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End of Year Revision Packet
Mathematics – Level I
SABIS® School Network
Page 1 of 30
Mathematics
Level I
End of Year Revision Packet
Highlights:
 Test for divisibility.
 Apply prime factorization to find the GCF and LCM of two or more numbers.
 Compare, add, subtract, multiply, and divide fractions, mixed numbers, and whole numbers.
 Represent repeating and terminating decimals as ratios of integers.
 Add, subtract, multiply, and divide decimals.
 Simplify numeric and algebraic expressions.
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Compare positive and negative numbers.
Define and recognize rational, natural numbers, whole numbers, and integers.
Contrast the decimal representation of rational numbers with those of irrational numbers.
Apply the axioms of real numbers to simplify expressions.
Provide counterexamples to show that subtraction & division are neither commutative nor
associative.

Add, subtract, multiply, and divide real numbers expressed as fractions or decimals, placing
special emphasis on working with negative numbers.
Simplify algebraic expressions involving like terms.
Simplify and evaluate expressions involving sums, differences, products, quotients, and
exponents.
Apply the rules of powers to simplify numeric and algebraic expressions.
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Find the solution set of a linear equation in one variable.
Solve linear inequalities by adding, subtracting, multiplying, or dividing.
Write algebraic expressions involving one or more operations or word phrases.
Write algebraic expressions for word phrases with two related variables involving concepts
such as area, perimeter, averages, quotients, products, differences, and sums.
Translate a word sentence into an algebraic sentence. Solve for the unknown.
Solve word problems on topics such as age, money, speed, and perimeter by translating the
words to math equations or inequalities and solving them.
Express the ratio of a and b as a fraction and as a decimal.
Express a ratio in its simplest form.
Find the unit ratio equivalent to a given ratio.
Find quantities given their ratios.
Use the cross-multiplication property to solve for an unknown in a proportion.
Solve word problems involving proportions.
Convert between percents, ratios, fractions, and decimals.
Given a% of b is c, solve for an unknown.
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End of Year Revision Packet
Mathematics – Level I
Page 2 of 30
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Solve word problems on profit, loss, discount, taxes, percent change, percent error, and
simple interest.
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Recognize a direct variation/proportionality. Given one instance, find another.
Graph direct variations.
Solve word problems on rate and speed.
Calculate rates of change.
Solve problems involving direct proportionality.
Estimate actual distances by reading a map.
Identify the x-intercept, y-intercept, and slope of a line of a linear relation.
Analyze graphs of linear relations.
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Calculate and use the measures of spread of a data set.
Interpret the data displayed in different types of graphs.
Use the visual properties of box-and-whisker plots to analyze sets of spread data.
Group and organize continuous data in a frequency table and draw a histogram.
Given a survey, identify the population and the size of a sample.
Identify a random sample and ways of sampling to ensure randomness. Understand the
benefit of sampling and collecting data properly.
Identify random and fair samples. Recognize that data represented in the absolute form, as
opposed to relative, might be misleading.
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Use organized lists, tables, and tree diagrams to list the different possible outcomes in a
given context.
Use the fundamental principle of counting to find the total number of outcomes in a given
context.
Use adequate counting techniques to find the probability of an event.
Analyze and solve word problems involving probability and counting.
Determine probabilities based on previous experience.
Discuss the consistency and reasonableness of experimental outcomes and predictions.
Judge the fairness of a game.
Interpret the result of a simulation.
SABIS® Proprietary
End of Year Revision Packet
Mathematics – Level I
Page 3 of 30
Vocabulary:
 Natural numbers, Whole numbers, Divisibility, Exponent, Base, Factor, Prime number,
Composite number, Relatively prime, Prime factorization, GCF, LCM, Multiple
 Fractions, Equivalent fractions, Comparing fractions, Improper fractions, Mixed numbers,
Reciprocal, Number line
 Decimal fractions, Tenths, Hundredths, Thousandths, Place value, Value
 Algebraic expression, Numeric expression, Variables, Substitution principle, Domain of a
variable, Formulas, Evaluate, Negative numbers, Integers, Real numbers, Rational numbers,
Order
 Axioms, Theorems, Axioms of equality, Reflexive axiom, Symmetric axiom, Transitive
axiom, Closure axioms of addition and multiplication, Commutative axioms of addition and
multiplication, Associative axioms of addition and multiplication, Identity axiom of
addition, Axiom of additive inverses, Identity axiom of multiplication, Axiom of
multiplicative inverses, Distributive axiom
 Numerical coefficient, Like/similar terms, Simplify, Property of the opposite of a sum,
Property of opposites in products, Product property of quotients
 Power, Absolute value, Order of operations
 Equation, Solution set, Solve, Inverse operations, Addition property of equality,
Multiplication property of equality, Inequality
 Word phrases, Plus, Added to, Incremented, Sum, Total, More than, Increased by, Minus,
Subtracted from, Less than, Decreased by, Reduced by, Diminished by, Difference, Times,
Multiplied by, Product, Double of, Twice, Divided by, Quotient, Unknown
 Ratio, Equivalent ratios, Unit ratios, Proportion, Cross-multiplication property
 Percentage, Profit, Loss, Discount, Tax, Percent Change, Errors in measurement, Simple
interest
 Direct variation, Proportionality, Table, Vary, Coordinate plane, Graph
 Rate, Unit pricing, Speed, Scale, Map
 Linear relation, Slope, Sign of a slope, Intercept
 Set of data, Mean, Mode, Median, Range, Line plot, Stem-and-leaf plot, Bar graph,
Box-and-whisker plot, Line graph, Circle graph, Histogram, Survey, Sampling, Bias
 Tree diagrams, Fundamental principle of counting, Probability, Fair, Simulation
SABIS® Proprietary
End of Year Revision Packet
Mathematics – Level I
Page 4 of 30
Level I Revision Exercises:
Ch. 1
Section 1
Essentials of Arithmetic
Natural Numbers
1. Which of the following numbers is divisible by 11?
a) 55,517
b) 194,370
c) 301,456
2. Test each of the following numbers for divisibility by 2, 3, 5, 9, and 11.
a) 13,310
b) 604,877
c) 2,277,000
3. Give the prime factorization of 75. What are the prime factors of 75?
4. Find the greatest common factor of each pair of numbers.
a) 16 and 32
b) 1 and 50
c) 48 and 80
5. Find the least common multiple of each pair of numbers.
a) 16 and 32
b) 1 and 50
c) 48 and 80
6. Two pieces of cloth measuring 45 cm and 60 cm are to be cut into the longest possible strips
of equal length. How long will the strips of cloth be?
7. a) Find the smallest number which when divided by either 11 or 13 leaves a remainder of 7.
b) Find the smallest number which when divided by 10, 14, and 20 leaves a remainder of 8 in
each case.
Section 2
Fractions
1. Fill in the missing number to make the fractions equivalent.
14 ?
72 ?
48 24



