Ratios, rates, proportions, and percent change

A ratio compares quantities in a specified order. A rate is a ratio with different units, and a unit rate has denominator $1$. Equivalent ratios result from multiplying both quantities by the same nonzero scale factor.

A proportional relationship has a constant unit rate $k$ and can be written $y =$ kx. Ratio tables, double number lines, equations, and graphs all represent the same multiplicative structure. Cross-products are a consequence of equality between two ratios, not a reason to ignore units.

Percent change compares the change with the original amount: $\frac{new - original}{original} \cdot 100\%$. A change from $40\%$ to $50\%$ is $a 10-percentage-point$ increase but $a 25\%$ relative increase because $10$ is one-fourth of the original $40$.

A ratio compares quantities multiplicatively. The ratio $3:5$ says the first quantity is $\frac{3}{5}$ of the second, not that their difference is $2$. Equivalent ratios arise by scaling both quantities by the same nonzero factor, so they preserve a constant multiplicative relationship. A ratio table makes the scale factor visible and helps distinguish proportional change from a fixed additive change.

A rate is a ratio of quantities with different units. Dividing by the second quantity creates a unit rate, such as $\frac{180 miles}{3}$ hours $= 60$ miles per hour. The compound unit is part of the answer and determines how the rate can be used. Multiplying $\frac{60 miles}{hour}$ by $5$ hours leaves miles; multiplying by $5$ miles would not model time. Unit cancellation therefore acts as a structural check on proportional calculations.

A proportion is an equation stating that two ratios are equal. Cross multiplication follows from multiplying both sides of $\frac{a}{b} = \frac{c}{d}$ by the nonzero product bd, producing ad $=$ bc. It is not a free-standing trick and is invalid when a denominator is zero. Often a scale-factor or unit-rate method is clearer than cross multiplication because it preserves the meaning of the quantities instead of compressing them immediately.

Percent change compares a change with the original amount: $\frac{new - original}{original}$. The original value is the reference whole, even when the new value is more convenient. An increase from $80$ to $100$ is $\frac{20}{80} = 25\%,$ while a decrease from $100$ back to $80$ is $-\frac{20}{100} = -20\%$. The percentages differ because the reference wholes differ. A percentage-point change instead subtracts two percent rates directly.

Proportional models must pass a constant-ratio test. If a table includes $(2, 7), (4, 14),$ and $(6, 20),$ the first two rows have $\frac{output}{input}$ ratio $3.5$ but the third does not, so the relationship is not proportional. A graph of a proportional relationship is a straight line through the origin. These tests prepare the transition from arithmetic rates to Algebra’s equations, tables, and graphs of linear relationships.

Proportions compare corresponding quantities, so the order inside every ratio must remain consistent. If $4$ notebooks cost $\$10,$ the ratios $\frac{4 notebooks}{\$10}$ and $\frac{n notebooks}{\$35}$ may be equated; so may $\frac{\$10}{4}$ notebooks and $\frac{\$35}{n}$ notebooks. Mixing one orientation with the other creates an equation about unlike rates. Label numerator and denominator units before solving. When the units match across the equality, cross-products inherit compatible compound units and the numerical equation represents the intended comparison.

Not every rate problem is proportional. A taxi fare with $a \$4$ starting fee plus $\$2$ per mile does not have a constant fare-to-mile ratio, because the fixed fee is present even at zero miles. A quantity $y$ is proportional to $x$ only when $y =$ kx for one constant $k$ and the relationship passes through $(0,0)$. Look for fixed starting amounts, minimum charges, or one-time fees before setting up a proportion. If such a quantity exists, the model is usually additive-plus-multiplicative and belongs to the later study of linear equations rather than a single ratio equality.

Scale drawings require the same unit on both sides of a scale ratio. A map scale of $1$ cm to $8$ km may be used directly only if map lengths are in centimeters and actual lengths are in kilometers; converting first avoids mixing centimeters with meters or miles. After solving, ask whether enlargement or reduction makes sense. A small map distance should correspond to a larger real distance, whereas a scale model may be smaller or larger depending on the stated factor.