The various pre-algebra topics are the most basic on the
ACT Math Test. You probably covered much of this material in your
middle school math classes. These topics are not conceptually difficult,
but they do have some nuances you may have forgotten along the way.
Also, since questions covering pre-algebra are often not that hard,
you should make sure you review properly to get these questions
The topics in this section appear roughly in order of
frequency on the Math Test. Number problems are usually the most
common pre-algebra questions on the test, while series questions
are usually the least common.
Factors, and Primes
Fractions, and Decimals
Median, and Mode
While “Multiples, Factors, and Primes” and “Divisibility
and Remainders” do not explicitly appear too frequently on the test,
the math behind them will help you answer number problems, so we’ve
included them at the top of the list.
We mentioned that the above list is only roughly ordered
by decreasing frequency. If it were in an exact order, percentages
would share the top billing with number problems; because we wanted
to keep related topics close together, we sacrificed a bit of precision.
On the ACT Math Test, number problems are word problems
that ask you to manipulate numbers. The math in number problems
is usually extremely simple. You are seldom asked to perform operations
that are more complicated than basic addition, subtraction, multiplication,
and division. Despite the simple operations, number problems can
be confusing because of their wording and because of the multiple
steps involved in answering them. Here’s an example of a typical
number problem on the Math Test:
A travels at 90 miles per hour and covers 360 miles. Train B covers
the same distance but travels at 60 miles per hour. How much longer
does it take Train B than Train A to cover that distance?
The first step in answering these questions
is to read carefully to make sure you know exactly what they are
asking. Because of the time pressure of the test, some students feel
as if the time they take to understand the question is wasted since
they aren’t actually doing any math. But taking a moment to ask
yourself what the question is asking is crucial.
Not only will you be more likely to get the question right if you
take a moment to make sure you understand it, but that little bit
of invested time will actually save you time later,
since you will be able to proceed with an understanding of what
you need to do.
The question above asks the difference in time it takes
the two trains to cover the same distance. Your first step should
be to figure out how long each train takes to travel 360 miles.
Once you’ve done that, you can subtract the smaller number from
the bigger number to get the difference in time.
The question gives you two pieces of information that
will help you figure out the trains’ times: the speed (miles per
hour) and the distance (miles). If you divide the distance by the
speed, you will cancel out the miles and end up with the hours:
done that, you’ll see that Train A travels for 4 hours and Train
B for 6 hours:
Multiples, Factors, and Primes
Multiples, factors, and primes appear quite frequently
on the ACT Math Test. You will rarely see a non-word problem covering
multiples, factors, and primes; this topic almost always appears
in word problem form.
While these questions are relatively easy, they can be
quite confusing simply because of the terminology they use. Below,
we give you the definition for each of these three mathematical
The multiple of a number is the product generated when
that number is multiplied by an integer. The first five multiples
are 7, 14, 21, 28,
The Least Common Multiple
The least common multiple (LCM) is the name given to the
lowest multiple that two particular numbers share. For example,
the multiples of 6 and 8 are:
- Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, 54,
. . .
- Multiples of 8: 8, 16, 24, 32, 40, 48, 56, 64, 72,
. . .
As the two lists show, 6 and 8 both
have 24 and 48 as multiples (they also
share many other multiples, such as 72, 96,
etc.). Because 24 is the lowest in value of these shared
multiples, it is the least common multiple of 6 and 8.
Being able to figure out the least common multiple of
two numbers can prove quite handy on the ACT, especially for questions
in which you have to add or subtract two fractions with unlike denominators
(we’ll explain when we talk about fractions).
A factor of a number is an integer that divides evenly
into the number. For example, 6, 4, 3
, and 2
all factors of 12
then, are related to multiples. A given number is a multiple of
all of its factors: 2
, so 12
is a multiple of both 2
The Greatest Common Factor
The greatest common factor (GCF) of two numbers is the
largest factor that the two numbers share. For example, the GCF
of 18 and 24 is 6, since 6 is
the largest number that is a factor of both 18 and 24.
To find all the factors of a number, write them down in
pairs, beginning with 1 and the number you’re factoring.
