Algebra · Basic
Percent
percent = (part / whole) × 100
Symbols
- part
- the amount you have
- whole
- the total amount
When to use it
Use when a question asks what percent one amount is of another.
Quick example
37 out of 45 is (37 / 45) × 100 = 82.22%.
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Algebra · Basic
percent = (part / whole) × 100
Use when a question asks what percent one amount is of another.
37 out of 45 is (37 / 45) × 100 = 82.22%.
Algebra · Basic
percent change = ((new - old) / old) × 100
Use when comparing an increase or decrease from one value to another.
From 40 to 50: ((50 - 40) / 40) × 100 = 25% increase.
Algebra · Intermediate
a / b = c / d
Use when two ratios are equal and one value is missing.
If 3/4 = x/20, then x = 15.
Algebra · Basic
m = (y2 - y1) / (x2 - x1)
Use when you know two points and need the steepness or direction of a line.
For (1, 2) and (5, 10), m = (10 - 2) / (5 - 1) = 8 / 4 = 2.
Algebra · Intermediate
y - y1 = m(x - x1)
Use when you know one point and the slope of a line.
With slope 2 through (3, 4): y - 4 = 2(x - 3).
Algebra · Basic
y = mx + b
Use when graphing a line or reading slope and intercept directly.
In y = 3x + 2, slope is 3 and y-intercept is 2.
Algebra · Intermediate
Ax + By = C
Use when a class asks for a line equation in standard form.
2x + 3y = 12 is standard form.
Algebra · Intermediate
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Use when finding the straight-line distance between two coordinate points.
For (1, 2) and (4, 6), d = sqrt(3^2 + 4^2) = 5.
Algebra · Basic
midpoint = ((x1 + x2) / 2, (y1 + y2) / 2)
Use when finding the point exactly halfway between two coordinates.
Between (2, 4) and (8, 10), midpoint = (5, 7).
Algebra · Intermediate
x = (-b ± sqrt(b^2 - 4ac)) / 2a
Use when solving a quadratic equation, especially when factoring is hard.
For x^2 - 5x + 6 = 0, roots are 2 and 3.
Algebra · Intermediate
D = b^2 - 4ac
Use to tell whether a quadratic has two real roots, one repeated root, or complex roots.
If D = 9, there are two real roots. If D = 0, there is one repeated root.
Algebra · Basic
a^m × a^n = a^(m+n)
Use when multiplying powers with the same base.
x^3 × x^4 = x^7.
Algebra · Advanced
log_b(x) = ln(x) / ln(b)
Use when a calculator does not have the exact log base you need.
log_2(8) = ln(8) / ln(2) = 3.
10 formulas with examples and links.
Geometry · Basic
A = s^2
Use when all four sides are equal.
If s = 5 cm, A = 25 cm^2.
Geometry · Basic
A = l × w
Use for rectangles and rectangular faces.
A 8 by 3 rectangle has area 24 square units.
Geometry · Basic
A = 1/2 bh
Use when you know a triangle's base and perpendicular height.
If b = 10 and h = 6, A = 30 square units.
Geometry · Basic
A = pi r^2
Use when finding the space inside a circle.
If r = 3, A = 9pi ≈ 28.27 square units.
Geometry · Basic
C = 2pi r = pi d
Use when finding the distance around a circle.
If diameter is 10, C = 10pi ≈ 31.42.
Geometry · Basic
a^2 + b^2 = c^2
Use only for right triangles.
If a = 3 and b = 4, c = 5.
Geometry · Basic
V = lwh
Use for boxes and rectangular solids.
A 2 by 3 by 4 prism has volume 24 cubic units.
Geometry · Intermediate
V = pi r^2 h
Use for cans, pipes, and cylinders.
If r = 3 and h = 10, V = 90pi ≈ 282.74.
Geometry · Intermediate
V = 1/3 pi r^2 h
Use for cones and cone-shaped solids.
A cone is one third of a cylinder with the same radius and height.
Geometry · Intermediate
V = 4/3 pi r^3
Use for balls and sphere-shaped solids.
