MATH WORKBOOK · You can do this because dividing by the same number on the top and bottom is the...

MATH WORKBOOK

Prep@Pingree Math Workbook 2

TABLE OF CONTENTS: Introduction 3 Week 1: Basics Fractions 4 Decimals 5 Percents 6 Homework Set #1 7 Week 2: More Basics Signed Numbers 8 Order of Operations 9 Homework Set #2 10 Week 3: Algebra One-Step Equations 12 Combining Like Terms 13 Distributing 13 Advanced Equations 14 Homework Set #3 15 Week #4: Advanced Algebra Square Roots 16 Exponent Rules 17 Proportions 19 Quadratic Equations 19 Homework Set #4 20 Week #5: Geometry Shapes: Area & Perimeter 21 Angles and Their Measure 22 Homework Set #5 23 Additional Practice Problems 24 Definitions and Rules to Know 27 Useful Fraction-Decimal-Percent Equivalents to “Just Know”

29

Prep@Pingree Math Workbook Revised, March 2017, H. Duren


INTRODUCTION:

MATH IS NOT A SPECTATOR SPORT! Math is a hands-on, trial and error, heads up, tackle it kind of subject. In order to truly excel, you need to actively engage in the learning process. In this class, you will be expected to practice active learning by taking notes, asking questions, participating willingly, doing practice problems in class and at home, and showing off your skills on periodic "learning opportunities" to assess your knowledge. This summer, we challenge you to keep an open mind and to practice your active study strategies daily. Taking on math as a full-contact sport may hurt occasionally, and you may hit a few roadblocks along the way, but the final outcome is always a win.


WEEK 1: BASICS

Introduction: Most people do not eat a WHOLE pizza. Nor do they grow in full inch increments. Little kids don't measure age in full years (They're not "five" they're "five and a half!"). We all have an ability to recognize parts of a whole. In fact, chances are you may have even learned how to manipulate these parts of a whole in a math class before. In this class, I hope that you will not just learn but master how to handle operations (addition, subtraction, multiplication, division) with these parts of a whole, represented mathematically as fractions, decimals and percents. Some RULES to live by: A. FRACTIONS –

1. To SIMPLIFY (or REDUCE) a fraction, divide the both the numerator (top) and denominator (bottom) by a number that divides evenly into each part. You can do this because dividing by the same number on the top and bottom is the same as dividing by ____!

E.g. To simplify 2114 , divide both parts by 7. So,

32

2114

=

2. To MULTIPLY fractions, simply multiply straight across OR _________________!

E.g. 114

5520

1110

52

==• OR =•1110

52

3. To DIVIDE fractions, remember to "Keep-Change-Flip!" That is, keep the first

fraction the same, change the operation to multiplication, and flip the second fraction. You can do this because division, by definition, is multiplying by the reciprocal!

E.g. 25

615

13

65

31

65

==•=÷

Key Word: A "reciprocal" is not simply "the flipped fraction". Rather, a reciprocal is a number such that, when multiplied by its own reciprocal, the product is 1.

E.g.: 52

and 25

are "reciprocals" BECAUSE 11010

25

52

==× !

*CHALLENGE: Explain why 0 has no reciprocal.


4. To ADD or SUBTRACT fractions, first you must find a common denominator (a number that is large enough so that all denominators are evenly divisible into the new common denominator). Then, you must write each fraction in terms of that common denominator. Finally, add or subtract the numerators (keep the common denominator) and reduce if possible. NEVER ADD THE DENOMINATORS!

E.g. 4023

4015

408

83

51

=+=+

IMPORTANT NOTE: For ALL cases, you should change any mixed number to an improper fraction before using the rules above. Also, it is perfectly acceptable to leave your answer in "improper fraction" form! (Your high school teachers will strongly prefer this over mixed numbers.) Other KEYS to Fractions: a. Anything divided by itself equals 1.

