SUMMER REVIEW PACKET 1 FOR STUDENTS ENTERING ALGEBRA 1

MA'AYANOT YESHIVA HIGH SCHOOL FOR GIRLS NAME:______________________________________

1

SUMMER REVIEW PACKET 1

FOR STUDENTS ENTERING ALGEBRA 1

Dear Students,

Welcome to Ma’ayanot. We are very happy that you will be with us in the Fall.

The Math department is looking forward to working with you and watching you learn. Your

teachers are very supportive and want you to understand and enjoy learning math.

This packet is designed to help you to review the skills that are essential as a foundation for

your algebra class. There are model problems which will help to guide you.

INSTRUCTIONS:

1) PRINT OUT PACKET

2) SHOW YOUR WORK IN THE SPACE PROVIDED IN THE PACKET.

3) NO CALCULATOR (UNLESS STATED IN THE PROBLEM)

4) FOR THE FRACTION PROBLEMS STARTING (p.15-22) do the ODD problems only.

5) HAND YOUR PACKET TO YOUR TEACHER ON THE FIRST DAY OF SCHOOL!

GOOD LUCK AND HAVE A GREAT SUMMER!


2

ORDER OF OPERATIONS

P.E.M.D.A.S.

PARENTHESES- this means operations inside the parentheses

EXPONENTS - raise number to exponents

MULTIPLICATION/DIVISION - all in order from left to right

ADDITION/SUBTRACTION - all in order from left from right

Example 1:

Example 2:

SHOW STEPS AS IN EXAMPLES. NO CALCULATOR

1) 60 ÷ ( 6+2-4)- 8-1 -6 2) 4-8+{ 12 ÷ 2 × 7-3 × 8}

3) [4-90 ÷ 3+8] × 1 × 2 × 7 4) 3 × 4 × (2-1 × 45) × 8 × 1


3

.

5) 1-[6 × 8-4+6-3]+ 8 6) 9+[8+14 ÷ 2+7-6] × 3

7) 6+[8 × 61 ÷ 2+9] × 7 × 5 8) 8+14 ÷ (5+1-1 × 5) × 3

9) [7 × 5 × 21 ÷ 1+7]- 2- 1 10) [7 × 30 ÷ 5 × 9 × 4+5]+8


4

Solving One-Step Equations

Solving equations means to get the variable by itself (isolate).

Note: The answer should look like (variable) = (some number), where the variable is never negative

Solve using addition and subtraction.

1)

16 7 Get the variable by itself. Right now 16 is being added to it.

16 16 Undo the addition by subtracting 16 from both sides.

23 Answer.

r

r

When solving equations, eliminate double signs.

As a general rule, replace “+ (- )” with “–” and “– (- )” with “+”.

2) ( 3) 8y

3 8

3 3

11

y

y

Solve using multiplication and division.

1)

5 60 Get the variable by itself. Right now -5 is being multiplied to it.

-5t 60 Undo the multiplication by dividing both sides by -5.

-

1

5 5

A r.2 nswet

t

2) 12 Since 4 is dividing x, multiply both sides by 4 to clear the fraction.4

x

(4) 12(4) 4

4 4x 48 The fours will cancel each other out. simplifies to 1x

4 4

48x

x

x

Subtraction Property of

Equality Subtracting the

same value from both

sides of the equation.

Addition Property of Equality

Adding the same value from both

sides of the equation.

Division Property

of Equality

Dividing the same value

from both sides of the

equation.

Multiplication Property

of Equality Multiplying

the same value from

both sides of the

equation.


5

MULTIPLY BOTH SIDES BY RECIPROCAL

2

18 To get rid of multiplying a fraction, multiply by the reciprocal.3

x

3 2 318 Multiply straight across.

2 3 2

54

2

2 7

x

x

x

Solve the Equation. (Write the Steps!)