a) 18 72
b) 216 ?
c) 84 7
2. Order the fractions from least to greatest.
8
7
12
8
5
6
1
4
3. Add. Express your answer in simplest form, or as a mixed number if necessary.
5 3
3 2
1 1



a) 8 5
b) 6 10
c) 6 3
4. Subtract. Express your answer in simplest form.
6 3
9 7


7
5
15
25
a)
b)
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7 5

8
12
c)
2
3
Mathematics – Level I
End of Year Revision Packet
Page 5 of 30
5. Add. Express your answer as a mixed number in simplest form.
1
1 3
2
5
7
3
1 2 4
2 3
4 1
12
20
a) 7 5
b) 25 15
c) 8
6. Subtract. Express your answer in simplest form.
6
3
1
5
7 2
5 2
12
7
a) 7
b) 12
7
c)
13
17
2
20
20
7. Multiply. Express your answer in simplest form, or as a mixed number if necessary.
3 5
4 7
17

6

36
a) 5 5
b) 8 7
c)
8
5
d) 15
72 32

f) 96 48
3 5

e) 50 2
8. Divide. Express your answer in simplest form, or as a mixed number if necessary.
1
4
3
1
3 1
1 2
3 5
5 1
7
3
a) 2
b) 7
c) 4 2
82
d)
3
4
6 1
e)
3
2  22
f) 4
3
5
2
1
5 pounds. Find the approximate weight,
9. An object whose mass is 1 kilogram weighs about
5
7
in pounds, of a television set whose mass is 8 kilograms.
5
1
1
1
10. Mona mixed 4 liters of orange juice with 8 liter of peach juice and 2 liters of carrot
1
juice. She plans to serve the mix in glasses of 4 liter. How many such glasses can she serve?
1
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End of Year Revision Packet
Section 3
Mathematics – Level I
Page 6 of 30
Representation of Decimals
1. Round to the nearest whole number.
a) 23.91
b) 23.42
c) 23.5
521
2. a) What digit is in the thousandths place of 1,000 ? Write the fraction in decimal form.
2
b) What digit is in the thousandths place of 1,000 ? Write the fraction in decimal form.
136
c) What digit is in the ten-thousandths place of 1,000 ? Write the fraction in decimal form up
to 4 decimal places.
3. What number is 2 hundredths more than 547.307?
Section 4
Operations With Decimals
1. Compute.
a) 1.3 + 5.5
b) 0.9 + 0.5
c) 41.4 + 23.99
d) 8 + 12.7 + 144.329
e) 65.82 + 7.981
f) 61 + 54.3 + 7.91
g) 0.75  0.34
h) 1.2  0.08
i) 0.3  0.02
j) 45.921  8.92
k) 54.27  7.05
l) 5.00  2.34
m) 4.2  4
n) 23.3  25
o) 8.4  17
p) 53.063  5
q) 15.025  29
r) 81.25  18
s) 70.89  3
t) 6.28  4
u) 0.904  8
v) 17.5  0.7
w) 6  0.12
x) 0.169  1.3
2. Find the price of 1.83 pounds of flour costing $0.85 per pound. Round your answer to the
nearest cent.
3. Divide. If the division does not terminate or repeat after three decimal places, stop and give
the answer rounded to two decimal places.
a) 30.3  9
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b) 42  18
c) 56.6 ÷ 14
Mathematics – Level I
End of Year Revision Packet
Section 5
Page 7 of 30
Simplifying Numerical Expressions
1. Simplify.
a) 30 ÷ 5 + 1 + 2 × 5 – 3 × 2
b) 30 ÷ 5 + [(1 + 2) × 5 – 3] × 2
c) 100 – [15 + 3 × (28 – 8)]
d) 100 – 15 + 3 × (28 – 8)
e) 20.5 – [20 – 5 × (10 ÷ 4)]
f) (25.5 – 20 – 5) × (10 ÷ 4)
g) 50 – 4 × 1.5
h) 33 × 22 – 54 ÷ 5
i) 92 – (102 – 4 × 7)
j) 34 + 24 × 5
k) 70 + 82 ÷ 23
11 1 2
 