122= and 1

145145

= and 1=xx and 1=

KaisyKaisy

b. Zero divided by anything equals 0.

030= and 0

234,110

= and 00=

xyz

c. Anything divided by zero is "undefined".

=06

undefined and =012x

undefined and =0

Mathundefined

B. DECIMALS –

1. To ADD or SUBTRACT decimals, you must line up the decimal points vertically. Then, add or subtract normally and drop the decimal point down in its original place.

E.g. _____7.334.2 =+ 2. To MULTIPLY decimals, multiply the numbers as if there were no decimals.

Then, count the number of decimal places you need in the answer by adding the number of decimal places in each of the two factors.

E.g. _____1.267.1 =•


C. PERCENTS – The word "percent" literally means "per 100" and is a representation of a fraction written

over a denominator of 100. So, 5% means 1005 (or

201 when completely reduced).

To convert a PERCENT to a DECIMAL: Move the decimal point two places to the left. (You can do this because moving the decimal is the same as dividing the number by 100! “Per-cent!”) E.g. 25% = 0.25 To convert a DECIMAL to a PERCENT: Move the decimal point two places to the right. E.g. 0.045 = 4.5%

To calculate Percent Change, use the following formula:

𝑛𝑒𝑤 𝑎𝑚𝑜𝑢𝑛𝑡 − 𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝑎𝑚𝑜𝑢𝑛𝑡𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝑎𝑚𝑜𝑢𝑛𝑡 ×100%

For example, if a baby who weighs 8lb at birth weighs 10lbs at 1 month of age, his percent weight increase would be:

10𝑙𝑏 − 8𝑙𝑏8𝑙𝑏 ×100% =

28×100% =

14×100% = 0.25×100% = 25%

Now, Try these:

Simplify:

a.) 125

32+ b.) 1

5÷275

c.) 9.088.1 − d.) 2.025.3 ×

Answers: a.) !"

!" , b.) !"

! , c.) 0.98 , d.) 0.65


Prep@Pingree Name: __________________________

Homework Set #1 1. Simplify to lowest terms:

a.) 204 b.)

4236 c.)

108090

2. Change to improper fractions:

a.) 311 b.)

2033 c.)

13122

3. Simplify and reduce all answers to lowest terms:

a.) 52

31+ b.)

321

43+ c.)

152

207−

d.) 6.18203.22 + e.) 25.8037.1306.0 ++ f.) 237.065712.25.63288.373

+

++

g.) 05.4702.42 − h.) 087.0876.0 − i.) 77.98.23 − j.) 8.21•3.7 k.) 87.98( ) 10.71( ) l.) 4.43.228 ×

m.) 75

43⋅ n.)

34

433 • o.)

91428

73

••

p.) ⎟⎠

⎞⎜⎝

⎛⎟⎠

⎞⎜⎝

⎛516

41

q.) 314

413 ⋅ r.)

4122

21×

s.) 151

31÷ t.)

312

415 ÷ u.)

2891

432 ÷

4. Write a decimal, fraction and percent representation for each of the following:

a.) "three quarters" b.) "one fourth" c.) "two tenths" 5. You notice that the price of a gallon of milk at Market Basket has recently changed from $5 to $5.75. By what percent has Market Basket increased their price? CHALLENGE ZONE!

a.) Summer

PPFunMath @

+ b.) yxy11121

÷


WEEK 2: MORE BASICS

Introduction: If not in a math class, you've likely already seen cases of negative numbers in every day life. For example, on a chilly winter day, you may note the temperature is -4°F. In some games, like golf, it is good to have a negative score. One place you do NOT want to see a negative number is in your bank account! On a number line, like the one below, numbers to the left of zero are negative where numbers to the right of zero are positive. Zero is neither positive nor negative!

