Arithmetic with Decimals

Algebra 1 Summer Packet

Sherwood High School

Summer 2013

General Information

·  The purpose of this packet is to review important skills that you have learned which are necessary to succeed in Algebra 1:

ð  Calculating Decimals

ð  Calculating Fractions

ð  Evaluating Expressions

ð  Plotting points in the Coordinate Plane

ð  Graphing Equations

ð  Order of Operations

ð  Operations with Signed Numbers (Integers)

ð  Solving Equations

ð  Solving Word Problems

ð  Rounding Numbers

Getting Help:

·  When completing the problems in this packet, you should show your work and do your best to complete the problems correctly.

·  During the summer, the answer key will be posted on the Sherwood website.

Enjoy your summer! We look forward to seeing you in the fall.

(If you have any questions, please contact the math office at (301) 924 – 3253.)

Arithmetic with Decimals

Adding Decimals

Example 1: Add 4.36 and 2.89

1 1 1 1 1

Step1 4.36 Step 2 4.36 Step 3 4.36

+2.89 +2.89 +2.89

5 25 7.25

Line up the decimal Add the tenths. Add the ones.

points. Add the Place a decimal point in the sum in

hundredths. line with those decimals above it.

Example 2: Add: 4.3 + 12.75 + 0.093

1 1

Step 1 4.3 Step 2 4.300 Step 3 4.300

12.75 12.750 12.750

+ 0.093 + 0.093 + 0.093

17.143

Line up the decimal points. Place zeros in the Add the the thousandths.

missing digits for Add the hundredths and ones.

place holders. Place a decimal point in the sum in

line with those decimals above it.

Subtracting Decimals

Example 1: 4.72 – 2.65

4.72 Line up the decimal points.

- 2.65 Subtract hundredths, tenths, and ones.

2.07 Place a decimal point in the answer in line with those

decimals above it.

Example 2: 7.6 – 4.362

7.600 Line up decimal points.

- 4.362 Place zeros in the missing digits for place holders.

3.238 Subtract.

Multiplying Decimals

Example 1: 0.90 x 1.2

0.90 ← 2 decimal places To multiply decimals, multiply as with

x 1.2 ← 1 decimal place whole numbers. The number of decimal

180 places in the product is the total number

0900 of decimal places in the factors.

1.080 ← 3 decimal places

Example 2: .32 x .004

.32 ← 2 decimal places When you multiply decimals, sometimes

x .004 ← 3 decimal places you need to write one or more zeros in

128 the product to equal the total number of

0000 decimal places in the factors.

.00128 ← 5 decimal places

Dividing decimals

Example 1: 2.48 ÷ 2

1.24

2 Rewrite 2.48 ÷ 2 as shown. Place the

- 2 decimal point in the quotient directly

04 above the decimal point in the dividend.

-4 Divide as with whole numbers.

08

-8

0

Add or subtract the following decimals.

1. 4.2 + 3.7 = 2. 0.09 + 3.6 = 3. 0.72 + 3.921 + 7.5 =

4. 7.41 – 5.63 = 5. 9.4 – 7.25 = 6. 17.365 – 12.19 =

Multiply or divide the following decimals.

7. 1.75 x 2.6 = 8. 0.53 x 0.008 = 9. 2.31 x 0.002 =

10. 45.6 ÷ 8 = 11. 1.3174 ÷ 2 = 12. 2826.6 ÷ 42 =

Arithmetic with Fractions

Adding and Subtracting Fractions

Example 1: +

If the denominators are the same, then add the numerators and keep the same denominator. Then simplify the answer & reduce, if possible.

+ = =

Example 2: –

The denominators are not the same. You must rename the

fractions to have the same denominator by finding the least

common denominator.

Step 1 Step 2 Step 3

= = =

– = – = – =

=

Find the least common Rename the fractions. Subtract the

denominator. numerators.

Multiplying Fractions

Example 3: x

Step 1 Multiply the numerators. x =

Step 2 Multiply the denominators. x =

Example 2: 5 x

Step 1 Rename the whole number x = = 3

using a fraction. Then follow

the example above. Finally

simplify answer by writing

it as a mixed numeral.

Dividing Fractions

To divide fractions, multiply the first fraction (dividend) by the reciprocal of the second fraction (divisor).