1. x + 2 = 10 2. 3x = -15

3. -4 + x = 12 4. 3

x

= -6

5. 15 = t – 2 6. 1

2x = 3

7. x – (-4) = 3 8. 3 = -x


6

9. w + 14 = -8 10. y + (-10) = 6

11. -11 = a + 8 12. 5h = 35

13. -2.3 = x + (-1.1) 14. -7 = -16 - k

15. -13m = 39 16. z + (-13) = -27

17. p - (-27) = 13 18. 41 = 32 - r


7

19) 67

2

x

21) 58

m

23) 3

2

4

3t

25) 4

3

6

5

x

20) 6

5x

22) 23

m

24) t5

3

3

2

26) 2

1

4

3x


8

Evaluate by substituting the given values of the variables:

1) m p ; use m , and p

2) ( p q)(); use p , and q

3) z(x y); use x , y , and z

4) k h; use h , and k

5 ) xy; use x , and y

6) y x ; use x , and y

7) 22 5

3

x y zfor x and y

x z

8) 5

2 53

x y zfor x and y

x z


9

FRACTIONS- Methods (these are methods and examples for each type of problem- just in case

you’re uncertain)

A. Changing a Mixed Number to an Improper Fraction

Mixed number – 4 3

2 (contains a whole number and a fraction)

Improper fraction - 3

14 (numerator is larger than denominator)

Step 1 – Multiply the denominator and the whole number

Step 2 – Add this answer to the numerator; this becomes the new numerator

Step 3 – Carry the original denominator over

Example #1: 3 8

1 = 3 × 8 + 1 = 25 =

8

25

Example #2: 4 9

4 = 4 × 9 + 4 = 40=

9

40

B. Changing an Improper Fraction to a Mixed Number

Step 1 – Divide the numerator by the denominator

Step 2 – The answer from step 1 becomes the whole number

Step 3 – The remainder becomes the new numerator

Step 4 – The original denominator carries over

Example 5

47 = 47 ÷ 5 or 5 47 = 9, remainder 2

Ans: 2

95


10

C. Reducing Fractions

Step 1 – Find a number that will divide into both the numerator and the

denominator

Step 2 – Divide numerator and denominator by this number

Example #1: 15

10 =

3

2 (because both 10 and 15 are divisible by 5)

Example #2: 8

4 =

2

1 (because both 4 and 8 are divisible by 4)

D. Raising Fractions to Higher Terms When a New Denominator is Known

Step 1 – Divide the new denominator by the old denominator

Step 2 – Multiply the numerator by the answer from step 1 to find the new

numerator

*Note: If the original number is a mixed number, convert it to an improper

fraction before raising to higher terms (see Example #2)

Example #1: 3

2 =

12 becomes

3

2 =

12

8 because 12 ÷ 3 = 4

and 2 × 4 = 8

Example #2: 2 5

1 =

20 becomes

5

11 =

20 becomes

5

11 =

20

44

because 20 ÷ 5 = 4 and 11 × 4 = 44


11

E. (1) Multiplying Simple Fractions

Step 1 – Multiply the numerators

Step 2 – Multiply the denominators

Step 3 – Reduce the answer to lowest terms

Example: 7

1 ×

6

4 =

42

4 which reduces to

21

2

E. (2) Multiplying Mixed Numbers

Step 1 – Convert the mixed numbers to improper fractions first

Step 2 – Multiply the numerators


Step 4 – Reduce the answer to lowest terms

Example: 23

1 × 1

2

1 =

3

7 ×

2

3 =

6

21 which then reduces to 3

2

1


12

F (1) Dividing Simple Fractions

Step 1 – Change division sign to multiplication

Step 2 – Change the fraction following the multiplication sign to its

reciprocal (flip the fraction around so the old denominator is the

new numerator and the old numerator is the new denominator)

Step 3 - Multiply the numerators


Step 5 – Change the answer to lowest terms

Example: 8

1 ÷

3

2 = becomes

8

1 ×

2

3 which when solved is

16

3

F(2). Dividing Mixed Numbers

Step 1 – Convert the mixed number or numbers to improper fraction

Step 2 – Change the division sign to multiplication

Step 3 – Change the fraction following the multiplication sign to its

reciprocal (flip the fraction around so the old denominator is the

new numerator and the old numerator is the new denominator)

Step 4 - Multiply the numerators



Example: 34

3 ÷ 2

6

5 = becomes

4

15 ÷

6

17 becomes

4

15 ×

17

6 =

which when solved is 24

15 ×

17

63 =

34

45 which simplifies to 1

34

11


13

G. Adding and Subtracting Fractions

Step 1 – Find a common denominator (a number that both denominators

will go into)

