Comprehensive Math Study Guide: Fractions, Decimals, Percentages, Ratios, Algebra, Geometry, Trigonometry, Statistics, and Conversions

What is a fraction?

A fraction looks like a/b, where a is the numerator (top number) and b is the denominator (bottom number).

What are the parts of a fraction?

The numerator is the top number, and the denominator is the bottom number.

How do you simplify a fraction?

Divide the numerator and denominator by the same number.

How do you add fractions with the same denominator?

Keep the denominator and add the numerators.

How do you subtract fractions with the same denominator?

Keep the denominator and subtract the numerators.

How do you add fractions with different denominators?

Find a common denominator, convert both fractions, add the numerators, and keep the denominator.

What is a common denominator?

A number that both denominators can divide into evenly.

How do you multiply fractions?

Multiply the numerators together and the denominators together.

How do you divide fractions?

Keep the first fraction, change ÷ to ×, and flip the second fraction.

How do you convert a mixed number to an improper fraction?

Multiply the whole number by the denominator, add the numerator, and keep the denominator.

How do you convert an improper fraction to a mixed number?

Divide the numerator by the denominator to get the whole number and the remainder as the numerator.

How do you add decimals?

Line up the decimal points, add zeros if necessary, and add normally.

How do you subtract decimals?

Line up the decimal points, add zeros if necessary, and subtract normally.

How do you multiply decimals?

Ignore the decimal points, multiply normally, then count total decimal places and place the decimal in the answer.

How do you divide decimals?

If the divisor has a decimal, move its decimal point to the right until it is a whole number, then move the decimal in the dividend the same number of spaces.

What are decimal place values?

0.1 = tenths, 0.01 = hundredths, 0.001 = thousandths, 0.0001 = ten-thousandths.

What does 'percent' mean?

Percent means 'per 100'.

How do you convert a percent to a decimal?

Divide it by 100.

How do you convert a decimal to a percent?

Multiple it by 100.

How do you convert a fraction to a percent?

Divide the numerator by the denominator and convert the decimal to percent.

How do you find 20% of a number?

Convert the percent to decimal and multiply.

How do you calculate percentage increase?

Find the difference, divide by the old number, and multiply by 100.

How do you calculate percentage decrease?

Find the difference, divide by the old number, and multiply by 100.

How do you increase a number by a percentage?

Convert the percentage to decimal, add 1, and multiply.

How do you decrease a number by a percentage?

Convert the percentage to decimal, subtract from 1, and multiply.

What is a ratio?

A ratio compares two quantities.

How do you simplify a ratio?

Divide BOTH numbers by the same number.

What is a proportion?

A proportion says two ratios are equal.

How do you solve a proportion?

Use CROSS MULTIPLICATION.

What is cross multiplication?

Multiply diagonally: a/b = c/d means a × d = b × c.

What is a variable?

A variable is a letter representing an unknown number.

What is a coefficient?

A coefficient is the number multiplied by a variable.

What is a constant?

A constant is a number without a variable.

What are like terms?

Terms with the SAME variables and SAME exponents.

How do you simplify like terms?

Add coefficients.

How do you distribute?

Multiply the number outside by EVERY term inside.

How do you simplify -2(x + 3)?

-2 × x = -2x and -2 × 3 = -6.

How do you solve x + 5 = 12?

Undo +5 with -5.

What operation undoes addition?

Subtraction.

How do you solve 2x + 5 = 15?

Remove the constant, then divide by the coefficient.

How do you solve 3x + 2 = x + 10?

Move variables to one side, then move constants.

What does factoring mean?

Factoring means breaking an expression into multiplication.

How do you find the GCF?

Find the largest number that divides evenly into ALL terms.

What are the basic exponent rules?

Add exponents when multiplying, subtract when dividing.

What is perimeter?

Perimeter = distance AROUND a shape.

What is area?

Area = space INSIDE a 2D shape.

What is the area of a rectangle?

A = L × W.

What is the perimeter of a rectangle?

P = 2L + 2W.

What is the area of a triangle?

A = 1/2 × base × height.

What are radius and diameter?

Radius = center to edge; Diameter = edge to edge through center.

What is the circumference of a circle?

Circumference = distance around a circle.

What is the area of a circle?

A = πr².

What is the volume of a cube?

V = s³.

What is the volume of a rectangular prism?

V = L × W × H.

What is the volume of a cylinder?

V = πr²h.

What is the formula for the volume of a cylinder?

V = πr²h

What is the difference between area and volume?

Area = 2D = square units; Volume = 3D = cubic units

What does basic trigonometry deal with?

Trigonometry connects the angles and sides of triangles.

What are the three basic trig functions?

SINE = sin, COSINE = cos, TANGENT = tan

What does SOH CAH TOA mean?

SOH: sin = Opposite / Hypotenuse; CAH: cos = Adjacent / Hypotenuse; TOA: tan = Opposite / Adjacent

How do you find the hypotenuse?

The hypotenuse is always opposite the right angle and is the longest side.

When do you use sine?

Use sin(θ) = opposite / hypotenuse.

When do you use cosine?

Use cos(θ) = adjacent / hypotenuse.

When do you use tangent?

Use tan(θ) = opposite / adjacent.

How do you find a missing angle?

Use inverse trig: sin⁻¹, cos⁻¹, tan⁻¹.