If r = 3, V = 36pi ≈ 113.1.
5 formulas with examples and links.
Trigonometry · Intermediate
sin(theta)=opp/hyp, cos(theta)=adj/hyp, tan(theta)=opp/adj
Use with right triangles when an angle and side relationship is needed.
If opposite = 3 and hypotenuse = 5, sin(theta) = 3/5.
Trigonometry · Advanced
sin^2(theta) + cos^2(theta) = 1
Use when simplifying trig expressions or finding sine from cosine.
If cos(theta) = 0.6, then sin^2(theta) = 0.64.
Trigonometry · Advanced
a / sin(A) = b / sin(B) = c / sin(C)
Use with non-right triangles when you know an angle-side pair.
If a/sin(A) is known, use it to solve another side or angle.
Trigonometry · Advanced
c^2 = a^2 + b^2 - 2ab cos(C)
Use with non-right triangles when two sides and the included angle are known.
If a, b, and C are known, solve for c.
Trigonometry · Basic
radians = degrees × pi / 180
Use when switching between degree and radian angle measures.
180 degrees = pi radians.
7 formulas with examples and links.
Statistics · Intermediate
weighted average = sum(value × weight) / sum(weights)
Use when scores or categories do not all count the same amount.
80 worth 40% and 95 worth 60% gives 80×0.4 + 95×0.6 = 89.
Statistics · Basic
mean = sum of values / count
Use when finding the arithmetic average of a data set.
Mean of 2, 4, 9 is 15 / 3 = 5.
Statistics · Basic
median = middle value after sorting
Use when the middle of a sorted data set matters more than outliers.
Sorted 2, 4, 9 has median 4.
Statistics · Basic
range = maximum - minimum
Use when measuring the spread from smallest to largest value.
For 2, 4, 9, range = 9 - 2 = 7.
Statistics · Advanced
s^2 = sum((x - xbar)^2) / (n - 1)
Use when measuring average squared spread in a sample.
Find each deviation from the mean, square it, add, then divide by n - 1.
Statistics · Advanced
s = sqrt(sum((x - xbar)^2) / (n - 1))
Use when describing how spread out sample data is.
Standard deviation is the square root of sample variance.
Statistics · Intermediate
z = (x - mu) / sigma
Use when measuring how many standard deviations a value is from the mean.
If x = 80, mean = 70, sigma = 5, z = 2.
2 formulas with examples and links.
Probability · Basic
P(event) = favorable outcomes / total outcomes
Use when all outcomes are equally likely.
Rolling a 6 on a fair die: P = 1 / 6.
Probability · Basic
P(not A) = 1 - P(A)
Use when it is easier to find the probability that something does not happen.
If chance of rain is 30%, chance of no rain is 70%.
7 formulas with examples and links.
Measurement / conversions · Basic
F = C × 9/5 + 32; C = (F - 32) × 5/9
Use when switching between weather temperature and science temperature units.
70°F is (70 - 32) × 5/9 = 21.1°C.
Measurement / conversions · Basic
1 in = 2.54 cm
Use when switching between US customary and metric length.
12 in × 2.54 = 30.48 cm.
Measurement / conversions · Basic
1 ft = 0.3048 m
Use when converting height, distance, or geometry measurements.
5 ft × 0.3048 = 1.524 m.
Measurement / conversions · Basic
1 mi = 1.609344 km
Use when converting road distance or speed units.
3 mi × 1.609344 = 4.828 km.
Measurement / conversions · Basic
1 lb = 0.45359237 kg
Use when switching weight or mass values into metric units.
150 lb × 0.45359237 = 68.04 kg.
Measurement / conversions · Intermediate
(1 ft)^2 = (0.3048 m)^2 = 0.09290304 m^2
Use when converting squared units. Square the length conversion factor.
Do not convert 1 ft² by multiplying by 0.3048. Use 0.3048².
Measurement / conversions · Intermediate
(1 ft)^3 = (0.3048 m)^3 = 0.028316846592 m^3
Use when converting cubed units. Cube the length conversion factor.
Do not treat cubic feet like feet. Cubed units need cubed factors.
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