Download Pre-Algebra Summer Work

Pre-Algebra Summer Work
Part 1
Order of Operations:
To evaluate expressions involving more than one operation, mathematicians
have agreed on a set of rules called the order of operations. Use the following
steps to solve problems involving more than one operation:
Step 1: Evaluate expressions inside grouping symbols such as ( ), { }, [ ], or a
fraction bar.
Step 2: Evaluate powers.
Step 3: Multiply and divide from left to right.
Step 4: Add and subtract from left to right.
Example: 3 + 22 – (5 – 2)
Step 1: 3 + 22 – 3
Simplify inside ( ).
Step 2: 3 + 4 – 3
Simplify exponents.
Step 3: 3 + 4 – 6
Step 4: 3 + 4 – 2
Multiply/Divide from left to right.
Multiply/Divide from left to right.
Step 5: 7 – 2 Add/Subtract from left to right.
Practice: Evaluate the following using order of operations.
1)
2) (6 + 4)
3) 33 – 5 + 2
4)
5)
2
2
6)
)–5+2
]
Decimals: Place Value and Rounding:
The value of each digit in a number depends on the position, or place, of the
digit within the number. The place values for the number 12345.6789 are listed
below.
1
ten thousands
2
thousands
3
hundreds
4
tens
5.
ones
6
tenths
7
8
9
hundredths thousandths ten thousandths
Practice: Give the place value of the 5 in each of the following
numbers.
1) 4561.23
2) 875.43
3) 87.533
4) 91.8756
To round a number means to approximate it to a given place. When rounding,
look at the digit to the right of the given place. If the digit to the right is less
than 5, round down by replacing all digits to the right with zeros. If the digit to
the right is 5 or greater, round up by adding one to the given digit and replacing
all digits to the right with zeros.
Example: Round 88.173 to hundredths place. The digit to the right of the 7 is a
3. Because
3 < 5, replace the 3 with a 0. 88.173 rounds to 88.170.
Practice:
1) 56.75 rounded to tens place.
2) 7251.041 rounded to hundredths place.
3) 9172.043 rounded to hundreds place.
4) 539.52 rounded to ones place.
Decimals: Adding, Subtracting, Multiplying and Dividing:
When adding or subtracting decimals, begin by lining up the decimal points.
Write zeros as place holders if necessary. Add or subtract as you would with
whole numbers being sure to place the decimal in the answer.
Example: Find the sum or difference.
1) 20 – 2.8 Rewrite in a vertical format:
20.0
-2.8
17.2
To multiply decimals, multiply as you would whole numbers, then place the
decimal point in the product. The number of decimal places in the product is
equal to the sum of the number of decimal places in the factors.
Example: 4.94 x 0.45
4.94
X 0.45
2470
19760
2.2230 - four decimal places
One strategy you can use to estimate a sum or difference is to round to the
place of the leading digit. The leading digit is the leftmost digit.
Example: 4.94 has a leading digit in ones place; round this number to 5.
0.45 has a leading digit in tenths place; round this number to 0.5.
Half of 5 is equal to 2.5. This answer is close to the actual answer
will provide you with an approximate answer.
To divide decimals, multiply both the divisor and the dividend by a power of 10
that will make the divisor a whole number. Then line up the decimal point in the
quotient with the decimal point in the dividend. You may need to write
additional zeros in a dividend to continue dividing. The zeros will not change the
value of the dividend.
Example: Find the quotient 7.848
0.24
_____
0.24 7.848 (move the decimal so the divisor is a whole number)
32.8
24 784.8
72
64
48
168
168
0 7.848
0.24 = 32.8
Practice: Add, subtract, multiply, or divide the following.
1) 4.1 + 2.3
2) 1.34 + 0.9
3) 84.34 + 67.23
4) 3.596 + 5.618
5) 2.6 – 0.9
6) 67.38 – 37.46
7) 4.956 – 1.234
8) 8.267 – 6.52
9) 1.2 x 2.3
10) 0.72 x 0.06
11) 8.52 x 3.5
12) 4.33 x 0.019
13)
14)
15)
16)
12.5
Fractions: Mixed Numbers and Improper Fractions:
A fraction consists of a denominator, which represents the number of pieces in a
whole, and a numerator, which represents how many of those pieces we have.
