MTH1W · Exam review

Mathematics Exam Review

A practice exam with questions from all 8 units, then the key facts from every lesson, unit by unit. For a deeper look at one unit, open its test review.

Unit test reviews

Practice exam

40 questions, the same number from each unit. Each one shows the lesson it comes from.

Unit 1: Number Sets: Natural to Real
1
Which number is irrational?
Unit 1: Numbers Across Cultures and Time
2
A Maya number shows one dot on the top level and a shell on the bottom level. What is its value in base 10?
Unit 1: Powers, Zero and Negative Exponents, Scientific Notation
3
Evaluate $3^{4}$ .
Unit 1: Powers, Zero and Negative Exponents, Scientific Notation
4
A sheet of paper 0.1 mm thick is folded in half 8 times, doubling its thickness each time. How thick is the folded stack, in millimetres?
mm
Unit 1: The Exponent Laws
5
Multiply $(2 \times 1 0^{3}) (4 \times 1 0^{5})$ .
Unit 2: Integers: Location, Direction and Change
6
Which of these numbers is the greatest?
Unit 2: Fractions, Unit Fractions and Measuring
7
The end of a nail is 6 sixteenth-gaps past the 0 mark on an imperial ruler. What is its length in simplest form?
Unit 2: Operations with Positive and Negative Fractions
8
How many pieces $1 \frac{1}{3}$ ft long can be cut from an 8 ft board? (Ignore the width of the saw cut.)
pieces
Unit 2: Ratios, Rates and Proportions
9
Match each example with what it is.
3 cups of flour to 2 cups of sugar
2.50 dollars per kilogram
1 cm on the map is 500 m on the ground
x/12 = 5/8
Unit 2: Percentages: Increase, Decrease, Taxes and Discounts
10
After a 20% discount, a pair of boots costs $64. What was the original price?
Unit 3: Writing Expressions from Patterns and Words
11
What is the coefficient of $y$ in $7 x - y + 4$ ?
Unit 3: Equivalent Expressions and Simplifying
12
Simplify $4 x + 7 - x + 2$ .
Unit 3: Solving Linear Equations
13
Solve $4 x - 7 = 21$ .
Unit 3: Solving Linear Equations
14
Is $x = 4$ a solution of $5 x - 3 = 2 x + 9$ ? Answer yes or no.
Unit 3: Equations in Context and Rearranging Formulas
15
Match each formula with the rearrangement asked for.
1. A. $r = \frac{C}{2 π}$
2. B. $s = \frac{d}{t}$
3. C. $s = \frac{P}{4}$
4. D. $w = \frac{A}{l}$
$A = lw$ , solved for $w$
$d = s t$ , solved for $s$
$P = 4 s$ , solved for $s$
$C = 2 π r$ , solved for $r$
Unit 4: Variables, Expressions and Output in Python
16
In Python, what does the line x = 7 do?
Unit 4: Variables, Expressions and Output in Python
17
What does print(5 ** 2 - 4) print?
Unit 4: Loops: Patterns and Tables of Values
18
Put these lines in order to make a program that adds 1 + 2 + 3 + 4 + 5 and prints the answer once.
1. 1for n in range(1, 6):
2. 2print(total) (not indented)
3. 3 total = total + n (indented)
4. 4total = 0
Unit 4: Conditionals, Inequalities and Altering Code
19
How many numbers does this program print?for x in range(1, 21): if 4 * x - 3 > 50: print(x)
numbers
Unit 4: Conditionals, Inequalities and Altering Code
20
The lesson's phone-plan program starts with the parameters fee_a = 20, rate_a = 5, fee_b = 40 and rate_b = 1. To model Plan A raising its rate to 6 dollars per GB, which single line should you change?
Unit 5: Linear and Non-Linear Relations
21
In each table, $x$ goes $0, 1, 2, 3$ . Which list of $y$ -values comes from a linear relation?
