Mathematics Knowledge: mixed review

A factor divides a number with no remainder (factors of 12: 1, 2, 3, 4, 6, 12, a finite list). A multiple is the number times a whole number (12, 24, 36, ... without end). Since 4 is a factor of 12, 12 is a multiple of 4.

A prime is a whole number greater than 1 whose only factors are 1 and itself. A composite has more than two factors. The number 1 counts as neither, and 2 is the one even prime.

3 or 9: the digit sum is divisible by 3 or 9. 4: the last two digits form a multiple of 4. 8: the last three digits form a multiple of 8. Example: 7,128 has digit sum 18 and ends in 128, so 3, 9, 4 and 8 all divide it.

6: the number must be even and pass the test for 3. 11: from the left, alternately subtract and add the digits; a result of 0 or a multiple of 11 means divisible. Example: 3,729 gives 3 − 7 + 2 − 9 = −11, so 11 divides it.

GCF: multiply only the shared primes, each at its lower power. LCM: multiply every prime that appears, each at its higher power. For $24=2^3\times 3$ and $18=2\times 3^2$: GCF = 6 and LCM = 72.

GCF: splitting things into the largest equal pieces or identical groups (cutting, packing kits). LCM: repeating cycles that must line up again (buses, blinking lights). The GCF is at most the smaller number; the LCM is at least the larger.

Three times the middle one, since n + (n + 1) + (n + 2) = 3n + 3. If the sum is 84, the middle integer is 84 ÷ 3 = 28, so the integers are 27, 28 and 29.

No. Simplify inside the bars first: |2 − 10| = |−8| = 8. Absolute value is distance from zero, so it is never negative, and the bars work like grouping symbols.

Same signs: add the absolute values and keep the common sign (−7 + (−5) = −12). Different signs: subtract the smaller absolute value from the larger and keep the sign of the larger (−12 + 5 = −7).

Add its opposite: a − b = a + (−b). So 5 − (−8) = 5 + 8 = 13 and −4 − 7 = −4 + (−7) = −11. Subtracting a negative always makes the result larger.

Work with the absolute values, then attach the sign: like signs give a positive result, unlike signs a negative one. Examples: (−6)(−4) = 24 and −42 ÷ 7 = −6.

Count the negative factors first. An even count gives a positive product; an odd count gives a negative one. (−1)(−2)(−3)(5) has three negatives, so it equals −30.

No. In $-3^2$ only 3 is squared and then negated, giving −9. In $(-3{)}^2$ the base is −3, giving 9. An odd power of a negative base is negative: $(-3{)}^3=-27$.

−3, because it lies to the right of −12 on the number line. Among negative numbers, the one with the larger absolute value is the smaller number.

Zero is neither positive nor negative. Any number times 0 is 0, and 0 divided by a nonzero number is 0, but division by 0 is undefined, so 7 ÷ 0 has no answer.

Change = final value − starting value. From −6°F to 11°F: 11 − (−6) = 17, a rise of 17°F. A negative result means the quantity decreased.

No. An exponent counts how many times the base is used as a factor: $3^4=3\times 3\times 3\times 3=81$. It is repeated multiplication, not base times exponent.

Same base: multiply by adding exponents, $a^m\cdot a^n=a^{m+n}$; divide by subtracting, $\frac{a^m}{a^n}=a^{m-n}$ with a ≠ 0. Example: $x^6\cdot x^2=x^8$ and $\frac{x^6}{x^2}=x^4$.

$(a^m{)}^n=a^{mn}$: multiply the exponents, so $(y^3{)}^4=y^{12}$. $(ab{)}^n=a^n{b}^n$: the outside exponent applies to every factor, so $(2x{)}^3=8x^3$.

For nonzero a, $a^0=1$ and $a^{-n}=\frac{1}{a^n}$. So $7^0=1$ and $4^{-2}=\frac{1}{16}$. A negative exponent makes a reciprocal, never a negative number.

The radical gives the principal, nonnegative square root: $\sqrt{64}=8$, not −8; square roots of negatives are not real. Cube roots of negatives are real and negative: $\sqrt[3]{-27}=-3$.