Step 2 – Raise each fraction to higher terms as needed

Step 3 – Add or subtract the numerators only as shown

Step 4 – Carry denominator over


Example #1: 2

1 +

8

7 = Common denominator is 8 because both 2 and

8 will go into 8

2

1 =

8

4

+ 8

7 =

8

7

8

11 which simplifies to 1

8

3

Example #2: 45

3 –

4

1 = Common denominator is 20 because both 4

and 5 will go into 20

45

3 = 4

20

12

– 4

1 =

20

5

420

7


14

Example #3: 2 8

1 = 2

8

1 = 12

8

1+

8

8 = 1

8

9

– 14

1 = 1

8

2 = 1

8

2 = 1

8

2

8

7 **


15

INSTRUCTIONS: YOU SHOULD DO THE ODD PROBLEMS ONLY IN THIS SECTION

( See previous pages for instructions on how to do each type of problem)

A. Write as an improper fraction.

1. 18

1 2. 4

5

1 3. 1

3

2 4. 2

16

3

5. 27

5 6. 2

16

1 7. 1

8

5 8. 3

5

4

9. 74

1 10. 5

3

2 11. 3

6

5 12. 6

2

1


16

B. Write as a mixed number.

1. 4

10 2.

2

19 3.

3

25 4.

8

9

5. 16

25 6.

4

35 7.

3

7 8.

8

21

C. Write in lowest terms.

1. 32

6 2.

35

21 3.

24

18 4.

15

12

5. 30

5 6.

27

9 7.

49

14 8.

32

8


17

9. 121

12 10. 2

20

16 11. 5

14

8 12. 3

25

10

D. Find the missing numerator by raising the fraction to higher terms.

1. 4

3 =

12

? 2.

16

7 =

64

? 3.

8

5 =

48

? 4.

9

5 =

72

?

5. 53

2 =

12

? 6. 1

5

4 =

10

? 7. 1

4

1 =

12

? 8. 2

5

3 =

10

?


18

E. Multiply.

1. 9

1 ×

2

1 = 2.

10

7 ×

5

2 = 3.

8

3 ×

7

2 = 4.

2

1 ×

16

3 =

5. 4

3 ×

3

2 = 6.

16

7 ×

3

4 = 7.

64

15 ×

12

1 = 8.

9

2 ×

9

5 =

9. 4

3 × 10 = 10. 1

2

1 ×

6

5 = 11.

16

3 ×

12

5 = 12. 14 ×

8

3 =


19

13. 2

1 × 1

3

1 = 14. 3

16

1 ×

5

1 = 15. 18 × 1

2

1 = 16. 16 × 2

8

1 =

17. 68

3 × 1

5

3 = 18. 2

3

2 × 4

8

3 = 19. 4

9

4 × 4

4

2 = 20. 3

8

1 × 2

5

2=

F. Divide as shown.

1. 2

1 ÷

4

1 = 2.

5

2 ÷

2

1 = 3.

3

8 ÷

3

2 = 4.

9

2 ÷

3

1 =


20

5. 4 ÷ 8

1 = 6. 8 ÷

5

4 = 7. 9 ÷

4

3 = 8.

5

6 ÷

5

4 =

9. 11

4 ÷

11

1 = 10.

7

2 ÷

9

5 = 11.

3

2 ÷ 4 = 12. 14 ÷

8

7 =

13. 15 ÷ 6

5 = 14. 8 ÷

4

3 = 15. 1

4

1 ÷ 1

2

1 = 16. 3

2

1 ÷ 5 =

17. 64

1 ÷ 2

2

1 18. 5

3

1 ÷ 2

3

2 = 19. 2

4

3 ÷ 1

8

1 = 20. 3

5

1 ÷ 1

7

5=


21

G. Add or subtract as shown.

1. 8

3 +

8

7 = 2.

3

2 +

4

3 = 3.

32

3 +

8

1 = 4.

5

3 +

6

5 =

5. 8

5 +

10

1 = 6.

8

3 + 1

4

1 = 7.

4

1 +

5

1 = 8. 2

8

1 + 1

4

1=

9. 18

5 +

16

13 = 10. 2

3

2 +

9

4 = 11.

10

9 –

16

3 = 12.

8

7 –

2

1 =


22

13. 16

11 –

4

1 = 14.

6

5 –

5

1 = 15.