Example 1: 2/3 ÷ 4/5 Example 2: 2/3 ÷ 4

x = = x = =

Perform the indicated operation with the following fractions. Don’t forget to simplify (reduce) your final answer to lowest terms!

13. – = 14. + = 15. + =

16. – = 17. – = 18. x =

19. x = 20. 6 x = 21. x 30 =

22. ÷ = 23. ÷ = 24. ÷ 2 =

Evaluating Expressions

Example

Evaluate the following expression when x = 5

Rewrite the expression substituting 5 for the x and simplify.

a.  5x = 5(5)= 25

b.  -2x = -2(5) = -10

c.  x + 25 = 5 + 25 = 30

d.  5x - 15 = 5(5) – 15 = 25 – 15 = 10

e.  3x + 4 = 3(5) + 4 = 19

Evaluate each expression given that: x = 5 y = -4 z = 6

1. 3x 5. y + 4

2. 2x2 6. 5z – 6

3. 3x2 + y 7. xy + z

4. 2 (x + z) – y 8. 2x + 3y – z

9. 5x – (y + 2z) 13. 5z + (y – x)

10. 14. 2x2 + 3

11. x2 + y2 + z2 15. 4x + 2y – z

12. 2x( y + z) 16.

Graphing

Points in a plane are named using 2 numbers, called a coordinate pair. The first number is called the x-coordinate. The x-coordinate is positive if the point is to the right of the origin and negative if the point is to the left of the origin. The second number is called the y-coordinate. The y-coordinate is positive if the point is above the origin and negative if the point is below the origin.

The x-y plane is divided into 4 quadrants (4 sections) as described below.

All points in Quadrant 1 has a positive x-coordinate and a positive y-coordinate (+ x, + y).

All points in Quadrant 2 has a negative x-coordinate and a positive y-coordinate (- x, + y).

All points in Quadrant 3 has a negative x-coordinate and a negative y-coordinate (- x, - y).

All points in Quadrant 4 has a positive x-coordinate and a negative y-coordinate (+ x, - y).

Plot each point on the graph below. Remember, coordinate pairs are labeled (x, y). Label each point on the graph with the letter given.

1. A(3, 4) 2. B(4, 0) 3. C(-4, 2) 4. D(-3, -1) 5. E(0, 7)

Example: F(-6, 2)

Determine the coordinates for each point below:

Example. ( 2 , 3 ) 6. (____, ____) 7. (____, ____)

8. (____, ____) 9. (____, ____) 10. (____, ____)

11. (____, ____) 12. (____, ____) 13. (____, ____)

Complete the following tables. Then graph the data on the grid provided.

Example: y = -2x - 3

X / Y
-3 / 3
-2 / 1
-1 / -1
0 / -3

14. y = x + 2

X / Y
0
1
2

15. y = 2x

X / Y
0
1
2
3

16. y = -x

X / Y
-3
-1
1
3

17. y = 2x - 3

18. y = x + 1

19.

20.

Order of Operations

To avoid having different results for the same problem, mathematicians have agreed on an order of operations when simplifying expressions that contain multiple operations.

1. Perform any operation(s) inside grouping symbols. (Parentheses, brackets

above or below a fraction bar)

2. Simplify any term with exponents.

3. Multiply and divide in order from left to right.

4. Add and subtract in order from left to right.

One easy way to remember the order of operations process is to remember the acronym PEMDAS or the old saying, “Please Excuse My Dear Aunt Sally.”

P - Perform operations in grouping symbols

E - Simplify exponents

M - Perform multiplication and division in order from left to right

D

A - Perform addition and subtraction in order from left to right

S

Example 1 Example 2

2 – 32 + (6 + 3 x 2) -7 + 4 + (23 – 8 ÷ -4)

2 – 32 + (6 + 6) -7 + 4 + ( 8 – 8 ÷ -4)

2 – 32 + 12 -7 + 4 + ( 8 - -2)

2 – 9 + 12 -7 + 4 + 10

-7 + 12 -3 + 10

= 5 = 7

Order of Operations

Evaluate each expression. Remember your order of operations process (PEMDAS).

