Fractions, Decimals, and Percentages
9 min read
Fractions, decimals, and percentages are three notations for one thing, and speed comes from moving between them without computing: knowing the standard equivalences cold and understanding how percentage changes compose. The composition rules, in particular, hide the asymmetries that trip up intuition.
The standard equivalences
Together with their multiples, these cover most fractions met in practice. Eighths and sixths repay memorising: 5/8 = 0.625 and 5/6 ≈ 0.833 should be instant.
Percentage changes multiply
A change of percent is multiplication by , and successive changes MULTIPLY their factors; they never add. Up 20 percent then up 30 percent is : up 56 percent, not 50. For small changes the factors' product is close to adding the percentages (the cross term is small), which is why the additive shortcut feels right and quietly degrades as changes grow: the estimation module returns to exactly when the shortcut is safe.
Down 20 percent then up 20 percent does not return to the start:
A 4 percent loss survives the round trip, and the asymmetry worsens with size: down 50 then up 50 leaves . Undoing a fall of percent requires a rise of percent: 25 to undo 20, 100 to undo 50. Multiplication, not addition, is the grammar of successive changes.
Percent of a percent, and base discipline
Expressions like "40 percent of 15 percent" resolve by multiplying: , six percent. The recurring source of error is the BASE: a percentage is always of something, and changing the base mid-calculation produces plausible nonsense. If 30 percent of a group are of type A, and 20 percent of type A have a property, then 6 percent of the GROUP have it, but the fraction of property-holders who are type A cannot be computed without knowing the other types' rates: the conditional probability lessons formalised exactly this discipline.
A rate moves from 4 percent to 5 percent. Two true statements:
The first measures the absolute change in the rate; the second the relative change. Both usages are standard, they differ by a factor of 25 here, and precise speech always says which is meant.
Quick check
A quantity falls 20 percent, then rises 25 percent. What is the net change in percent? Answer as a signed number.
Quick check
What is 35 percent of 40 percent, in percent? Answer as a decimal.
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