m) 12 4 3
l) 53 – 72 ÷ 2
11 1 2
 
n) 12 4 3
11 1 2
 
o) 12 4 3
1 1 1 1 1
   
p) 3 3 3 9 9
4 1 1
  3
q) 5 2 2
9
1
 2   10
8
r) 4
2
Section 6
Algebraic and Numerical Expressions
1. Evaluate each expression for x = 2 and y = 5.
a) x + 2y
b) y + 2x
d) 4y + (2x + 6)2
e) 2y  (4x + 2)
c) xy + 7
 x 
 
f) xy + 25  y 
2. Let the domain of w be {3, 8, 12, 14, 17}.
What are the possible values of each of the following expressions?
a) w + 4
b) 3w
c) 2w – 2
Chapter Summary
Chapter Test
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TB read pages 52-55
TB pages 56-58
Mathematics – Level I
End of Year Revision Packet
Ch. 2
Section 1
Page 8 of 30
The Set of Real Numbers
Negative Numbers
1. Write the numbers in order from least to greatest.
a) 0, +1, 2, 1, +2
b) 14, +31, 41, 13, +34, 43
c) +0.75, 1.25, +1.25, 1.75, +2.25
d) 0.05, +0.5, 0.8, 0.75, +0.95, +0.55
1
1
1
1
1
1
1
 ,  ,  3 ,  5 ,  3 ,  , 1
2
2
2
2
2
2
e) 2
2. Simplify.
a) – (+5.3)
Section 2
b) – (–15)
  19  
    
c)   2  
d) – [– (–10)]
Rational Numbers
1. Verify whether each of the following is a decimal fraction by writing an equivalent fraction
whose denominator is a power of 10.
2
4
9
1
a) 7
b) 125
c) 30
d) 64
Section 4
Axioms of Real Numbers
1. Simplify.
a) 2b[4(b + 3) + 5(4 + 3b) + 6b] + 4[5b + 3(2b + 7)]
b) 7[3x2 + (x2 + x + 5)] + 2[2(x2 + x + 7) + 2x]
2 x 3x x
 
4 2
c) 7
3
1
 6x  4   2  8x 
2
d) 2
Chapter Summary
Chapter Test
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TB read pages 86-88
TB pages 89, 90
Mathematics – Level I
End of Year Revision Packet
Ch. 3
Section 1
Page 9 of 30
Operations on Real Numbers
Addition of Real Numbers
1. Add.
a) 15 + (20)
b) 8 + (8)
c) 24 + (24)
d) 18 + 11
e) 7 + 7
f) –10 + 3
g) 10 + –3
h) –10 + –3
i) 3 + (–5.18)
 3  1
    
l)  2   2 
j) –2 + (–8.1)
 4
   0
m)  2 
k) –7 + 3.18
 4 4
 
n)  5  5
2. Mercury is liquid at room temperature. It melts at about –39C. If the temperature of a mass of
mercury starts at –55C and increases by 23C, does this mass of mercury melt?
Section 2
Subtraction of Real Numbers
1. Simplify.
a) 12 – (5)
b) 10 – (10)
c) 7 – (7)
d) 12 – 14
e) 8 – 8
f) –10 – 4
g) 10 – –4
h) –10 – –4
i) 13 – 6
j) 13 – (–6)
k) –13 – (–6)
l) –15 – (–8)
m) 0.8 – 2.6
n) 1.2 – 8
o) –1.9 – (–0.2)
p) 6 – 7.5
q) 1.5 – 13
r) 0.9 – 2
s) –7.5 – 2.5
t) –6.87 – (–1.1)
2. The price of a share of a certain stock is $18 on Friday. By Monday, it shows a change of —5.
Use a number line to find the new price of this share. Did the price of the share increase or
decrease?
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Mathematics – Level I
End of Year Revision Packet
Section 3
Page 10 of 30
Multiplication of Real Numbers
1. Simplify.
a) 4(8)
b) 6(12)
c) 3(11)
d) 5(28)
e) 6(2)(5)
f) 5(9)(4)
g) 3(3)(14)
h) 17(3)(1)
i) (7)(5)
j) 8(5)(5)
k) 10(0)
l) (1)(2)
m) (–11)(0.8)
n) 2.5 × 4
o) (2)(1.5)
p) (–0.25) 14
q) (6)(4.5)
r) 1.5(1.5)
s) 3(0.4)
t) 7(0.7)
2. An amusement park bought a Ferris Wheel for $93,000. 45,213 people paid $2 each to ride the
wheel in its first year of operation. Ignoring the cost of operation, did the park recover the cost
of the wheel? If so, how much excess money did the park make?
Section 4
Division of Real Numbers
1. Simplify.
a) 51  (–17)
b) (–48)  6
c) 121  (11)
d) (–335)  (5)
 5   7 
   
g)  8   8 
e) 20  (10  8)
 5  1
 1    3 
h)  9   9 
f) 26  (13)
3
4

3
i) 4
2. What is the average of six negative integers and their opposites? Explain how you got your
answer.
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End of Year Revision Packet
Section 5
Mathematics – Level I
Page 11 of 30
Simplifying Numerical Expressions
1. Simplify.
a) (8 + 15) + 5
b) (7 + 3)  4
c) (12 + 6)  2
d) 9 + [14 + (12)]
e) 21 + [81 + (31)]
f) 30 + [14 + 16]
g) (8 + 15) + 5
h) (7 + 3)  4
i) (12 + 6)  2
j) (7 – 23) × 5 + 9
k) 4 (–3 × 5)2 – 103
l) 62 ÷ (10 – 8)2 + 3 ×(–8)
m) 45  (72  2 – 28 ÷ 4)
n) (45
– 23  5)3 ÷ 4 1
10 1 14


