HESI A2A2
HESI MATH
Math STUDY GUIDE
Study Guide
Introduction to Fractions
A fraction is a mathematical tool which helps us to count in terms of equal parts of a whole.
ଷ ଶ ଷ
For example, a recipe requires 2 cups of milk, Mike ate of a cake, Sarah run of a mile, John repaid
ସ ସ
ସ
of a loan.
ହ
A fraction is any number of equal parts.
For example, if we cut a small size sausage pizza into 4 equal slices, then 3 such slices of the pizza
are the same as:
3 out of the 4 equal slices of the pizza,
or three fourth of the pizza,
ଷ
or ସ of the pizza.
As we can see from the above example, a fraction has two components, which are called numerator
and denominator.
ଵ
For example, if we consider of a cake, then 1 is the numerator and 4 is the denominator.
ସ
A denominator tells us a number of equal parts that make up a whole while a numerator tells us how
many of them we are considering.
For example, when we say , we are talking about 7 equal parts such that 4 of them make up a whole;
ସ
in other words, we divided a whole into 4 equal parts and now considering 7 such parts.
Both numerator and denominator of a fraction are integers; however, denominator cannot be zero (we
cannot divide a whole into zero parts).
ଶ
Exercise 1. The cost of the vacation trip is $4,500 and we are required to pay of that amount when
ହ
we reserve the trip. How much we need to pay during the reservation?
ଶ
Solution. All we need to do is to calculate ହ of the cost of the trip.
ଶ
Since the denominator of the fraction is 5, we understand that the cost is divided into 5 equal amounts:
ହ
$4,500 = $900 + $900 + $900 + $900 + $900
ଶ
The numerator of the fraction ହ
(which is two) tells us how many of those equal amounts we have to
pay to reserve the trip:
$900 + $900 = $1,800
Answer. So, we need to pay $1,800.
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, ଶ
Exercise 2. There are 24 students in a class. If of the students are girls, then how many boys are
ଷ
there in the class?
Solution. Let’s calculate the number of girls. Then if we subtract the number of girls from the total
number of students, we will get the number of boys.
ଶ
The denominator of the fraction ଷ
(which is three) tells us that the class is divided into three groups of
equal numbers of students:
24 = 8 + 8 + 8
ଶ
The numerator of the fraction ଷ (which is two) tells us that two of these groups are girls:
8 + 8 = 16
Answer. So, the number of boys is 24 − 16 = 8.
Finally, we would like to give a different interpretation of a fraction. We said that a fraction is any number
of equal parts; equivalently, a fraction can be thought of as a division of two integers.
For example,
ଶ
(two fifth) is the same as 2 divided by 5,
ହ
ଵ
ଷ
(one third) is the same as 1 divided by 3,
ହ
(five seventh) is the same as 5 divided by 7,
ସ
(six fourth) is the same as 6 divided by 4.
Summary
A fraction is any number of equal parts
݊
݉, ݊ are integers ݉ > 0
݉
݊ is numerator ݉ is denominator
݊ is how many such parts we are considering ݉ is in how many parts a whole is divided
A fraction as a division of two integers
݊ divided by ݉ ݉, ݊ are integers ݉ > 0
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,Addition and Subtraction of Simple and Mixed Fractions
We will consider 4 cases in each of which we will need a different approach of adding/subtracting two
fractions. Let’s summarize these cases in the following table.
Adding or Subtracting Two Fractions Simple Fractions Mixed Fractions
With common denominator Case 1 Case 2
With different denominators Case 3 Case 4
Case 1: In this case, we will learn how to add and subtract two simple fractions with common
denominators.
A simple fraction is a fraction which is written in the form , where ݉, ݊ are integers and ݉ ≠ 0.
ଷ ିଶ ହ
For example, , , , are all simple fractions.
ସ ଵଵ ଵ
When we need to add or subtract two fractions with common denominator, we just need to add or
subtract numerators.
ଷ ଵ ଷ ଵ
For example, given the two fractions and . What is the sum + =?
ହ ହ ହ ହ
ଵ ଷ
We can see that the fractions and have the common denominator which is 5. So, we just need to
ହ ହ
add the numerators of these fractions in the following way:
3 1 3 + 1 4
+ = =
5 5 5 5
ଷ ଵ
Similarly, we can find the difference of and as follows:
ହ ହ
3 1 3 − 1 2
− = =
5 5 5 5
ହ
Exercise 3. What is ଵଶ + ଵଶ
=?