Unit 5: Rate of Change and Initial Value
22
Match each representation to the place where you find the initial value.
Equation $y = 5 x + 12$
Graph
Table that includes x = 0
Story
Unit 5: Comparing Two Linear Relations
23
Plan P costs $C = 6 g + 18$ and Plan Q costs $C = 2 g + 42$ dollars for $g$ GB. For how many GB do they cost the same?
GB
Unit 5: Special Lines, xy = k and Inequalities
24
What is the $y$ -intercept of $x - y = 7$ ? Give the $y$ -coordinate.
Unit 5: Translating, Reflecting and Rotating Lines
25
A line $y = a x$ (with $a \neq = 0$ ) and its image after a 90° rotation about the origin have slopes whose product is always…
Unit 6: Data in Our Lives: Collection, Use and Consequences
26
A city wants to know how residents get to work. It surveys people waiting at subway stations. What is the main problem?
Unit 6: Measures of Centre, Spread and Box Plots
27
Find the mean of 4, 7, 7, 9, 13.
Unit 6: Scatter Plots, Correlation and Lines of Best Fit
28
You study how the number of hours a part-time worker works affects their weekly pay. Which variable belongs on the x-axis?
Unit 6: Scatter Plots, Correlation and Lines of Best Fit
29
Put the steps for making a prediction from a scatter plot in order.
1. 1Substitute the new x-value and state whether it is interpolation or extrapolation
2. 2Choose two points on the line and find its equation
3. 3Draw a line of best fit through the middle of the points
4. 4Plot the data with the independent variable on the x-axis
5. 5Describe the direction, strength and form of the correlation
Unit 6: The Mathematical Modelling Cycle
30
Why should a model be tested with data that was NOT used to build it?
Unit 7: Angles, Parallel Lines and Polygons
31
What is the complement of a $2 7^{\circ}$ angle?
Unit 7: Triangle and Circle Properties in Design
32
Two sides of a triangle are 5 cm and 12 cm. The third side must be longer than how many centimetres?
cm
Unit 7: The Pythagorean Theorem
33
A right triangle has legs 9 cm and 12 cm. How long is the hypotenuse?
cm
Unit 7: How Changing Dimensions Changes Area and Volume
34
Each edge of a cube is multiplied by 4. By what number is the volume multiplied?
Unit 7: Volume: Prisms, Pyramids, Cylinders and Cones
35
Put the steps in order to find how many litres a cylindrical tank with radius 40 cm and height 2 m holds.
1. 1Find the base area: π × 40² = 1600π cm²
2. 2Divide by 1000 to get litres: about 1005 L
3. 3Multiply by the height in centimetres (2 m = 200 cm): 1600π × 200 = 320 000π cm³
4. 4Write the formula for a cylinder: V = πr²h
Unit 8: Making Financial Decisions
36
Canada's annual average CPI was 157.1 in 2023 (2002 = 100). What does this mean?
Unit 8: Appreciation and Depreciation
37
A bike costs $600 and depreciates by $75 each year (straight-line). What is it worth after 4 years?
$
Unit 8: Interest and the Cost of Borrowing
38
How much simple interest does $800 earn in 3 years at 5% per year?
$
Unit 8: Interest and the Cost of Borrowing
39
Using the rule of 72, about how long does money take to double at 8% per year compounded annually?
Unit 8: Building and Adjusting a Budget
40
What is the name for savings kept aside to cover unexpected costs, such as a broken phone or a lost shift?