   5  
p) 21 3 25  2 
o) 10
– (35 – 52) + (120 ÷ 22)3
7 1 1 1
   
q) 10 3  5 2 
4  1 3  10 9
1    
7
r)  2 4  21 15
1 1 1 1
 1   
3
2  5  10
s)
Section 6
Simplifying Algebraic Expressions
1. Simplify.
a) (2.25x + 7.25) + (1.25x + 0.25)
b) [2.2 + (3.8m)] + (8.8 + 4.2m)
c) [9.6y + (3.3)] + [2.9 + (12.5y)]
d) [3w  (92.7)] + (92.7  3w)
e) (10a  3)  14a
f) 7  (7b  14)
g) (12x + 4)  (4x + 2)
h) (6y  10)  (9y  15)
2. Evaluate
 p  q   r 
for p = –4.5, q = –8, and r = –3.
3. Write each expression as a product of two factors, one of which is −5.
a) 50x – 15y
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b) –30x – 95
Mathematics – Level I
End of Year Revision Packet
Section 7
Page 12 of 30
Powers
1. Simplify.
a) 6  52 + 64  42  15(90)
b) 7  24  23  4 + 7(60)
2
4
c) 2  3  5  2
53  52  43  4  12  10 
0
d)
e) 64  24  33  10
2. Simplify, assuming that x, y, m, and z are non-zero real numbers.
2
4
3
x2
z3
 x2
5
y

2
y
a)
b)
c)
 
 
Chapter Summary
Chapter Test
Ch. 4
Section 1
2
 z 2
TB read pages 145, 146
TB pages 147-149
Linear Equations and Inequalities in One Variable
Equations
1. Carlos scored 7 points higher on a test than Peter. If Peter’s score is represented by a and
Carlos’ score is represented by b, write an equation that relates the scores Carlos and Peter
achieved.
2. Find the solution set of the equation over the domain D.
If the equation has no solution, indicate so.
a) 3a = 81, D = {3, 4}
Section 2
b) 3a = 81, D ={3, 4}
Solving Equations of the Form x + a = b
1. Solve and check your answer.
a) x + 2.3 = 4
b) x + 1.5 = 10
c) 25 = x  125
d) 17 = n  127
e) 12.8 = r + 12.3
f) 1.35 = s + 5.65
g) 0.36 = x  0.47
2 2
x 
5 3
j)
h) 16 = c + 9
3 7
y 
8 12
k)
i) 32 = d + 23
7
9
x
10
l) 15
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End of Year Revision Packet
Section 3
Mathematics – Level I
Page 13 of 30
Solving Equations of the Form ax = b
1. Solve and check your answer.
a) 16x = 100
b) 10x = 72
d) 1.2x = 6
1
x  2
g) 2
e) 3x = 5
1
1 r  15
h) 2
c) 4x = 3
f) 5x = 14
1
5
1 x
6
i) 3
2. Karen bought 7 identical books for her classmates. She paid $42.35 for all the books. Write an
equation and solve it to find the price of one book. Check your answer.
Section 4
Solving Equations of the form n(ax + b) = c
1. Solve the equation. Check your answer.
a) 2y + 3 = 17
b) 15m  12 = 48
c) 3x + 9 = 6
d) 1.2x  6 = 12
e) 3x  1.4 = 1
f) 2y + 5 = 8.6
g) 0.4x  0.6 = 2
1
x 5  6
j) 3
h) 1.25x  0.5 = 2
i) 2.35y + 1.3 = 6
1
2
1
5
2
3
x   2
n  
x3 0
6
5
4
k) 5
l) 2
m) 3
2. Solve and explain. Check your answer.
a) 2(x + 3) = 7
b) 7(x  4) = 27
c) 12(3  5x) = 5
d) 42 = 9(2x  7)
e) 23 = 3(x + 7)
g) 2 + 5(1  3x) = 8
h) 9(x  4) = 27
f) 5(1  2x) = 6
4
  7  2c    12
i) 7
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Mathematics – Level I
End of Year Revision Packet
Section 5
Page 14 of 30
Solving Linear Equations in One Variable
1. Solve and check.
a) 3a  (6  a) = 36
b) 6b + (8  3b) = 35
c) 24 = 6c + 2c  8
d) 4[2z  3(6  4z)]  12(z  1) = 4
e) 3d + 7 = 7d + 3
f) 2s  6 = 9s + 15
4
1
z  z 9
5
h) 5
1
8
2 1
x   x
3
3 4
j) 2
g) 8n  15 = n  43
2
2
w 1  
3
i) 3
 2
3 1 
 3
k)

