Name: ______Teacher: ______
FINAL REVIEW PACKET MATH 7A
Unit 1: Rational Numbers
1) Name the largest negative integer. ______
2) Name the smallest positive integer. ______
Absolute Valuemeasures the distance a number is from zero on the number line.
The symbol for absolute value is “| |.”
Evaluate.
3) |102| - |-2|4) |102 --2| 5) -|10| - |-2|6) |-36 - 4|
7) -12 -208) -18 + 30 9) 0 --14 10) -21 - (-14)
11) -9(-11) 12) (-15)(7)13) 14) -90 ÷-15
15) -(32)16) -3217) -(-32) 18) (-3)2
Write a number sentence and evaluate.
19) A dolphin swam to a depth of 110 feet below sea level. Then, it rose 85 feet. What was
the dolphin’s final depth?
20) The temperature outside was 22˚F. The wind chill made it feel like -8˚F. Find the
difference between the real temperature and the apparent temperature.
21) The temperature one morning in was –16oF. By the afternoon, the temperature had
risen 9oF. What was the temperature in the afternoon?
Evaluate each expression, using the correct order of operations.
22) 6 + 9 ÷ 3 1023) (15 - 7) 6 + 224)
Evaluate each expression if:a=3, b=6, and c=-5.
25) -a - c26) -b2 + c327) 5b – 2c28)
State whether the following answers will be zero or undefined.
29) 30) 31) 0 ÷ 2232) 22 ÷ 0
Unit 2: Expressions, Equations & Inequalities
Term – a part of an expression that is separated by a "plus" or "minus" sign.
Ex: 3x + 4y → 3x is a term & 4y is a term
Coefficient – a number in front of a variable
Ex: 4n → 4 is the coefficient and n is the variable
Constant Term –a term that has a number but no variable. Ex: 5, 7, 100, 2,000
Like Terms– terms with the EXACT same variables and EXACT same exponents
Examples: 5y and 6y5x2 and 6x210 and -2
Non-examples: 5x and 3y 2x and 3-4x and 3x2
List the terms, like terms, coefficient(s), and constant(s) for the following expressions.
Remember, the sign in front of the number goes with the number.
1) 5x + 2y – x + 3y – 72)-4a – 10c + 8 – 2a + 7
Terms: ____, ____, ____, ____, ____Terms: ____, ____, ____, ____, ____
Like Terms: ____ and ____; ____ and ____Like Terms: ____ and ____; ____ and ____
Coefficient(s): ____, ____, ____, ____Coefficient(s): ____, ____, ____
Constant(s): ____Constant(s): ____, ____
Distributive Property states: a(b + c) = ab + ac or a(b - c) = ab - ac
Two steps in simplifying an expression:
Step 1: Get rid of parenthesis by using the Distributive Property.
Step 2: Combine like terms.
Simplify each expression.
3) -7(3 + 4x) + 2(4 + 5x)4) 10 - 6(3x + 2) + 9x
5) (-19x + 24) + (9x - 13)6) (12x - 17) - (-7x + 9)
7) (10x - 20) - 19x - 58) 0.5(-30x - 24y) + 34 - 16
9) x + 5 + x – 710) x - y - x - y
Factoring
The first step to factoring is to find the GCF of the terms:
The second step to factoring is to factor out the GCF.
- First write the GCF, then begin your parenthesis.
- To figure out what goes inside the parenthesis, divide each term by the GCF
- Remember the final answer will look like the distributive property.
Example:Factor the expression 10x + 20
Step 1: Find the GCF
Factors of:10: 1, 2, 5, 10
20: 1, 2, 4, 5, 10, 20
Step 2: Factor
10 ( x + 2)
Find the Greatest Common Factor (GCF) of each pair of terms.
11) 25xand 30y12) 3xand 21xy13) 4y and 1614) 12y and 28xy
Factor each expression. Remember, when you factor you are dividing each term by the GCF. Your final answer should look like the Distributive Property.
15) x – xy16) –15m + 5017) 18n + 2418) 21xy – 28y
Simplify each expression, THEN factor (write it as a product of two factors)
19) 8x + 14 – 2x + 420) 6x + 15y + 12y + 3x21) 8x– 2(3x– 4) + 2
LAWS OF EXPONENTS
(Remember, these shortcuts only work if the bases are the same (x is the base)
Multiplication of Exponents: x2 x5 = x2+5 = x75 54 = 51+4 = 55
If the bases are the same: KEEP the base and ADD the exponents.