Key Definition! The “Absolute Value” of a number is the number’s distance from zero on a number line. That is, the absolute value of -3 is 3 because -3 is a distance of 3 places from zero on a number line. The symbol for absolute value is the number between two vertical lines.

e.g.: −3 = 3 Some RULES to live by:

1. To ADD numbers with the SAME SIGN (both positive or both negative), add the absolute value of each number, then keep the common sign.

E.g. 853 −=−+− and 6.44.32.1 =+

To ADD numbers with OPPOSITE SIGNS (one positive, one negative), subtract the number with the smaller absolute value from the number with the larger absolute value and keep the sign of the number with the larger absolute value.

E.g. 3 5 2− + = + and 1 3 4 9 53 4 12 12 12

⎛ ⎞ ⎛ ⎞+ − = + − = −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

Note: Subtraction, by definition, is the same thing as adding a negative (or adding the opposite) number. So, 45 − is the same thing as saying )4(5 −+ . And −3 — 5 is the same thing as saying −3 + (−5). Use this definition first, then apply the rules above to subtract signed numbers!

Some past P@P students have shown us a trick to remember this! It is: “L.A., U.S.” (Like signs, Add! Unlike signs, Subtract!)


2. When MULTIPLYING or DIVIDING numbers with the SAME SIGN, the

product/quotient is always POSITIVE.

E.g. 48412 +=−×−

When MULTIPLYING or DIVIDING numbers with OPPOSITE SIGNS, the product/quotient is always NEGATIVE.

E.g. 914

37

32

73

32

−=•−=÷−

3. When performing multiple operations in one expression, you must follow a

certain ORDER OF OPERATIONS (AKA: GEMS!)

The ORDER OF OPERATIONS is:

G ________________ E ________________ M _______________ S ________________ NOTE: Some of you or your friends may refer to this as PEMDAS, which is also fine, but remember: Multiplication and Division (as are Addition and Subtraction) are identically ranked in the hierarchy and are completed from LEFT to RIGHT! E.g. ( ))16(35 2 −⋅+ Grouping symbols – complete the inside first. ( )535 2 ⋅+= Exponents ( )595 ⋅+= Multiplication/Division ( )455+= Subtraction/Addition 50=

Sing to the tune of "Row-Row-Row Your Boat" "SAME SIGNS: ADD AND KEEP, DIFFERENT SIGNS: SUBTRACT!

KEEP THE SIGN OF THE LARGER NUMBER, THEN YOU'LL BE EXACT!"



Homework Set #2 Complete the following WITHOUT a calculator.

1. Place in order from least to greatest:

1,34,

53,

27,

57,2,

35

−−−

2. Simplify:

a.) )10(42 −++− b.) 63)27()130(125 +−+−+

c.) 396323 +−−+ d.) 0356.1284.0 −

e.) 1.185.1 −•− f.) 6)4(312)18( −+−

g.) 232 +− h.) ( )20)5(83 −−−

i.) 207%31 of j.) ( )23533675 −+−

k.) 817

534 −+ l.)

2891

432 ÷−

m.) 524

43+− n.)

( )26412)15(49

÷−÷+

3. Fill in the blank with <, > or = . Show work to support your conclusion.

a.) 3262_____)4(

431 +−−+

b.) 2.06.3_____85.202.0 −−+

c.) 25.681.3_____35.74851 ++−

CHALLENGE ZONE!

Replace the ? with = or ≠ to make a true statement. Show work to support your conclusion.

( )[ ] ⎭

⎬⎫

⎩⎨⎧

−+÷+−

34)624()3(8)2(252 ? [ ] ⎟

⎠

⎞⎜⎝

⎛−−

÷−+35212)630()52(


WEEK 3: ALGEBRA

Introduction: Algebra is a useful way to model situations when we don’t have all the info. Often, its study centers around one letter: x. In algebra, x, called a "variable" is used to represent any unknown quantity. Whenever a situation arises where a quantity is unknown, it is useful to write (and then solve) an equation (or two!) involving a variable (or variables) to determine the unknown(s).