Pull out the largest perfect-square factor using $\sqrt{ab}=\sqrt{a}\sqrt{b}$: $\sqrt{75}=\sqrt{25}\sqrt{3}=5\sqrt{3}$. Perfect squares to know: 4, 9, 16, 25, 36, 49, 64, 81, 100, ...

A number written as $c\times 10^n$ with $1\le c<10$. Large numbers get positive n and small numbers negative n: 48,000 = $4.8\times 10^4$ and 0.0062 = $6.2\times 10^{-3}$.

Multiply (divide) the front numbers and add (subtract) the exponents, then adjust so the front is at least 1 and under 10: $(4\times 10^5)(5\times 10^2)=20\times 10^7=2\times 10^8$.

Terms are the parts joined by + or −: $7a^2$, $-4b$ and 9. Coefficients multiply the variables: 7 and −4 (the sign belongs to the term). The constant is the term with no variable: 9.

Like terms have identical variables raised to identical powers, such as $3x{y}^2$ and $-5x{y}^2$. Add the coefficients and keep the variable part: $-2x{y}^2$. Terms like $x^2$ and x cannot be combined.

$a(b+c)=ab+ac$: multiply the outside factor by every term inside. A leading minus flips every sign inside: $8-(2y-5)=8-2y+5=13-2y$.

Write $(-4{)}^2+3(-4)=16-12=4$. Without parentheses you might get −16 for the square. Put every substituted value in parentheses, then follow the order of operations.

Multiply each term of one binomial by each term of the other (first, outer, inner, last), then combine like terms: $(x+4)(x-2)=x^2-2x+4x-8=x^2+2x-8$.

$a^2-b^2=(a-b)(a+b)$. Example: $4x^2-25=(2x-5)(2x+5)$. A sum of two squares, such as $x^2+9$, does not factor this way.

No. $(a+b{)}^2=a^2+2ab+b^2$ and $(a-b{)}^2=a^2-2ab+b^2$; the middle term 2ab is easy to drop. Test a = 1, b = 2: 9 versus 5.

Factor top and bottom, then cancel only common factors, never single terms across + or −: $\frac{x^2-4}{x-2}=x+2$, valid only for x ≠ 2, since the original denominator cannot be zero.

Do the same thing to both sides: add, subtract, multiply or divide both sides by the same number. Never multiply both sides by zero (that wipes out the solution, leaving 0 = 0), and never divide by zero, which is undefined.

Undo the operations in reverse order using inverse operations: subtract 3 first to get 5x = 35, then divide by 5 to get x = 7. Check: 5(7) + 3 = 38.

Dropping the negative sign. Divide both sides by −4, sign included: x = −7. Writing x = 7 gives the opposite of the correct answer. Check: −4(−7) = 28.

Subtract the smaller x term from both sides to keep a positive coefficient: subtract 4x to get 5x − 2 = 18, add 2 to get 5x = 20, so x = 4.

Multiply every term on both sides by the LCD (or a power of ten for decimals), constants included. For $\frac{x}{3}+\frac{x}{4}=7$, multiply by 12: 4x + 3x = 84, so x = 12.

A false leftover such as 5 = 9 means the equation is a contradiction with no solution. A true leftover such as 3 = 3 means it is an identity: every real number is a solution.

'Is' means equals and 'less than' subtracts in reverse order: 2n − 4 = 18. Then 2n = 22 and n = 11.

Treat the other letters as numbers and use inverse operations: subtract 2w, then divide by 2, giving $l=\frac{P-2w}{2}$.

Only when you multiply or divide both sides by a negative number: −3x > 12 becomes x < −4. Adding, subtracting, or multiplying or dividing by a positive number keeps the direction.

No. You divide by positive 5, so x < −4. A negative number sitting on one side does not cause a flip; only a negative multiplier or divisor does.

Open circle for strict signs < and >: the boundary is not a solution. Closed, filled circle for ≤ and ≥: the boundary is included. Then shade the side that makes the inequality true.

Left endpoint, then right. A bracket means included, a parenthesis means excluded, and infinity always takes a parenthesis. $-1\le x<5$ is [−1, 5), and x > 2 is $(2,\infty )$.

