math workbook - pingree schoolpingree/mathworkbook.pdf · prep@pingree math workbook ... then, you...

$: MATH WORKBOOK - Pingree SchoolPingree/MATHWorkbook.pdf · Prep@Pingree Math Workbook ... Then, you must write each fraction in terms of ... as many key phrases and their mathematical$
MATH WORKBOOK

Holly O'Donohue

Prep@Pingree Math Workbook (7th) 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


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, I 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.

- Ms. O'Donohue


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.

E.g. To simplify 21

14, divide both parts by 7. So,

3

2

21

14

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

E.g. 11

4

55

20

11

10

5

2 OR

11

10

5

2

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.

E.g. 2

5

6

15

1

3

6

5

3

1

6

5

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

E.g.: 5

2 and

2

5are "reciprocals" BECAUSE 1

10

10

2

5

5

2 !

Note: By definition, in all cases "division" is nothing more than multiplying by the 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. 40

23

40

15

40

8

8

3

5

1

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.

12

2 and 1

145

145 and 1

x

x and 1

Kaisy

Kaisy

b. Zero divided by anything equals 0.

03

0 and 0

234,11

0 and 0

0

xyz

c. Anything divided by zero is "undefined".

0

6undefined and

0

12xundefined 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 100

5(or

20

1when completely reduced).

To convert a PERCENT to a DECIMAL: Move the decimal point two places to the left. 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%

Now, Try these:

Simplify:

a.) 12

5

3

2 b.)

75

5

5

1 c.) 9.088.1 d.) 2.025.3


Prep@Pingree Name: __________________________

Homework Set #1 1. Reduce to lowest terms:

a.) 20

4 b.)

42

36 c.)

1080

90

2. Change to improper fractions:

a.) 3

11 b.)

20

33 c.)

13

122

3. Simplify and reduce all answers to lowest terms:

a.) 5

2

3

1 b.)

3

21

4

3 c.)

15

2

20

7

d.) 6.18203.22 e.) 25.8037.1306.0 f.) 237.065712.2

5.63288.373

g.) 05.4702.42 h.) 087.0876.0 i.) 77.98.23

j.) 7.32.8 k.) 7.109.87 l.) 4.43.228

m.) 7

5

4

3 n.)

3

4

4

33 o.)

9

1428

7

3

p.)

5

16

4

1 q.)

3

14

4

13 r.)

4

122

2

1

s.) 15

1

3

1 t.)

3

12

4

15 u.)

28

91

4

32

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

a.) "three quarters" b.) "one fourth" c.) "two tenths" 5. Extra Credit:

a.) Summer

PP

Fun

Math @ b.)

yxy

11121


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! Positive and negative numbers are collectively referred to as "signed numbers" and they are "opposites". 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.

Some RULES to live by:

1. To ADD numbers with the SAME SIGN (both positive or both negative), add the numbers ignoring the signs, 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 smaller number from the larger number and keep the sign of the larger number.

E.g. 3 5 2 and 1 3 4 9 5

3 4 12 12 12

Note: Subtraction, by definition, is the same thing as adding a negative (or opposite) number. So, 45 is the same thing as saying )4(5 .

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!"


2. When MULTIPLYING or DIVIDING numbers with the SAME SIGN, the product/quotient is POSITIVE.

E.g. 48412

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

E.g. 9

14

3

7

3

2

7

3

3

2

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

certain ORDER OF OPERATIONS (AKA: PEMDAS, AKA: BEDMAS if you are Canadian!)

The ORDER OF OPERATIONS is:

P ________________ E ________________ M _______________ D _______________ A _______________ S ________________ NOTE: Although it may look like there are six steps to the order of operations, there are really only four. M/D (D/M) and A/S (S/A) are completed from LEFT to RIGHT!

E.g. )16(35 2 Parentheses – complete the inside first.

535 2 Exponents

595 Multiplication/Division

455 Addition/Subtraction

50



Homework Set #2 Complete the following WITHOUT a calculator.

2. Place in order from least to greatest:

1,3

4,

5

3,

2

7,

5

7,2,

3

5

3. 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.) 8

17

5

34 l.)

28

91

4

32

m.) 5

24

4

3 n.)

264

12)15(49

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

a.) 3

262_____)4(

4

31

b.) 2.06.3_____85.202.0

c.) 25.681.3_____35.748

51

5. Shopping Problem: Tanya and Tiffany go shopping at the Loop. They stop by their favorite store which is having a sale. Each buys a shirt for 30% off its original price of $19.50 and a skirt for 25% off its original price of $28.00. How much money did they spend in all?

6. Extra Credit:

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

34)624(

)3(8)2(252 ?

35

212)630()52(


WEEK 3: ALGEBRA

Introduction: Algebra is really all about one letter: x. In algebra, x, called a "variable" is used to represent some unknown quantity. Whenever a situation arises where a quantity is unknown, it is useful to write (and then solve) an equation involving variables to determine the unknown.

BUT FIRST! Before we go too far into algebra, we should know some common but important English words that represent math. I've begun my brainstorm below. Add as many key phrases and their mathematical equivalents as you can to my list…

TRANSLATION: English Mathematics "is" = "some number" x "of" "and" + "per" 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"


Some RULES to live by: The GOAL of solving algebraic equations is 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 6x

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

E.g. 213x // 3 24x

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 3

1

41x

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

66x 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 and the 2 is called the .

Now, "LIKE TERMS" are terms that have the SAME VARIABLE(s) and the SAME EXPONENT(s).

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

To COMBINE LIKE TERMS: add the coefficients. (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 2x ).

