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Ma’ayanot Yeshiva High School Summer Review Packet-2018 R. BERNSTEIN
Page 1 of 29
SUMMER REVIEW PACKET 2
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
4) FOR THE FRACTION PROBLEMS STARTING (p.11-16) 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!
THE MATH DEPARTMENT
Ma’ayanot Yeshiva High School Summer Review Packet-2018 R. BERNSTEIN
Page 2 of 29
Objective: Write an algebraic expression to represent unknown quantities. • A variable is a symbol, usually a letter, used to represent a number. • Algebraic expressions are combinations of variables, numbers, and at least one operation. Sum: addition Ex: x+ 5 is “the sum of a number and 5 “ or “5 added to a number” Difference : subtraction Ex: x-12 is “ the difference between a number and 12” or “12 less than a number” but 12 –x is “ the difference between 12 and a number” or “12 decreased by a number” Product: multiplication Ex: 20x is “the product of 20 and a number”
Quotient: Division Ex: 20
x is “the quotient of 20 and a number” or “ 20 divided by a number”
But 20
x is “ quotient of a number and 20” or “ a number divided by 20”
In each case, let x= the number. Be careful with difference and quotient!
1.) a number plus 2
1
2.) the sum of a number and 73
3.) the difference of 21 and a number
4.) the difference of a number and 21
5) a) Five less than a number
6.) Five decreased by a number
7) The quotient of 20 and a number
8) Twenty divided by a number
9) The quotient of a number and 20
10) A number divided by 20
10. John is 5 years younger than Bill; If john’s age is x, what is Bill’s age in terms of x
10. Bill is 5 years younger than John. If John’s age is x, what is bill’s age in terms of x?
Ma’ayanot Yeshiva High School Summer Review Packet-2018 R. BERNSTEIN
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Objective: Evaluate an algebraic expression. • A variable is a symbol, usually a letter, used to represent a number. • Algebraic expressions are combinations of variables, numbers, and at least one operation. • Multiplication in algebra can be shown as 4n or 4 x n • The variables in an algebraic expression can be replaced with any number. • Once the variables have been replaced, you can evaluate, or find the value of, the algebraic expression. Examples: Evaluate 44 + n if n= 9 44 + n original expression 44 + 9 replace the variable with it’s value 53 solution
1.) Evaluate 150 + n if n = 15 2.) Evaluate 12n if n = 9
3.) Evaluate 15n + 19 if n = 3
1
4.) Evaluate 30n if n = 2.5
5.)Evaluate 24n k if n = 6 and k = 8
6.)Evaluate nk – 2b + 8 if b = 1.5, k = 8, and n = 7
Ma’ayanot Yeshiva High School Summer Review Packet-2018 R. BERNSTEIN
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Unit: KNOWLEDGE of ALGEBRA, PATTERNS, and FUNCTIONS Objective: Evaluate numeric expressions using order of operations. • A numerical expression is a combination of numbers and operations. • The Order of Operations tells you which operation to perform first so that everyone gets the same final answer. • The Order of Operations is: Parentheses, Exponents, Multiplication or Division (left to right), and Addition or
Subtraction (left to right.) Examples:
48 (3 + 3) – 22 original expression
48 6 - 22 P simplify the expression inside the parentheses
48 6 – 4 E exponents 8 – 4 M,D multiplication and division from left to right 4 A, S addition and subtraction from left to right Show your steps. No calculator!
1.) (8 + 1) x 12 – 13
2.) 13 x 4 – 72 8
3.) 88 – 16 x 5 + 2 – 3
4.) 100 52 x 43
5.) 45 9 – 3 + 2 x 3
6.) (52 + 33) x (81 + 9) 10
Ma’ayanot Yeshiva High School Summer Review Packet-2018 R. BERNSTEIN
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Unit: solve 1 step equation with addition and subtraction Objective: Determine the unknown in a linear equation (addition & subtraction). • Addition equations: Subtract the same number from each side of the equation so that the two sides remain equal. • Subtraction equations: Add the same number to each side of the equation so that the two sides remain equal. Examples: b + 3 = 6 original equation b – 8 = 4 original equation - 3 - 3 subtract 3 from each side +8 +8 add 4 to each side b + 0 = 3 solution b + 0 = 12 solution b = 3 simplify b = 12 simplify
1.)
g + 5 = 12
2.)
s – 12 = 29
3.)
m + 3.5 = 10.5
4.)
k – 5.5 = 8.5
5.)
w + 6.25 = 22
6.)
g – 3.75 = 49.75
Ma’ayanot Yeshiva High School Summer Review Packet-2018 R. BERNSTEIN
Page 6 of 29
Unit: Solve 1 step equation with multiplication and division Objective: Determine the unknown in a linear equation (multiplication & division). • In a multiplication equation, the number by which a variable is multiplied is called the coefficient. In the multiplication
equation 2x = 8, the coefficient is 2. • Multiplication equations: Divide both sides by the coefficient so that the two sides remain equal.