BUT FIRST! Before we go too far into algebra, we should know some common but important English words that represent math. A brainstorm is begun already below. Add as many key phrases and their mathematical equivalents as you can to this list before moving on!

TRANSLATION BRAINSTORM: English à Mathematics

Now, translate the following phrases from English to Mathematics. "4 plus some number" "ten and some number is 2" "three times some number" "the quotient of thirty and some number is 2" Answers: a.) 4 + 𝑥, b.) 10 + 𝑥 = 2, c.) 3 ∙ 𝑥 𝑜𝑟 3𝑥 , d.) !"

!= 2

"is" à = "some number" à x "of" à "and" à + "per" à


When an expression involving x is used in a math sentence, that sentence is called an “equation”. We will discuss techniques for solving equations shortly… BUT FIRST! We must understand what it means when the directions say “Solve for x.” In math, the word “solve” means to find the value(s) of the variable that makes the equation true! Some RULES to live by: In order to solve algebraic equations, you will work to _______________ the variable by doing ________________ operations and _________________ the equation. A. INVERSE/OPPOSITE OPERATIONS:

To undo ADDITION of a number to x, you must SUBTRACT the same number from both sides of the equation.

E.g. 104 =+x // 4− 6=x

To undo SUBTRACTION of a number from x, you must ADD the same number to both sides of the equation.

E.g. 213 =−x // 3+ 24=x

To undo MULTIPLICATION of a number by x, you must DIVIDE both sides by the number attached to the x (or multiply by the reciprocal!).

E.g. 1233 =x // 3÷ OR 31

•

41=x

To undo DIVISION of x by a number, you must MULTIPLY both sides by the number x is divided by.

E.g. 116=

x // 6•

66=x NOTE: The answer to an exercise where directions read "Solve for x" should always be written as "x = ____".


B. COMBINING LIKE TERMS: First, some vocabulary… In the expression 23x , the 3 is called the , the x is called the base and the 2 is called the .

"LIKE TERMS" have the SAME ________________ and the SAME __________________.

E.g. 28x and 211x are "like terms" because they have the same base (x) and the same exponent (2).

To COMBINE LIKE TERMS:________________________. (Keep the variable(s) and exponent(s) the same.) E.g. 222 19118 xxx =+

C. DISTRIBUTING: Just like in English, to "distribute" means to hand out. In algebra, distributing is used to simplify expressions which cannot be simplified by Order of Operations alone. For example, the expression ( )23 +x cannot be simplified by Order of Operations since the items inside the parentheses are not "like terms". In order to simplify this expression, you must DISTRIBUTE the 3 to the terms inside the parentheses via multiplication (since that is the operation between the 3 and the 2+x ). It looks like this: ( ) 6323323 +=⋅+⋅=+ xxx NOW TRY THESE: Simplify: a.) 10623 +−+ xx b.) ( )532 −+ x c.) ( )623 +− xx d.) ( ) ( )2118 −−+ xx

Answers: a.) −3𝑥 + 8 , b.) 3𝑥 − 13 , c.) 𝑥 − 12 , d.) 7𝑥 + 90


ADVANCED EQUATIONS:

1. Two-Step Equations: To solve an equation with two steps, first "undo" the operation which is farthest from the variable. Then, solve the equation as you would a regular, one-step equation. E.g. 26143 =−x // 14+ 403 =x // 3÷

340

=x

2. Equations with Variables on Both Sides: To solve an equation with variables on both sides, first gather all terms with x on one side and all constants on the other side (combining like terms as necessary). Then, solve the equation. E.g. 7253 −=+ xx // x2− 75 −=+x // 5− 12−=x 3. Equations with Distributing: To solve an equation with distributing, distribute first. Then, solve as you would using the methods above. E.g. ( ) 12325 =− x //distribute 121510 =− x // 10− 215 =− x // 15−÷

152

−=x

4. Inequalities: To solve an inequality, solve the equation as you would using the methods above, substituting the inequality symbol (>, <, ≥, ≤) for the normal = sign… UNLESS you multiply or divide both sides by a negative quantity. In that case, you must switch the direction of the sign!