Key facts by unit

Unit 1: Number Sets and Powers

Number Sets: Natural to Real

$N \subset W \subset Z \subset Q \subset R$ ; the irrational numbers are real but sit outside $Q$ .
Rational: can be written as $\frac{a}{b}$ with integers $a$ , $b$ and $b \neq = 0$ . Its decimal terminates or repeats.
Irrational: its decimal never ends and never repeats (e.g. $2$ , $π$ ).
For a whole number $n$ , $n$ is rational only when $n$ is a perfect square.
Density: between any two different rational numbers there is another, such as their average $\frac{a + b}{2}$ .
The running totals of $\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots$ approach the limit 1 without ever passing it; the endless sum equals 1.

Numbers Across Cultures and Time

In base $b$ , the places are worth $1, b, b^{2}, b^{3}, \dots$ ; e.g. base 60 places are 1, 60, 3600.
Babylonian base 60 survives in 60 seconds per minute and 60 minutes per hour; their zero was a placeholder sign (by about 300 BCE).
Maya numerals: dot = 1, bar = 5, shell = 0, stacked with ones at the bottom and twenties above (base 20).
Brahmagupta (628 CE) gave rules for zero and negative numbers as numbers; Hindu–Arabic base 10 digits were popularized in Europe by Fibonacci (1202).
Chinese counting rods (Nine Chapters, Liu Hui 263 CE): red rods positive, black rods negative.
Golden ratio $φ = \frac{1 + 5}{2} \approx 1.618$ , irrational; ratios of neighbouring Fibonacci numbers approach it.

Powers, Zero and Negative Exponents, Scientific Notation

$a^{n}$ means $n$ factors of $a$ multiplied together (for a natural number $n$ ); $a^{1} = a$ .
$(- 2)^{4} = 16$ but $- 2^{4} = - 16$ : without brackets, the exponent applies before the negative sign.
$a^{0} = 1$ for any $a \neq = 0$ .
$a^{- n} = \frac{1}{a ^{n}}$ for $a \neq = 0$ ; a negative exponent gives a reciprocal, not a negative number ( $2^{- 3} = \frac{1}{8}$ ).
Scientific notation: $a \times 1 0^{n}$ with $1 \leq a < 10$ and $n$ an integer; positive $n$ for large numbers, negative $n$ for numbers between 0 and 1.

The Exponent Laws

Product law: $a^{m} \times a^{n} = a^{m + n}$ (same base: keep it, add exponents).
Quotient law: $\frac{a ^{m}}{a ^{n}} = a^{m - n}$ for $a \neq = 0$ .
Power of a power: $(a^{m})^{n} = a^{mn}$ .
Power of a product: $(ab)^{n} = a^{n} b^{n}$ ; e.g. $(2 x)^{3} = 8 x^{3}$ .
The laws apply to multiplying and dividing powers, never to adding them: $x^{2} + x^{3} \neq = x^{5}$ .

Unit 2: Rational Numbers and Proportional Reasoning

Integers: Location, Direction and Change

Integers: $\dots, - 2, - 1, 0, 1, 2, \dots$ ; zero is neither positive nor negative.
On a number line the number further right is greater, so $- 2 > - 7$ .
Change $=$ final $-$ initial; a positive change is an increase, a negative change is a decrease.
Subtracting a negative is adding its opposite: $a - (- b) = a + b$ .
One negative sign makes a fraction negative: $\frac{- a}{b} = \frac{a}{- b} = - \frac{a}{b}$ ; two negatives make it positive: $\frac{- a}{- b} = \frac{a}{b}$ (for $b \neq = 0$ ).
Rate of change $= \frac{change}{time taken}$ ; its sign tells you the direction.

Fractions, Unit Fractions and Measuring

A unit fraction is $\frac{1}{n}$ ; the fraction $\frac{a}{n}$ is $a$ copies of $\frac{1}{n}$ .
For positive unit fractions, a larger denominator means a smaller piece: $\frac{1}{16} < \frac{1}{8}$ .
Equivalent fractions: $\frac{a}{b} = \frac{a \times k}{b \times k}$ for any $k \neq = 0$ .
A typical imperial ruler has 16 gaps per inch; read a length by counting sixteenths past the last inch mark, then simplify.
1 in $= 2.54$ cm exactly; 1 cup is usually taken as 250 mL in Canadian recipes.