• In a division equation, the number by which the variable is divided is called the divisor. In the division equation 4
x,
4 is the divisor. • Division equations: Multiply both sides of the equation by the divisor so that the two sides remain equal. Examples:
4b = 16 original equation 6
m = 11 original equation
4 4 divide both sides by 4 6 x 6
m = 11 x 6 multiply each side by 6
1b = 4 solution 1m = 66 solution b = 4 simplify m = 66 simplify
1.) 7x = 63
2.)
9
k = 8
3.)
5b = 3.55
4.)
7
n = 5.55
5.)
12m = 84.72
6.)
13
p = 2.67
Ma’ayanot Yeshiva High School Summer Review Packet-2018 R. BERNSTEIN
Page 7 of 29
Unit: NUMBER RELATIONSHIPS and COMPUTATION Objective: Read, write, and represent integers. Examples: Integer: Any number from the set {… -3,-2,-1,0,1,2,3…} Write an integer to describe each situation EX: a height increase of 3 inches The word increase represents positive. Answer: 3 or +3. EX: 50 feet below sea level The word below represents negative. Answer: - 50
1.) Write an integer to describe: The stock market increased 75 points
2.) Write an integer to describe: A loss of 15 yards
3.) Write an integer to describe the situation: Nancy owes her friend $10
4.) Write an integer to describe: Frederick is located 290 feet above sea level.
5.) Write an integer to describe: The temperature was 3° below zero
6.) Write an integer to describe: The 6th grade has 12 fewer students than last year
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
Integers less than zero
are negative integers Integers greater than zero
are positive integers
Negative integers are written with a - sign
Positive integers can be written with or without a + sign Zero is neither nor positive
Ma’ayanot Yeshiva High School Summer Review Packet-2018 R. BERNSTEIN
Page 8 of 29
FRACTIONS- Methods
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
Ma’ayanot Yeshiva High School Summer Review Packet-2018 R. BERNSTEIN
Page 9 of 29
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
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
Ma’ayanot Yeshiva High School Summer Review Packet-2018 R. BERNSTEIN
Page 10 of 29
E. (2) Multiplying Mixed Numbers
Step 1 – Convert the mixed numbers to improper fractions first
Step 2 – Multiply the numerators
Step 3 – Multiply the denominators
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
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 4 – Multiply the denominators
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
Step 5 – Multiply the denominators
Step 6 – Change the answer to lowest terms
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
Ma’ayanot Yeshiva High School Summer Review Packet-2018 R. BERNSTEIN
Page 11 of 29
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
Step 5 – Change the answer to lowest terms
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
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 **
Ma’ayanot Yeshiva High School Summer Review Packet-2018 R. BERNSTEIN
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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
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
Ma’ayanot Yeshiva High School Summer Review Packet-2018 R. BERNSTEIN
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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
9.. 121
12 10. 2
20
16 11. 5
14
8 12. 3
25
10
Ma’ayanot Yeshiva High School Summer Review Packet-2018 R. BERNSTEIN
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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
?
Ma’ayanot Yeshiva High School Summer Review Packet-2018 R. BERNSTEIN
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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 =
13. 2
1 × 1
3
1 = 14. 3
16
1 ×
5
1 = 15. 18 × 1
2
1 = 16. 16 × 2
8
1 =
Ma’ayanot Yeshiva High School Summer Review Packet-2018 R. BERNSTEIN
Page 16 of 29
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 =
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 =
Ma’ayanot Yeshiva High School Summer Review Packet-2018 R. BERNSTEIN
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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=
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=
Ma’ayanot Yeshiva High School Summer Review Packet-2018 R. BERNSTEIN
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9. 18
5 +
16
13 = 10. 2
3
2 +
9
4 = 11.
10
9 –
16
3 = 12.
8
7 –
2
1 =
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=
Ma’ayanot Yeshiva High School Summer Review Packet-2018 R. BERNSTEIN
Page 19 of 29
Converting between decimals and fractions and percents: please read this carefully
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!
Ma’ayanot Yeshiva High School Summer Review Packet-2018 R. BERNSTEIN
Page 20 of 29
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%
There's more on next page- read it all carefully!
Ma’ayanot Yeshiva High School Summer Review Packet-2018 R. BERNSTEIN
Page 21 of 29
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%
Ma’ayanot Yeshiva High School Summer Review Packet-2018 R. BERNSTEIN
Page 22 of 29
Do the following and show work: Please do all problems on this page
1 Write these decimals as fractions:
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% = …………. 12% = …………. 67% = …………. 17.5% = ………….
Ma’ayanot Yeshiva High School Summer Review Packet-2018 R. BERNSTEIN
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4 Write these percentages as fractions:
20% = …………. 75% = …………. 5% = …………. 30% = ………….
40% = …………. 15% = …………. 24% = …………. 35% = ………….