E.g. 7412 −>− x // 2−

941

−>− x //41

−÷ OR 4−• (NOTE: The Catch!)

36<x



Homework Set #3 1. Simplify:

a.) xxx 7.19.05 ++ b.) 10)2(35 +++ xx c.) )49117( x−−

d.) )7.04.0(8.23 +−+ xx e.) [ ])5(324 xx −++−

2. Solve the following equations for x:

a.) 67 =−x b.) 1518 =+x c.) 99 =−x

d.) 44 −=+− x e.) x+−= 83 f.) 734 =+x

g.) 124 −=+− x h.) 1832

=x i.) x−−= 83

j.) 103=

x k.) xx 52 −=+− l.) 16412 −=− x

m.) xx 53 = n.) 24)2(3 =−− x o.) 1213)32(5 =−− x

p.) 5432

−=−x q.) 0

235

=− x r.) 42)1(3 +=+− xxx

3. Solve the following inequalities:

a.) 31712 +−<−x b.) 83 ≤− x c.) 43 >−+− x

d.) 1734 −≥+x e.) 11435 <− x f.) 14

435 +≤ x

4. Fill in the blank, using algebra to support your conclusions.

a.) 12 is 5% of ____. b.) 130% of $60 is ____.

c.) %3266 of ____ is 104. d.) 5 is ____% of 24.

5. Jersey Problem: The starting offensive line for the varsity hockey team consists of 3 players. Each player wears a jersey with a consecutive even number. The sum of the numbers on the jerseys is 144. Let x represent the smallest of the three jersey numbers. Write an equation (in terms of x) that models this situation.

CHALLENGE ZONE!

a.) Solve hprV 2

31

= for h.

b.) Solve ( )3295

−= FC for F.


WEEK 4: ADVANCED ALGEBRA

Introduction: In the last chapter, you explored the basics of algebra as well as some more advanced applications of solving algebraic equations and simplifying algebraic expressions. In this chapter, we will continue to expand our knowledge of algebra and tackle some more advanced topics including simplifying square roots, the rules of exponentiation, and solving basic proportions and quadratic equations.

Some RULES to live by: A. SQUARE ROOTS – (AKA: Radicals)

The expression x is pronounced "the square root of x", "radical x" or simply "root x". When you see the symbol, ask yourself the question "What number times itself will give me the number inside the symbol?" For example, in expression 636 = because 3666 =• .

NOW TRY THESE: Simplify: a.) 16 b.) 121 c.) 4− d.) 4−

Answers: a.) 4 , b.) 11 , c.) 𝑛𝑜𝑛𝑟𝑒𝑎𝑙! , d.) −2


B. EXPONENT RULES – We became familiar with exponents during our study of the Order of Operations. For example, we learned that 42 means to multiply 2 by itself a total of 4 times.

So, 16222224 =⋅⋅⋅= Some vocabulary: In the expression 42 , the 2 is called the "base", the 4 is called the "exponent" and the entire expression 42 is called a "power".

Now, we must master the rules of exponents so that we may simplify complicated expressions involving multiple bases and exponents. 1. When multiplying exponents of the same base, you ADD the exponents and keep

the base the same. nmnm xxx +=⋅

E.g. 62424 xxxx ==⋅ + Now try: =⋅ 63 xx

2. When dividing exponents of the same base, you SUBTRACT the exponents and keep the base the same.

nmn

m

xxx −=

E.g. 6282

8

xxxx

== − Now try: =10

30

xx

3. When raising a power to a power, you MULTIPLY the exponents.

( ) nmnm xx ⋅= E.g. ( ) 124343 xxx == ⋅ Now try: ( ) =35x 4. When raising more than one base to a common power, you DISTRIBUTE the

exponent.