We’ll factor 24 for this example. So, 1 and 24 are
both factors of 24. Next, try every integer greater
than 1 in increasing order. Here are the factor pairs
we find for 24: 1 and 24, 2 and 12, 3 and 8,
and 4 and 6.
You know you’ve found all the factors of a number when
the increasing integer in each pair exceeds the decreasing integer.
For example, after you found that 4 was a factor of 24 and 5 was
not, you would see that 6, the next factor of 24,
had already been included in a pair of factors. Thus, all the factors
have been found.
As you might imagine, factoring a very large number can
get pretty involved. But don’t worry—that kind of extensive factoring
won’t be asked of you on the test.
A prime number is divisible by only 1 and
itself (the number 1 itself is not considered prime).
For example, 17 is prime because it is divisible by
only 1 and 17. The first few primes, in
increasing order, are:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37,
41, 43, 47, 53, . . .
Another form of factorization is called prime factorization.
Prime factorization expresses an integer as the product of a series
of prime numbers.
To find the prime factorization of a number, divide it
and all of its factors until every integer remaining is prime. This
group of prime numbers is the prime factorization of the original
integer. Let’s find the prime factorization of 36 as
As you may already have noticed, there is more than one
way to find the prime factorization of a number. We could have first
example, and then determined the prime factorization from there.
So don’t worry—you can’t screw up. No matter which path you take,
you will always get the same result—that is, as long as you do your arithmetic
Just for practice, let’s find a couple more prime factorizations:
Since the only factors of 41 are 1 and 41, it is
a prime number. In other words, 41 is its own prime factorization.
Relatively Prime Numbers
Two numbers are called relatively prime if
they share no common prime factors (i.e., if their GCF is 1
This doesn’t necessarily mean, however, that each number is itself prime.
For instance, 8
relatively prime, because they have no common primes in their prime
neither number is prime. It is a good idea just to know the definition
of relatively prime numbers, in case the concept pops up on the
Divisibility and Remainders
Divisibility and remainders are also popular subjects
for pre-algebraic number problems on the ACT Math Test. As with
multiples, factors, and primes, you will probably not see basic
problems on divisibility and remainders, but the topic will appear
in relatively complicated word problems.
A number (x) is divisible
by another number (y) if, when x is
divided by y, the answer is a whole
number. For example, 6 is divisible by 3 because 6/3
= 2, and 2 is a whole number. However, 6 is
not divisible by 4, because 6/4 = 1
which is not a whole number. Another way of describing 6/4 is
to say that you can make one complete division with a remainder of 2.
To check divisibility, it is always possible
to do the division by hand and see whether the result
is a whole number. However, if the number we are dividing is large,
this becomes very difficult. There are some divisibility rules that
make this task much easier—these rules allow us to determine whether
a number is divisible by another number without having to carry
out the division.
All whole numbers are divisible by 1.
numbers with a ones digit of 0, 2, 4, 6, and 8 are
divisible by 2.
number is divisible by 3 if its digits add up to a
number divisible by 3. For example, 6,711 is
divisible by 3 because 6 + 7 + 1 + 1 = 15,
and 15 is divisible by 3.
number is divisible by 4 if its last two digits are
divisible by 4. For example, 78,052 is divisible by 4 because 52 is
divisible by 4. But 7,850 is not divisible by 4 because 50 is
not divisible by 4.
number is divisible by 5 if it ends in 0 or 5.
number is divisible by 6 if it is even and also divisible
There are no rules for 7.
number is divisible by 8 if its last three digits are
divisible by 8. For example, 905,256 is divisible by 8 because 256 is
divisible by 8. But 74,513 is not divisible
by 8 because 513 is not divisible by 8.
number is divisible by 9 if its digits add up to a
number divisible by 9. For example, 1,458 is
divisible by 9 because 1 + 4 + 5 + 8 = 18 and 18 is
divisible by 9.
number is divisible by 10 if it ends in 0.
Two Notes: (1) Because a number divided by
itself always yields 1, a number is always divisible by itself.
For example, 7 is divisible by 7, and 8,374 is
divisible by 8,374. (2) No number is divisible
by a number greater than itself.