c  2  c  2

1
1
3 
 7 1 
d   6  d   5  d 
2
2 
 5 2 
l) 4

5x  3 6 x  1

3
m) 2
Section 7
2 x  1 3x  1

5
n) 4
Solving Linear Inequalities in One Variable
1. Solve and graph the solution set of each of the following inequalities.
a) x + 4 > 10
b) x + 2  5
c) x – 5 < 9
e) 3n < 15
f) 6n > 12
g) 3 < 6n
d) 6  x + 1
n 1

h) 4 2
x 2 1
 
k) 3 3 2
x 1 1
 
l) 6 3 2
3x 3 7
 
i) 4 5 10
4  3 
j)
x
7
2. a) In 2 years, Nick’s age will be more than 14. Let x denote Nick’s age now. Write an
inequality based on the given information.
b) In 3 years, Julia will be younger than 12 years old. Let x represent Julia’s age. Write an
inequality based on the given information.
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Ch. 5
Section 1
Mathematics – Level I
Page 15 of 30
Problem Solving
Writing Algebraic Expressions for Word Phrases
1. Write an algebraic expression for each phrase.
a) n is an even integer, give the next even integer.
b) n is an even integer, give the first odd integer after n.
c) h is an odd integer, give the next odd integer.
d) h is an odd integer, give the first even integer after h.
e) y is subtracted from fourteen
f) Five less than x
g) y is diminished by two
h) x decreased by sixteen
i) A half of y
j) One nth of d
k) Four fifths of c
l) y enlarged by a factor of 7
m) Fifty-nine plus four ninths of m
n) The opposite of 3 plus x is divided by 9
o) The quotient of nine by a number is incremented by four
p) Nine is divided by the product of a number with two
q) The perimeter of a rectangle of length 7 and width w
2. The sum of two numbers is 10. If x represents one number, write an expression in terms of x
for the other number increased by 12.
3. The difference of two numbers is 20. If x is the larger number, write an expression in terms of
x for 2 times the difference between the smaller number and 10.
4. The area of a rectangle is y square units. If the width is 12 units, write an expression in terms
of y for the length of the rectangle.
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Section 2
Mathematics – Level I
Page 16 of 30
Word Problems
1. Write an algebraic sentence for each word sentence.
a) Six increased by three sevenths of a number is 8.
b) If 15 is subtracted from two thirds Mark’s weight, the result is 30.
c) The average of 13 and a number is less than twice the number.
d) The number of hours h increased by 13 equals 56.
2. Write an algebraic sentence and solve. Check your answer.
a) When 13.6 is added to three times a number, the result is 73.9. Find the number.
b) A number decreased by 16 equals 105. Find the number.
c) Five times a whole number is more than 92. Find the possible values of this number.
3. Linda scored 72 on her first algebra test. What must she score on the second test if her average
score on both tests is to be 81?
4. A movie theater has a certain number of rows of seats.
a) Let n represent the number of rows.
b) If each row has 25 seats, write an expression that represents the total number of seats in
the theater.
c) If exactly two of the rows have 4 seats fewer than the others, write an expression that
represents the total number of seats in the theater.
d) If it is known that the total number of seats in the theater as described in part c) is 517,
then how many rows of seats does the theater have?
5. The width of a rectangle is 3 cm less than its length. If the perimeter of the rectangle is 30 cm,
find the dimensions of the rectangle.
6. A number of passengers were on a bus. At the first stop, 6 more passengers got on the bus, and
no one got off. At the second station, as many passengers got on the bus as there were on it.
No one got off. At the last stop, 24 passengers got out of the bus and there were no passengers
left. How many passengers were on the bus before the first stop?
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Ch. 6
Section 1
Page 17 of 30
Variations
Ratios
1. A school has 735 students and 30 teachers. What is the student-teacher ratio? Express your
answer as a unit ratio. In your own words, what does this unit ratio represent?
2. A grocer sells 5 pounds of apples for $4.00. What is the ratio of dollars to pounds? Express
your answer as a unit ratio. In your own words, what does this unit ratio represent?
3. Two out of every five students in a certain school wear glasses. How many students in this
school wear glasses if the total number of students is 1,650?
4. The lengths of the sides of a triangle are in the ratio 2:4:5. The perimeter of the triangle is
44 cm. Find the length of each side.
5
7
5. A recipe calls for 9 cup of orange juice and 12 cup of pineapple juice. What is the ratio of
orange juice to pineapple juice in this recipe? Express your answer in the form a:b where a
and b are whole numbers with no common factors.
Section 2
Direct Variation/Proportionality
In 1 – 8, solve for x.
1 x
4 36


