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 Multiply/Divide from left to right. Step 4: 3 + 4 – 2 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) 2

3) 33 – 5 + 2

4)

5)

6) 2) – 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 2 3 4 5. 6 7 8 9 ten thousands thousands hundreds tens ones tenths 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 tenths 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. Simplify.

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: = and =

Step 3: - =

Step 4: The answer is already in lowest terms.

Practice: Add or subtract the following fractions. Simplify.

1)

2)

3)

4)

5)

6)

To multiply fractions, follow the following steps:

Step 1: Reduce both fractions and cross cancel if possible. Step 2: Multiply numerators together. Step 3: Multiply denominators together. Step 4: Reduce if necessary.

Example: x =

Practice: Multiply the following fractions. Simplify.

1) x

2) x

3) x

4) x

5) 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. Simplify.

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 = 0.75 100 = 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.

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

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 Perimeter = 7252

Perimeter = 24in

2) Pentagon

Perimeter = 5273

Perimeter = 31in

5in

7in

7in

7in

7in

5in 5in

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: Rectangle

Base = 7 inches

Height = 2 inches

A = bh

A = (7)(2)

A = 14 square inches

2in

7in

To find the area of a triangle, we use the following formula:

AREA = BASE X HEIGHT or A = bh

Example: Find the area of the following shape: Base = 8 inches

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.

8in

3in

diameter

radius

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 = A = r2

C = (3.14)(8) A = (3.14)(4)2

C = 25.12 feet A = 50.24 square feet

4ft

Practice: Find the area and perimeter of the following figures: 1 2

3

4

3.8in

8.2 in

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) 11

2) 11

3) 18

4) -2

5) 17

6) 26

Decimals: Place Value and Rounding:

1) Hundreds

2) Ones

3) Tenths

4) Thousandths

1) 60

2) 7251.04

3) 9200

4) 540

Decimals: Adding, Subtracting, Multiplying and Dividing:

1) 6.4

2) 2.64

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 Decimal

52% .52

37.5% .375

45% 0.45

125% 1.25

88.8% .8

73% .73

240% 2.4

Percents: Finding a percent of a number:

1) 9 2) 47 3) 19 4) 13 5) 15 6) 70 7) 63 8) 75

Geometry: Area and Perimeter:

1) P = 24 inches A = 31.16 square inches 2) P = 24 cm A = 24 square cm 3) P = 6 cm A = 9 square cm

4) P = 21 in A = 20 square in 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

Pre-Algebra Summer Work Part 2

Order of Operations: Solve the following.

a) b)

c) d)

e) f)

g) h)

Decimals: Place Value and Rounding: Give the place value of the underlined digit; then round the number to that place.

a) 4561.23 b) 87.344

c) 875.43 d) 91.8756

e) 0.598 f) 63.9721

g) 15213.04 h) 213.392

b) 356.418 j) 214986.32

Decimals: Adding, Subtracting, Multiplying and Dividing: Perform the indicated operation.

a) 4.2 + 1.9 b) 8.6 – 3.45

c) 9.3 0.6 d)

e) 18.24 + 22.09 f) 8.21 – 5.19

g) 15.2 h)

c) 5.6 + 9.2 j) 5.1 – 2.67

d)

Fractions: Mixed Numbers and Improper Fractions: Change the mixed numbers to improper fractions and the improper fractions to mixed numbers.

a) b)

c) d)

e) f)

g) h)

i)

Fractions: Adding, Subtracting, Multiplying, and Dividing: Perform the indicated operation.

a) b)

c) d)

e) f)

g) h)

i) j)

k)

Fractions: Adding, Subtracting, Multiplying, and Dividing Mixed Numbers: Perform the indicated operation.

a) b)

c) d)

e) f)

g) h)

Percents: Fractions and decimals and percents: Complete the following chart.

Percent Fraction Decimal

125%

0.52

0.056

240%

Percents: Finding a Percent of a Number: Solve by finding the percent of the number.

a) 4% of 16 b) 132% of 78

c) 0.06% of 31 d) 25% of 140

e) 120% of 85 f) 40% of 300 g) 25% of 28 h) 75% of 76

Geometry: Area and Perimeter: Find the perimeter and area of the following shapes. a) b)

c) d) e) f)