5 Write these decimals as percentages:
0.25 = …………. 0.5 = …………. 0.7 = …………. 0.07 = ………….
0.45 = …………. 0.09 = …………. 0.4 = …………. 0.375 = ………….
6 Write these fractions as percentages:
101 = …………. 5
1 = …………. 109 = …………. 4
3 = ………….
54 = …………. 20
17 = …………. 31 = …………. 3
2 = ………….
Ma’ayanot Yeshiva High School Summer Review Packet-2018 R. BERNSTEIN
Page 24 of 29
Complete this table.
Fraction Decimal Percentage
101
51
103
52
21
53
107
54
109
41
43
Ma’ayanot Yeshiva High School Summer Review Packet-2018 R. BERNSTEIN
Page 25 of 29
Fill the gaps in the table.
YOU ARE NOT REQUIRED TO COMPLETE THIS PAGE, BUT PLEASE DO IF YOU HAVE TIME.
Percentage
Fraction Decimal
10%
0.2
103
40%
0.5
53
70%
0.8
109
25%
Ma’ayanot Yeshiva High School Summer Review Packet-2018 R. BERNSTEIN
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Unit: NUMBER RELATIONSHIPS and COMPUTATION Objective: Identify and determine equivalent forms of proper fractions as decimals, percents, and ratios. Key Concept: Ratio: a comparison of two numbers A ratio can be written in 3 ways: a:b a to b or a b
EX: Write the ratio as a fraction simplest form: 4 wins to 6 losses
Since the ratio can be written as: 6
4 we can the simplify to
3
2 or 2:3 or 2 to 3
1.) Write the ratio as a fraction simplest form: 12 boys to 15 girls
2.) Write the ratio as a fraction simplest form: 20 books to 24 magazines
3.) Write the ratio as a fraction simplest form: 10 circles to 15 triangles
4.) Write the ratio as a fraction simplest form: 8 cups to 2 servings
5.) Write the ratio as a fraction simplest form: 50 cars to 100 trucks
6.) Write the ratio as a fraction simplest form: 9 pencils to 11 pens
Ma’ayanot Yeshiva High School Summer Review Packet-2018 R. BERNSTEIN
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Objective: Compare and order fractions and decimals
A. Ordering fractions only: 1) rewrite each fraction as an equivalent fraction
using the LCD 2) Compare the numerators
EX: order the fractions 12
7;
8
3;
2
1 from least to greatest
1) LCD of 2, 8, and 12 is 24
2) 24
12
2
1
24
14
12
7
24
9
8
3
3) Comparing the numerators:
12
7
2
1
8
3
SHOW WORK
B. Ordering fractions and decimals: 1) Change the fractions to decimals 2) Compare the decimals
EX: order the numbers 0.3; ;8
3 and 0.38
from least to greatest
1) 375.08
3
2) Compare the decimals:
0.3 < 0.375 < 0.38
Therefore: 38.08
33.0
1.)
Order the fractions 4
3;
6
5;
3
2 from least to greatest
2.)
Order the numbers 0.78; ;4
3 and 0. 8 from least to greatest
3.)
Order the fractions 6
5;
10
7;
5
3 from least to greatest
4.)
Order the numbers ;10
3 ;5
1 and 0.25 from least to greatest
5.) Order the fractions 6
5;
9
5;
2
1 from least to greatest
6.)
Which number has the greatest value? 0.94; ;20
19 or 25
24
40-
40
56
60
24
375.0
000.38
Ma’ayanot Yeshiva High School Summer Review Packet-2018 R. BERNSTEIN
Page 28 of 29
Unit: NUMBER RELATIONSHIPS and COMPUTATION Objective: Determine 10, 20, 25, or 50 percent of a whole number. Example: Determine 25% of 40
Method 1: Write a proportion and solve
25
100 40
1
4 40
1 10
4 10 40
10
40 40
10
x
x
x
x
so x
Therefore 25% of 40 is 10.
Method 2: Change the percent to a decimal and multiply
25%= 0.25
0.25 X 40 = 10.00
Therefore 25% of 40 is 10.
1.) Determine 20% of 65.
Show using both methods Method 1: Method 2:
2.) Determine 50% of 120. Show using both methods Method 1: Method 2:
3.) Determine 25% of 20.
4.) Determine 10% of 35.
40 X 0.25 200 +800 10.00
Ma’ayanot Yeshiva High School Summer Review Packet-2018 R. BERNSTEIN
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5.) 20% of the 250 students ate pizza for lunch. How many students ate pizza?
6.) Nadia saved 10% on her CD purchase. If the CD originally cost $24.90, how much did she save?
7) Juan answered 25
24questions correctly on his quiz.
What percent of the questions did he get correct?
8) 78% of the class completed their homework last night. What fraction of the class completed their homework?
YOU'RE DONE!
HAND IT IN TO YOUR TEACHER ON THE FIRST DAY OF CLASS