Unit 8: Ratio, Proportion, Indices & Logarithms - Basics of Statistics and Mathematics

41 key vocabulary flashcards covering core terms, properties, and rules from Unit 8 on Ratio, Proportion, Indices, and Logarithms.

Ratio

A comparison of two like quantities by division, written a : b or a/b.

Simplified Ratio

A ratio that has been reduced to lowest whole-number terms by dividing both terms by their GCD.

Part-to-Part Ratio

A ratio that compares two distinct parts of a whole (e.g., boys : girls).

Part-to-Whole Ratio

A ratio that compares one part to the entire set (e.g., red apples : total apples).

Antecedent

The first term (numerator) of a ratio a : b.

Consequent

The second term (denominator) of a ratio a : b.

Proportion

An equality of two ratios, written a : b :: c : d or a/b = c/d.

Unitary Method

Procedure of finding the value of one unit first, then any required number of units.

Compound Ratio

The single ratio obtained by multiplying two or more ratios term-wise.

Duplicate Ratio

The ratio of the squares of two quantities, (a² : b²).

Triplicate Ratio

The ratio of the cubes of two quantities, (a³ : b³).

Sub-duplicate Ratio

The ratio of the square roots of two quantities, (√a : √b).

Sub-triplicate Ratio

The ratio of the cube roots of two quantities, (³√a : ³√b).

Invertendo

Proportion property: if a/b = c/d, then b/a = d/c.

Alternando

Proportion property: if a/b = c/d, then a/c = b/d.

Componendo

Proportion property: if a/b = c/d, then (a + b)/b = (c + d)/d.

Dividendo

Proportion property: if a/b = c/d, then (a − b)/b = (c − d)/d.

Componendo and Dividendo

Combined property: if a/b = c/d, then (a + b)/(a − b) = (c + d)/(c − d).

Mean Proportional (Geometric Mean)

For a, b > 0, the value x satisfying a/x = x/b; x = √ab.

Third Proportional

For a, b, the number c such that a : b :: b : c; c = b²/a.

Fourth Proportional

For a, b, c, the number d such that a : b :: c : d; d = bc/a.

Exponent / Index

The power to which a base is raised, indicating repeated multiplication.

Base (Exponentiation)

The number that is multiplied by itself as indicated by the exponent.

Power

The complete exponential expression, e.g., aⁿ, or sometimes synonymous with exponent.

Law of Exponents – Product Rule

aᵐ · aⁿ = aᵐ⁺ⁿ (same base, add exponents).

Law of Exponents – Quotient Rule

aᵐ / aⁿ = aᵐ⁻ⁿ (same base, subtract exponents).

Law of Exponents – Power of a Power

(aᵐ)ⁿ = aᵐⁿ.

Law of Exponents – Product to a Power

(ab)ᵐ = aᵐ bᵐ.

Zero Exponent Rule

For a ≠ 0, a⁰ = 1.

Logarithm

The inverse operation of exponentiation; log_b x equals the exponent y such that bʸ = x.

Common Logarithm

Logarithm with base 10, written log x or log₁₀ x.

Natural Logarithm

Logarithm with base e (≈ 2.718), written ln x or log_e x.

Characteristic (Logarithm)

The integer part of a common logarithm, indicating the order of magnitude.

Mantissa (Logarithm)

The fractional part of a common logarithm, giving the precise digits after the characteristic.

Logarithm – Product Rule

logb(xy) = logb x + log_b y.

Logarithm – Quotient Rule

logb(x/y) = logb x − log_b y.

Logarithm – Power Rule

logb(xᵏ) = k · logb x.

Change of Base Formula

logb x = logk x / log_k b, allowing conversion to any base k.

Logarithmic Function

Function of the form y = log_b x (b > 0, b ≠ 1) mapping positive x to real y.

Substrahendo

Proportion technique: if p/q = r/s, then (p − r)/(q − s) equals the same ratio.