Summer Mathematics Prep
Students Entering AFDA
Chesterfield County Public Schools
Department of Mathematics
SOLUTIONS
NOTE: There are many ways to solve problems in mathematics. Only one method is
included for each problem in this document.
Part I: Adding, subtracting, and multiplying fractions
1)
3 1
4 2



10 10 10 5
2)
5 4 25 24 49
19
 


1
6 5 30 30 30
30
Find a common denominator and rewrite each fraction using
the common denominator. Add the fractions and write as a mixed number.
3)
12 9
3


17 17 17
4)
3 2 15 8
7
 


4 5 20 20 20
Find a common denominator and rewrite each fraction using
the common denominator. Subtract the fractions and reduce.
5)
4 3
12
1
 

52 51 2652 221
6)
8 5 40
5
 

14 8 112 14
Chesterfield County Public Schools
Department of Mathematics
Page 1 of 10
May 2010
7)
1
1
1
cups of walnuts, cups of pecans, and cups of
3
4
6
almonds. What is the total number of cups of the three types of nuts that will be
used in this recipe?
A recipe for cookies uses
1 1 1 4
3
2
9 3
  
 


3 4 6 12 12 12 12 4
Find a common denominator and rewrite each fraction using
the common denominator. Add the fractions and reduce.
Part II: Probability
8)
Mike has a bag that contains 3 white marbles, 4 green marbles, and 8 blue
marbles. If Mike reaches in the bag, without looking, what is the probability he
will pick a green marble?
P(Mike will select a green marble) =
9)
The silverware drawer contains 15 forks, 16 spoons, and 14 knifes. If Rebecca
randomly selects one item from the drawer, what is the probability she will pick a
fork?
P(choosing a fork) =
10)
number of green marbles 4

total number of marbles 15
number of forks
15 1


total number of silverware 45 3
Greg and Melody are rolling 2 fair dice. What is the probability that Greg will
roll a sum of 7? What is the probability that Melody will roll a sum of 1?
P(Greg rolling a 7) =
total number of ways to roll a sum of 7 6 1


number of total sums of two dice
36 6
P(Melody rolling a 1) =
Chesterfield County Public Schools
Department of Mathematics
total number of ways to roll a sum of 1 0

0
number of total sums of two dice
36
Page 2 of 10
May 2010
Part III: Solving Equations
11) 2 x  12  40
2 x  52 add 12 to both sides 
x  26 (divide both sides by 2)
12) 3 x  20  2 x  55
5 x  20  55 add 2x to both sides 
5 x  35 subtract 20 from both sides 
x  7 divide both sides by 5
13)
2
x  1  15
3
2
x  14 subtract 1 from both sides 
3
3

x  21  multiply both sides by 
2

14) 2( x  3)  3(2 x  8)
2 x  6  6 x  24 (distribute the 2 and the 3)
6  4x  24 subtract 2x from both sides 
 18  4 x subtract 24 from both sides 
 18
 x divide both sides by 4
4
9
 x reduce
2
15) 3 x  7  22 : Do not solve this equation. Instead, write (make-up) a story problem
that would lead a student to write an equation identical to the one provided according to
your story. Answers will vary
Chesterfield County Public Schools
Department of Mathematics
Page 3 of 10
May 2010
Part IV: Factoring. Factor each expression completely.
16) 6 x  20 factor out GCF
 2(3x  10)
17) 4 x 2  8 x factor out GCF
 4 x( x  2)
18) x 2  5 x  6
 ( x  2)( x  3)
19) x 2  6 x  8
 ( x  2)( x  4)
20) x 2  16
 ( x  4)( x  4)
21) The area of a rectangular playground can be defined by the expression x 2  12 x  64 .
What are the possible dimensions of the playground?
x 2  12 x  64
Area  length  width
x 2  12 x  64  length  width
factoring x 2  12 x  64
Area  ( x  4)( x  16)
length  ( x  4)
width  ( x  16)
Chesterfield County Public Schools
Department of Mathematics
Page 4 of 10
May 2010
Part V: Solving quadratic equations.
22) ( x  3)( x  2)  0
( x  3)  0  x  2  0
x  3
x2
23) ( x  5)( 2 x  1)  0
( x  5)  0 (2 x  1)  0
1
x  5
x
2
24) x 2  8 x  7  0
( x  1)( x  7)  0
( x  1)  0 ( x  7)  0
x  1
x  7
Part VI: Graphing functions.
2
25) y  x  1
3
y

2
The y - intercept is 1 and the slope is .
3
You start by graphing the y - intercept at (0,1).
To graph your next point, from the y - inetecept,
you go up 2 units (rise) and forward 3 units(run).
Repeat this process until you have enough points to
















x










draw the line. Your teacher will have you use at
least 3 points.






