5.9 Fundamental Percentage Calculations

Fundamental Percentage Calculations covers the three basic questions that arise from the part-percent-whole relationship—finding the part, finding the whole, or finding the percentage itself—along with the calculations for expressing an increase or decrease numerically, the terminology distinguishing an original value from its resulting value after a change, the direction such a change represents, the compounding effect of multiple successive percentage changes, and the verification techniques used to confirm these calculations.

Percentage of a Quantity

Percentage of a quantity finds the part represented by applying a given percent to a given whole, calculated by converting the percent to a decimal and multiplying it by the whole, such as finding 25% of 80 by computing 0.25 × 80 = 20.

Part from Percentage and Whole

Part from percentage and whole applies the general relationship part = percent × whole to solve directly for the part whenever the percentage and the whole are both known, following the same calculation as percentage of a quantity.

Whole from Part and Percentage

Whole from part and percentage solves the same relationship for the whole whenever the part and the percentage are known, dividing the part by the percentage expressed as a decimal, such as finding the whole when 20 is known to be 25% of it by computing 20 ÷ 0.25 = 80.

Percentage from Part and Whole

Percentage from part and whole solves the same relationship for the percentage whenever the part and the whole are both known, dividing the part by the whole and converting the resulting decimal to a percent, such as finding what percentage 20 is of 80 by computing 20 ÷ 80 = 0.25, or 25%.

Numerical Percentage Increase

Numerical percentage increase computes the resulting value after growth by a given percentage, found by multiplying the original value by 1 plus the percentage expressed as a decimal, such as increasing 80 by 25% by computing 80 × 1.25 = 100.

Numerical Percentage Decrease

Numerical percentage decrease computes the resulting value after shrinking by a given percentage, found by multiplying the original value by 1 minus the percentage expressed as a decimal, such as decreasing 80 by 25% by computing 80 × 0.75 = 60.

Original and Resulting Quantity Distinction

Original and resulting quantity distinction separates the value that existed before a percentage change, the original quantity, from the value that exists after that change has been applied, the resulting quantity, a distinction essential for correctly identifying which value serves as the base in any percentage increase, decrease, or comparison calculation.

Percentage Change Direction

Percentage change direction identifies whether a described change represents an increase or a decrease, determining whether the calculation should add to or subtract from 1 when computing the multiplying factor applied to the original quantity, with the described language of the situation—"increased by," "grew," "decreased by," "dropped"—signaling which direction applies.

Successive Percentage Changes

Successive percentage changes computes the compounded result of applying two or more percentage changes one after another, each applied to the resulting quantity produced by the previous change rather than to the original quantity each time, meaning a 10% increase followed by a 10% decrease does not return to the original value, since the second change is applied to a different base than the first.

Percentage Calculation Verification

Percentage calculation verification confirms a computed part, whole, or percentage by substituting the result back into the original part-percent-whole relationship and checking that the equation holds true, or by independently recomputing the same value using a different one of the three fundamental calculations and confirming both approaches agree.

Together, these fundamental calculations establish a complete toolkit for percentage problems: the single part-percent-whole relationship rearranged to solve for any one of its three quantities, extended to compute percentage increases and decreases relative to a correctly identified original quantity, compounded correctly across successive changes, and confirmed through direct verification against the original relationship.