Ratios, Rates & Proportions

A ratio compares two quantities by division --- it tells you how much of one thing there is for each amount of another. If a bowl holds $3$ apples and $5$ oranges, the ratio of apples to oranges is “$3$ to $5$.”

There are three ways to write the same ratio:

• with the word to:   $3$ to $5$
• with a colon:   $3:5$
• as a fraction:   $\frac{3}{5}$

Order matters! The ratio of apples to oranges ($3:5$) is different from oranges to apples ($5:3$). To simplify a ratio, divide both numbers by their greatest common factor, just like reducing a fraction.

Opposite, adjacent and hypotenuse are named from the angle.

Two ratios are equivalent if they simplify to the same ratio. You build an equivalent ratio by multiplying or dividing both parts by the same number --- exactly how you make equivalent fractions.

A ratio table lists several equivalent ratios in a row, which makes patterns easy to see and missing values easy to fill in.

Opposite, adjacent and hypotenuse are named from the angle.

A rate is a ratio that compares two quantities with different units, such as miles and hours, or dollars and pounds. A unit rate is a rate with a denominator of 1: how much of the first quantity for exactly one of the second, like miles per hour or dollars per pound. To find a unit rate, divide the first quantity by the second.

A unit price is a unit rate for cost (dollars per item or per unit). Comparing unit prices tells you the best buy --- the lower unit price is the better deal.

A proportion is an equation stating that two ratios are equal, like $\frac{2}{3} = \frac{8}{12}$. When one part is unknown, we solve for it using cross-multiplication: in a true proportion, the two diagonal products are equal.

Multiply across the diagonals, set the products equal, then divide to isolate the unknown.

Many everyday problems are really proportions in disguise: recipes, prices, speeds, and mixtures. To solve one:

• Write a ratio from the known facts, labeling the units.
• Set it equal to a second ratio in the same order, using a letter for the unknown.
• Cross-multiply and solve.

A scale tells how a drawing, map, or model relates to the real object. It is written as a ratio, such as “$1$ inch $= 20$ miles” or $1:18$. Every scale problem is a proportion:

Set up the scale ratio, then cross-multiply to find the missing distance --- either the real distance from a drawing, or the drawing size from a real distance.

Two figures are similar when they have the same shape but not necessarily the same size --- one is a scaled copy of the other. In similar figures, corresponding sides are proportional: they all share the same scale factor. To find a missing side, set up a proportion of corresponding sides and solve.

When two quantities stay in a constant ratio, we say $y$ varies directly with $x$:

Here $k$ is the constant of proportionality. Double $x$ and $y$ doubles too --- the graph is a straight line through the origin.

When instead the two quantities have a constant product, $y$ varies inversely with $x$:

Now doubling $x$ halves $y$. More workers means less time; faster speed means fewer hours for the same trip. The tell-tale sign of inverse variation is that the product stays fixed while the ratio does not.

Opposite, adjacent and hypotenuse are named from the angle.

A ratio can compare three or more quantities at once, such as $2:3:4$. Think of it as splitting a whole into parts: $2:3:4$ uses $2+3+4 = 9$ equal parts. To share an amount in a given ratio, divide by the total number of parts to find one part, then multiply.

You can also combine two separate two-term ratios into one. If $A:B = 2:3$ and $B:C = 6:5$, first rescale so the shared term $B$ matches. Multiply the first ratio by $2$ to get $A:B = 4:6$; now $B$ is $6$ in both, giving $A:B:C = 4:6:5$.

In the proportion $\frac{a}{b} = \frac{b}{c}$, the repeated middle term $b$ is called the mean proportional (or geometric mean) of $a$ and $c$. Cross-multiplying gives

This is different from the ordinary (arithmetic) average $\frac{a+c}{2}$. For example, the mean proportional of $4$ and $9$ is $\sqrt{4 \cdot 9} = \sqrt{36} = 6$, whereas their arithmetic average is $6.5$.

The golden ratio appears when a length is split so that the whole is to the larger part as the larger part is to the smaller part. Writing the larger part as $a$ and the smaller as $b$:

Its exact value is $\varphi = \dfrac{1+\sqrt{5}}{2}$. The golden ratio has a remarkable self-referencing property: $\varphi^2 = \varphi + 1$ and $\dfrac{1}{\varphi} = \varphi - 1$. It shows up in art, architecture, and the spiral arrangement of leaves and seeds in nature.

A plain average treats every item equally, but a weighted average counts some values more heavily according to a ratio of “weights” (amounts). For two groups,

where each $v$ is a value and each $w$ is how much of it there is. Mixture problems --- blending solutions of different concentrations, or averaging test scores of different weights --- are exactly this idea. The result always lands between the two values, pulled toward whichever group is larger.

When two workers (or pipes, or machines) act together, add their rates, not their times. If one finishes a job in $a$ hours and the other in $b$ hours, their combined time $t$ satisfies

A close cousin appears with average speed over equal distances. If you drive one leg at speed $v_1$ and an equal-length leg at $v_2$, the average speed is not $\frac{v_1+v_2}{2}$ but the harmonic mean:

You spend more time at the slower speed, so the true average is dragged below the plain average.

When a figure is enlarged by a scale factor $k$, its lengths multiply by $k$ --- but its area multiplies by $k^2$ and its volume by $k^3$:

This is why doubling a photo's dimensions uses four times the ink, and why a scale model at $1:10$ holds only $\frac{1}{1000}$ of the real object's volume. It is the most-missed idea in ratios: side ratios and area (or volume) ratios are not the same number.