Chesterfield County Public Schools
Department of Mathematics
Page 5 of 10
May 2010

26) y   x  2
y
 -1
The y - intercept is - 2 and the slope is  1 or  .
1
You start by graphing the y - intercept at (0,-2).
To graph your next point, from the y - inetecept,
you go down 1 unit (rise) and forward 1 unit(run).
Repeat this process until you have enough points to
draw the line. Your teacher will have you use at
least 3 points.

















x

















27) y  (x  3)(x  3)
Intercept form : y  a ( x  p )( x  q )
Find the x - coordiante of the vertex
p  q 33 0
x

 0
2
2
2
Using the x - coordinate, find the y coordinate of the vertex
y  (0  3)(0  3)  (3)(3)  9
vertex (0,-9)
Find the x - intercepts
( x  3)  0 ( x  3)  0
x  3
x3
Plot the intercepts and the vertex and connect them using a smooth
curve to form a parabola

Chesterfield County Public Schools
Department of Mathematics

y















x












Page 6 of 10
May 2010





28) y  x 2  4x  3
Standard form : y  ax 2  bx  c
Find the x - coordiante of the vertex
y


-b
4

2
x
2a 2(1)
Make a table of values, choosing x - values
to the left and right of the vertex







x
0







y
x







3

1
0
2 1
3 0
4 3





Plot the points and connect them using a smooth
curve to form a parabola
Part VI: Solving and Writing Systems of Linear Equations
29) y  x  6
y  3 x  2
Solve by graphing
y






Graph the first equation. Graph the second
equation. Where the two line intersect
is the answer, (2,-4)











x

















Chesterfield County Public Schools
Department of Mathematics
Page 7 of 10
May 2010





30) y  4 x
Solve by Substitution:
x y 9
Use : x  y  9
x  ( 4 x )  9
y  4(3)
 3x  9
x  3
y  12
Therefore the solution is (-3,12)
31) 18 x  15 y  36
 18 x  22 y  78
Solve by Elimination: (Linear Combination)
18 x  15 y  36
 18 x  22 y  78
7 y  42
y  6
18 x  15(6)  36
18 x  90  36
18 x  54
x  3
Therefore the solution is (-3, -6)
32)
8 x  y  1
Solve using any method:
 10 x  2 y  5
16 x  2 y  2  multiply the top equation by 2 to eliminate the y's 
10 x  2 y  5
6x  3
x
1
2
1
8    y  1
2
4  y  1
 y  5
y5
1 
Therefore the solution is  ,5 
2 
Chesterfield County Public Schools
Department of Mathematics
Page 8 of 10
May 2010
Write (set-up) a system of equations to represent the scenario described.
Do not solve the system.
33) The total cost of 10 gallons of regular gasoline and 15 gallons of premium gasoline
is $32.75. Premium costs $0.20 more per gallon than regular. What is the cost per gallon
of each type of gasoline?
a) assign variables to describe the situation
Let r = regular gasoline
Let p = premium gasoline
b) write a linear system to model the situation
10r  15 p  32.75
p  .20  r
Part VII: Scatter Plots and Making Predictions
34. Describe the relationship shown in the scatter plot as positive, negative, or no
relationship.
y






x







No relationship
Chesterfield County Public Schools
Department of Mathematics
Page 9 of 10
May 2010
35.
Alternative-fueled vehicles: The table below shows the number of y (in
thousands) of alternative-fueled vehicles in the United States x years after 1997.
Graph the data points and describe the type of relationship that exists.
1
2
3
4
5
6
7
X 0
Y 280 295 322 395 425 471 511 548
y











x








Positive relationship
36.How many bags of popcorn must be sold in order to make a profit of $100.00?
For every 20 bags of popcorn sold they make $10. So, in order to make $100.00 they
must sell 200 bags of popcorn
profit
10 100


# of bags 20
x
10 x  2000
x  200
Chesterfield County Public Schools
Department of Mathematics
Page 10 of 10
May 2010