1. 3 21
2. 5 x
x 19

3. 7 14
x 11

4. 12 24
3 x2

8
5. 4
1
3

7. 4  x 5
12
1

8. x  13 2
2
6

6. 9 18x
9. The ratio of boys to girls in Grade 8 is 5 to 3. How many girls are there in Grade 8 if there are
80 students in Grade 8?
10. Fill in the blank entries in each table, given the formula of the relation between the two
variables.
a) y = 12x
x
–3
2
1
0
1
y
11. Assume y varies directly as x.
a) y =  27 when x =  3. Find y when x = 7.
b) y = 5.5 when x = 11. Find y when x = 94.
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12. Write an equation for the relation between x and y in each of the graphs below.
a)
b)
7
6
5
4
3
2
y
7
6
5
4
3
2
Section 3
y
7
6
5
4
3
2
1
1
0
c)
1
2 3 4 5 6 7 x
0
y
1
1
2 3 4 5 6 7 x
0
1
2 3 4 5 6 7 x
Applications
1. Amy has a choice between two sizes of bottled orange juice in a supermarket. One is 750 mL
and sells for $1.50 while the other is 1 L and sells for $2.00. Which is cheaper per mL, the
750 mL bottle or the 1 L bottle?
2. Flower shop A sells tulips at the rate of $5.40 per dozen.
Flower shop B sells tulips at the rate of $6.00 per dozen.
Tia went to the less expensive store and Mia went to other one. If each spent $9.00 buying
tulips, how many fewer tulips did Mia get than Tia?
3. Job A pays $950 for 180 hours of work. Job B pays $6.50 for each hour’s work. Which job
pays a higher salary? Explain.
1
3
4. Valeria can walk 2 a mile in 4 of an hour.
a) What is Valeria’s pace in miles per hour?
2
b) If Valeria can keep walking at the same pace, what distance can she walk in
1
2 hours?
5. Car A travels 50 miles on 2 gallons of fuel. Car B uses 3 gallons of fuel to travel 70 miles.
a) How many miles can car A travel per gallon of fuel? How many gallons of fuel does it
take car A to travel 1 mile?
b) How many gallons of fuel does it take car B to travel 1 mile? How many miles can car B
travel per gallon of fuel?
c) With regards to fuel consumption, which one of the two cars is more economical?
6. The scale on a map shows that 5 centimeters = 4 kilometers.
a) What number of centimeters on the map represents an actual distance of
5 kilometers?
b) What is the actual number of kilometers that is represented by 4 centimeters on the map?
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7. Jana measured the distance between two cities on a map and found it to be 6 inches. What was
the
1 actual distance between the two cities if the scale on the map was
2 inch:8 miles?
Section 4
Percent
1. Write each percentage as a ratio.
a) 0.25%
b) 35%
c) 180%
2. Write each percentage as a decimal.
a) 65%
b) 35%
c) 0.75%
3. Write each percentage as a fraction.
a) 25%
b) 0.5%
c) 0.05%
4. Five out of 14 number 1 hits on the billboard charts this year were by female vocalists. What
percent of the number 1 hits were by female vocalists?
5. A professional basketball player missed 41 free throws out of 820. What percentage of the
shots did he miss?
6. A jeweler buys an adornment for $11,000. He sells it later for $10,900. What percent of the
initial investment is the loss?
7. What is the final price of an item on sale if the discount is 5% and the marked price is $120?
8. During a sale, the price of every item is reduced by 15%. An item is priced at $19.55 after the
reduction. What is its original price?
9. Fred bought a coat at a 40% discount. If he paid $140 for the coat, what was the original
price?
10. a) A store owner paid $40 for a jacket. She marked up the price of the jacket by 25% to
determine its selling price. What is the selling price of the jacket?
b) A customer buys a shirt that has an original selling price of $50. The shirt is discounted by
30%. The customer must pay a 5% sales tax on the discounted price of the shirt. What is
the total amount the customer pays for the discounted shirt?
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11. Nancy got a 15% pay raise this year. If her new salary is $4,200 a month, what was her
monthly salary before the raise?
12. Michelle deposited $3,000 in bank A at an annual interest rate of 4% and $2,400 in bank B at
an annual interest rate of 3.5%. How much did Michelle earn in interest in 1 year?
13. If $5,000 is invested for 3 years at an annual simple interest rate of 4%, what is the interest
earned during the 3-year period?
14. A department store places an item at 50% off. A week later, the item remains unsold, so the
store manager decides to take an additional 40% of the sale price. What percent of the
original price is the final price after both discounts are applied?
15. Items at a clothing store are on sale for 60% of the original price. Simona gets an additional
15% discount on the sale price for being part of the staff. If the orginal price of a jacket is
$200 and no sales tax applies, how much will Simona pay for the jacket?
16. To convert from kilograms to pounds, Emilia doubles the mass and adds 20% of the
Section 5
Patterns and Rules
1. Consider the sequence: 1, 3, 6, 10, 15, 21, …
a) What is the rule for generating this sequence?
b) What are the next three terms of this sequence?
Section 6
Linear Relations
1. In each of the following, given the slope and y-intercept of a line, write its equation in the
form y = mx + b.
1
a) Slope = 2 and y-intercept = –5
b) Slope = 3 and y-intercept = –5
c) Slope = 0 and y-intercept = –11
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Ch. 9
Section 1
Mathematics – Level I
Page 21 of 30
Statistics
Measures of Center and Variation
1. Miguel scored 78, 92, and 88 on three math exams. What must Miguel’s average on the next
two exams be so that his overall average on all five exams is 80?
2. Find the mode for each set of data.
a) {12, 12, 12, 15, 15, 15, 15, 20}
b) {10, 10, 10, 10, 10}
3. Find the median for each set of data.
a) {3, 6, 8, 10, 21}
b) {6, 8, 12, 16, 21, 32}
4. Find the mean absolute deviation for each set of data.
a) {3, 6, 8, 10, 21}
b) {6, 8, 12, 16, 21, 32}
5. A company tested 10 light bulbs from each of two brands for durability. The results are listed
in the table below.
Life Expectancy of a Light Bulb (in hours)
Brand A
890, 880, 800, 950, 20, 900, 880, 860, 830, 850
Brand B
840, 820, 880, 860, 790, 750, 780, 780, 750, 710
a) What is the mean life expectancy for the 10 light bulbs tested from each brand? Show your
work.
b) Based on the 10 bulbs tested, what is the median life for each of the brands?
c) Based on your answers in a) and b), which brand do you think is better? Give reasons for
your choice.
6. Ten college graduates received the following salaries, in thousands of U.S. dollars:
29, 34, 35, 36, 37, 38, 40, 41, 225, 300. Name the outliers in the set of data and describe their
effects on the mean and on the median.
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Graphical Representation of Data
1. The line plot shows the scores Tamika received on her math exams.
15
16
17
18
19
a) What is the range of Tamika’s scores on the five exams?
b) What is the mode of Tamika’s scores on the five exams?
c) What is the median of Tamika’s scores on the five exams?
d) What is the average of Tamika’s scores on the five exams?
e) Tamika will take one more math exam. Is it possible for Tamika to end up with an average
of 18?
2. The stem-and-leaf plot shows the number of students who attended the school play in
14 performances.
Stem
Leaf
6 4 4 7 9
7 0 2 5 8 8 8
Key
7 | 2 represents 72
8 4 6 7 7
a) What is the range of the number of students who attended the 14 performances?
b) What is the mode of the number of students who attended the 14 performances?
c) What is the median number of students who attended the 14 performances?
d) After the last two performances, the median of the number of students that attended all 16
performances was the same as that of the first 14. What must be true about the number of
students that attended the last two performances?
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3. Consider the following survey of 20 students from the U.S. and 20 students from Canada
