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Proportions and Ratios

Definition of Ratio

A ratio is a relationship between two values. For instance, a ratio of 1 pencil to 3 pens would imply that there are three times as many pens as pencils. For each pencil there are 3 pens, and this is expressed in a couple ways, like this: 1:3, or as a fraction like 1/3. There do not have to be exactly 1 pencil and 3 pens, but some multiple of them. We could just as easily have 2 pencils and 6 pens, 10 pencils and 30 pens, or even half a pencil and one-and-a-half pens! In fact, that is how we will use ratios -- to represent the relationship between two numbers.

Definition of Proportion

A proportion can be used to solve problems involving ratios. If we are told that the ratio of wheels to cars is 4:1, and that we have 12 wheels in stock at the factory, how can we find the number of cars we can equip? A simple proportion will do perfectly. We know that 4:1 is our ratio, and the number of cars that match with those 12 wheels must follow the 4:1 ratio. We can setup the problem like this, where x is our missing number of cars:

$$ \frac{4}{1}=\frac{12}{x}$$

To solve a proportion like this, we will use a procedure called cross-multiplication. This process involves multiplying the two extremes and then comparing that product with the product of the means. An extreme is the first number (4), and the last number (x), and a mean is the 1 or the 12.

To multiply the extremes we just do \(4 * x = 4x\). The product of the means is \(1 * 12 = 12\). The process is very simple if you remember it as cross-multiplying, because you multiply diagonally across the equal sign.

You should then take the two products, 12 and 4x, and put them on opposite sides of an equation like this: \(12 = 4x\). Solve for x by dividing each side by 4 and you discover that \(x = 3\). Reading back over the problem we remember that x stood for the number of cars possible with 12 tires, and that is our answer.

It is possible to have many variations of proportions, and one you might see is a double-variable proportion. It looks something like this, but it easy to solve.

$$ \frac{16}{x}=\frac{x}{1} $$

Using the same process as the first time, we cross multiply to get \(16 * 1 = x * x\). That can be simplified to \(16 = x^2\), which means x equals the square root of 16, which is 4 (or -4). You've now completed this lesson, so feel free to browse other pages of this site or search for more lessons on proportions.

Ratios and Proportions Calculator

Use the tool below to convert between fractions and decimal, or to take a given ratio expression and solve for the unknown value.


A rate compares 2 quantities that have different types of units, such as kilometers per hour or miles per gallon. A ratio compares quantities with the same type of units by dividing the quantity in the numerator by the quantity in the denominator, as in a fraction. A proportion states that 2 ratios are equivalent, such as 3/6 = ½.


Rates are often used when calculating a relationship between 2 different types of measurements. For example, a mile is a measure of length or distance, and a gallon is a measure of volume. A question such as “How many miles per gallon does the new car get?” really asks “How long of a distance per gallon can the new car travel?” That is important, because it is an indirect estimate of how much it will cost the owner to operate the car. Suppose the car gets 30 miles to a gallon, and its tank has a capacity of 11 gallons. That means that it can travel about 330 miles on a tank of gas. The mileage per gallon is often an important selling point for new cars.


While a rate compares different types of units, a ratio compares the same types of units in a fraction. In symbol language, a ratio can be written as a: b or a/b, when b is ≠ 0. Suppose that the sale price is 40% off the regular price, and the regular price is 20.00. 40% of $20.00 is $8.00, so 20-8 = $12.00.


Ratios and proportions are closely related, because a proportion is simply an equation of two ratios. One of the ways to estimate a ratio is by using proportions. In finding a value for the sale price, several different properties were used. If the regular price were $20.00, 40% of that price would also be equal to 4(10%), and 10% of $20.00 is $2.00. So 4($2.00) = $8.00, and $20.00 -8.00 = 12.00. Similarly, $20.00/1 times 40/100 = 800.00/100, or $8.00. In that example, all the values are known. If one value is unknown, a proportion can be solved by using cross products, so that 8/20 = 40/100. In this case, 20·40 =800, and 8·100 =800. In algebraic language, If a/b =c/d, and b≠0;   d ≠0, then ad =bc.


Scale drawings and scale models use ratios and proportions in 2 and 3 dimensional applications. Suppose a map has the scale ¼ in = 10 miles. It is about 2 ¾ in between 2 cities in a state. In other words, ¼ /11/4 or 1/11 = 10x or x = 110. Similarly, many scale model vehicles are made on a scale of 1:18, so that anything that is 2 inches long on the model will be 2(18) or 36 inches long on the real thing.

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