Division of Exponents: = 89-5= 84= x2-5 = x-3 = (remember no negative exponents)
If the bases are the same: KEEP the base and SUBTRACT the exponents.
Power to a Power: (x2)5 = x(2)(5) = x10
A power raised to another power: KEEP the base and MULTIPLY the exponents.
Write each expression using exponents.
22) 2 2 2 223) s s s s s s s24) a a b a b a a
Simplify using the Laws of Exponents. Express your answer using POSITIVE EXPONENTS.
Ex. -4x3(7x5) = -4 x3 7 x5 = -4 7 x3x5 = -28x8
25) 3-226) 4-3 4227) x-5 x-3
28) 27 2229) 42 4430) 102 103
31) k8 k32) a4c6(a2c)33) 2w2x 5w3x4
34) 3x3 7x335) 4y4(-4y3)36) (-6x7)(5x2)
37) 7y3 6y38) (-2w7z4)(-8w3 z2)39) (-3x2y3)(2xy4)
40) (a2c)441) (x3y4)242) (m2n3)3
43) (2xy)444) (-3x4y7)345) (10xy5)2
46) 47) 48) 49)
50) 51) 52) 53)
Scientific Notation
Scientific Notation is when we rewrite a number as a PRODUCT of 2 factors.
Factor #1:
Must be greater than or equal to 1 AND less than 10.
Factor #2:
Must be a power of 10 . Numbers greater than 1 have positive exponents, numbers less than one have negative exponents.
Ex: Write 24,000 in scientific notation.EX: Write 0.00045 in scientific notation
Scientific notation is 2.4 x 104 Scientific notation is 4.5 x 10-4
Standard Form:
Remember the exponent tells you: How many places to move the decimal.
Positive exponents are numbers greater than or equal to 1 .
Negative exponents are small numbers, numbers less than 1, decimals.
Ex:Write 2.03 106 in standard form.
The exponent is a positive 6 so you move the decimal 6 places to the right.
Standard form is 2,030,000
Ex: Write 3.2 10-8 in standard form.
The exponent is a negative 8 so you move the decimal 8 places to the left.
Standard form is0.000000032
Write each number in scientific notation.
54) 6,59055) 4,733,80056) 2,204,000,000
57) 0.2958) 0.0000057159) 0.0008331
Write each number in standard form.
60) 6.7 10161) 6.1 10462) 1.6 103
63) 2.91 10-564) 8.651 10-765) 3.35 10-1
Compare using or .
66) 3.7 107 _____ 8.5 10467) 7.5 103 _____ 9.42 103
68) 9.5 10-6 _____ 3.7 10-269) 9.75 10-4 _____ 3.5 10-6
Find the product. Write your answer in scientific notation.
70) (2 106) (3 10-4)71) (4 x 106)(2 x 103)
Find the quotient. Write your answer in scientific notation.
72) ( 8.5 x 104) ÷ (1.7 x 102)73) ( 4.4 105)
(4 10-7)
The Real Number System
ALL the numbers we worked with this year are REAL NUMBERS. That means every number we worked with was either RATIONAL or IRRATIONAL.
RATIONAL Numbers are numbers that CAN be written as fractions.
COUNTING NUMBERS 1, 2, 3… also known as NATURAL NUMBERS
WHOLE NUMBERS 0, 1, 2, 3…
INTEGERS …, -5,-4,-3,-2,-1, 0, 1, 2, 3…
FRACTIONS ALREADY A FRACTION!!
TERMINATING DECIMALS 0.13
REPEATING DECIMALS 0.333…
PERFECT SQUARES 7
IRRATIONAL Numbers are numbers that CANNOT be written as fractions.
PI 3.1415926…
NON-PERFECT SQUARES
NON-TERMINATING NON-REPEATING DECIMALS 0.12112111211112…
WHAT ARE PERFECT SQUARES? A number is a perfect square if its square root is a whole number. That is, the number is equal to a number times itself.
FOR EXAMPLE: 25 = 5 5 AND 25 = -5 -5therefore, 25 IS A PERFECT SQUARE.
Name ALL the sets of numbers to which each number belongs.