Operations with Positive and Negative Fractions

Add or subtract: $\frac{a}{d} \pm \frac{b}{d} = \frac{a \pm b}{d}$ , after rewriting with a common denominator $d$ .
Multiply: $\frac{a}{b} \times \frac{c}{d} = \frac{a c}{b d}$ (cancel common factors first).
Divide: $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$ , for $c \neq = 0$ .
Mixed number to improper fraction: $p \frac{q}{r} = \frac{p r + q}{r}$ ; and $- p \frac{q}{r} = - \frac{p r + q}{r}$ .
Sign rules for multiplying and dividing: same signs give a positive result, different signs give a negative result.

Ratios, Rates and Proportions

Equivalent ratios: $a : b = (ka) : (kb)$ for any $k \neq = 0$ .
Unit rate $= \frac{amount}{number of units}$ ; the lowest unit price is the best buy.
A proportion states two ratios are equal: $\frac{a}{b} = \frac{c}{d}$ .
Cross-multiplying: if $\frac{a}{b} = \frac{c}{d}$ (with $b, d \neq = 0$ ), then $a d = b c$ .
Map scale $1 : n$ : real distance $=$ map distance $\times n$ (then convert units).

Percentages: Increase, Decrease, Taxes and Discounts

$r %$ of an amount $= \frac{r}{100} \times$ amount.
Percent change $= \frac{new - old}{old} \times 100%$ (old value not zero; positive is an increase, negative a decrease).
Increase by $r %$ : multiply by $1 + \frac{r}{100}$ . Decrease by $r %$ : multiply by $1 - \frac{r}{100}$ .
Ontario HST: total $=$ price $\times 1.13$ ; pre-tax price $=$ total $\div 1.13$ .
Successive percent changes combine by multiplying the multipliers, not by adding the percents.

Unit 3: Algebraic Expressions and Equations

Writing Expressions from Patterns and Words

A variable stands for a number that can change; a coefficient multiplies a variable; a constant is a term with no variable.
An expression has no equals sign ( $3 n + 2$ ); an equation says two things are equal ( $3 n + 2 = 20$ ).
The sign in front of a term belongs to it: in $4 - 3 x$ the coefficient of $x$ is $- 3$ . The coefficient of $x$ alone is $1$ , and of $- x$ is $- 1$ .
For a pattern that grows by the same amount each stage: expression = (change) $n$ + (stage 0 value), where stage 0 value = stage 1 value − change.
"5 less than $n$ " is $n - 5$ ; "twice $n$ " is $2 n$ ; "the square of $n$ " is $n^{2}$ ; "triple the sum of $n$ and 6" is $3 (n + 6)$ .

Equivalent Expressions and Simplifying

Like terms have the same variables with the same exponents; collect them by adding coefficients: $3 x + 5 x = 8 x$ .
Distributive property: $a (b + c) = ab + a c$ . Multiply every term inside the brackets.
A negative outside brackets changes every sign inside: $- 2 (x - 4) = - 2 x + 8$ and $- (x + 3) = - x - 3$ .
Two expressions are equivalent if they are equal for every value of the variable. One value where they differ proves they are not equivalent.
A zero pair ( $+ 1$ and $- 1$ ) has value $0$ and can be removed.

Solving Linear Equations

Whatever you do to one side of an equation, do to the other side (and to every term on that side).
Undo operations in reverse order: for $3 x + 2 = 11$ , subtract 2 first, then divide by 3.
With variables on both sides, gather the $x$ terms on one side and the constants on the other.
Clear fractions by multiplying every term on both sides by the lowest common denominator; keep each numerator in brackets.
Verify by substituting into the original equation: the left side and the right side must give the same number.
If the $x$ terms cancel: a false statement (like $3 = 5$ ) means no solution; a true statement (like $2 = 2$ ) means every $x$ is a solution.