It looks like this: 6323323 xxx

NOW TRY THESE: Simplify:

a.) 10623 xx b.) 532 x

c.) 623 xx d.) 2118 xx


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

3

40x

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

12x

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

15

2x

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. 74

12 x // 2

94

1 x //

4

1 OR 4 (NOTE: The Catch!)

36x



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.) 1518x c.) 99 x

d.) 44 x e.) x 83 f.) 734 x

g.) 124 x h.) 183

2x i.) x 83

j.) 103

x k.) xx 52 l.) 16

4

12 x

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

p.) 54

32

x q.) 0

2

35

x r.) 42)1(3 xxx

3. Solve the following inequalities:

a.) 31712 x b.) 83 x c.) 43 x

d.) 1734 x e.) 114

35 x f.) 14

4

35 x

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

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

c.) %3

266 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. What are the numbers of the three jerseys? Show your work by providing an equation and answers.

6. Extra Credit:

a.) Solve hprV 2

3

1 for h.

b.) Solve 329

5 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


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.

nm

n

m

xx

x

E.g. 628

2

8

xxx

x Now try:

10

30

x

x

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.

nnnyxxy OR n

nn

y

x

y

x

E.g. 3333822 xxx Now try:

4

2

y

x


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

n

n

xx

1

E.g. 3

3 1

xx Now try: 14x

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

10 x

E.g. 130 Now try: 0

4x

** 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

4x

x

e.) 210352 yxyx


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. x

3

4

11 // cross-multiply Now try:

3

10

5

2

x

4311 x 1211 x // 11

11

12x

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.) 5

2

2

3 x c.)

x

x 20

5




a.) 144 b.) 64

1 c.) 400

d.) 281 e.)

16

49 f.)

2

5

2

g.) 81.0 h.) 36 i.) 916

j.) 916 k.) 94 l.) 94

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

p.) 8

1

8

1 q.) 72 xx r.) 14 x

s.) 42 43 xx t.) 2

10

4

16

x

x u.) 043 32 xx

v.) xxx 222 w.) 423xy x.) 895

874

)3(

)3(

c

c

y.) 66 25 mm z.)

25

32

ttu

tuut

2. Solve for x:

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

d.) 613 2 x e.) 7

5

3

x f.)

2

49

2

2

x

g.) 3

5

4

13

x h.)

7

38

x

x i.)

5

13

7

2

xx

3. Extra Credit: 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.) 3

21

4

3 b.)

48

12

16

243 c.)

23

5

7

y

x

y

x 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.) c.)


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.

5. Extra Credit: Speed Problem: Tia runs for 30 minutes at a rate of 20

3miles per hour.

How far has she run? Show your work by providing an equation and answer.


Math Basics: Additional Practice Problems I. FRACTIONS:

a.) 4

1

2

1 b.)

10

6

100

19 c.)

5

3

20

17 d.)

5

3

25

21

e.) 3

2

5

21 f.)

2

9

18

5 g.)

5

4

8

3 h.)

5

12

15

21

i.) 8

5

6

5 j.)

12

3

8

7 k.)

5

4

9

5 l.)

4

32

4

13

m.) 8

11

2

1 n.) 2

48

13 o.)

8

7

8

7 p.)

3

1

8

52

q.) 4

3

8

31 r.)

4

13

2

13 s.)

4

1527

3

2 t.)

5

8

8

51

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.) 9

8

4

1 f.) 623.1005.341

g.) )62.3(3.10 h.) 5

31

8

3

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.) 1503592 h.) 10523

2 i.) 52352

VI. ADVANCED BASICS – SQUARE ROOTS:

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

e.) 81 f.) 81 g.) 4

9 h.) 2169

VII. MIXED PRACTICE:

a.) 6.072.3 b.) )45(36 c.) )85(7

d.) 5

4

4

3 e.)

4

13

6

55 f.) 325

g.) 64 h.) 9

4

2

1

4

3 i.)

6

14

3

12

j.) 930 k.) 85% of ______ is 68 l.) 4

1

12

11

m.) 100 n.) 400 o.) 24)42(3

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

s.) 2

3

2

t.)

33

2515

5

3 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 30184

12115

3

1

II. BASIC EQUATIONS:

a.) 1110 x b.) 1263 x c.) 123

2x d.) 11

5

x

e.) 153 x f.) 5

3

2

1x g.) 104 x h.) 12

5

3 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.) 22103

2 x f.) 5.134 x

g.) 5327 xx h.) 2385 xx i.) 2346 xxx

j.) 105

23

x k.) xx 95 l.) 17133 xxx

IV. PROPORTIONS/QUADRATIC EQUATIONS:

a.) x

10

4

3 b.)

63

2 x c.)

7

3

2

35

x d.)

6

5

6

1

3

2

x

e.) 812 x f.) 24123 2 x g.) 46

12

x

h.) 04

103 2

x

VI. SIMPLIFYING EXPONENTS:

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

4x e.) xx 35 2

f.) 73

5

yx

yx g.)

3

4

53

y

x h.)

8

4

2

y

x

y

x i.)

15

35

4

4

x

x 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:

Subtraction –

Division – Absolute Value – Reciprocal – Perimeter – Area – Variable – Coefficient – Exponent – Circumference – "Like Terms" –


2. COMPLETE THE RULE:

To add numbers with different signs: To add numbers with the same sign:

To multiply decimals:

To multiply fractions:

To divide fractions:

To add/subtract fractions:

To solve an equation with 2 steps:

To solve an equation with variables on both sides:

To solve an inequality:

To solve a proportion: To combine like terms:

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