Proportions

What a proportion is

A proportion is a statement that two ratios are equal. From the last two lessons you know that $3 : 6$ is the same comparison as $1 : 2$. That is because dividing both parts of $3 : 6$ by $3$ lands on $1 : 2$. Writing that equality down is a proportion:

Because a ratio can be written as a fraction whenever its second number is not zero, and every ratio in this lesson has one, the same proportion is usually written with two equal fractions. That fraction form is the one we will work in:

Read it as ”$1$ is to $2$ as $3$ is to $6$.” A proportion is true only when the two ratios really are equivalent. If they are not equivalent, the statement is simply false, the same way $4 = 5$ is false.

Telling whether two ratios are in proportion

Since the two sides are just fractions, deciding whether a proportion is true is the same job as deciding whether two fractions are equal. You already have two ways to do that from the Fractions chapter, and both work here.

Start with $6 : 8$ and $9 : 12$. The ratio $6 : 8$ reduces to $3 : 4$ (divide both parts by $2$), and $9 : 12$ also reduces to $3 : 4$ (divide both parts by $3$). They share the same simplest form, so

is a true proportion. That is the first test: simplify each ratio, then see whether the two land on the same lowest terms.

The second test scales one ratio to match the other. Multiplying both parts of $6 : 8$ by $1.5$ gives $6 \times 1.5 = 9$ and $8 \times 1.5 = 12$, which is the second ratio exactly. Whenever multiplying both parts of one ratio by the same factor gives the other ratio, the two are equivalent.

Cross-multiplication, and why it works

Simplifying is easy when the numbers are friendly, but it gets fiddly when they are not. There is a test that works on any two ratios and needs only multiplication.

Simplifying already showed that $\frac{6}{8} = \frac{9}{12}$ is true. Multiply along each diagonal instead, the top of one side times the bottom of the other:

The two diagonal products are equal. They are called the cross-products. The picture shows where that name comes from. It draws the proportion $\frac{1}{2} = \frac{3}{6}$, with the two products running along its diagonals and crossing in the middle.

The match is not luck. Over the shared denominator $8 \times 12 = 96$, the ratio $\frac{6}{8}$ becomes $\frac{72}{96}$ and $\frac{9}{12}$ becomes $\frac{72}{96}$. Those two numerators are the diagonal products, and two equal ratios over one denominator must have equal numerators.

The same argument works for any proportion, not just this one pair of numbers. Writing it once for every possible proportion needs a letter in each of the four spots, $a$, $b$, $c$, and $d$, each one standing for whatever number happens to sit there. For any proportion, the same two diagonals give the same test:

The proportion is true exactly when these two cross-products are equal, $a \times d = b \times c$. The test carries a condition you have to check: $b$ and $d$ must not be zero. Their product $b \times d$ is then not zero either. That nonzero product is the common denominator each ratio gets rewritten over, which is where the rule comes from.

Finding a missing amount

The real power of a proportion is that you can know three of the four numbers and find the fourth. That holds for the positive amounts this lesson works with, such as cups, miles and dollars. This is the everyday use: you know one ratio and one matching amount, and you want the other. Set the two ratios equal, keeping the same kind of quantity in matching positions, then find the missing number with arithmetic you already have.

Both ratios must list their quantities in the same order. If the left ratio is cups of water over scoops of powder, the right ratio must also be cups of water over scoops of powder. Mixing the order (water over scoops on one side, scoops over water on the other) builds a false proportion and gives a wrong answer.

Method 1: scale the ratio

If one ratio’s known part is a whole-number multiple of the other’s, the fastest route is the equivalent-ratio scaling from the Ratios lesson. Find the scale factor from the pair you know, then apply that same factor to the other part.

Method 2: cross-multiply, then divide

Scaling is neat, but it only stays simple when the known parts divide evenly. When they do not, cross-multiplication handles any proportion. The idea uses the rule you just proved: in a true proportion the cross-products are equal. So multiply the two numbers on the diagonal you fully know, which gives the value of the other cross-product. Then divide that value by the remaining known number to recover the missing one.

Take a missing numerator, and call the unknown amount $n$. That $n$ is a variable standing for the amount you are finding, and here you solve for it by multiplying and dividing:

The cross-products of a true proportion are equal, so $n \times 4$ must equal $6 \times 10$. The right side is a product of two known numbers:

So $n$ multiplied by $4$ gives $60$:

The number that does that is found by dividing $60$ by $4$, since division undoes multiplication:

The missing amount is $15$. Every proportion with one blank yields to the same two steps. Multiply the diagonal you know, then divide by the number diagonally opposite the blank.

Proportions in the real world

Most proportion problems arrive as a word problem, and the work is in the setup. The reliable habit is to write a sentence ratio first, with its units. Then build the second ratio in the same order and put the blank where the unknown belongs.

Common mistakes