Equations in Context and Rearranging Formulas

Word problems: define the variable (with units), write the equation, solve, check in the situation, answer in a sentence.
To rearrange a formula, use the same inverse operations as solving an equation, treating the other letters like numbers.
Undo in reverse order: $P = 2 l + 2 w$ becomes $w = \frac{P - 2 l}{2}$ (subtract $2 l$ , then divide by 2).
$d = s t$ rearranges to $t = \frac{d}{s}$ and $s = \frac{d}{t}$ ; $C = 2 π r$ rearranges to $r = \frac{C}{2 π}$ .
A rearranged formula with a variable in the denominator only works when that variable is not zero.

Unit 4: Coding with Python

Variables, Expressions and Output in Python

An assignment name = expression works out the right side using the current values, then stores the result in the variable on the left. The left side must be a variable name, not a calculation.
= stores a value; == compares two values and gives True or False.
Operators: +, -, * (multiply), / (divide, decimal answer), // (divide and round down), % (remainder), ** (power).
Order of operations: brackets, then **, then multiplication and division from left to right, then addition and subtraction from left to right. Python has no implied multiplication.
A variable keeps its value until a new assignment changes it. Changing one variable does not recalculate variables that were worked out from it earlier.
A parameter is a value that stays fixed while the program runs but can be edited to model a new situation, like $m$ and $b$ in $y = m x + b$ .

Loops: Patterns and Tables of Values

for x in range(a, b): runs the indented body once for each integer x from a up to, but not including, b.
range(n) gives 0, 1, …, n − 1 (that is n values, for a positive whole number n). range(a, b, step) counts by step; a negative step counts down.
A table of values from a loop has three steps: choose the x-values with range, calculate y with the rule, print both.
An accumulator starts at 0 before the loop, has each new amount added inside the loop, and is printed after the loop.
When x goes up by 1 on each pass, a constant change in y means the relation is linear, and that change is the coefficient of x (the $m$ in $y = m x + b$ ).

Conditionals, Inequalities and Altering Code

Comparison operators: <, >, <= ( $\leq$ ), >= ( $\geq$ ), == ( $=$ ), != ( $\neq =$ ). Each gives True or False.
In an if, elif, else chain Python runs the block under the first True condition, or the else block if none is True, so exactly one block runs.
A loop with an if inside tests an inequality for every value in the range; the printed values are the solutions among the values tested, not all solutions.
> and < leave out the boundary value; >= and <= include it. "At least 50" is >= 50.
To alter a program for a new situation, change its parameters (fees, rates), its range, or its conditions, then predict and check the new output.

Unit 5: Linear and Non-Linear Relations

Linear and Non-Linear Relations

A linear relation has a straight-line graph; in a table with equal $x$ -steps, its first differences are all equal.
First difference = next $y$ -value $-$ current $y$ -value, found between consecutive rows.
With equal $x$ -steps, first differences that change mean the relation is non-linear and the graph curves.
A positive constant first difference gives a rising line, a negative one a falling line, and $0$ a horizontal line.
If the $x$ -steps are unequal, compare $\frac{change in y}{change in x}$ for each pair of rows instead.

Rate of Change and Initial Value

Initial value: the value of the dependent variable when the independent variable is $0$ ; on a graph, where the line meets the vertical axis.
Rate of change $= \frac{change in dependent variable}{change in independent variable},$ with units such as dollars per km.
A linear relation can be written as $y = (rate of change) x + (initial value) .$
Direct variation: initial value $0$ , $y = k x$ ( $k \neq = 0$ ), graph through $(0, 0)$ . Partial variation: initial value not $0$ , $y = m x + b$ with $m \neq = 0$ and $b \neq = 0$ .
A positive rate of change means the quantity grows (line rises); a negative rate means it shrinks (line falls).