A remainder is the number that remains after x
been divided by y
. If y
evenly into x
, the remainder of
zero. A remainder will always be smaller than the number that is
doing the dividing. For instance, if you divide 22
your answer is 4
with a remainder of 2
Percentages, Fractions, and Decimals
Percentage problems appear frequently on the ACT Math
Test. Because percentages are essentially fractions and decimals,
our review of percentages will begin with a review of fractions
and decimals. While questions dealing specifically with fractions
and decimals per se are rare on the ACT Math Test,
knowing more about them will aid your understanding of the more
common questions about percentages.
Although you may not see a fraction problem on the Math
Test (or, at most, you’ll see one or two), you should still review
your knowledge of fractions, as they form the basis for percentages,
a favorite topic of the ACT.
A fraction describes a part of a whole. The number on
the bottom of the fraction is called the denominator, and it denotes
how many equal parts the whole is divided into. The number on the
top of the fraction is called the numerator, and it denotes how
many of the parts we are taking. For example, the fraction
“three of four equal parts,” 3
being the numerator
being the denominator. You can also think of
fractions as similar to division. In fact, 3
same value as
The ACT may indirectly test your ability to add, subtract,
multiply, and divide fractions. Questions that deal more directly
with fractions will probably test your ability to reduce and compare
Adding and Subtracting Fractions
There are two different types of fractions that you may
have to add or subtract: those with the same denominator and those
with different denominators.
If fractions have the same denominator, adding them is
extremely easy. All you have to do is add up the numerators:
Subtraction works similarly. If the denominators of the
fractions are equal, then you simply subtract one numerator from
If the fractions do not have equal denominators, the process
is somewhat more involved. The first step is to make the denominators
the same. To set the denominators of two fractions as equal, find
the least common denominator (LCD), which is simply the Least Common
Multiple of the two denominators. For example, 18 is
the LCD of 1/
since 18 is the smallest multiple of both 6 and 9.
Setting the denominators of two fractions equal to one
another is a two-step process. First, find the LCD. Second, write
each fraction as an equivalent fraction with the LCD as the new
denominator, remembering to multiply the numerator by the same multiple
as the denominator. For example, if you wanted to add
you would do the following:
First, find the LCD:
Factor the denominators: and
the LCM of the denominators:
LCD is 36.
Once you’ve found the LCD, write each fraction as an equivalent
fraction with the LCD as the new denominator. Multiply the denominator
of the first fraction by an integer to get the LCD. Multiply the
numerator by the same integer.
The new first fraction is, therefore,
Multiply the denominator and numerator of the second fraction
by an integer to get the LCD. Multiply the numerator by the same
The new second fraction is, therefore,
Now that the fractions have the same denominator, you
can quickly add the numerators to get the final answer: 15
+ 16 = 31, so the answer is 31
Multiplying fractions is quite easy. Simply multiply the
numerators together and multiply the denominators together, as seen
in the example below:
Multiplication and division are inverse operations. It
makes sense, then, that to perform division with fractions, all
you have to do is invert (flip over) the dividing fraction and then multiply:
Note that just as multiplication by a fraction smaller
than one results in a smaller product, division
by a fraction smaller than one results in a larger product.
If you encounter fractions involving large, unwieldy numbers,
such as 18/
102, the best move is
usually to see if the fraction can be reduced to smaller numbers.
The fastest way to simplify a fraction is to divide both
the numerator and denominator by their greatest common factor. In
the case of 18/
102, the GCF of 18 and 102 is 6,
leaving you with 3/
17. With your knowledge
of divisibility rules, you should be able to see that both the numerator
and denominator are divisible by 6. Had you not immediately
seen that 6 was the greatest common factor, you could
have divided both numbers by 2 and gotten
From there, it would have been pretty obvious that both the numerator
and denominator are also divisible by 3, yielding
The ACT might also present you with variables in fraction
form and ask you to reduce them. You can reduce these variable fractions
as long as you can find like factors in both the numerator and denominator.
For example, to reduce this fraction,
you merely have to notice that all of the terms in both
the numerator and denominator contain 2 as a factor.
Dividing 2 out of the fraction, you get:
The rare fraction problem you see may ask you to compare
two fractions. If either the denominators or the numerators
of the two fractions are the same, that comparison is easy. For example,
obviously greater than 5
/9, just as
greater than 5/
17. Just remember,
if the numerators are the same, the greater fraction is the one
with the smaller denominator.
