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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!
MA'AYANOT YESHIVA HIGH SCHOOL FOR GIRLS NAME:______________________________________
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
MA'AYANOT YESHIVA HIGH SCHOOL FOR GIRLS NAME:______________________________________
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
MA'AYANOT YESHIVA HIGH SCHOOL FOR GIRLS NAME:______________________________________
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.
MA'AYANOT YESHIVA HIGH SCHOOL FOR GIRLS NAME:______________________________________
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
MA'AYANOT YESHIVA HIGH SCHOOL FOR GIRLS NAME:______________________________________
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
MA'AYANOT YESHIVA HIGH SCHOOL FOR GIRLS NAME:______________________________________
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
MA'AYANOT YESHIVA HIGH SCHOOL FOR GIRLS NAME:______________________________________
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
MA'AYANOT YESHIVA HIGH SCHOOL FOR GIRLS NAME:______________________________________
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
MA'AYANOT YESHIVA HIGH SCHOOL FOR GIRLS NAME:______________________________________
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
MA'AYANOT YESHIVA HIGH SCHOOL FOR GIRLS NAME:______________________________________
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 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
MA'AYANOT YESHIVA HIGH SCHOOL FOR GIRLS NAME:______________________________________
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 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 FOR GIRLS NAME:______________________________________
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
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
MA'AYANOT YESHIVA HIGH SCHOOL FOR GIRLS NAME:______________________________________
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 **
MA'AYANOT YESHIVA HIGH SCHOOL FOR GIRLS NAME:______________________________________
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
MA'AYANOT YESHIVA HIGH SCHOOL FOR GIRLS NAME:______________________________________
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
MA'AYANOT YESHIVA HIGH SCHOOL FOR GIRLS NAME:______________________________________
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
?
MA'AYANOT YESHIVA HIGH SCHOOL FOR GIRLS NAME:______________________________________
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 =
MA'AYANOT YESHIVA HIGH SCHOOL FOR GIRLS NAME:______________________________________
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 =
MA'AYANOT YESHIVA HIGH SCHOOL FOR GIRLS NAME:______________________________________
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=
MA'AYANOT YESHIVA HIGH SCHOOL FOR GIRLS NAME:______________________________________
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 =
MA'AYANOT YESHIVA HIGH SCHOOL FOR GIRLS NAME:______________________________________
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=
MA'AYANOT YESHIVA HIGH SCHOOL FOR GIRLS NAME:______________________________________
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!
MA'AYANOT YESHIVA HIGH SCHOOL FOR GIRLS NAME:______________________________________
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%
MA'AYANOT YESHIVA HIGH SCHOOL FOR GIRLS NAME:______________________________________
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%
MA'AYANOT YESHIVA HIGH SCHOOL FOR GIRLS NAME:______________________________________
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% = ………….
MA'AYANOT YESHIVA HIGH SCHOOL FOR GIRLS NAME:______________________________________
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..
MA'AYANOT YESHIVA HIGH SCHOOL FOR GIRLS NAME:______________________________________
28
Percentage
Fraction Decimal
10%
0.2
103
40%
0.5
53
70%
0.8
109
25%
MA'AYANOT YESHIVA HIGH SCHOOL FOR GIRLS NAME:______________________________________
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
MA'AYANOT YESHIVA HIGH SCHOOL FOR GIRLS NAME:______________________________________
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
MA'AYANOT YESHIVA HIGH SCHOOL FOR GIRLS NAME:______________________________________
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
MA'AYANOT YESHIVA HIGH SCHOOL FOR GIRLS NAME:______________________________________
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?
MA'AYANOT YESHIVA HIGH SCHOOL FOR GIRLS NAME:______________________________________
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?
MA'AYANOT YESHIVA HIGH SCHOOL FOR GIRLS NAME:______________________________________
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