about their favorite fruit.
Favorite Fruit
Number of Students
Fruit
U.S.
Canada
Apple
8
6
Peach
2
4
Orange
6
5
Banana
4
5
Plot the data using a bar graph. Label your graph appropriately.
4. Refer to the box-and-whisker plot shown below to answer the questions.
Girls
Boys
0
10
20
30
40
50
60
70
80
90
100
Grade Distribution for 100 Girls and 100 Boys
a) How many boys scored between 30 and 65?
b) How many girls scored between 40 and 90?
c) Below what value did 75% of all girls score?
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5. The chart below shows the salary distribution of 200 randomly picked female employees and
200 randomly picked male employees. The notation [a, b) stands for all numbers between a
and b, including a and excluding b.
Salary Distribution
Percentage of employees
40%
30%
20%
10%
0
< 20
[20 – 30)
[30 – 40) [40 – 50)
Salary in $1,000
> 50
Male employees
Female employees
a) How many female employees have a salary less than $40,000?
b) How many male employees have a salary greater than $40,000?
c) How many more female employees than male employees are in the $20,000 to $30,000
range?
6. A group of 20 students were asked to choose their favorite food from the following list:
hamburger, tacos, and pizza.
The results are shown in the table below.
Favorite Food
Food
Number of Votes
Hamburger
10
Tacos
4
Pizza
6
Draw a circle graph displaying the data. Give the measure of the angle formed in each of the
sectors. Show your work.
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Histograms
1. A survey concerning the duration of telephone calls was conducted. Fifty calls were chosen at
random. The table below lists the duration of each of the calls in minutes.
2
20
14
33
48
7
1
25
14
4
15
12
3
37
32
33
8
22
7
14
3
4
17
42
38
2
24
4
18
32
1
31
25
36
16
23
1
2
9
38
11
45
27
31
26
28
2
18
8
30
a) Tally the durations of the calls using intervals of 10 minutes each. Start from 0.
b) Draw a histogram.
2. Denzel tossed four coins 200 times and noted the number of heads appearing each time he
tossed the coins. He summarized his findings in the table below.
Outcomes of Tossing Four Coins
Number of
heads
Frequency
Cumulative
Frequency
0
10
10
1
52
62
2
84
3
42
4
12
a) Find the missing entries in the table under the “Cumulative Frequency” heading.
b) In your own words, what does it mean to have a cumulative frequency of 62 corresponding
to 1 heads?
c) For what percentage of the 200 times did the outcome of tossing the four coins show 2
heads?
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Page 26 of 30
Sampling and Sample Proportions
1. The sample size plays a key factor in designing a successful survey.
For populations in the millions, 1%  2% of the population is a good sample size.
For populations in the hundreds, 10%  15% of the population is a good sample size.
What is a good sample size to use if you want to survey the students in your school about their
favorite hobby? Describe a way of choosing the participants for the survey.
2. Give the sample size and classify the data collected in each of the following cases as a random
sample or not. Give reason(s) for your answers. Describe a way of randomly choosing the
participants for each of the surveys where you think the sample is not random.
a) Twenty names were randomly chosen from the California telephone book. The chosen
numbers were called and the respondents were asked whether or not they support enforcing
the seatbelt law. Of the 20 attempted calls, 18 responded.
b) The first 200 students to arrive at school were asked if they prefer that the school day
begins at 8:00 A.M. or at 8:30 A.M. The student body of the school has 1,008 students.
c) Every fifth person who enters a mall was asked about his/her favorite store at the mall. At
the mall entrance, 980 people were surveyed.
Bias and Misleading Data
1. The adjacent graph gives the
impression that the circulation of the
State News magazine doubled from
2008 to 2009. Explain why this graph
is misleading.
State News Circulation
Numbers, in thousands
Section 5
56
54
52
50
2008
2009
Year
2. A random sample of 50 men and 50 women were picked from a population consisting of 200
men and 800 women. They were asked if they support more spending for a new daycare
center. Is there any source of bias in the study? If so, what is it?
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Ch. 10
Section 1
Mathematics – Level I
Page 27 of 30
Counting and Predicting Chances
Methods of Listing and Counting
1. A person has 6 shirts, 5 pairs of pants, and 4 pairs of shoes. An outfit is made up of any one of
the shirts, any one of the pairs of pants, and any one of the pairs of shoes. How many different
outfits are there?
2. Books in a library are labeled by two letters followed by a nonzero digit. How many different
labels can be formed?
3. Products are assigned codes that consist of two letters followed by two different digits. How
many different codes can be formed?
Section 2
Probability
1. The cards shown below are placed on a table top facing down in random order and a card is
picked.
a) How many different possible outcomes are there?
b) Are the different outcomes equally likely to occur?
c) What is the probability the letter “C” is picked?
2. A pair of dice are rolled and the sum of the numbers rolled is observed.
The possible outcomes are 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12.
a) What is the probability of rolling a sum of 2 or 3?
b) What is the probability of rolling a sum greater than 3?
3. One marble is randomly picked from a bag containing four red marbles and four green
marbles. The color of the marble is recorded and the marble is placed back in the bag. What
is the probability of getting two marbles of the same color?
4. A wheel is divided into 36 sectors. 20 of the sectors are marked with the number 5. If the
wheel is spun, what is the probability it will stop at a sector marked by the number 5, given
that it is equally likely to stop at any of the 36 sectors?
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5. Refer to the diagram below to answer the question.
20 m
60 m
20 m
Seeds are spread out evenly in the rectangular lot. What is the probability the first bud will
blossom in the triangular region?
Section 3
Experimental Probability
1. A spinner awarded prizes 20 times out of the 50 times it was spun. What is the experimental
probability the spinner will award a prize the next time it is spun?
2. One hundred light bulbs were tested and 6 were found to be defective. What is the probability
a light bulb purchased from the same manufacturer is not defective?
3. A bag contains a collection of marbles. Each time a marble is taken out, its color is recorded
and then placed back in the bag before the next marble is taken out. This experiment is
repeated 100 times and the outcomes are summarized in the table below.
Red
10
Green
7
White
8
Blue
75
There are 50 red marbles in the bag.
a) What is the estimated number of blue marbles in the bag?
b) What is the estimated total number of marbles in the bag?
4. One hundred bags of sugar labeled 5 kg were weighed and the findings were recorded.
Measured weight Number of bags
[4.90 – 4.95)
12
[4.95 – 5.00)
48
[5.00 – 5.05)
33
[5.05 – 5.10)
7
If a similar bag is weighed, what is the probability that its weight is less than 5 kg?
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Page 29 of 30
Probability Models
1. Anna wants to estimate the number of people at a mall that will use the ATM machine. Based
on past data, the probability that a person who is selected at random in the mall uses the ATM
machine is 0.4. Anna designs a simulation to estimate the probability that exactly two of five
people selected at a mall will use the ATM machine. For the simulation, Anna uses a number
generator that generates random numbers.