If the two fractions don’t lend themselves to easy comparison,
there is still a quick and easy method that will allow you to make
the comparison: cross multiplication. To do this, multiply the numerator
of each fraction by the denominator of the other. Write the product
of each multiplication next to the numerator you used to calculate
it. The greater product will be next to the greater fraction. For
Since 35, the greater product, is written
next to 5/8,
that is the greater fraction.
Decimals are simply another way to express fractions.
To get a decimal, divide the numerator of a fraction by the denominator.
For example, if you take the fraction 2
/5 and divide 2 by 5, you
get 0.4. Therefore the decimal 0.4 is
equal to 2/
Questions testing decimals almost never appear on the
ACT. If decimal numbers do appear and you have to add, subtract,
multiply, or divide them, the best thing to do is to use a calculator.
Percentage problems always make an appearance on the ACT
Math Test. You will probably see at least two per test. Percentages
are just another way to talk about a specific type of fraction.
Percent literally means “of 100.” If you have 25% of
all the money in the world, that means you have
the world’s money.
Let’s take the question “4 is what percent
of 20?” This question presents you with a whole, 20,
and then asks you to determine how much of that whole 4 represents
in percentage form, which means “of 100.” To come to
the answer, you have to set up an equation that sets the fraction 4/20 equal
If you then cross multiply to solve for x,
you get 20x = 400, meaning x =
20. Therefore, 4 is 20% of 20.
You also might realize that instead of working out all this cross
multiplication, you could simply do the following:
Important Percentage Terms
Percentage terminology can be a little tricky, so here
is a short glossary of terms:
- Percent more: if one person
has 50% more children than a second person, then that
first person has the same amount as the second person, plus 50% of
the amount the second person has.
- Percent increase: percent increase means
the same thing as percent more. If the price of some item increases 10%,
the new price is the original plus 10% of that original—in
other words, 110% of the original.
- Percent decrease: the opposite of percent
increase. This term means you subtract the specified percent of
the original value from that original.
Sometimes students see these terms and figure out what
the 10% increase or decrease is, but then forget to
carry out the necessary addition or subtraction. Here’s a sample
ACT percentage problem:
shirt originally cost $20, but during a sale its price was reduced
by 15%. What is the current price of the shirt?
In this question, you are told the whole, $20,
and the percentage, 15%, and you need to figure out
the part. You can therefore quickly set up the equation (once you
are comfortable with percentages you might be able to skip this
step of setting up the equation and move straight to solving for x):
You can find x by multiplying 20 by .15 to
see what the change in price was:
Once you know the price change, you then need to subtract
it from the original price, since the question asks for the reduced
price of the shirt.
The answer is D. Notice
that if you had only finished the first part of this solution and
had looked at the answer choices, you might have seen that $3 hanging
out at answer A like a big affirmation of correctness and been tempted
into choosing it without finishing the question. You could also
solve this problem in one step by realizing that if the sale price
was 15% lower than the original,
it was 85% of the original. Therefore, $0.85($20)
Some ACT questions will ask you to determine
a percent of a percent. Take this question:
original price of a banana in a store is $2. During a sale, the
store reduces the price by 25% and Joe buys the banana. Joe then
meets his friend, Sam, who is almost faint with hunger. Seeing an
opportunity, Joe raises the price of the banana 10% from the price
at which he bought it, and sells it to Sam. How much does Sam pay?
In this question, you are asked to determine the cumulative
effect of two percentage changes. The key to solving this type of
problem is to realize that each percentage change is dependent on
the last. In other words, you have to work out the effect of the
first percentage change, come up with a value, and then use that
value to determine the effect of the second percentage change.
In the problem asked above, you would first find 25% of
the original price.
Now subtract that $.50 from the original
Then we find 10% of $1.50:
Therefore, Sam buys the banana at a price of $1.50
+ $.15 = $1.65.
When you are working on a percentage problem that involves
a series of percentage changes, you should follow the same procedure
you would for one single percentage change at each stage of the
series. For the first percentage change, figure out what the whole is,
calculate the percentage of the whole, and make sure to perform
addition or subtraction, if necessary. Then take the new value and
put it through these same steps for the second percentage change.
