Ratios and Proportions

• A ratio is a comparison of two quantities
• Ratios can be written in several ways e.g. a to b, a:b, or a/b
• Ratios should be simplified to their lowest terms
• A proportion is a statement that two ratios are equal, e.g. a/b = c/d

Simplifying Ratios

• Divide both parts of the ratio by their greatest common factor (GCF)
• If the ratio involves fractions, multiply both parts by the least common denominator (LCD) to eliminate the fractions
• If the ratio involves decimals, multiply both parts by a power of 10 to eliminate the decimals
• Ensure both parts of the ratio are in the same units before simplifying

Sharing in a Given Ratio

• To divide a quantity in a given ratio, first find the total number of parts in the ratio
• Divide the total quantity by the total number of parts to find the value of one part
• Multiply the value of one part by each number in the ratio to find the size of each share

Direct Proportion

• Two quantities are in direct proportion if they increase or decrease together, and their ratio remains constant
• If y is directly proportional to x, then y = kx, where k is the constant of proportionality
• To solve direct proportion problems, find the constant of proportionality using the initial values, then use it to find the unknown value

Inverse Proportion

• Two quantities are in inverse proportion if one increases as the other decreases, and their product remains constant
• If y is inversely proportional to x, then y = k/x, where k is the constant of proportionality
• To solve inverse proportion problems, find the constant of proportionality using the initial values, then use it to find the unknown value

Functions

• A function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output
• The input is called the argument or independent variable, and the output is called the value or dependent variable
• Functions are often denoted by letters such as f, g, or h
• The notation f(x) represents the value of the function f at the input x
• The domain of a function is the set of all possible input values
• The range of a function is the set of all possible output values

Evaluating Functions

• To evaluate a function at a specific value, substitute the value for the variable in the function's expression
• Simplify the expression to find the value of the function at that point
• E.g., if f(x) = 2x + 3, then f(4) = 2(4) + 3 = 11

Composite Functions

• A composite function is a function that is formed by applying one function to the results of another
• If f and g are functions, then the composite function f(g(x)) is obtained by substituting g(x) for x in f(x)
• The order of composition matters; f(g(x)) is generally not the same as g(f(x))

Sequences

• A sequence is an ordered list of numbers, called terms
• Sequences can be finite or infinite
• Each term in a sequence is denoted by a subscript, e.g., a₁, a₂, a₃, ...

Arithmetic Sequences

• An arithmetic sequence is a sequence where the difference between consecutive terms is constant
• This constant difference is called the common difference (d)
• The nth term of an arithmetic sequence is given by aₙ = a₁ + (n - 1)d, where a₁ is the first term

Geometric Sequences

• A geometric sequence is a sequence where the ratio between consecutive terms is constant
• This constant ratio is called the common ratio (r)
• The nth term of a geometric sequence is given by aₙ = a₁ * r^(n-1), where a₁ is the first term