Slope, y-Intercept and y = mx + b

Slope: $m = \frac{rise}{run} = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}$ for any two points with $x_{1} \neq = x_{2}$ .
Positive slope: the line rises left to right. Negative: it falls. Zero: horizontal. Vertical lines: slope undefined.
The $y$ -intercept $b$ is the $y$ -coordinate of the point where the line crosses the $y$ -axis, $(0, b)$ .
Slope-intercept form: $y = m x + b$ , where $m$ is the slope (rate of change) and $b$ is the $y$ -intercept (initial value).
To find $b$ from a point and the slope, substitute both into $y = m x + b$ and solve for $b$ .

Comparing Two Linear Relations

The point of intersection of two lines is the point that lies on both: the same input gives the same output in both relations.
To find it with algebra, set the two expressions equal (for example $5 g + 30 = 2 g + 45$ ), solve, then substitute to find the shared value.
On one side of the intersection one relation has the smaller output; on the other side, the other relation does.
Different slopes: exactly one intersection. Same slope, different y-intercepts: parallel, no intersection. Same slope and same y-intercept: the same line.
Always check that the intersection makes sense in the context (no negative times, whole numbers where only whole numbers are possible).

Special Lines, xy = k and Inequalities

$x = k$ is a vertical line through $(k, 0)$ with undefined slope; $y = k$ is a horizontal line through $(0, k)$ with slope $0$ .
$x + y = k$ rearranges to $y = - x + k$ (slope $- 1$ ); $x - y = k$ rearranges to $y = x - k$ (slope $1$ , $y$ -intercept $- k$ ).
For $a x + b y = k$ with $a$ , $b$ and $k$ not zero: $x$ -intercept $\frac{k}{a}$ (set $y = 0$ ), $y$ -intercept $\frac{k}{b}$ (set $x = 0$ ).
$x y = k$ (with $k \neq = 0$ ) is non-linear: $y = \frac{k}{x}$ , a curve with two branches that never touches either axis.
Inequality graphs: solid boundary for $\leq$ or $\geq$ , dashed for $<$ or $>$ ; shade the side where a test point (not on the boundary) makes the inequality true.

Translating, Reflecting and Rotating Lines

Translating $y = a x$ up $k$ units gives $y = a x + k$ (down $k$ : $y = a x - k$ ); the slope does not change.
Reflecting $y = a x$ in the $x$ -axis or in the $y$ -axis gives the same line, $y = - a x$ .
Rotating $y = a x$ ( $a \neq = 0$ ) by 90° about the origin gives $y = - \frac{1}{a} x$ ; the two slopes multiply to $- 1$ .
Point rules: up $k$ : $(x, y + k)$ ; $x$ -axis reflection: $(x, - y)$ ; $y$ -axis reflection: $(- x, y)$ ; 90° counterclockwise rotation: $(- y, x)$ .
To check an answer, transform one point of the original line and substitute it into the new equation.

Unit 6: Data and Mathematical Modelling

Data in Our Lives: Collection, Use and Consequences

Big data is data with huge volume, high velocity (arrives fast) and wide variety (many types).
Data moves through a lifecycle: collect, store, analyse, use. Each stage carries privacy or bias risks.
A sample is biased if it systematically differs from the population; a larger biased sample is still biased.
Meaningful consent means knowing what is collected, why, and who it is shared with, and being able to refuse.
On a truncated axis, compare the real values, not the bar heights: the real ratio is $value_{A} \div value_{B}$ .

Measures of Centre, Spread and Box Plots

Mean $= \frac{sum of values}{n}$ ; median = middle of the sorted list; mode = most frequent value.
Range = greatest value − least value. $IQR = Q_{3} - Q_{1}$ .
Quartile method: $Q_{1}$ and $Q_{3}$ are the medians of the lower and upper halves; when $n$ is odd, leave the median out of both halves.
Outlier fences: lower $= Q_{1} - 1.5 \times IQR$ , upper $= Q_{3} + 1.5 \times IQR$ . Values outside the fences are outliers.
Box plot: box from $Q_{1}$ to $Q_{3}$ , line at the median, whiskers to the least and greatest non-outliers, outliers as dots.

