Make the whole one hundred

If 5 of 20 items are red, red is 5/20 = 1/4 of the whole. Scaling the whole to 100 gives 25/100, written 25%. The percent sign expresses a rate per hundred.

You do not need to physically increase twenty items to one hundred. You are expressing the comparison per hundred. The fraction 1/4, decimal 0.25 and percentage 25% represent the same ratio. Twenty-five percent is not the decimal twenty-five.

To express the ratio as a percentage, divide the part by the whole, multiply by 100, and attach %. Here 5 ÷ 20 = 0.25 and 0.25 × 100 = 25, giving 25%. Name the whole before calculating.

The same five red items make 50% of ten but 25% of twenty. The part alone does not determine the percentage. Likewise, 50% of ten is five while 50% of twenty is ten: equal percentages need not mean equal amounts.

To find 30% of a whole, multiply by 30/100. Thirty percent of forty is 40 × 0.3 = 12 items. This lesson uses disjoint parts within a whole, so values range from 0% to 100%. Other comparisons, such as growth, can exceed 100%.

The hundred-cell grid below represents a reference scaled to one hundred. Each cell does not necessarily mean one original item. Change part and whole to distinguish preserving a count from preserving a ratio, then continue to direct proportion.