Real, Irrational, Rational, Integer, Whole, Counting/Natural
74) -8______, ______, ______
75) ______, ______, ______, ______, ______
76) ______, ______
77) 7. ______, ______
78) ______, ______
79) -______, ______, ______
80) 0.25 ______, ______
81) ______, ______
82) 5______, ______, ______, ______, ______
Answer each of the following with ALWAYS, SOMETIMES or NEVER true.
83) Integers are______rational numbers.
84) Real Numbers are ______irrational numbers.
85) Whole numbers are ______integers.
86) Rational numbers are ______irrational numbers.
87) List the first 15 Perfect Squares.
___, ___, ____, ____, ____, ____, ____, ____, ____, ____, ____, ____, ____, ____, ____
The opposite of squaring a number is finding a square root. A square root of a number is one of its two equal factors. EX. Since 3 3 = 9, a square root of 9 is 3.
Since -3 -3 = 9, a square root of 9 is -3.
Remember there are positive roots and negative roots. Be sure you know which root you are looking for. When solving for a variable there will ALWAYS be 2 solutions. Read carefully to see if you need to reject the negative root.
indicates thepositive, or principal square root of 64. Therefore, = 8.
- indicates the negativesquare root of 121. Therefore, - = -11.
± indicates BOTHpositiveand negative square roots of 225. Therefore, ± = ±15
Find each square root.
88) = _____89) ±= _____90) -= _____
Find the 2 consecutive whole numbers each non-perfect square is between.
91) is between _____ and _____92) is between _____ and _____.
93) is between _____ and _____94) is between _____ and _____
Solve for x:
95) x2 = 4996) x2 + 10 = 9197) -5 + 2x2 = 45
Evaluate each cube root. (What number times itself three times makes the inside number?)
98) 99) 100)
Solving Equations
Step 1: Get rid of any parenthesis by using the Distributive Property.
Step 2: Combine Like-Terms on the same side of the equal sign.
(Same Side Use Same Operation)
Ex. -5x + 2x + 12 = -10x +16 + 17
-3x +12 = -10x + 33
Step 3: Get all variables on one side & constants on the other side.
(Opposite Sides Use Opposite Operations)
Ex. -3x + 12 = -10x + 33
+10x = +10x
7x + 12 = 33
-12 = -12
7x = 21
Step 4: Solve for the Variable
Ex. 7x = 21
7 7
x = 3
Solve the following equations. Show all work ALGEBRAICALLY!
101) 3x + 4 = 7102) -4 = x – 2103) -5 = x– 14
104) 19x – 12 = 24x – 22105) 4x + 2x + 2 = 26106) 9x + 1 – 7x = –17
Solve and check:
107) -4(x + 5) = 35 + 25108) 12x + 8 - 4x = 2(x + 16)
Solve each literal equation for x.
109) 3x – q = 5q110) 9x - 24a = 6a + 4x
111) r = 5(x + 2y)112) y = 2x + z
Solve Algebraically. Don’t forget your let statement.
113) It costs $7.50 to enter a petting zoo. Each cup of food to feed the animals is $2.50. If
you have $12.50 to spend, how many cups of food can you buy?
114) You are selling chocolates that you have made for $3 each. You spent $45 on materials.
How many chocolates must you sell to make a profit of $105?
115) You want to buy a bicycle that costs $280. Your parents agree to pay $100 and you have
to pay for the rest. You can save $20 a week. How many weeks will it take you to save
enough money?
INEQUALITIES
“Greater Than” “Greater Than or Equal To”
“Less Than” “Less Than or Equal To” “Not Equal To”
True or False:
4 4 False 4 is not greater than 4
4 4 True 4 is not greater than 4, however, 4 is equal to 4.
GRAPHING INEQUALITIES
You can only graph an inequality on a number line if the VARIABLE is BY ITSELF.
You solve inequalities the same way you solve equations. Remember, whatever you do to one side of the inequality you must do the same thing to the other side.
*When you multiply/divide both sides of the inequality by a negative number you need to FLIP the sign.*
Use an Open Circle “ “ for “Greater than” or “Less than”
Use a Closed Circle “ “for “Greater than or equal to” or “Less than or equal to”
Solve and graph on a number line.
116) 0.5(x + 12) 9 117) -2x + 7 17
118) -10x – 2 8x -20119) –8x + 12x – 3 21
When translating. . .