Any number from 0 through 3 represents a person who uses the ATM machine.

Any number from 4 through 9 represents a person who does not use the ATM machine.
For each trial, Anna generates 5 numbers. Anna ran 20 trials of the simulation and recorded
the results in the following table:
45193
51236
43590
66125
83709
13742
74908
25104
31254
36789
19203
25786
80967
81825
94087
71876
42865
14023
80659
06374
a) In the simulation, one result was “80967.” What does this result simulate?
b) Use the results of the simulation to estimate the probability that 2 of 5 people selected at a
mall will use the ATM machine.
c) Make a table of all the possible outcomes of the experiment and use it to find the theoretical
probability that 2 of 5 people selected at a mall will use the ATM machine. Compare it with
the result you obtained in part b).
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2. The table below contains 200 random digits from 0 to 9.
2
3
1
2
6
0
8
6
4
3
8
4
2
1
8
3
5
7
2
4
3
6
6
9
6
2
9
0
2
0
0
1
2
4
8
9
4
6
6
6
6
6
3
9
6
8
5
2
9
3
7
7
7
0
2
6
5
2
4
4
0
6
4
3
9
8
7
1
6
0
0
6
6
3
4
3
5
8
9
5
2
9
5
6
1
5
6
3
2
7
0
9
3
0
8
8
6
6
4
5
8
4
5
1
7
3
9
3
0
0
8
7
4
4
3
2
6
3
5
9
1
6
5
1
7
5
9
5
0
7
8
2
5
8
4
4
4
3
4
2
0
5
8
9
1
9
7
0
1
7
7
0
6
1
8
0
3
2
2
6
1
5
1
4
1
9
7
9
6
5
0
3
0
7
9
2
1
5
8
7
9
8
7
9
5
7
8
0
2
2
8
8
1
9
7
0
4
8
3
0
a) Explain how you can use the table to simulate an experiment with two outcomes, success
and failure with the probability of success 0.65. Use the table to simulate the experiment 50
times.
b) Is the experimental probability close to the theoretical probability?
Chapter Summary
Chapter Test
SABIS® Proprietary
TB read page 200
TB pages 201–203