The numerator is above the fraction bar and the denominator is below the
fraction bar.
Fractions come in many forms. One form is a mixed number. A mixed number
consists of a whole number with a fraction part. When changing a mixed
number to an improper fraction, multiply the denominator of the fraction by the
whole number, add answer to the numerator and use as your new numerator
over the existing denominator.
Example: 4
4 x 8 + 5 = 37 (new numerator) The denominator is still 8.
4
=
Another type of fraction is an improper fraction. An improper fraction has a
numerator which is greater than its denominator. When changing an improper
fraction to a mixed number, divide the numerator by the denominator to
represent the whole number. Your remainder becomes the numerator of your
new fraction.
Example:
Since 31 divided by 6 is 5 with 1 left over, the new mixed number
will be 5 .
Practice: Write the mixed number as an improper fraction.
1) 1
2) 4
3) 7
4) 12
5) 15
Practice: Write the improper fraction as a mixed number.
1)
2)
3)
4)
5)
Fractions: Adding, Subtracting, Multiplying, and Dividing:
To add fractions with a common denominator, write the sum of the numerators
over the denominator. To subtract fractions with a common denominator, write
the difference of the numerators over the denominator. Always write your
answer in simplest form by reducing if possible.
Example:
This will reduce to
Practice: Find the sum or difference.
1)
+
2)
+
3)
+
4)
+
5)
-
6)
-
7)
-
8)
-
Not all fraction problems will have common denominators. Oftentimes, you will
need to add or subtract fractions that have unlike or different denominators. In
this case, you will need to find a common denominator for your fractions before
you add or subtract. Use the following steps to add/subtract fractions with
unlike denominators:
Step 1: Find the LCD.
Step 2: Rewrite both fractions as equivalent fractions with the LCD as the
denominator.
Step 3: Add or subtract, keeping the denominator the same.
Step 4: Reduce to lowest terms.
Example:
-
Step 1: The LCD of 8 and 12 is 24.
Step 2:
=
Step 3:
-
and
=
=
Step 4: The answer is already in lowest terms.
Practice: Add or subtract the following fractions.
1)
2)
3)
4)
5)
6)
To multiply fractions, follow the following steps:
Step
Step
Step
Step
1:
2:
3:
4:
Reduce both fractions and cross cancel if possible.
Multiply numerators together.
Multiply denominators together.
Reduce if necessary.
Example:
x
=
Practice: Multiply the following fractions.
1)
x
2)
x
3)
x
4)
5)
x
x
Dividing by a fraction is the same as multiplying by its reciprocal. Change the
division sign to multiplication and flip the second fraction. Follow the rules for
multiplication.
Example:
=
x
=
Practice: Divide the following fractions.
1)
2)
3)
4)
5)
Fractions: Adding, Subtracting, Multiplying, and Dividing Mixed
Numbers:
There are two ways to add mixed numbers. One – change mixed numbers to
improper fractions first, and then add. Two – follow these steps:
1) Rewrite fractions with the least common denominator.
2) Add the fraction parts together.
3) Add the whole parts together.
4) If the fraction is improper, convert to a mixed number and combine with
the whole number.
Example:
____________
To subtract, multiply, or divide mixed numbers, change to improper fractions
first and then use the rules previously covered.
Example:
______________
Practice: Simplify each of the following:
1)
5)
9)
2)
6)
10)
3)
7)
11)
4)
8)
12)
Percents:
Recall that the word percent means “per hundred.” A percent is a ratio whose
denominator is 100. The symbol for percent is %.
Percents: Fractions and decimals and percents:
 To change a fraction to a percent, divide the numerator by the
denominator then move the decimal two places to the right or multiply by
100.
 To change a decimal to a percent, move the decimal two places to the
right or x by 100.
 To change a percent to a decimal, move the decimal two places to the left
or divide by 100.