●Identify the key words
●Translated in the exact order they are read
●Switch the order ONLY when you read: “less than”, “more than”, “fewer than” , “subtracted from” and “taken away from”
●Place parentheses around sums and differences
Translate the following statements into INEQUALITIES.
120) Four less than twice a number, x is greater than 17. ______
121) Three more than 5 times a number, n is no less than 38.______
122) The quotient of a number, x and 2 increased by 7 is at most 40.______
123) The elevators, e, in an office building have been approved for no more than 3600 pounds.
______
124) An assignment, a, requires at least 45 minutes. ______
125) While shopping, Abby, a, can spend at most $50. ______
Unit 3: Ratios & Proportionality
We use proportions to solve for missing information, to solve word problems, conversion problems, scale problems and so much more. The key to using proportions correctly is to
BE CONSISTENT!!!!
Express each rate as a unit rate.
1) 6 pounds lost in 12 weeks ______2) $800 for 40 tickets ______
Solve.
3) 4)
5) The distance between two cities on a map is 3.2cm. If the scale on the map is
1cm=50km,find the actual distance between the two cities.
6) A 14-ounce energy drink contains 10 teaspoons of sugar. How much sugar is on one
ounce of the drink?
7) In 4 minutes Benny swims 6 laps. What is this rate in minutes per lap?
8) Find the UNIT PRICE to tell which is the better buy. Show all work and explain.
20 pounds of pet food for $14.99 OR 50 pounds of pet food for $37.99
9) Find the UNIT PRICE to tell which is the better buy. Show all work and explain.
2 DVDs for $26.50 OR 3 DVDs for $40.00
10)
PROPORTIONAL RELATIONSHIPS
What is a proportional relationship? A Proportional relationship can be recognized in a table, on a graph, or by using an equation.
11) You can buy 4 tickets for $16. This is a proportional relationship.
#of tickets / 1 / 2 / 4 / 6 / 8Cost ($) / 8 / 16
12) Fill in the missing values in the table above.
13) What is the constant of proportionality in the table? ______
14) Write an equation that relates the cost, c, to the number of tickets, t. ______
15) Graph the data on the grid below:
Remember your LABELS!!!
Use the following graph about the number of cups to the number of brownies to answer the following questions.
16) Does the graph represent a proportional or non-proportional relationship? Explain.
______
______
______
17) What is the constant of proportionality? ______
18) How many brownies will 4 cups make? ______
19) If b represents the number of brownies and c represents the number of cups write an
equation to show the relationship between the number of brownies and the number of cups
used.
20) Using the above equation, how many brownies will 18 cups make? ______
21) You can buy 3 apps on your iPhone for $2.97 or 5 apps for $4.95. Write an equation to
show the relationship between the number of apps, a and the total cost, c.
22) If 5 out of 21 people have green eyes. How many people in a room of 512 would you expect
to have green eyes?
23) A car company found that 15 out of every 1230 cars have a problem with their headlights.
If a dealership bought 375 cars from the car company, how many cars should they expect
to have headlight problems?
Each of the following isa pair of similar figures. Solve for the missing side.
24) 25)
1
PERCENTS
= or =
Percent Increase or DecreasePercent Error
amount of increase/decrease ramount of error r
original amount 100 actual amount 100
Simple Interest: Interest=PrincipalxRatexTime (I=PRT)
- I = Interest, The ($) amount of interest that is owed or earned.
- P = Principal, the amount of money that was borrow, saved or invested.
- R = Rate, the percent of interest.*Make sure to convert your % to a decimal
- T = Time, time is always in years.
Complete the following chart.
Fraction / Decimal / Percent/ 26) / 27)
28) / 2.15 / 29)
30) / 31) / 4%
/ 32) / 33)
34) / 0.02 / 35)
36) / 37) / 28%
Solve Algebraically. One way to solve each of the following word problems is using the percent proportion. There are quicker methods, but beware, they also require a higher level of understanding.
38) Mary sold $192 worth of greeting cards. If she received 25% commission on her sales,
how much commission did she earn?
39) Jenny bought a pair of boots priced at $85. If the boots were on sale for 15% off the
regular price, how much did Jenny pay for the boots?
40) The regular price of a bicycle is $99.50. If sales tax is 7.5%, how much is the bicycle
including sales tax?
41) There are 350 people at a luncheon. If 12% of the people will win a door prize, how many
door people will win a door prize?