Solution. Since the denominators are both equal to 12, we just need to add the numerators of the given
fractions:
5 7 12
+ = =1
12 12 12
Answer. So, we have obtained:
5 7
+ = 1
12 12
ଵ ଽ
Exercise 4. What is ଵଵ − ଵଵ
=?
Solution. We have equal denominators; so, we just need to take difference of the numerators of the
given fractions as follows:
10 9 10 − 9 1
− = =
11 11 11 11
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, Answer. So, we have obtained:
10 9 1
− =
11 11 11
ଷ
Exercise 5. Two books are lying one on top of the other. If one of the books is inch thick and the other
ହ
ସ
book is inch thick, then what is the thickness of these two books together?
ହ
Solution. To find the total thickness, we need to add together the thickness of each of books:
3 4 3 + 4 7
+ = =
5 5 5 5
Answer. The total thickness is inches.
ହ
Exercise 6. Two brothers decided to buy 2 large pepperoni pizzas for the dinner. Each of the two pizzas
ହ
was cut into eight equal slices. If Mike was able to eat of one pizza and John of the other pizza, then
଼ ଼
how much of the 2 pizzas is left? Express your answer in terms of fractions.
Solution.
Step 1. Let’s first find how much of pizza did two brothers ate together.
ହ
Mike ate of one pizza and John ate of the other pizza. In total, they ate together:
଼ ଼
5 7 5 + 7 12
+ = =
8 8 8 8
of pizza.
Step 2. Now we can calculate how much of pizza is left. Note that 2 pizzas when each is cut into 8 equal
slices, we have 16 slices in total. So, 2 pizzas can be expressed in terms of fractions as follows:
16
2 =
8
ଵ ଵଶ
If we take the difference of and , then we will get how much of pizza is left:
଼ ଼
16 12 16 − 12 4
− = =
8 8 8 8
ସ
Answer. So, of pizza is left and we know that this is just 4 slices.
଼
Summary (Case 1)
Adding (or subtracting) two simple fractions with common denominators
ܽ ܾ
± =? Need to add (or subtract) numerators
݉ ݉
ܽ ܾ ܽ ± ܾ
± =
݉ ݉ ݉
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HESI MATH
Math STUDY GUIDE
Study Guide
Introduction to Fractions
A fraction is a mathematical tool which helps us to count in terms of equal parts of a whole.
ଷ ଶ ଷ
For example, a recipe requires 2 cups of milk, Mike ate of a cake, Sarah run of a mile, John repaid
ସ ସ
ସ
of a loan.
ହ
A fraction is any number of equal parts.
For example, if we cut a small size sausage pizza into 4 equal slices, then 3 such slices of the pizza
are the same as:
3 out of the 4 equal slices of the pizza,
or three fourth of the pizza,
ଷ
or ସ of the pizza.
As we can see from the above example, a fraction has two components, which are called numerator
and denominator.
ଵ
For example, if we consider of a cake, then 1 is the numerator and 4 is the denominator.
ସ
A denominator tells us a number of equal parts that make up a whole while a numerator tells us how
many of them we are considering.
For example, when we say , we are talking about 7 equal parts such that 4 of them make up a whole;
ସ
in other words, we divided a whole into 4 equal parts and now considering 7 such parts.
Both numerator and denominator of a fraction are integers; however, denominator cannot be zero (we
cannot divide a whole into zero parts).
ଶ
Exercise 1. The cost of the vacation trip is $4,500 and we are required to pay of that amount when
ହ
we reserve the trip. How much we need to pay during the reservation?
ଶ
Solution. All we need to do is to calculate ହ of the cost of the trip.
ଶ
Since the denominator of the fraction is 5, we understand that the cost is divided into 5 equal amounts:
ହ
$4,500 = $900 + $900 + $900 + $900 + $900
ଶ
The numerator of the fraction ହ
(which is two) tells us how many of those equal amounts we have to
pay to reserve the trip:
$900 + $900 = $1,800
Answer. So, we need to pay $1,800.
2|Page
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, ଶ
Exercise 2. There are 24 students in a class. If of the students are girls, then how many boys are
ଷ
there in the class?
Solution. Let’s calculate the number of girls. Then if we subtract the number of girls from the total
number of students, we will get the number of boys.
ଶ
The denominator of the fraction ଷ
(which is three) tells us that the class is divided into three groups of
equal numbers of students:
24 = 8 + 8 + 8
ଶ
The numerator of the fraction ଷ (which is two) tells us that two of these groups are girls:
8 + 8 = 16
Answer. So, the number of boys is 24 − 16 = 8.
