Ratios, Rates, and Proportions

Ratios Compare, Fractions Divide

A ratio compares quantities of the same kind: "$3$ parts red to $2$ parts blue," written $3 : 2$ or $\tfrac{3}{2}$. The crucial idea is that a ratio fixes only the relative sizes, not the actual amounts. $3 : 2$ could mean $3$ and $2$, or $30$ and $20$, or $3k$ and $2k$ for any positive $k$. That hidden multiplier $k$ is the engine behind almost every ratio question.

Part-to-Part vs. Part-to-Whole

Keep the two flavors straight. If a class has boys and girls in the ratio $3 : 2$, then boys are $3$ parts, girls $2$ parts, and the whole is $3 + 2 = 5$ parts. So boys make up $\tfrac{3}{5}$ of the class — a part-to-whole fraction — even though the given part-to-part ratio was $3 : 2$. Turning a part-to-part ratio into "what fraction of the total" means dividing by the sum of the parts.

Solving with the Multiplier

Assign the common multiplier $k$ and translate any given total. If coins are pennies : nickels : dimes $= 3 : 5 : 7$ and there are $90$ coins total, then $3k + 5k + 7k = 15k = 90$, so $k = 6$. The number of dimes is $7k = 42$. This single technique — parts times $k$ equals total — solves the large majority of ratio word problems.

Combining Ratios Through a Shared Term

When two ratios share a quantity, scale them so the shared term matches, then read across. Given red : blue $= 2 : 3$ and blue : green $= 4 : 5$, make the blue values equal. The blue terms are $3$ and $4$, whose LCM is $12$:

$\text{red} : \text{blue} = 2 : 3 = 8 : 12, \qquad \text{blue} : \text{green} = 4 : 5 = 12 : 15.$

Now red : blue : green $= 8 : 12 : 15$, so red : green $= 8 : 15$. The naive answer $2 : 5$ (just pairing the outer numbers) is wrong.

Direct and Inverse Proportion

Two quantities are directly proportional when their ratio is constant: $\tfrac{y}{x} = k$, so doubling $x$ doubles $y$. They are inversely proportional when their product is constant: $xy = k$, so doubling $x$ halves $y$. Recognizing which one governs a problem is half the battle:

Rates and Unit Rates

A rate is a ratio of quantities with different units — miles per hour, dollars per pound, widgets per machine-hour. Reduce to a unit rate to compare or to scale. If $6$ machines make $240$ widgets in $4$ hours, the rate per machine-hour is $\tfrac{240}{6 \times 4} = 10$ widgets; then $9$ machines in $5$ hours make $9 \times 5 \times 10 = 450$ widgets. The master equation for motion is

$\text{distance} = \text{rate} \times \text{time}.$

Combined Work

When two agents work together, add their rates, not their times. If pipe $A$ fills a tank in $3$ hours (rate $\tfrac{1}{3}$ per hour) and pipe $B$ in $6$ hours (rate $\tfrac{1}{6}$), together they fill $\tfrac{1}{3} + \tfrac{1}{6} = \tfrac{1}{2}$ of the tank per hour, so the job takes $2$ hours. Averaging the two times ($4.5$ hours) is a classic trap.

A Reliable Game Plan

• Introduce the multiplier $k$, write parts $\times k =$ total, and solve.
• Part-to-whole? Divide a part by the sum of the parts.
• Combining ratios? Scale to a common shared term, then read across.
• Rates? Reduce to a unit rate; for motion use distance $=$ rate $\times$ time.
• Working together? Add rates, then invert to get the time.