Percentage Calculations: Part, Base, and Rate

Multiply the base by the percentage rate and divide by 100: W=G·p/100. For 15% of €84, calculate 84·15/100=€12.60. The percentage part is the concrete amount represented by the rate. Write the percentage either as 15/100 or as the decimal 0.15, but do not apply both conversions. A useful check is W/G: 12.60/84=0.15, which is 15% again. Also make sure the base and the part use the same unit. Write the values with units before calculating; this keeps clear whether you are comparing money, people, or another reference quantity.

Rearrange W=G·p/100 for G: G=100W/p. If €24 is exactly 15% of an amount, then G=100·24/15=€160. Check in the forward direction: 15% of €160 is €24. The base always means the whole amount to which the percentage refers. In a word problem it is not necessarily the first amount mentioned, so identify which value represents 100%. The rate p cannot be zero because the rearranged formula divides by p. When two different rates appear, check whether both use the same base or apply to amounts changed in sequence.

The base G is the whole and represents 100%. The rate p describes the share in percent, while the part W is the corresponding amount in the base’s unit. For 12 out of 80 people, G=80, W=12, and p=12/80·100=15%. A quick check is W/G=p/100. First ask, “Out of which whole?” This prevents using the wrong amount as the 100% reference in discounts, surveys, or mixtures. The names describe each quantity’s role, not the order in which it appears in a problem. The rate is a ratio and independent of the base unit; the part takes the unit of G.

The second change applies to the already changed price. After a 20% discount, the remaining factor is 0.80; another 10% discount multiplies it by 0.90. Together, 0.80·0.90=0.72 of the original price remains, so the total discount is 28%. An item starting at €100 then costs €72. Percentage points can be added when two rates use the same base; successive percentage changes must instead be combined by multiplying their factors. This is the same idea used in the growth formula. For reverse calculations, a small table with 100% for the base and the known part in the second row can help.

Yes. p compares a part with the selected base; the part does not have to be smaller than the whole. If an amount grows from 40 to 100, the final amount is 250% of the starting amount, while the increase is 150% of the starting amount. These statements answer different questions. For the final amount use new/old·100; for the percentage change use (new−old)/old·100. State whether you mean the new total or only the change. Negative reference values require extra care beyond the usual school interpretation. Round only after calculating, especially with multiple changes; early rounding can shift the final result.