( ) nnn yxxy = OR n

nn

yx

yx

=⎟⎟⎠

⎞⎜⎜⎝

⎛

E.g. ( ) 3333 822 xxx == Now try: =⎟⎟⎠

⎞⎜⎜⎝

⎛42

yx


5. When raising a base to a negative exponent, take the reciprocal of the base and make the exponent positive.

nn

xx 1

=−

E.g. 33 1

xx =− Now try: =−14x

6. Anything raised to the power of zero equals 1.

10 =x E.g. 130 = Now try: ( ) =04x

** Challenge: Using the rules of exponentiation, PROVE that 10 =x . (Hint: You only need to use one of the rules above to do this.)

NOW TRY THESE:

Simplify:

a.) 57 xx ⋅ b.) ( )57x c.) ( )422 yx−

d.) ( )2

3

4xx− e.) ( ) ( )210352 −⋅ yxyx

Answers: a.) 𝑥!" , b.) 𝑥!" , c.) 16𝑥!𝑦! , d.) !

!"!!, e.) 𝑥!𝑦!"


C. PROPORTIONS – An equation where one fraction is set equal to another fraction is called a PROPORTION. To solve a proportion, CROSS-MULTIPLY then set the products equal and solve the resulting equation.

E.g. x3

411

= // cross-multiply Now try: 310

52=

+x

4311 ⋅=⋅ x 1211 =x // 11÷

1112

=x

D. QUADRATIC EQUATIONS –

An equation where the variable x is squared is called a QUADRATIC EQUATION. To solve a basic quadratic equation, you must first isolate the variable and its exponent. Then, undo the squaring by taking the square root of both sides. E.g. 3003 2 =x // 3÷ 1002 =x // }10,10{ −=x

Notice that both 10 and -10 are solutions to the equation 3003 2 =x . NOW TRY THESE: Solve for x:

a.) 5012 =+x b.) 52

23 x= c.)

xx 205=

Answers: a.) 𝑥 = ±7 , b.) 𝑥 = !"

! , c.) 𝑥 = ±10




a.) 144 b.) 641 c.) 400−

d.) 281 e.) 1649 f.)

2

52⎟⎠

⎞⎜⎝

⎛

g.) 81.0 h.) 36− i.) 916 +

j.) 916 + k.) 94 ⋅ l.) 94 ⋅

m.) 2)3(− n.) 23− o.) "twenty squared"

p.) 81

81⋅ q.) 72 xx ⋅ r.) 14 −x

s.) 42 43 xx ⋅ t.) 2

10

416xx u.) ( ) 043 32 xx ⋅

v.) xxx 222 ⋅⋅ w.) ( )423xy− x.) 895

874

)3()3(

cc

y.) 66 25 −⋅ mm z.) ( ) ( )( ) 25

32

ttutuut

2. Solve for x:

a.) 1212 =x b.) 3003 2 =x c.) 1442 2 =−x

d.) 613 2 −=− x e.) 75

3=

x f.) 249

2

2

=x

g.) 35

413=

+x h.) 738

=−xx i.)

513

72 −=

− xx

CHALLENGE ZONE! My calculator tells me that 2-2 = 0.25. Use the rules of exponentiation to help explain why.


WEEK 5: GEOMETRY

Introduction: Basic geometry is the branch of mathematics concerned primarily with two things: SHAPES and ANGLES. In high school, you will explore a lot more about geometry but for now, a solid understanding of a few basics is sufficient. Let's start with shapes first. Do your best at defining the key terms below. Then, give an example of a situation in every day life where you would see each. Definitions:

Perimeter – E.g. Area –

E.g.

Now, Some RULES to live by:

SHAPE PICTURE AREA PERIMETER

Rectangle

Square

Triangle

Circle

* Note that the perimeter of a circle has a special name. It is called the ___________________.