42) Jen’s bill at a restaurant before tax and tip is $22. If tax is 5.25% and she wants to
leave 15% of the bill including the tax for a tip, how much will she spend in total?
43) A $300 mountain bike is discounted by 30%, and there is a 8% sales tax. Find the final
cost of the mountain bike.
44) Michael borrowed $1750 from his brother to buy a computer. He agreed to repay the
money in 2½ years at 8.75% interest. How much interest will he pay? What is the total
amount of money Michael will have to repay his brother?
45) Michelle started a bank account that earns 12.25% interest. After 1½ years, she earned
$147 in interest. How much money did Michelle start her bank account with?
46) When John got his puppy, she weighed 8 pounds. Now that she is one year old, her weight
is 60 pounds. What is the percent increase in the puppy’s weight?
47) Irene thinks she has the space for a 45-inch-wide bookcase. It turns out that she only
has space for a 40-inch-wide bookcase. What is the percent error in Irene’s
measurement?
48) The planners of a school carnival estimate that they will sell 500 hot dogs. They only sell
400. What is the percent error in their estimate?
49) If the original price of a sweater was $75.99. The sweater is now on sale for $62.50,
what was the percent decrease in the price of the sweater? (Round to the nearest 0.1%)
Unit 4: Statistical Analysis & Probability
PROBABILITY
There are 2 kinds of events: Independent Events and Dependent Events.
Independent Events – 1st event does not affect the 2nd event
Dependent Events – 1stevent does affect the 2nd event
Probability is a ratio, a comparison of 2 numbers.
EXPERIMENTAL PROBABILITY =
A box contains 5 green pens, 3 blue pens, 8 black pens, and 4 red pens. A pen is picked at random. Answer questions 1-9 using the above information.
1) P(blue or red) ______2) P(gold) ______3) P(green) ____
4) P(green, blue, black, or red)______
5) P(blue andgreen)with replacement ______
6) P(green and red) with replacement ______
7) P(green and red) without replacement ______
8) P(blue, black, and green) without replacement ______
9) P(black and blue) with replacement ______
10) A spinner has four sections: a tree, a flower, a cloud and a sun. Jordan spins the pointer
on the spinner 20 times. The pointer lands on the tree 3 times, the flower 9 times, the
cloud 2 times and the sun 6 times.
A) Based on the experiment, what is the probability the spinner lands on the sun?
B) If Jordan spins the spinner 150 times, how many times should she expect it to land on
the sun?
11)
12) Whitney has a choice of a floral, plaid, or striped blouse to wear with a choice of tan,
black, navy, or white skirt. How many different outfits can she make?
13) You flip 3 coins. How many possible outcomes are there?
STATISTICS
14) Find the MEAN, MEDIAN, MODE, and RANGE of the following set of data: Show work.
15, 12, 21, 18, 25, 11, 17, 19, 20
15) Find the median, upper and lower quartiles and the extremes, and then create a box plot
to display the following data set. Remember your labels!!
1, 3, 5, 4, 3, 6, 2, 7, 6, 4
16) The double box plot shows the costs of MP3 players at two different stores.
On average which department store charges more for an MP3 player?
The following box whiskers diagrams represents how many minutes it take a cleaning service to clean the same house. Answer questions 17 – 20below based on the data from the diagram.
17) On average, what service appears to clean the house faster? ______explain why?
______
18) Which service would you say is more consistent?______explain why? ______
______
19) What percent of the time did service B finish under 41 mins? ______
20) Whatpercentof the time did service A clean at least 34 mins? ______
Unit 5: Geometry
ANGLE RELATIONSHIPS
Complementary angles – Two angles are complementary if the sum of their angle measures is 90
Supplementary angles –Two angles are supplementary if the sum of their angle measures is 180
Vertical Angles – congruent angles formed by 2 intersecting lines. They are opposite each other
Alternate interior angles – interior angles on opposite sides of the transversal.( if lines are parallel)
Alternate exterior angles – exterior angles on opposite sides of the transversal. ( if lines are parallel)
Corresponding angles – hold the same position on 2 different lines. (congruent if lines are parallel)
21) Find the complement of a 40 angle. ______
22) Find the supplement of a 55 angle. ______
23) Two angles are vertical angles. If one of the angles measure 135, what is the
measure of the other angle?
Solve for x ALGEGBRAICALLY, and then find the measure of each angle.
24)25)26)
Use the diagram below and the given information to answer questions 27 – 36.