Scatter Plots, Correlation and Lines of Best Fit

Independent variable on the $x$ -axis, dependent variable on the $y$ -axis.
Describe correlation by direction (positive, negative, none), strength (strong, weak) and form (linear, non-linear).
The correlation coefficient $r$ is between $- 1$ and $1$ ; values near $\pm 1$ mean a strong linear pattern, values near $0$ mean little or no linear pattern.
Line of best fit: from two points on the line, $m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}$ , then substitute one point into $y = m x + b$ to find $b$ .
Interpolation predicts inside the data's range; extrapolation predicts outside it and is less reliable.
Correlation alone does not prove that one variable causes the other.

The Mathematical Modelling Cycle

A mathematical model is a simplified mathematical description (equation, graph or table) of a real situation, built on assumptions.
The modelling cycle: ask a question, make assumptions and choose variables, plan and collect data, build the model, test it, report. Revise when the test fails.
Choose the model from the data's shape: constant change means linear; change that keeps shrinking or growing means a non-linear model.
Test a model with data that was not used to build it, and compare predicted and measured values.
Every report states the assumptions and the limits of the model, including where its predictions become impossible.

Unit 7: Geometry and Measurement

Angles, Parallel Lines and Polygons

Complementary angles add to $9 0^{\circ}$ ; supplementary angles add to $18 0^{\circ}$ ; vertically opposite angles are equal.
When a transversal crosses parallel lines: corresponding angles are equal (F), alternate angles are equal (Z), co-interior angles add to $18 0^{\circ}$ (C).
Interior angle sum of a polygon with $n$ sides (sides not crossing): $S = 18 0^{\circ} (n - 2)$ .
Each interior angle of a regular $n$ -gon is $\frac{18 0 ^{\circ} ( n - 2 )}{n}$ ; each exterior angle is $\frac{36 0 ^{\circ}}{n}$ .
The exterior angles of a convex polygon, one at each corner, add to $36 0^{\circ}$ .

Triangle and Circle Properties in Design

Triangle inequality: the two shortest sides must add to more than the longest side.
Exterior angle of a triangle $=$ sum of the two opposite interior angles.
In a triangle, the longest side is opposite the largest angle and the shortest side is opposite the smallest angle.
On the same arc: central angle $= 2 \times$ inscribed angle, and all inscribed angles are equal.
An angle inscribed in a semicircle (standing on a diameter) is $9 0^{\circ}$ .
The perpendicular from the centre to a chord bisects the chord; the perpendicular bisector of any chord passes through the centre.

Units and Measurement Systems

Metric prefixes are powers of 10: kilo $= 1000$ , centi $= 0.01$ , milli $= 0.001$ . Each step down the ladder multiplies by 10; each step up divides by 10.
Square the length factor for area and cube it for volume: $1 m^{2} = 10000 cm^{2}$ , $1 m^{3} = 1000000 cm^{3}$ .
$1 cm^{3} = 1 mL$ , $1 L = 1000 cm^{3}$ , $1 m^{3} = 1000 L$ .
Exact: $1 in = 2.54 cm$ , $1 ft = 12 in$ , $1 mi = 1.609344 km$ , $1 lb = 0.45359237 kg$ .
Multiply by a conversion factor equal to 1 (like $\frac{2.54 cm}{1 in}$ ) so the old unit cancels; round only at the end.

The Pythagorean Theorem

In a right triangle with legs $a$ , $b$ and hypotenuse $c$ : $a^{2} + b^{2} = c^{2}$ .
The hypotenuse is opposite the right angle and is the longest side.
Hypotenuse: $c = a^{2} + b^{2}$ . Leg: $a = c^{2} - b^{2}$ .
Converse: if $a^{2} + b^{2} = c^{2}$ with $c$ the longest side, the triangle is right-angled. If $a^{2} + b^{2} < c^{2}$ it is obtuse; if $a^{2} + b^{2} > c^{2}$ it is acute.
Common triples: 3, 4, 5; 5, 12, 13; 8, 15, 17; 7, 24, 25, and any positive whole-number multiple of them.