Ratios and Proportions
On the typical ACT Math Test, you’ll see a couple of problems
dealing with proportions or ratios.
Ratios can look a lot like fractions, and they are related
to fractions, but they differ in important ways. Whereas a fraction
describes a part out of a whole, a ratio compares two separate parts
of the same whole.
A ratio can be written in a variety of ways. Mathematically
it can appear as 3/1 or as 3:1. In words,
it should be written out as the ratio of three to one. Each of these
three forms of this ratio means the same thing: there are three
of one thing for every one of another. If you have three red marbles
and one blue marble, then the ratio of red marbles to blue marbles
is 3:1. For the ACT, you must remember that ratios
compare parts to parts, rather than parts to a whole. For example:
every 40 games a baseball team plays, it loses 12 games. What is
the ratio of the team’s losses to wins?
This ratio question is a little tricky because the information
is stated in terms of whole to part, but the question asks for a
part to part answer. The problem tells you that the team loses 12 of
every 40 games, but it asks you for the ratio of losses
to wins, not losses to games.
So the first thing you have to figure out is how many times the
team wins in 40 games:
The team wins 28 of every 40 games.
So for every 12 losses, the team has 28 wins,
or 12:28. You can reduce this ratio by dividing both
sides by 4, to get 3 losses for every 7 wins,
or 3:7. Answer C is correct. However,
if you didn’t realize that losses to games was part to whole, you
might have just reduced the ratio 12:40 to 3:10,
and then picked choice A.
If you have a ratio of 3 red marbles to 1 blue
marble, that doesn’t necessarily mean that you have exactly 3 red
marbles and 1 blue one. It could also mean that you
have 6 red and 2 blue marbles, or that
you have 240 red and 80 blue marbles.
In other words, ratios compare only relative size.
In order to determine how many of each color of marbles you actually
have, you need to know how many total marbles you have in addition
to knowing the ratio.
The ACT will occasionally ask questions testing your ability
to figure out a quantity given the ratio between items and the total
number of all the items. For example:
have red, blue, and green marbles in the ratio of 5:4:3, and you
have a total of 36 marbles. How many blue marbles do you have?
The information given states that for each group of 5 red
marbles, you have a corresponding group of 4 blue marbles
and a group of 3 green marbles. The ratio therefore
tells you that out of every 12 marbles (since 12
= 5 + 4 + 3), 4 of them will be blue. The question
also tells you that you have 36 total marbles.
Since we know that the ratio will not change no matter
how many marbles you have, we can solve this problem by setting
up a proportion, which is an equation that states that two ratios
are equal. In this case, we are going to equate 4:12 and x:36,
with x being the number of blue marbles
that we would have if we had 36 total marbles. To do
math with proportions, it is most useful to set up the proportions
in fraction form:
Now you just need to isolate x by
Mean, Median, and Mode
The arithmetic mean, median, and mode are all different
ways to describe a group, or set, of numbers. On the ACT, you’ll
most likely see questions dealing with the arithmetic mean, but
you should be prepared for median and mode questions as well.
Arithmetic Mean (a.k.a. Average)
The arithmetic mean, which is also called the average,
is the most important and most commonly tested of these three mathematical
concepts. The basic rules for finding an average are not very complicated.
To find the average of a set of n numbers,
you need to find the sum of all the numbers and divide that sum
For example, the mean of the set 9, 8, 13, 10 is:
Many ACT problems about mean will be straightforward,
giving you a bunch of numbers and asking you to find their average.
But some problems will be presented in a more roundabout fashion.
For instance, the ACT might give you three numbers of a four--number
set as well as the average of that set, and ask you to find the
fourth number, like so:
the average of four numbers is 22, and three of the numbers are
7, 11, and 18, then what is the fourth number?
To solve this type of problem, you have to realize that
if you know the average of a group, and also know how many numbers
are in the group, you can calculate the sum of the numbers in the
group. In the question asked above, you know that the average of
the numbers is 22
and that there are four numbers.