NOW TRY THESE:

Use the given information to write the exact AREA and PERIMETER of each shape: a.) b.) A = A = P = C = s = 4ft

More RULES to live by:

1. The angles in a triangle add to 180°.

E.g. so, °=°+°+ 18020100x //CLT °=°+ 180120x // 120−

°= 60x 2. Straight angles (supplementary angles) add to 180°.

E.g. so, °=++ 1802xxx //CLT °= 1804x // 4÷

°= 45x 3. Opposite/Vertical angles are equal.

E.g. so, °= 1233x // 3÷

°= 41x

(Figures not drawn to scale.)




a.) 321

43

−+ b.) 4812

16243

÷ c.) 23

5

7 −

⎟⎟⎠

⎞⎜⎜⎝

⎛

yx

yx d.) ( ) xxx 332 320 ⋅⋅

e.) )36.0(6.33.2 −+− f.) ( )[ ] ( )[ ]5843926 −+−−−− yyyy

2. Solve for x:

a.) b.) P = 20ft

c.) A = 36in2 d.)


3. Find the perimeter and area of the figures below:

a.) b.)


4. Banquet Problem: Prep@Pingree's new banquet table is in the shape of a rectangle. It is 14 inches longer than it is wide. If the area of the table is 735 in2, what are the length and width of the table? Show your work by providing an equation and answers.

CHALLENGE ZONE! Find the perimeter and area of the triangle seen here.


Math Basics: Additional Practice Problems

I. FRACTIONS:

a.) 41

21+ b.)

106

10019

+ c.) 53

2017

− d.) 53

2521

+

e.) 32

521 − f.)

29

185• g.)

54

83÷ h.)

512

1521 •

i.) 85

65+ j.)

123

87− k.)

54

95÷ l.)

432

413 −

m.) 811

21+ n.) 2

4813

• o.) 87

87÷ p.)

31

852 •

q.) 43

831 − r.)

413

213 − s.)

