HESI A2 Maths Study Guide Questions and Answers Download To Score A

Introduction to Fractions

A fraction is a mathematical tool which helps us to count in terms of equal parts of a whole.

For example, a recipe requires 2     cups of milk, Mike ate     of a cake, Sarah run     of a mile, John repaid of a loan.

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, or     of the pizza.

As we can see from the above example, a fraction has two components,  which are called numerator and denominator.

For example, if we consider       of a cake, then 1 is the numerator and 4 is the denominator.

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;

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).

Exercise 1. The cost of the vacation trip is $4,500  and we are required to pay      of that amount when 

we reserve the trip. How much we need to pay during the reservation?

Solution. All we need to do is to calculate       of the cost of the trip.

Since the denominator of the fraction      is 5, we understand that the cost is divided into 5 equal amounts:

 $4,500 = $900 + $900 + $900 + $900 +  $900

 The numerator of the fraction       (which is two) tells us how many of those equal amounts we have to pay to reserve the trip:

 Exercise 2. There are 24 students in a class. If      of the students are girls, then how many boys are 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.

The denominator of the fraction       (which is three) tells us that the class is divided into three groups of equal numbers of students:

24  =  8  +  8  +  8

The numerator of the fraction       (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,

   (two fifth) is the same as 2 divided by 5,

   (one third) is the same as 1 divided by 3,

   (five seventh) is the same as 5 divided by 7,

   (six fourth) is the same as 6 divided by 4.

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.

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.

For example,     ,    ,        ,     are all simple fractions.

When we need to add or subtract two fractions  with common  denominator,  we just need to add or subtract numerators.

For example, given the two fractions       and     . What is the sum       +        = ?

We can see that the fractions        and      have the common denominator  which is 5. So, we just need to

add the numerators of these fractions in the following way:

3       1       3  +  1      4

5  +  5  =

5      = 5

Similarly, we can find the difference of     and     as follows:

3       1       3  −  1      2

−      =               =

5       5            5          5

Exercise 3. What is    

+         =?

5         7

+

12       12

12

=        = 1

12

Answer. So, we have obtained:

5         7

+         =  1

12       12

Exercise 4. What is   ! −    "   =?

Solution. We have equal denominators;  so, we just need to take difference of the numerators of the given fractions as follows:

10        9

−

11       11

10  −  9       1

=                  =

11          11

Answer. So, we have obtained:

10        9         1

11  −  11  = 11

Exercise 5. Two books are lying one on top of the other. If one of the books is     inch thick and the other

book is     inch thick, then what is the thickness of these two books together?

Solution. To find the total thickness, we need to add together the thickness of each of books:

3       4

+

5       5

Answer. The total thickness is     inches.

3  +  4      7

=               =

5          5

Exercise 6. Two brothers decided to buy 2 large pepperoni pizzas for the dinner. Each of the two pizzas

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.

Mike ate     of one pizza and John ate     of the other pizza. In total, they ate together:

#                                                                               #

5       7       5  +  7      12

of pizza.

8  +  8  =

8      =  8

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

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

Answer. So,     of pizza is left and we know that this is just 4 slices.

#

Case 2: In this case, we will learn how to add and subtract mixed fractions with common denominators.

A mixed fraction is a sum of integer and a simple fraction.

For example, 2  +     , 3  +     , 5  +     , 6  +       are mixed fractions.

"

To simplify notation, mixed fractions are written as follows:

2  +        = 2   ,

3  +        =  3   ,

5  +        =  5   ,

6  +        =  6   .

"           "

When the numerator of a simple fraction is bigger than its denominator, we call such fraction improper.

For example,    , ",      ,       are improper fractions since their numerators are bigger than their denominators.

"

We can convert mixed fraction into improper fraction and vice versa.

For example, let’s convert the mixed fraction 5     into improper fraction. To be able to add integer with a

fraction, we need to convert integer into a fraction as follows:

15

5  =

3

Indeed, 15 divided by 3 is equal to 5. Now, we can convert the mixed fraction 5     into its equivalent

improper fraction form by adding two fractions as follows:

1                 1

5     =  5  +

3                 3

15        1

=         +

3         3

15  +  1

=        3

16

=

3

As a shorthand, we did the following:

1        '5×3)  +  1     16

5     =                         =

3                 3                3

To convert the improper fraction         back to its mixed form, we just need to divide 16 by 3. Then the

quotient would be the integer component and the remainder would be the numerator of the resulting mixed fraction:

If we divide 16 by 3, then the quotient is 5 and the remainder is 1. So, we get:

16                 1          1

=  5  +     =  5

3                   3          3

Exercise 7. Convert the mixed fraction 7     into its equivalent improper fraction form and convert the

"

improper fraction         into its equivalent mixed fraction form.

Solution. Converting 7     into improper fraction form:

"

5

7

9

Converting        into mixed fraction form:

'7×9)  +  5      68

=                         =

9                 9

(if we divide 17 by 5, then the quotient is 5 and the remainder is 2)

17                 2          2

5   =  3  +  5 =  3 5

Answer. After conversion, 7    =  #  and     = 3  

"       "

To add (or subtract) two mixed fractions with common denominators, we need to convert mixed fractions into improper fractions and then add (or subtract) numerators.

For example, given the two mixed fractions 3     and 2   . What is the sum 3      +  2      =?

Notice that we need to add to mixed fractions with common denominators. We can compute the sum in two steps as follows:

Step 1. Convert mixed fractions into their equivalent improper fraction forms.

Similarly,

2        '3×7)  +  2     23

3     =                         =

7                 7                7

1      '2×7)  +  1      15

2     =                        =

7                7                 7

Step 2. Add simple fractions.

2          1

3     +  2

7          7

23       15

=         +

7         7

23  +  15       38

=         7        =   7

We can leave the answer as an improper fraction or we can convert it into mixed fraction if needed. So, we have obtained:

2          1        38

3 7  +  2 7  =   7

Exercise 8. A cake recipe requires 2     of cups of milk and 1     of melted butter. If these are the only

liquids, then how much liquid is in the recipe?

Solution. The only liquids are 2     of cups of milk and 1     of melted butter; so, we need to compute:

1          1

2     +  1     = ?

3          3

Step 1. Convert mixed fractions into improper fractions.

1        2 ⋅ 3  +  1      7

2     =                     =

3               3              3

Similarly,

1        1 ⋅ 3  +  1      4

1     =                     =

3               3              3

Step 2. Add simple fractions.

1          1        7       4

2     +  1     =      +

3          3        3       3

7  +  4      11

=                =

3             3

Answer. So, we have obtained:

1          1        11

2     +  1     =

3          3         3