, 8
divides
evenly
into
21
twice
(8
x
2
=
16),
with
a
remainder
of
5
(21 – 16
=
5).
Therefore, , is
equal
to 
. -
When converting a fraction to a decimal, divide the numerator by the denominator. For example
,
=
3 ÷ 4,
or
.75. Place value refers to the value of a digit in a number relative to its position. Starting from the left of the decimal point, the values of the digits are ones, tens, hundreds, and so on. Starting to the right of the decimal point, the values of the digits are tenths, hundredths, thousandths, and so on.
P: First, do the operations within the parentheses, if any.
E: Next, do the exponents, if any.
M: Next, do the multiplication, in order from left to right.
D: Next, do the division, in order from left to right.
A: Next, do the addition, in order from left to right.
S: Finally, do the subtraction, in order from left to right.
Understanding Squares and Square Roots
In
mathematics,
a square
is the
product
of any
number
multiplied
by itself,
and
is expressed
as a2 = n.
A square
root
is written
as 
,
and
is the
non-negative
value a that
fulfills
the
expression a2 = n.
For
example,
the
square
root
of 25
would
be written
as 
.
The
square
root
of 25
is 5,
and
5-squared,
or 52,
equals
5 x
5, or
25.
A number
is considered
a perfect
square
when
the
square
root
of that
number
is a
whole
number.
So,
25 is
a perfect
square
because
the
square
root
of 25
is 5,
which
is a
whole
number.
When you square a fraction, simply calculate the square of both the numerator and the denominator.
-
=
-
Understanding Exponents
When a whole number is multiplied by itself, the number of times it is multiplied is referred to as the exponent. As shown above with square roots, the exponent of 52 is 2 and it signifies 5 x 5. Any number can be raised to any exponential value.
76 = 7 x 7 x 7 x 7 x 7 x 7 = 117,649.
Remember that when you multiply a negative number by a negative number, the result will be a positive number. When you multiply a negative number by a positive number, the result will be a negative number. These rules should be applied when working with exponents, too.
-32 = -3 x -3
-3 x -3 = 9
-33 = -3 x -3 x -3
(–3 x –3) = 9
9 x –3 = -27
The basic rules of exponents follow:
-
am x an = a(m+n)
-
(am)n = amn
-
(ab)m = am x bm
-
-
a0 = 1, when a ≠ 0
-
,
when a ≠ 0 -
,
when b ≠ 0
Understanding Ratios and Proportions
A ratio is the relationship between two quantities expressed as one divided by the other.
For
example:
Kate
works
2 hours
for
every
3 hours
that
Darleen
works.
This
can
be expressed
as 
or
2:3,
and
is known
as a
part-to-part
ratio.
If you
compared
the
number
of hours
that
Kate
works
in one
week
to the
total
number
of hours
that
every
employee
works
in one
week,
that
would
be a
part-to-whole
ratio.
A proportion is an equation in which two ratios are set equal to each other.
For example: Kate worked 30 hours in one week and earned $480. If she received the same hourly rate the next week, how much would she earn for working 25 hours that week?
-
;solve
for x -
30x = 12,000
-
x = 400
Kate would earn $400 that week.
Understanding Percent and Percentages
A percent is a fraction whose denominator is 100. The fraction is equal to 55%.
Percentage problems often deal with calculating an increase or a decrease in number or price.
For example: A jacket that originally sells for $90 is on sale for 35% off. What is the sale price of the jacket (not including tax)?
-
35% of $90 =
-
-
.35 x $90 = $31.50
The discount is equal to $31.50, but the question asked for the sale price. Therefore, you must subtract $31.50 from $90.
-
$90.00 – $31.50 = $58.50
You could also more quickly solve a problem like this by recognizing that, if the jacket is 35% off of the regular price, the sale price must be equal to 100% – 35%, or 65% of the original price.
-
65% of $90 =
-
-
.65 x $90 = $58.50.
Understanding Simple Probability
Probability
is used
to measure
how
likely
an event
is to
occur.
It is
always
between
0 and
1; an
event
that
will
definitely
not
occur
has
a probability
of 0,
whereas
an event
that
will
certainly
occur
has
a probability
of 1.
To determine
probability,
divide
the
number
of outcomes
that
fit
the
conditions
of an
event
by the
total
number
of outcomes.
For
example,
the
chance
of getting
heads
when
flipping
a coin
is 1
out
of 2,
or 
.
There
are
2 possible
outcomes
(heads
or tails)
but
only
1 outcome
(heads)
that
fits
the
conditions
of the
event.
Therefore,
the
probability
of the
coin
toss
resulting
in heads
is or
.5.
When two events are independent, meaning the outcome of one event does not affect the other, you can calculate the probability of both occurring by multiplying the probabilities of each of the events together. For example, the
probability
of flipping
3 heads
in a
row
would
be x x ,
or 
.
Understanding Number Lines and Sequences
A number line is a geometric representation of the relationships between numbers, including integers, fractions, and decimals. The numbers on a number line always increase as you move to the right and decrease when you move to the left. Number line questions typically require you to determine the relationships among certain numbers on the line.
TIP
Remember that the distance between points on the number line does not always have to be measured in whole units. Sometimes the distance between points is a fraction. For example: The distance on the number line between 1 and 1.5 is .5, or .
An
arithmetic
sequence
is one
in which
the
difference
between
consecutive
terms
is the
same.
For
example,
2, 4,
6, 8...,
is an
arithmetic
sequence
where
2 is
the
constant
difference.
In an
arithmetic
sequence,
the nth
term
can
be found
using
the
formula an = a1 +
(n –1)d,
where d is
the
common
difference.
A geometric
sequence
is one
in which
the
ratio
between
two
terms
is constant.
For
example, ,
1, 2,
4, 8...,
is a
geometric
sequence
where
2 is
the
constant
ratio.
With
geometric
sequences,
you
can
find
the nth
term
using
the
formula an = a1(r)n – 1,
where r is
the
constant
ratio.
Typically, if you can identify the pattern or the relationship between the numbers, you will be able to answer the question. Following is an example of a sequence question similar to one you might find on the PSAT:
0,1,2,0,1,2,...
The numbers 0, 1, and 2 repeat in a sequence, as shown above. If this pattern continues, what will be the sum of the ninth and twelfth numbers in the sequence?
-
To solve this problem, simply recognize that the third number in the sequence is 2, which means that both the ninth number and the twelfth number in the sequence will also be 2. Therefore, the sum of the ninth and twelfth numbers is 4.
Understanding Absolute Value
The absolute value of a number is indicated by placing that number inside two vertical lines. For example, the absolute value of 10 is written as follows: | 10 |. Absolute value can be defined as the numerical value of a real number without regard to its sign. This means that the absolute value of 10, | 10 |, is the same as the absolute value of -10, | -10 |, in that they both equal 10. Think of it as the distance from -10 to 0 on the number line, and the distance from 0 to 10 on the number line...both distances equal 10 units, as shown below:
Following is an example of how to solve an absolute value question:
|3 – 5| =
-
|–2| = 2
Understanding Mean, Median, and Mode
The arithmetic mean refers to the average of a set of values. For example, if a student received grades of 80%, 90%, and 95% on three tests, the average test grade is 80 + 90 + 95 ÷ 3. The median is the middle value of an ordered list. If the list contains an even number of values, the median is simply the average of the two middle values. It is important to put the values in either ascending or descending order before selecting the median. The mode is the value or values that appear the greatest number of times in a list of values.