CJMC Math Formula Flashcards

Vocabulary-style flashcards covering essential formulas and definitions from arithmetic, number theory, algebra, geometry, probability, and kinematics as presented in the CJMC Math Formula Flashcards transcript.

Order of Operations

Brackets $\rightarrow$ exponents $\rightarrow$ multiplication/division from left to right $\rightarrow$ addition/subtraction from left to right.

Divisibility Rule for $2$

An integer is divisible by $2$ if its last digit is $0$, $2$, $4$, $6$, or $8$.

Divisibility Rule for $3$

An integer is divisible by $3$ if the sum of its digits is divisible by $3$.

Divisibility Rule for $4$

An integer is divisible by $4$ if the number formed by its last two digits is divisible by $4$.

Divisibility Rule for $5$

An integer is divisible by $5$ if its last digit is $0$ or $5$.

Divisibility Rule for $6$

An integer is divisible by $6$ if it is divisible by both $2$ and $3$.

Divisibility Rule for $8$

An integer is divisible by $8$ if the number formed by its last three digits is divisible by $8$.

Divisibility Rule for $9$

An integer is divisible by $9$ if the sum of its digits is divisible by $9$.

Divisibility Rule for $10$

An integer is divisible by $10$ if its last digit is $0$.

Divisibility Rule for $11$

An integer is divisible by $11$ if the alternating sum of its digits is a multiple of $11$, including $0$.

Division-with-Remainder Formula

$\text{Dividend} = \text{divisor} \times \text{quotient} + \text{remainder}$, where the remainder is at least $0$ and less than the divisor.

Relationship Between GCD and LCM

$\gcd(a,b) \times \text{lcm}(a,b) = a \times b$ for positive integers $a$ and $b$.

LCM via Prime Factorization

The method of finding the least common multiple by using the highest power of each prime appearing in either number.

GCD via Prime Factorization

The method of finding the greatest common divisor by using the lowest power of each prime common to both numbers.

Consecutive Integers (Algebraic Form)

Algebraic representation written as $n, n+1, n+2, n+3, \dots$

Fraction of a Quantity

Found by multiplying the quantity by the fraction.

Percentage Formula

$$\text{Percentage} = (\text{part} \div \text{whole}) \times 100\%$$

$p\%$ of a Number $x$

$$(p \div 100) \times x$$

Percentage Change

$$(\text{change} \div \text{original value}) \times 100\%$$

Percentage Discount

$$\text{Discount} = \text{original price} \times \text{discount rate}$$

Price After a Discount

$$\text{Original price} \times (1 - \text{discount rate as a decimal})$$

Value After a Percentage Increase

$$\text{Original value} \times (1 + \text{increase rate as a decimal})$$

Ratio Division of Total $T$ in Ratio $a:b$

The shares given by $\frac{aT}{a+b}$ and $\frac{bT}{a+b}$.

Equivalent Fraction

A fraction made by multiplying or dividing its numerator and denominator by the same nonzero number.

Arithmetic Sequence ($n^{\text{th}}$-Term Formula)

$a_n = a_1 + (n-1)d$, where $d$ is the common difference.

Arithmetic Sequence (Sum Formula)

$$S_n = \frac{n(a_1+a_n)}{2}$$

Geometric Sequence ($n^{\text{th}}$-Term Formula)

$a_n = a_1 r^{n-1}$, where $r$ is the common ratio.

$n^{\text{th}}$ Square Number

$$n^2$$

Difference Between Consecutive Square Numbers

$$(n+1)^2 - n^2 = 2n+1$$

Sum of the First $n$ Positive Integers

$$\frac{n(n+1)}{2}$$

Sum of the First $n$ Positive Odd Integers

$$n^2$$

Sum of the First $n$ Positive Even Integers

$$n(n+1)$$

Distributive Property

$$a(b+c) = ab+ac$$

Difference-of-Squares Identity

$$a^2-b^2 = (a-b)(a+b)$$

Linear Equation Solution ($ax+b=c$)

$x = \frac{c-b}{a}$, where $a \neq 0$.

Area of a Rectangle

$$A = lw$$

Perimeter of a Rectangle

$$P = 2l+2w$$

Area of a Square

$$A = s^2$$

Perimeter of a Square

$$P = 4s$$

Area of a Triangle

$$A = \frac{bh}{2}$$

Area of a Parallelogram

$$A = bh$$

Area of a Trapezoid

$A = \frac{(a+b)h}{2}$, where $a$ and $b$ are the parallel sides.

Circumference of a Circle

$$C = 2\pi r = \pi d$$

Area of a Circle

$$A = \pi r^2$$

Interior Angle Sum of a Triangle

$$180^\circ$$

Sum of Angles on a Straight Line

$$180^\circ$$

Sum of Angles Around a Point

$$360^\circ$$

Pythagorean Theorem

$a^2+b^2=c^2$ for a right triangle, where $c$ is the hypotenuse.

Volume of a Rectangular Prism

$$V = lwh$$

Surface Area of a Rectangular Prism

$$SA = 2(lw+lh+wh)$$

Interior Angle Sum of an $n$-Sided Polygon

$$(n-2) \times 180^\circ$$

Interior Angle of a Regular $n$-Sided Polygon

$$\frac{(n-2) \times 180^\circ}{n}$$

Exterior Angle Sum of a Convex Polygon

$$360^\circ$$

Complementary Angles

Two angles whose measures add to $90^\circ$.

Supplementary Angles

Two angles whose measures add to $180^\circ$.

Scaling in Similar Figures

If the length scale factor is $k$, perimeters scale by $k$ and areas by $k^2$.

Circle Diameter-Radius Relation

$$d = 2r$$

Fundamental Counting Principle

Multiply the number of choices at each successive step.

Permutation Formula

$\frac{n!}{(n-r)!}$, representing the number of ways to choose and arrange $r$ distinct objects from $n$ without repetition.

Factorial ($n!$)

$n! = n(n-1)(n-2)\dots 2\times 1$, and $0! = 1$.

Probability of an Event (Equally Likely Outcomes)

$$P(\text{event}) = \text{favourable outcomes} \div \text{total outcomes}$$

Complement Rule (Probability)

$$P(\text{not } A) = 1-P(A)$$

Probability of Mutually Exclusive Events

$$P(A \text{ or } B) = P(A)+P(B)$$

Probability of Independent Events

$$P(A \text{ and } B) = P(A) \times P(B)$$

Mean

$$\text{Mean} = \text{sum of values} \div \text{number of values}$$

Total (from Mean)

$$\text{Total} = \text{mean} \times \text{number of values}$$

Expected Value (Equally Likely Outcomes)

Add all outcome values and divide by the number of outcomes.

Distance Formula

$$\text{Distance} = \text{speed} \times \text{time}$$

Speed Formula

$$\text{Speed} = \text{distance} \div \text{time}$$

Time Formula

$$\text{Time} = \text{distance} \div \text{speed}$$

Relative Speed (Opposite Directions)

Add their speeds.

Relative Speed (Same Direction)

Subtract the slower speed from the faster speed.