'And' needs both conditions, so the solution is the overlap: x > 0 and x ≤ 4 gives 0 < x ≤ 4. 'Or' needs at least one, so the solution joins both sets, as in x < −2 or x > 3.

Less than: one between-statement, −b < x − a < b. Greater than: two pieces joined by or, x − a < −b or x − a > b. Example: |x − 3| ≤ 5 gives −2 ≤ x ≤ 8.

Round down for a maximum (at most, cannot exceed) and up for a minimum (at least). 9 + 8n ≤ 70 gives n ≤ 7.625, so at most 7; 5t ≥ 32 gives t ≥ 6.4, so at least 7.

Apply each step to all three parts: add 1 to get −6 ≤ 2x < 10, then divide by 2 to get −3 ≤ x < 5. Dividing all parts by a negative reverses both symbols.

The ordered pair (x, y) that makes both equations true at the same time; on a graph it is the point where the two lines cross. List x first, then y.

When one variable is already alone, or has coefficient 1 or −1 and is easy to isolate. For y = x + 2 and 3x + y = 14: 3x + x + 2 = 14, so x = 3 and y = 5.

Add the equations when a variable has opposite coefficients, subtract when they are equal, or scale one equation first. x + y = 9 and x − y = 3: adding gives 2x = 12, so x = 6 and y = 3.

Multiplying only the targeted term instead of every term on both sides, and subtracting only the first term instead of every term, constants included. Both break the equality.

Exactly one (the lines cross), none (the lines are parallel), or infinitely many (both equations describe the same line). It can never have exactly two.

In y = mx + b form: different slopes mean one solution; equal slopes with different intercepts mean none; equal slopes and intercepts mean infinitely many. Elimination ending in 0 = 0 means infinite; 0 = 4 means none.

Larger $=\frac{S+D}{2}$ and smaller $=\frac{S-D}{2}$. With sum 40 and difference 12, the numbers are 26 and 14.

Not always. Combining the equations may give it directly: 2x + y = 11 plus x + 2y = 10 gives 3x + 3y = 21, so x + y = 7. Always answer exactly what is asked.

Each input has exactly one output. {(1, 3), (2, 3)} is a function, since inputs may share an output; {(4, 1), (4, 7)} is not, since input 4 has two outputs. On a graph, use the vertical line test.

f(x) names the output at input x; it is not f times x. f(3) = 2(3) − 5 = 1. To find the input for a given output, write rule = output and solve for x.

Domain: allowed inputs. Range: outputs produced. Exclude inputs that make a denominator zero ($\frac{1}{x-5}$ excludes 5) or a square root's inside negative ($\sqrt{x+1}$ needs x ≥ −1).

$m=\frac{y_2-y_1}{x_2-x_1}$, change in output over change in input. From (2, 5) to (6, 17): m = 12/4 = 3. In f(x) = mx + b, b is the output when x = 0.

$a_n=a_1+(n-1)d$, where $a_1$ is the first term and d the common difference. For 7, 11, 15, ...: $a_{10}=7+9(4)=43$. Using n instead of n − 1 overshoots by d.

$S_n=\frac{n(a_1+a_n)}{2}$: the average of the first and last terms times the number of terms. 3, 7, 11, ..., 23 has 6 terms, so the sum is 6(26)/2 = 78.

$a_n=a_1\cdot r^{n-1}$, where r is the common ratio (any term divided by the one before). For 3, 6, 12, ...: $a_6=3\cdot 2^5=96$. Growth has r > 1; decay has 0 < r < 1.

Work inside out: compute g(x), then feed that result into f. With $f(x)=x^2$ and g(x) = x + 1: f(g(2)) = f(3) = 9, but g(f(2)) = g(4) = 5, so order matters.

Acute: more than 0° and less than 90°. Right: exactly 90°. Obtuse: between 90° and 180°. Straight: exactly 180°. Reflex: between 180° and 360°.

Complementary angles total 90°; supplementary angles total 180°. The complement of 34° is 56°, and its supplement is 146°. Memory aid: C comes before S, as 90 comes before 180.