 To change a percent to a fraction, write the percent as the numerator and
use 100 as the denominator. Reduce, if necessary.
Example:
Change
to a percent.
.75
4 3.00
= 0.75
28
20
20
0
Change 0.875 to a percent.
0.875 = 87.5%
Change 45% to a decimal.
45.0% = .45
Change 60% to a fraction.
100 = 75%
Practice: Complete the following chart.
Percent
Fraction
Decimal
52%
0.45
1.25
73%
Percents: Finding a Percent of a Number:
To find the percent of a number, change the percent to a decimal (see above)
and multiply the number by the decimal.
Example: Find the percent of the number.
1) 25% of 36
0.25 x 36 = 9
2) 70% of 70
0.70 x 70 = 49
3) 50% of 14
0.50 x 14 = 7
Practice: Find the percent of the number.
1) 75% of 12
2) 50% of 94
3) 20% of 95
4) 10% of 130
5) 30% of 50
6) 40% of 175
7) 70% of 90
8) 25% of 300
Graphing:
When a horizontal number line and a vertical number line cross at 0, a
coordinate plane is formed. Each number line is now called an axis. The
horizontal number line is called the x-axis, and the vertical number line is called
the y-axis. The point where they intersect is called the origin. These two axes
divide a plane into four quarters called quadrants. The quadrants are numbered
with roman numerals in a counter-clockwise direction.
On the coordinate plane, instead of graphing single numbers, we can now graph
pairs of numbers, which are called coordinate pairs. Every ordered pair has two
coordinates, an x-coordinate and a y-coordinate. The x-coordinate is always
listed first and the y-coordinate is always listed second. This should be easy to
remember because an ordered pair is always in alphabetical order (x,y). To
graph these points, start at the origin. Move on the x-axis to the right for a
positive number or to the left for a negative number as the x-coordinate
indicates. From there, move up for a positive number or down for a negative
number as the y-coordinate indicates.
Example: Plot the points: A (4, -2) [right 4, down 2]
B (6, 4) [right 6, up 4]
C (-3, -8) [left 3, down 8]
D (-7, 5) [left 7, up 5]
Practice: Plot the following points. Label each point with the letter
beside it.
A (3, 10)
F (6, 7)
B (0, 2)
G (-8, -1)
C (-3, -2)
H (1, -1)
D (-8, 4)
I (0, 0)
E (5, 9)
Geometry: Area and Perimeter:
The perimeter of a shape is the distance around the shape. This can be found
by adding up the lengths of every side in a polygon. Perimeter describes a
length and can be measured in units such as inches, feet, centimeters, etc…
Example: Find the perimeter of the following shapes.
1) Rectangle
2) Pentagon
3) Triangle
The word area describes the measurement of a flat region with two dimensions.
Area is measured in square units, such as square inches, square feet, square
centimeters, etc…
To find the area of a square, rectangle, or parallelogram, we use the following
formula:
AREA = BASE x HEIGHT or A = bh
The height is always perpendicular to the base.
Example: Find the area of the following shape:
Base = 7 inches
Height = 2 inches
A = bh
A = (7)(2)
A = 14 square inches
To find the area of a triangle, we use the following formula:
AREA =
BASE X HEIGHT or A =
Example: Find the area of the following shape:
Base = 8 inches
bh
Height = 3 inches
A = ½ bh
A = ½ (8)(3)
A = 12 square inches
The perimeter of a circle is called the circumference. The point at the exact
center of a circle is called the center point. The diameter of a circle is a line that
goes through the center point and has endpoints on the circle. The radius is a
line that has one endpoint on the center point and the other endpoint on the
circle. The radius of a circle is always one half of the diameter.
The number pi or
can be used to calculate the perimeter and area of circles.
We approximate
to be 3.14 as a decimal and
as a fraction. To find the
circumference of a circle, we use the following formula:
CIRCUMFERENCE =
DIAMETER or C =
To find the area of a circle, we use the following formula:
AREA =
RADIUS
RADIUS or A =
r2
Example: Find the circumference and area of the following circle.