How Changing Dimensions Changes Area and Volume

If every length is multiplied by the scale factor $k$ : perimeter $\times k$ , area and surface area $\times k^{2}$ , volume $\times k^{3}$ .
$k > 1$ is an enlargement, $0 < k < 1$ is a reduction.
If only some dimensions change, multiply the factors of the dimensions that change; for a cylinder, the radius counts twice because it is squared.
Working backwards for similar shapes: area factor $= k^{2}$ , so $k = area factor$ ; volume factor $= k^{3}$ , so $k$ is the number that cubes to the volume factor.

Volume: Prisms, Pyramids, Cylinders and Cones

Prism or cylinder: $V = B h$ , where $B$ is the area of the base. Cylinder: $V = π r^{2} h$ .
Pyramid or cone: $V = \frac{1}{3} B h$ . Cone: $V = \frac{1}{3} π r^{2} h$ .
A pyramid holds $\frac{1}{3}$ of a prism with the same base and height; a cone holds $\frac{1}{3}$ of a cylinder with the same base and height.
$h$ is the perpendicular height, not the slant height.
Use one length unit throughout. $1 cm^{3} = 1 mL$ , $1000 cm^{3} = 1 L$ , $1 m^{3} = 1000 L$ .

Unit 8: Financial Literacy

Making Financial Decisions

Inflation is a general rise in prices over time; it lowers the purchasing power of each dollar.
Canada's CPI (Statistics Canada) prices one basket of goods and services; the 2002 cost is set to 100.
Inflation rate $= \frac{CPI _{new} - CPI _{old}}{CPI _{old}} \times 100% .$
Price in another year $\approx$ price $\times \frac{CPI _{new}}{CPI _{old}}$ ; round money to the nearest cent at the end.
Yearly inflation rates combine by multiplying (1.068 × 1.039), not by adding.
Compare options on the same basis: same time period, same amount, same taxes.

Appreciation and Depreciation

Straight-line depreciation: $V = P - d t$ (same dollar loss $d$ each year; graph is a straight line).
Yearly straight-line loss with a salvage value: $d = \frac{P - S}{n}$ .
Percent depreciation: $V = P (1 - r)^{t}$ (same percent loss each year; curve that flattens).
Appreciation: $V = P (1 + r)^{t}$ (same percent gain each year; curve that gets steeper).
Losing $r$ per year means multiplying by $1 - r$ ; losing 15% means $\times 0.85$ , not $\times 0.15$ .

Interest and the Cost of Borrowing

Simple interest: $I = P r t$ and $A = P + I$ (interest on the original principal only; $t$ in years).
Compound interest: $A = P (1 + i)^{n}$ , with $i$ the rate per period and $n$ the number of periods.
Monthly compounding: $i = \frac{yearly rate}{12}$ and $n = 12 \times$ years.
Simple interest grows in a straight line; compound interest grows by a constant multiplier and curves upward.
Higher rate, longer time and a smaller down payment all raise the total cost of borrowing.
Credit cards: pay the full balance by the due date to use the grace period of at least 21 days on new purchases.

Building and Adjusting a Budget

Gross pay = hourly wage × hours; net pay = gross pay − deductions (income tax, CPP, EI). Budget with net pay.
Monthly pay $= \frac{weekly pay \times 52}{12}$ (about 4.33 weeks per month).
Percent of income $= \frac{amount}{net income} \times 100%$ ; all the categories together make 100%.
Fixed expenses stay the same each month; variable expenses change with your choices; savings are kept on purpose.
A balanced budget: net income = expenses + savings. Income above spending is a surplus; below is a deficit.
When income falls or a cost rises, find the shortfall first, then adjust variable spending and savings until the budget balances.