This means that the four numbers, when added together, must equal
. Now, from the information given in the problem and
our own calculations, we know three of the four numbers in the set
and the total sum of the numbers in the set:
Solving for the unknown number is easy: all you have to
do is subtract 7, 11, and 18 from 88 to
get 52, which is the answer.
The median is the number whose value is in the middle
of the numbers in a particular set. Take the set 6, 19, 3,
11, 7. If we arrange the numbers in order of value, we get:
When we list the numbers in this way, it becomes clear
that the middle number in this group is 7, making 7 the
The set we just looked at contained an odd number of
items, but in a set with an even number of items it’s impossible
to isolate a single number as the median. Let’s add one number to
the set from the previous example:
In this case, we find the median by taking the two numbers
in the middle and finding their average. The two middle numbers
in this set are 7 and 11, so the median
of the set is (7 + 11)/2 = 9.
The mode is the number within a set that appears
most frequently. In the set 10, 11, 13, 11, 20, the
mode is 11 since that number appears
twice and all the others appear just once. In a set where all the
numbers appear an equal number of times, there is no mode.
A typical ACT Math Test asks one question on probability.
To begin to deal with these questions, you first have to understand
what probability is:
For example, let’s say you’re on a game show and are shown
three doors. Behind one door there is a prize, while behind the
other two doors sit big piles of nothing. The probability that you
will choose the door with the prize is 1
out of the total three possibilities there is one chance to pick
the lucrative door.
Here’s an example of a probability question:
has 3 green marbles, 2 red marbles, and 5 blue marbles. If all the
marbles are dropped into a dark bag, what is the probability that
Joe will pick out a green marble?
There are three ways for Joe to pick a green marble (since
there are three different green marbles), but there are 10 total
possible outcomes (one for each marble in the bag). Therefore, the
probability of picking a green marble is:
When you calculate probability, always be careful to divide
by the total number of possible outcomes. In the last example, you
may have been tempted to leave out the three chances of picking
a green marble from the total possibilities, yielding the equation P =
If you did that, you’d be wrong.
The absolute value of a number is its magnitude, regardless
of sign. Absolute value is indicated by two vertical lines that
surround the number: |5|and |–5|, for
example. The absolute value of positive five is equal to five: |5|
= 5. The absolute value of negative five is also equal to
five: |–5| = 5. Simply remove the sign before the number
to produce its absolute value.
On the ACT, you will generally be asked to do a simple
addition, subtraction, multiplication, or division problem using
the absolute values of numbers. For example,
Remember that the vertical lines mean you simply ignore
the sign, so the question actually looks like this: 4 + 2
Exponents and Roots
At most, you’ll see one problem on the ACT Math Test dealing
with exponents or roots. It’s quite likely you won’t see any, but
you’re still doing yourself a favor by preparing for them.
Exponents are a shorthand method of describing how many
times a particular number is multiplied by itself. To write
exponent form, we would simply count how many threes were being
multiplied together (in this case, five), and then write
is stated as “three to the fifth
Raising an Exponent to an Exponent
Occasionally, a question might ask you to raise a power
to a power, in the following format:
such cases, multiply the exponents:
If you have an expression involving a variable, like
you raise it to the third power, then you would write
simplify this expression, you would multiply the exponents and raise 2
the third power; the end result would be
basic calculators have an exponent or
key. Be sure to know how to use this function on your calculator
before the test.
The square root of a number is the number that, when squared
(multiplied by itself), is equal to the given number. For example,
the square root of 16
perfect square is a number whose square root is an integer.
The sign denoting a square root is
use the previous example,
Again, be sure
to find and know how to use the square-root function, or
on your calculator.
The cube root of a number is the number that, when cubed
(raised to the third power), is equal to the given number. The cube
root of 8
The sign denoting a cube root is
Series questions are pretty rare on the ACT. Every once
in a while they do pop up, though. A series is a sequence of numbers
that proceed one after another, according to some pattern. Usually
the ACT will give you a few numbers in a series and ask you to specify
what number should come next. For example,
is a series in which each number is multiplied by –2 to
yield the next number; 32 is the next number in the
series. This type of question asks you to be able to recognize patterns
and then apply them. There isn’t one tried-and-true way to find
a pattern. Just think critically, and use your intuition and trial