41527

32

•• t.) 58

851 •

II. DECIMALS:

a.) 623.1005.341 − b.) 273.082.0 +

c.) 999.922.22 − d.) 009.0039.0 •

e.) 208.051.35 • f.) 9.99845.086.304.2 +++

g.) 38.08.3 • h.) 18.3586.93042.38 ++

i.) 177.08.43 • k.) 9999.0005.2 −

l.) 623.1005.341 − m.) 7.3251.10 •

III. SIGNED NUMBERS:

a.) )11(5 −+ b.) 410 −− c.) 125•−

d.) 2.101.3 +− e.) 98

41

−• f.) 623.1005.341 +−

g.) )62.3(3.10 −+ h.) 531

83÷

− i.) 031.035.2 −•−

j.) 2517 − k.) 6.325.4 −− l.) 7.3251.10 +−


IV. PERCENTS: (fill in the blank.)

a.) 80% of 40 is ______ b.) 72% of 6 is ______ c.) 38% of 45 is ______

g.) "three fourths" is ______% h.) 0.35 is _______ %

i.) 7 is 140% of ______ j.) ______% of 90 is 8.1

k.) 15 out of 48 is _____% k.) 5 is _____% of 24

V. ORDER OF OPERATIONS:

a.) 23÷ 36 + 18 b.) 42 25 × c.) ( )253289 ×−

d.) ( ) 93528 4 ÷•×+ e.) 1501215 2 +• f.) 6237 4 ÷×+

g.) ( ) 150359 2 −+× h.) ( )[ ] 10523 2 −+− i.) ( ) 52352 ⋅+÷

VI. ADVANCED BASICS – SQUARE ROOTS:

a.) 100 b.) 144 c.) 25 d.) 64.0

e.) 81− f.) 81− g.) 49 h.) ( )2169

VII. MIXED PRACTICE:

a.) 6.072.3 × b.) )45(36 −+ c.) )85(7 −

d.) 54

43

−÷ e.) 413

655 + f.) 325

g.) 64 h.) 94

21

43

×× i.) 614

312 −+

j.) 930 −•− k.) 85% of ______ is 68 l.) 41

1211

−

m.) 100− n.) 400− o.) 24)42(3 ÷×−

p.) 23.06.3807 − q.) 81.0 r.) 0342.6257.1 +−

s.) 2

32⎟⎠

⎞⎜⎝

⎛ t.) 332515

53

•−• u.) )4(455 2 −+−


Algebra Basics: Additional Practice Problems

I. DISTRIBUTING/COMBINING LIKE TERMS:

a.) xx 95 + b.) xyxy 89 − c.) xx 1154 +−−

d.) ( )235 −+ xx e.) ( )43213 +− x f.) ( ) xx 41467 +−−

g.) ( ) ( )634328 +−+ xx h.) ( ) ( )327 −−+ xx i.) ( ) ( )xx 3018412115

31

+−−

II. BASIC EQUATIONS:

a.) 1110 −=−x b.) 1263 =x c.) 1232

=x d.) 115

−=x

e.) 153 =−+x f.) 53

21=+x g.) 104 =− x h.) 12

53

−=− x

III: ADVANCED EQUATIONS (VARS ON BOTH SIDES, 2-3 STEP EQUATIONS, DISTRIBUTING):

a.) 19782 =+− xx b.) 326543 =+++ xxx c.) ( ) 43526 =+− xx

d.) ( ) xx 7640 +−= e.) 221032

−=+− x f.) 5.134 =− x

g.) ( ) ( ) 5327 −=−−+ xx h.) 2385 −=+ xx i.) ( ) ( )2346 +=+− xxx

j.) 10523=

+x k.) xx 95 = l.) ( ) 17133 −=−+ xxx

IV. PROPORTIONS/QUADRATIC EQUATIONS:

a.) x10

43= b.)

632 x−= c.)

73

235=

−x d.) 65

61

32

=+x

e.) 812 =x f.) 24123 2 =+x g.) 4612

=−x h.) 0

4103 2

=+x

VI. SIMPLIFYING EXPONENTS:

a.) 86xx b.) ( )86x c.) ( )532x d.) 0

4x e.) xx 35 2 +

f.) 73

5

yxyx g.)

3

4

53⎟⎟⎠

⎞⎜⎜⎝

⎛

yx h.)

8

4

2

⎟⎟⎠

⎞⎜⎜⎝

⎛−

yx

yx i.) ( )

15

35

44xx j.) 012 243 xxx ⋅⋅ −


Definitions and Rules to Know Now that you have completed your study at Prep@Pingree, test your knowledge of the rules and definitions you've learned by writing each in your own words.

1. DEFINE:

a. Subtraction –

b. Division – c. Absolute Value – d. Reciprocal – e. Perimeter – f. Area – g. Variable – h. Coefficient – i. Exponent – j. Circumference – k. "Like Terms" –


2. COMPLETE THE RULE:

a. To add numbers with different signs: b. To add numbers with the same sign:

c. To multiply decimals:

d. To multiply fractions:

e. To divide fractions:

f. To add/subtract fractions:

g. To solve an equation with 2 steps:

h. To solve an equation with variables on both sides:

i. To solve an inequality:

j. To solve a proportion: k. To combine like terms:


Useful FRACTION – DECIMAL – PERCENT Equivalents to “Just Know”

FRACTION DECIMAL %

1100

120

110

19

18

320

15

29

14

310

13

720

38

25

49

920

12

1120

59

35

58

1320

23

710

34

79

45

1720

78

89

910

1920

𝑎 𝑎 ,𝑎 ≠ 0

MATH WORKBOOK · You can do this because dividing by the same number on the top and bottom is the...

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Transcript of MATH WORKBOOK · You can do this because dividing by the same number on the top and bottom is the...