1. 6 + 4 – 2 ∙ 3 = 4. (-2) ∙ 3 + 5 – 7 =

2. 15 ÷ 3 ∙ 5 – 4 = 5. 29 – 3 ∙ 9 + 4 =

3. 20 – 7 ∙ 4 = 6. 4 ∙ 9 – 9 + 7 =

7. 50 – (17 + 8) = 16. (12 – 4) ÷ 8 =

8. 12 ∙ 5 + 6 ÷ 6 = 17. 18 – 42 + 7 =

9. 3(2 + 7) – 9 ∙ 7 = 18. 3 + 8 ∙ 22 – 4 =

10. 16 ÷ 2 ∙ 5 ∙ 3 ÷ 6 = 19. 12 ÷ 3 - 6 ∙ 2 – 8 ÷ 4 =

11. 10 ∙ (3 – 62) + 8 ÷ 2 = 20. 6.9 – 3.2 ∙ (10 ÷ 5) =

12. 32 ÷ [16 ÷ (8 ÷ 2)] = 21. [10 + (2 ∙ 8)] ÷ 2 =

13. 180 ÷ [2 + (12 ÷ 3)] = 22. ¼(3 ∙ 8) + 2 ∙ (-12) =

14. 5 + [30 – (8 – 1)2] = 23. 3[10 – (27 ÷ 9)] =

11 - 22 4 – 7

15. 5(14 – 39 ÷ 3) + 4 ∙ 1/4 = 24. [8 ∙ 2 – (3 + 9)] + [8 – 2 ∙ 3] =

Operations with Signed Numbers

Adding and Subtracting Signed Numbers

Adding Signed Numbers

Like Signs / Different Signs
Add the numbers & carry the sign / Subtract the numbers & carry the sign of the larger number
( + ) + ( + ) = + ( +3 ) + ( +4 ) = +7 / ( + ) + (– ) = ? ( +3 ) + (–2 ) = +1
( – ) + (– ) = – (– 2 ) + (– 3 ) = ( – 5 ) / ( – ) + ( + ) = ? ( –5 ) + ( + 3 ) = –2

Subtracting Signed Numbers

Don’t subtract! Change the problem to addition and change the sign of the second number.

Then use the addition rules.

( +9 ) – ( +12 ) = ( +9 ) + ( – 12) / ( +4 ) – (–3 ) = ( +4 ) + ( +3 )
( – 5 ) – ( +3 ) = ( – 5 ) + ( – 3 ) / ( –1 ) – (– 5 ) = ( –1 ) + (+5)

Simplify.

1.  9 + -4 = 7. 20 – - 6 =

2.  -8 + 7 = 8. 7 – 10 =

3.  -14 – 6 = 9. -6 – -7 =

4.  -30 + -9 = 10. 5 – 9 =

5. 14 – 20 = 11. -8 – 7 =

6. -2 + 11 = 12. 1 – -12 =

Multiplying And Dividing Signed Numbers

If the signs are the same, If the signs are different,

the answer is positive the answer is is negative

Like Signs / Different Signs
(+ ) ( + ) = + ( +3 ) ( +4 ) = +12 / ( + ) ( – ) = – ( +2 ) ( – 3 ) = – 6
(– ) (– ) = + ( – 5 ) ( – 3 ) = + 15 / ( – ) ( + ) = – ( –7 ) ( +1 ) = –7
(+ ) / ( + ) = + ( +3 ) / ( +4 ) = +12 / ( + ) / ( – ) = – ( +2 ) / ( – 3 ) = – 6
(+ ) / ( + ) = + ( +3 ) / ( +4 ) = +12 / ( – ) / ( + ) = – ( –7 ) / ( +1 ) = –7

Simplify.

1.  (-5)(-3) = 7. -7 =

-1

2.  -6 = 8. (3) (-4) =

2

3.  (2)(4) = 9. 8 =

-4

4.  -12 = 10. (-2)(7) =

-4

5.  (-1)(-5) = 11. -20 =

-1

6. -16 = 12. (2)(-5) =

8

Solving Equations

To solve an equation means to find the value of the variable. We solve equations by isolating the variable using opposite operations.

Example:

Solve.

3x – 2 = 10

+ 2 + 2 Isolate 3x by adding 2 to each side.