How do you find the mean?

Mean = average; Add all numbers and divide by the number of values.

How do you find the median?

Median = middle number; Put numbers in order and find the middle.

What is the mode?

Mode = number that appears most often.

How do you find the range?

Range = largest - smallest.

What is standard deviation?

Standard deviation tells you how spread out the data is around the mean.

How do you calculate population standard deviation?

Find the mean, subtract the mean from each number, square each difference, find the average of those squared differences, and take the square root.

What is the difference between sample and population standard deviation?

Population: divide by N; Sample: divide by n - 1.

What are the most important metric prefixes?

kilo = 1,000; hecto = 100; deka = 10; BASE = 1; deci = 0.1; centi = 0.01; milli = 0.001.

What are common metric units of length?

1 kilometer = 1,000 meters; 1 meter = 100 centimeters; 1 meter = 1,000 millimeters.

How do you convert between metric units?

Move the decimal based on the units; right for smaller units, left for larger units.

What are common U.S. length conversions?

12 inches = 1 foot; 3 feet = 1 yard; 5,280 feet = 1 mile.

What are common U.S. weight conversions?

16 ounces = 1 pound; 2,000 pounds = 1 ton.

What are common U.S. liquid volume conversions?

3 teaspoons = 1 tablespoon; 16 tablespoons = 1 cup; 8 fluid ounces = 1 cup; 2 cups = 1 pint; 2 pints = 1 quart; 4 quarts = 1 gallon.

What is the conversion from teaspoons to gallons?

1 teaspoon → 1 tablespoon → 1 cup → 1 pint → 1 quart → 1 gallon

What are useful metric/U.S. length conversions?

1 inch ≈ 2.54 centimeters, 1 foot ≈ 30.48 centimeters, 1 yard ≈ 0.9144 meters, 1 mile ≈ 1.609 kilometers, 1 kilometer ≈ 0.621 miles

What are useful metric/U.S. weight conversions?

1 kilogram ≈ 2.205 pounds, 1 pound ≈ 0.454 kilograms, 1 ounce ≈ 28.35 grams

What are useful metric/U.S. volume conversions?

1 U.S. gallon ≈ 3.785 liters, 1 liter ≈ 0.264 gallons, 1 U.S. cup ≈ 236.6 mL, 1 fluid ounce ≈ 29.57 mL

What should I do when I see fractions?

ADD/SUBTRACT: Get common denominator, add/subtract numerators, keep denominator, simplify. MULTIPLY: Top × top, bottom × bottom. DIVIDE: Keep → change → flip.

What should I do when I see decimals?

ADD/SUBTRACT: Line up decimal points. MULTIPLY: Ignore decimals, multiply, count decimal places, put decimal back. DIVIDE: Move divisor decimal until it is a whole number, move dividend decimal the same amount, divide.

What should I do when I see percentages?

Convert X% to decimal, then multiply: X% × Y.

How do I calculate percentage change?

Find CHANGE, divide by ORIGINAL, multiply by 100. Formula: CHANGE ÷ ORIGINAL × 100.

What should I do when I see a proportion?

Set up equal fractions, then cross multiply: a/b = c/d, a × d = b × c, then solve for the variable.

What should I do when I see an equation?

Simplify both sides, combine like terms, move variables to one side, move constants to the other, divide by coefficient, check your answer.

What are the formulas for area and perimeter of a rectangle?

Area = L × W, Perimeter = 2L + 2W.

What are the formulas for area and circumference of a circle?

Area = πr², Circumference = 2πr.

What should I memorize about adding signed numbers?

Same signs: Add the numbers and keep the sign. Different signs: Subtract and keep the sign of the number with the larger absolute value.

What happens when multiplying or dividing negative numbers?

Same signs → POSITIVE, Different signs → NEGATIVE.

What are the fraction rules to memorize?

ADD/SUBTRACT: Common denominator first. MULTIPLY: a/b × c/d = ac/bd. DIVIDE: a/b ÷ c/d = a/b × d/c.

What are the percentage formulas to memorize?

Percent of a number: decimal percent × number. Percentage change: change ÷ original × 100. Increase: original × (1 + rate). Decrease: original × (1 - rate).

What are the geometry formulas to memorize?

Rectangle: A = LW, P = 2L + 2W. Triangle: A = 1/2bh. Circle: A = πr², C = 2πr. Cube: V = s³. Rectangular prism: V = LWH. Cylinder: V = πr²h.

What are the trigonometry formulas to memorize?

SOH: sin θ = opposite/hypotenuse, CAH: cos θ = adjacent/hypotenuse, TOA: tan θ = opposite/adjacent.

What are the most important things to memorize in math?

Fractions: ADD/SUBTRACT → COMMON DENOMINATOR, MULTIPLY → TOP × TOP, BOTTOM × BOTTOM, DIVIDE → KEEP, CHANGE, FLIP.

What are the U.S. length conversions?

12 in = 1 ft, 3 ft = 1 yd, 5,280 ft = 1 mi.

What are the U.S. weight conversions?

16 oz = 1 lb, 2,000 lb = 1 ton.

What are the U.S. volume conversions?

3 tsp = 1 tbsp, 16 tbsp = 1 cup, 2 cups = 1 pint, 2 pints = 1 quart, 4 quarts = 1 gallon.