Finally, we would like to give a different interpretation of a fraction. We said that a fraction is any number
of equal parts; equivalently, a fraction can be thought of as a division of two integers.
For example,
ଶ
(two fifth) is the same as 2 divided by 5,
ହ
ଵ
ଷ
(one third) is the same as 1 divided by 3,
ହ
(five seventh) is the same as 5 divided by 7,
ସ
(six fourth) is the same as 6 divided by 4.
Summary
A fraction is any number of equal parts
݊
݉, ݊ are integers ݉ > 0
݉
݊ is numerator ݉ is denominator
݊ is how many such parts we are considering ݉ is in how many parts a whole is divided
A fraction as a division of two integers
݊ divided by ݉ ݉, ݊ are integers ݉ > 0
3|Page
© 2017 All Rights Reserved. www.ReadyForTheHESI.org
,Addition and Subtraction of Simple and Mixed Fractions
We will consider 4 cases in each of which we will need a different approach of adding/subtracting two
fractions. Let’s summarize these cases in the following table.
Adding or Subtracting Two Fractions Simple Fractions Mixed Fractions
With common denominator Case 1 Case 2
With different denominators Case 3 Case 4
Case 1: In this case, we will learn how to add and subtract two simple fractions with common
denominators.
A simple fraction is a fraction which is written in the form , where ݉, ݊ are integers and ݉ ≠ 0.
ଷ ିଶ ହ
For example, , , , are all simple fractions.
ସ ଵଵ ଵ
When we need to add or subtract two fractions with common denominator, we just need to add or
subtract numerators.
ଷ ଵ ଷ ଵ
For example, given the two fractions and . What is the sum + =?
ହ ହ ହ ହ
ଵ ଷ
We can see that the fractions and have the common denominator which is 5. So, we just need to
ହ ହ
add the numerators of these fractions in the following way:
3 1 3 + 1 4
+ = =
5 5 5 5
ଷ ଵ
Similarly, we can find the difference of and as follows:
ହ ହ
3 1 3 − 1 2
− = =
5 5 5 5
ହ
Exercise 3. What is ଵଶ + ଵଶ
=?
Solution. Since the denominators are both equal to 12, we just need to add the numerators of the given
fractions:
5 7 12
+ = =1
12 12 12
Answer. So, we have obtained:
5 7
+ = 1
12 12
ଵ ଽ
Exercise 4. What is ଵଵ − ଵଵ
=?
Solution. We have equal denominators; so, we just need to take difference of the numerators of the
given fractions as follows:
10 9 10 − 9 1
− = =
11 11 11 11
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, Answer. So, we have obtained:
10 9 1
− =
11 11 11
ଷ
Exercise 5. Two books are lying one on top of the other. If one of the books is inch thick and the other
ହ
ସ
book is inch thick, then what is the thickness of these two books together?
ହ
Solution. To find the total thickness, we need to add together the thickness of each of books:
3 4 3 + 4 7
+ = =
5 5 5 5
Answer. The total thickness is inches.
ହ
Exercise 6. Two brothers decided to buy 2 large pepperoni pizzas for the dinner. Each of the two pizzas
ହ
was cut into eight equal slices. If Mike was able to eat of one pizza and John of the other pizza, then
଼ ଼
how much of the 2 pizzas is left? Express your answer in terms of fractions.
Solution.
Step 1. Let’s first find how much of pizza did two brothers ate together.
ହ
Mike ate of one pizza and John ate of the other pizza. In total, they ate together:
଼ ଼
5 7 5 + 7 12
+ = =
8 8 8 8
of pizza.
Step 2. Now we can calculate how much of pizza is left. Note that 2 pizzas when each is cut into 8 equal
slices, we have 16 slices in total. So, 2 pizzas can be expressed in terms of fractions as follows:
16
2 =
8
ଵ ଵଶ
If we take the difference of and , then we will get how much of pizza is left:
଼ ଼
16 12 16 − 12 4
− = =
8 8 8 8
ସ
Answer. So, of pizza is left and we know that this is just 4 slices.
଼
Summary (Case 1)
Adding (or subtracting) two simple fractions with common denominators
ܽ ܾ
± =? Need to add (or subtract) numerators
݉ ݉
ܽ ܾ ܽ ± ܾ
± =
݉ ݉ ݉
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