Opposite (vertical) angles are equal, and side-by-side angles form linear pairs that total 180°. If one angle is 47°, the four angles are 47°, 133°, 47° and 133°.

Adjacent angles filling a straight line total 180°; non-overlapping angles that surround a point total 360°. Angles of 100°, 75°, 120° and x around a point give x = 65°.

Corresponding angles, alternate interior angles and alternate exterior angles are all equal. If one angle of a corresponding pair is 64°, its partner is also 64°.

No. Same-side (consecutive) interior angles are supplementary, adding to 180°. If one is 105°, the other is 75°.

Only two: four equal acute angles and four equal obtuse angles, with acute + obtuse = 180°. If one angle is 52°, every angle is either 52° or 128°.

No; figures may not be to scale. Trust only stated facts and markings: a small square for 90°, matching arrows for parallel lines, tick marks for equal parts. Equal vertical angles never prove lines parallel.

The three interior angles total 180°. Each exterior angle equals the two non-adjacent interior angles added together. With remote angles 45° and 70°, the exterior angle is 115° and the adjacent interior angle is 65°.

No. The sum of the two shorter sides has to exceed the longest side, but 5 + 7 = 12 < 13. With two sides a < b, the third side lies strictly between b − a and a + b.

$A=\frac{1}{2}bh$, where b is the base and h is the height perpendicular to that base (not a slanted side); the answer is in square units. Base 10 cm and height 6 cm give 30 cm².

In a right triangle, $a^2+b^2=c^2$, with legs a and b and hypotenuse c (the longest side, opposite the right angle). Legs 6 and 8 give $c=\sqrt{100}=10$. For a missing leg, subtract the squares.

3-4-5, 5-12-13, 8-15-17 and 7-24-25, plus any whole-number multiple of them, such as 6-8-10 or 10-24-26. Spotting one gives the missing side without square roots.

45-45-90: $1:1:\sqrt{2}$, so hypotenuse = leg $\times \sqrt{2}$. 30-60-90: $1:\sqrt{3}:2$; the short leg is half the hypotenuse and the long leg is short leg $\times \sqrt{3}$. Hypotenuse 14: legs 7 and $7\sqrt{3}$.

With scale factor k, every length (sides, perimeter, heights) is multiplied by k, but area is multiplied by $k^2$. Doubling every side of a 12 cm² triangle gives 48 cm², not 24 cm².

Valid: SSS, SAS, ASA, AAS, and HL for right triangles. Not valid: SSA, which can fit two different triangles, and AAA, which fixes the shape but not the size.

$S=(n-2)\times 180^{\circ }$, because diagonals from one vertex split the polygon into n − 2 triangles. A hexagon has 720°, and every quadrilateral has 360°.

Exterior $=\frac{360^{\circ }}{n}$; interior $=\frac{(n-2)\times 180^{\circ }}{n}$, which equals 180° minus the exterior angle. Regular octagon: exterior 45°, interior 135°.

Subtract it from 180° to get the exterior angle, then divide 360° by that, since the exterior angles of a convex polygon total 360°. Interior 150° gives exterior 30°, so n = 12.

$D=\frac{n(n-3)}{2}$: each vertex connects to n − 3 non-neighbors, and each diagonal is counted twice. A pentagon has 5 diagonals; a decagon has 35.

Opposite sides parallel and equal, opposite angles equal, neighboring angles summing to 180°, and diagonals that bisect each other. Rectangles add equal diagonals; rhombi add perpendicular diagonals; squares have both.

$A=\frac{1}{2}(b_1+b_2)h$: the average of the two parallel bases times the perpendicular height. Bases 5 and 9 with height 4 give 7 $\times$ 4 = 28 square units.

$A=\frac{1}{2}{d}_1{d}_2$, using the full lengths of the two perpendicular diagonals. Diagonals of 12 and 8 give 48 square units.

84 square units, from A = bh with the perpendicular height. Multiplying by the slanted side (getting 108) is the classic trap; a slanted side is always longer than the height.

$C=2\pi r=\pi d$, where r is the radius and d the diameter; it is a length in linear units. A radius of 9 in gives $18\pi$ in, about 56.5 in.

