Ratio ($a:b$)

A ratio compares two quantities $a,b$, written $a:b$ and corresponding to division $a/b$ when $b\ne 0$. Equal ratios of corresponding lengths characterize the scaling of similar triangles.

For example, $6:4$ and $3:2$ specify the same ratio because $6/4=3/2$. Multiplying both entries by the same nonzero scale preserves the quotient: $(ca)/(cb)=a/b$. This is useful when comparing mixtures or scaling a recipe; doubling both ingredient amounts preserves their ratio, while adding the same amount to both generally does not. For quantities with the same physical units, their ratio is dimensionless, connecting this elementary calculation with dimensional analysis.

A percentage expresses a ratio in hundredths: $p$ as a fraction is $100p\%$. For example, $18/60=0.3=30\%$. Comparing two percentages requires specifying whether the change is an absolute difference in percentage points or a relative change divided by the initial value.

A percentage point is one hundredth on a probability or proportion scale. A change from $20\%$ to $25\%$ is $5$ percentage points, while the relative increase is $(25-20)/20=25\%$. These answer different questions and have different denominators.