8

7 –

10

3 = 16. 1

2

1 –

32

3 =

17. 56

5 – 2

9

3 = 18. 3

3

2 – 1

8

7 = 19. 2

4

1 –

6

5 = 20. 4

6

5 – 1

2

1=


23

Converting between decimals and fractions and percents: please read this carefully

NOTE: THIS IS A REVIEW OF CONVERSIONS BETWEEN DECIMALS, FRACTIONS AND PERCENTS. READ THE PAGES HERE TO MAKE SURE YOU KNOW THE METHODS.

There are many different methods to convert between decimals, fractions and percents, so we must understand the meaning

1. Percent (%) means per hundred

so 52% = 52/100 which means as a fraction it is 52

100 which reduces to

13

25

and as a decimal it is 52 100 .52

2. A) To multiply by 100, move decimal point 2 places to the right

To divide by 100, we can move the decimal point over 2 places to the left

So .52 x 100 = 52 .735 x 100 = 73.5 12.3 x 100 = 123

52/ 100 = .52 73.5/ 100 = .735 123/100 = 12.3

B) To multiply by, decimal point right 1 digit

To divide by 10, move left by 1 digit

.3 x 10 = 3 .35 x 10 = 3.5 27.2 x 10 = 272

3/10 = .3 3.5/10= .35 272/10= 2.72

See next page!


24

3. Conversion between fractions and decimals

A. From fraction to decimal : divide numerator by denominator

Ex: 3

3 5 .65

B. From decimal to fraction: write as a fraction using the place value of the last digit, then simplify if possible

Ex: 41

.41100

4 2

.410 5

279

.2791000

3

.03100

4. Conversion between decimals and percents

A. From percent to decimal

Since percent means per hundred, it means divided by 100

So 23

23% .23100

(remember there’s an easy way to divide by 100)

5.2

5.2% .052100

B. From decimal to percent:

You can think of this as the reverse of the above, multiply by 100

So . .23 23% .512 51.2%


25

There's more on next page- read it all carefully!

5) Conversion between fraction and percents

A. From percent to fraction, since percent means per hundred , write the number over 100 and simplify the fraction:

2323%

100

55 1155%

100 20

14.214.2%

100 but we don’t want a decimal in the fraction so change to

14.2 10 14214.2%

100 10 1000 which can be simplified to

71

50

B) From fraction to percent, we can either set up a proportion

2

5 100

x and cross multiply 2 100 5x , and x=40, so it’s 40%

Or we can convert to a decimal first and change to a fraction

22 5 .4

5 and multiply by 100 to get 40%


26

Do the following and show work: Please do all problems on this page

1 Write these decimals as fractions in simplest form:

0.3 = …………. 0.5 = …………. 0.6 = …………. 0.02 = ………….

0.05 = …………. 0.25 = …………. 0.36 = …………. 0.125 = ………….

2 Write these fractions as decimals:

107 = ………………. 5

1 = ………………. 52 = ………………. 4

3 = …………………

87 = ………………. 3

2 = ………………. 209 =………………. 25

7 = …………………

3 Write these percentages as decimals:

3% = …………. 30% = …………. 25% = …………. 80% = ………….

8.72 % = …………. 12.3% = …………. .4 % = …………. 17.5% = ………….


27

Do all problems on this page

4 Write these percentages as fractions: Don’t leave decimals in your answers

20% = …………. 75% = …………. 5% = …………. 30% = ………….

40% = …………. 1.5% = …………. 2.4% = …………. 35% = ………….

5 Write these decimals as percentages:

0.25 = …………. 0.5 = …………. 0.7 = …………. 0.07 = ………….

0.45 = …………. 0.09 = …………. 2.4 = …………. 0.375 = ………….

6 Write these fractions as percentages:

101 = …………. 5

1 = …………. 109 = …………. 4

3 = ………….

54 = …………. 20

17 = …………. 31 = …………. 3

2 = ………….

Complete this table..


28

Percentage

Fraction Decimal

10%

0.2

103

40%

0.5

53

70%

0.8

109

25%


29

RATIOS , RATES, AND PROPORTIONS- What's the difference?

