Proportion Formula: a/b = c/d, Cross Multiplication, and Both Notations

The doubling gives the picture. Enlarge a photo from 2 inches to 6 inches, and every feature keeps step. What held between two widths on the small print still holds on the big one. Ratios that move together like this are exactly what the proportion formula records.

Ray's New Higher Algebra (1866, p. 239) states the core in one line: proportion is an equality of ratios. When two ratios are equal, their four numbers are said to be proportional. Written with fractions, the same fact gives the proportion formula its standard shape:

$\frac{a}{b}=\frac{c}{d},\qquad b\neq 0,\ d\neq 0$

Read it aloud the textbook way: $a$ is to $b$ as $c$ is to $d$. Three names do all the work:

Ray (p. 240) fixes the vocabulary: the first and last terms are the extremes, the second and third are the means. Every rule on this page is a sentence about those four slots.

A single ratio can be written several ways, and Ray lists the forms side by side (p. 239): proportion is written in two ways - with a double colon, $a : b :: c : d$, or with an equals sign between the ratios, $a : b = c : d$. Both spell the same proportion formula.

Write the Following Ratio Using Two Other Notations

Textbook exercises put it exactly that way. Write the following ratio using two other notations: $\frac{3}{5}$.

• Colon notation. Replace the fraction bar with a colon: $3 : 5$.
• Word notation. Read it out and keep the words: 3 to 5.

All three forms name one number. A proportion formula simply claims that two of these notations point at the same value - $3 : 5 :: 6 : 10$, for instance, because $\frac{3}{5}=\frac{6}{10}$. Whichever notation a problem arrives in, the proportion formula is the referee, and the product rule below checks its claim.

Ray states Proposition I on page 240: in every proportion, the product of the means is equal to the product of the extremes. His own numbers show it: $2 : 6 :: 5 : 15$, and indeed $6 \times 5 = 2 \times 15$.

The algebra is one line. Start from the ratio form and multiply both sides by $bd$:

$\frac{a}{b}=\frac{c}{d}\quad\Longrightarrow\quad \frac{a}{b}\cdot bd=\frac{c}{d}\cdot bd\quad\Longrightarrow\quad ad=bc$

The way classrooms say it: each denominator travels across the equals sign and multiplies the opposite numerator. So the proportion formula $\frac{a}{b}=\frac{c}{d}$ has a product twin, $ad=bc$ - and this product form is the standard equation for proportion work, because it has no fractions left in it.

The two forms carry identical information - one proportion formula, two costumes. Whichever one a problem hands you, the other is one multiplication away.

Most problems hide one of the four terms. Proportional equations answer by unwrapping the product form: if $ad=bc$, then each term equals the product of the other three divided by its partner across the X.

A quick demonstration shows the proportion formula at its shortest. Solve $\frac{x}{12}=\frac{3}{4}$. Cross multiply: $4x = 36$, so $x = 9$. The proportion formula needs exactly two moves every time - multiply across, then divide by the number left next to the letter. Proportional equations with a sum in a denominator, such as $\frac{x}{x+4}=\frac{3}{5}$, open the same way; the distributive step is the only newcomer.

One habit keeps proportional equations honest: label the units first. Cups over cups, miles over miles - the proportion formula compares like with like.

Problem. Solve the proportion formula $\frac{m}{72}=\frac{5}{9}$.

Step 1, cross multiply. Each denominator multiplies the opposite numerator: $9m = 72 \times 5 = 360$.

Step 2, divide. $m = \frac{360}{9} = 40$.

Step 3, check the answer in the proportion formula. $\frac{40}{72}=\frac{5}{9}$ after reducing by 8 - the answer stands.

Problem. A rice recipe uses 5 cups of water for every 2 cups of rice. How much water goes with 6 cups of rice?

Step 1, name the unknown. Let $w$ be the cups of water.

Step 2, set up the proportion formula with units aligned - cups of water over cups of rice on both sides:

$\frac{w}{6}=\frac{5}{2}$

Step 3, cross multiply. $2w = 30$, so $w = 15$.

Step 4, answer in context. 15 cups of water go with 6 cups of rice - in the proportion formula, the setup is where care lives.

Scenario reworded with the food setting changed.

Problem. Solve the proportional equation $\frac{7}{c}=\frac{49}{30}$.

Step 1, cross multiply. $49c = 7 \times 30 = 210$.

Step 2, divide. $c = \frac{210}{49} = \frac{30}{7}$.

Do not expect whole numbers: a proportion formula may absolutely produce a fraction, and $\frac{30}{7}$ is the complete answer the proportion formula gives here. As a decimal it is about $4.29$ - check $\frac{7}{30/7}=\frac{49}{30}$ for reassurance.

The fractional-answer pattern is everyday work for the proportion formula.