Diameter = 8 feet
Radius = 4 feet
= 3.14
C=
C = (3.14)(8)
C = 25.12 feet
A=
r2
A = (3.14)(4)2
A = 50.24 square feet
Practice: Find the area and perimeter of the following figures:
Word Problems:
1) Emily took her three best friends to J P Licks for ice cream. Ice cream
cones cost $3.75, sodas cost $1.99 and cookies cost $0.99. With her
allowance money, Emily bought herself and each of her friends and ice
cream cone, a soda and a cookie. How much did she spend?
2) Kelly ran
miles on Monday, 3 times that amount on Tuesday, and
miles on Wednesday. How far did she run?
3) At the end of a party, there were
pies left over. Four people were
fighting over taking the pie home (it was quite good!) so they decided to
split it up evenly. How much pie should each person take home so that
everyone has the same amount?
4) Seven girls ran the 45-meter dash. Tara ran it in 9.8 seconds, Heather ran
it in 8.65 seconds, Amy ran it in 7.43 seconds, and Becky ran it in 2.36 less
than Tara. Laura fell down, but got up and finished. It took her twice as
long as Heather to finish. Karen took 1 second longer than Amy to finish,
and Ellen ran it in 9.79 seconds. Find each girl’s time and put them in
order from first place to seventh place.
5) To get ready for the school year, you decide to go to the local bookstore
to get some supplies. You purchase 5 notebooks that cost $0.75 each, 8
pens that cost $1.25 each, and 3 pencils that cost $0.99 each. The store
charges a 5% sales tax on all items purchased. How much did you spend
including tax?
6) Find the perimeter of a rectangle with one side that is 14 inches long, and
an area of 259 square inches.
7) Find the area of a square with a perimeter of 54.4 cm.
8) A circle has a radius of 7 cm. Find its area and circumference.
Practice Problem Answer Key
Order of Operations:
1)
2)
3)
4)
5)
6)
11
11
18
-2
17
26
Decimals: Place Value and Rounding:
1)
2)
3)
4)
Hundreds
Ones
Tenths
Thousandths
1)
2)
3)
4)
56.8
7251.04
9200
540
Decimals: Adding, Subtracting, Multiplying and Dividing:
1) 6.4
2) 8.64
3) 151.57
4) 9.214
5) 1.7
6) 29.92
7) 3.722
8) 1.747
9) 2.76
10) .0432
11) 29.82
12) .08227
13) 5
14) 2.1
15) 1.3
16) 23.14
Fractions: Mixed Numbers and Improper Fractions:
1)
2)
3)
4)
5)
1)
2)
3)
4)
5)
Fractions: Adding, Subtracting, Multiplying and Dividing:
1)
2)
3)
4)
5)
6)
7)
8)
1)
2)
=
3)
4)
5)
6)
1)
2)
3)
4)
5)
1)
2)
3)
4)
5)
Fractions: Adding, Subtracting, Multiplying and Dividing Mixed
Numbers:
1)
2)
3)
4)
5)
6)
7)
8)
9)
10)
11)
12)
Percents: Fractions and decimals and percents:
Percent
Fraction
52%
.52
37.5%
.375
45%
0.45
125%
1.25
88.8%
.8
73%
.73
5.6%
0.056
Percents: Finding a percent of a number:
1)
2)
3)
4)
9
47
19
13
Decimal
5)
6)
7)
8)
15
70
63
75
Graphing:
Geometry: Area and Perimeter:
1)
2)
3)
4)
P
P
P
P
=
=
=
=
24 “
12cm
25.12 cm
21 ft
A
A
A
A
=
=
=
=
31.16 square inches
12 square cm
50.24 square cm
40 square feet
Word Problems:
1) $26.92
2) 20
3)
4) Amy = 7.43
Becky = 7.44
Karen = 8.43
Heather = 8.65
Ellen = 9.79
Tara = 9.8
Laura = 17.3
5) $17.56
6) 65 inches
7) 184.96 square cm
8) C = 43.96 cm A = 153.86 square cm