$A=\pi r^2$: square the radius, then multiply by $\pi$; the answer is in square units. A 10 m diameter means r = 5, so $A=25\pi \approx 78.5$ m². Using the diameter makes it 4 times too big.

Use the fraction $\frac{\theta }{360^{\circ }}$ of the whole: arc $=\frac{\theta }{360^{\circ }}\times 2\pi r$, sector $=\frac{\theta }{360^{\circ }}\times \pi r^2$. Radius 9, angle 40°: fraction 1/9, arc $2\pi$, sector $9\pi$.

Diameter and circumference are multiplied by k, but area is multiplied by $k^2$. Tripling the radius makes the area 9 times as large; halving it leaves one fourth of the area.

$A=\pi (R^2-r^2)$, where R is the outer radius and r the inner radius. R = 5 and r = 3 give $16\pi$. Squaring the difference of the radii (only $4\pi$) is wrong.

A tangent touches the circle at one point and is perpendicular to the radius drawn there, forming a right triangle. Radius 8 and a point 17 from the center give a tangent segment of 15.

An inscribed angle (vertex on the circle) is half its intercepted arc, while a central angle equals its arc. An arc of 110° gives an inscribed angle of 55°. An inscribed angle standing on a diameter is 90°.

One circumference per full turn: distance = turns $\times$ C, and turns = distance ÷ C. A wheel of diameter 3 ft rolling $300\pi$ ft makes $\frac{300\pi }{3\pi }=100$ turns.

$A=\frac{1}{2}(b_1+b_2)h$: average the two parallel bases $b_1,b_2$ and multiply by the perpendicular height h. Bases 5 and 9 with height 6 give 42 square units.

$V=Bh$, where B is the area of one base and h is the distance between the bases. For a cylinder $B=\pi r^2$, so $V=\pi r^{2}h$. Answer in cubic units.

It is one third: $V=\frac{1}{3}Bh$ for a pyramid and $V=\frac{1}{3}\pi r^{2}h$ for a cone. Pointed solids carry the one-third.

$V=\frac{4}{3}\pi r^3$ (cubic units) and $SA=4\pi r^2$ (square units). For r = 6, $V=288\pi$.

$SA=2\pi r^2+2\pi rh$: two circular ends plus the curved side (a rectangle whose width is the circumference $2\pi r$). Open top: drop one $\pi r^2$.

Volume uses the vertical height h; surface area uses the slant height l. They are linked by a right triangle, e.g. for a cone $l^2=r^2+h^2$.

Area (and surface area) is multiplied by $k^2$; volume by $k^3$. Doubling all dimensions gives 4 times the surface and 8 times the volume.

Square the length factor for area and cube it for volume: $1{\text{ ft}}^2=144{\text{ in}}^2$, $1{\text{ ft}}^3=1728{\text{ in}}^3$, $1{\text{ yd}}^3=27{\text{ ft}}^3$. Also 1 liter = 1000 cubic centimeters.

Numbered counterclockwise from upper right: I (+, +), II (-, +), III (-, -), IV (+, -). A point with a zero coordinate lies on an axis, in no quadrant.

Across the x-axis: (x, y) becomes (x, -y). Across the y-axis: (x, y) becomes (-x, y). Across the line y = x the coordinates swap.

$d=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2}$, from the Pythagorean theorem. Look for triples like 3-4-5 or 5-12-13 to skip the square root.

$M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$. If M and endpoint A are known, the other endpoint is $B=2M-A$, coordinate by coordinate.

$m=\frac{y_2-y_1}{x_2-x_1}$, rise over run. Subtract in the same order on top and bottom.

Horizontal (y = c): slope 0. Vertical (x = c): run is 0, so slope is undefined, not zero.

m is the slope (rate of change); b is the y-intercept, the point (0, b) where the line crosses the y-axis, i.e. the value of y when x = 0.

Parallel lines have equal slopes (different intercepts). Perpendicular slopes are negative reciprocals: $m_1\times m_2=-1$, so slope 3 pairs with $-\frac{1}{3}$.