Ratios

A ratio is a comparison of two quantities that have the same units. You can express a ratio in any one of

the following ways:

18

5 18:5 18 to 5

Example: If there are 8 boys and 21 girls then

8

21

boys

girls

21

8

girls

boys

8

29

boys

total

21

29

girls

total

RATIOS

A. Write each ratio as a fraction in lowest terms. (NO CALCULATOR)

1) 2 to 4 2) 3 to 12

3) 6 :18 4) 22 : 330


30

Rates: A rate is a comparison of two quantities that have different units. Rates are usually expressed

in the fractional form.

Example: Francine paid $16 for her 12-month subscription to Better Homes and Gardens magazine.

Express as a rate.

$16.00 = $4.00

12 magazines 3 magazines

If Francine wants to know how much she pays for each (1) magazine, she can divide $4 by 3

magazines. This will give her the price per magazine (also called the unit rate).

Answer: $1.33/magazine

3

B. Write each of the following rates as a unit rate. You may use a calculator.

1) 3 Tbsp sugar 2) 135 pitches

2 cups of milk 45 strikes

3) 128 miles 4) 2250 pencils

4 hours 18 boxes

5) $450 6) 2500 meters

18 shares 15 seconds


31

PROPORTIONS

A proportion is a statement of 2 equal ratios

Example: 25 3

25 :50 3: 650 6

also written which must be true since each is in a ratio of 1:2

To solve a proportion means to find a value which gives a true proportion.

Solve each proportion and state the answer in simplest form. Show work.

1) 6 : 8 = n : 12 2 ) 2 = 8

7 n

3 n = 11 4) 4 : n = 6 : 9

6 3

5) 2 ½ : 3 ½ = n : 2 6). 1: 2 = n : 9

69) 4 to 8 = 15 to n 70) 18 : n = 3 :11


32

D. Solve by using a proportion. Round answers to the nearest hundredth if necessary.

(YOU MAY USE A CALCULATOR, BUT SHOW THE WAY YOU SET UP THE PROBLEM. CALCULATOR SHOULD ONLY BE

USED AT THE END)

1) You jog 3.6 miles in 30 minutes. At that rate, how long will it take you to jog 4.8 miles?

2). An airplane flies 105 miles in ½ hour. How far can it fly in 1 ¼ hours at the same rate of speed?

3) What is the cost of six filters if eight filters cost $39.92?

4) If one gallon of paint covers 825 sq. ft., how much paint is needed to cover 2640 sq. ft.?

5) A map scale designates 1” = 50 miles. If the distance between two towns on the map is 2.75 inches, how many

miles must you drive to go from the first town to the second?


33

Percent Increase and Decrease

Percent Increase 100

Amount of Increase x

original price (percent means per hundred!)

Percent Decrease 100

Amount of Decrease x

original price

EXAMPLES

Percent Increase Percent Decrease

The price of gasoline increased from $2.10 per

gallon to $2.50 per gallon What is the percent

increase in the price?

Let x = % increase

2.42 2.10

2.42 100

.32

2.10 100

(2.10) .32(100)

2.10 32

32 / 2.10 15.24

% 15.24%

x

x

x

x

x

so increase

A jacket is on sale for $80. If the original price

is $100, what is the percent decrease in the

price?

Let x = % decrease

120 80

120 100

40

120 100

1

3 100

3 100

100 / 3 33.3

% 33.3%

x

x

x

x

x

so decrease

For the following, you may use a calculator, but show all work and use the calculator for the final step.

Make sure to declare your variable (Let x= )

1) Lane saved $8 by buying her video game on sale. If the game was on sale for 25% off the original price,

what the original price of the game?

2) Jasmine took the hem out of her skirt. As a result, she increased the length of her skirt by 14%. The original length of her skirt was 16 inches. What was the final length of her skirt?


34

3) A survey two years ago showed that 120 students in the school were able to use computers at home. A new survey shows that the number of students with access

to computers at home has increased by 15%. How many students now have access to computers?

4) In a survey taken two weeks ago, 146 students said they would vote for Megan for class president. Today 163 students said they would vote for her. What was the approximate percent increase in the number of students voting for Megan?

5) Mike weighed 160 pounds last year. This year he weighs 200 pounds. What is the percent increase in his weight?

6) Jeff weighs 144 pounds now, but last year he weighed 180 pounds. What is his percent decrease?

YOU'RE DONE!

HAND IT IN TO YOUR TEACHER ON THE FIRST DAY OF CLASS

Documents

SUMMER REVIEW PACKET 1 FOR STUDENTS ENTERING ALGEBRA 1