Ratio Calculator

1. What is a ratio
2. How to solve ratios
3. Multiplication, division, and reduction of ratios
4. Odds as ratios
5. Aspect ratio calculation
6. Ratios in geometry

    What is a ratio

A ratio is a mathematical relationship between two numbers that expresses how many times the first number contains the second. More specifically, it is a proportional relationship. For example, the ratio between cars on the street and people in them would be 1:5 (1 to 5) if each car can fit 5 people and all cars are loaded to the maximum. If there are just 2 persons in each car, the ratio would be 1:2. A ratio can also be between things in a container, area, or subgroups in larger groups. For example, if you have 10 fruits and 3 of them are apples and 7 are oranges, the ratio between apples and oranges is 3:7. The ratio of apples to all fruits will be 3:10, which is often written as a fraction: 3/10 in this example.

The numbers in a ratio can be integers (whole numbers), decimal numbers, or fractions. When the two quantities are measured in the same unit, the ratio output from our ratio calculator is a dimensionless number. If the units are different, then it is called a rate, e.g. in productivity calculations, if it takes 1 person 11 days to complete a given project, the daily productivity rate is 1:11 (he or she can complete 1/11-th of the work in a day), while the monthly productivity, assuming 22 working days a month, is 2:1 (he or she can complete 2 projects in 1 month).

Ratios and therefore ratio calculators are often used in business metrics, accounting and investment decision-making to calculate metrics like profit margin, markup, ROI, return on assets, and conversion rate among many others.

Complex ratios

There are more complex ratios which contain more than two values in a single notation. For example, if you have 3 apples, 5 oranges and 2 bananas, the ratio between them is 3:5:2. These are useful when describing the relationship of dimensions in three-dimensional space, among other things.

    How to solve ratios

Ratio equations usually involve two ratios on both sides of an equation, with three numbers from the ratios given and the forth being solved for. Solving ratio equations is all about finding which number, multiplied by the one number given on the side of the missing number, would result in the same relationship as the one on the other side of the equation.

You can use simple equation rules: multiplying or dividing both sides of the equation in order to leave the missing variable alone on one side of the equation. For example, solving x:2 = 8:4 for x would be equivalent to solving x / 2 = 8 / 4, requiring us to multiply both sides by 2, to get to x = 8/4 x 2 = 2 x 2 = 4.

In our ratio solver / calculator, just leave the part of the ratio you are solving for empty, then press "Calculate" to get what number needs to be there to balance the equation.

    Multiplication, division, and reduction of ratios

If you multiply each member of a ratio by the same number, the ratio remains the same. For example, the ratio 1:3, when multiplied by 4, becomes 4:12, and in it the proportions between the quantities remain the same. Using the same logic, if you divide each quantity in a ratio by the same number, the number remains the same. These are examples of ratio scaling.

Reducing a ratio is done the same way as simplifying fractions. It requires finding the smallest integer that preserves the ratio, which is the greatest common divisor, also known as a common factor. If a ratio is given as 4:12, the GCD of 4 and 12 is 4. We divide the value on each side by 4 to get 1:4, which is the most convenient form of the ratio to use, which can be written as the fraction 1/4.

    Odds as ratios

Odds, or probabilities of events occurring or not occurring, are expressed as ratios. Ratios usually add up to 1 or 100%, for example 0.3:0.7 or 30:70, but most often they are presented in a form convenient for apprehension. In this case that would be 3:7 (30/10 and 70/10, and 0.3 x 10 , 0.7 x 10). Such ratios are often used in gambling and other games of chance, for example the chance for you to win the US Powerball lottery jackpot (2018 rules) are 1 in 292,201,338, or 1:292,201,338, as can be seen on our lottery number generator page. Ratios of this kind are also how odds in sports betting are presented.

    Aspect ratio calculation

Ratios are also often used when one needs to calculate the aspect ratio of an image or video, for scaling maps or models, as well as in sizing cooking recipes. For example, some old TV sets had an aspect ratio of 4:6 (width by height), while others had a ratio of 2.35:1. New LCD, OLED, and other TVs and screens usually have a ratio of 16:9, which is in fact a compromise between the two.

Models of cars and buildings are often made in ratios of 1:20 to 1:40, while for maps the scaling is often much higher, for example 1:1,000,000 which means that 1 cm on the map would be 1,000,000 cm in real life (10 kilometers).

    Ratios in geometry

Other than maps, ratios are very common in geometry. One of the most famous ratios in the world is denoted by the Greek letter π (pi) and is the ratio between the diameter of a circle and its circumference. It even has its own day of celebration: March 14 (3.14.YYYY)! It is used in the calculation of the area and circumference of circles and ovals, as well as the volume of spheres and spherical bodies. Similarly, trigonometric functions are also ratios: the sine, cosine and tangent function are simply expressions of the ratios between two sides of a triangle, relative to a specified angle.

Plotting ratios on charts

Ratios can be visually represented using lengths or areas, and rarely - as volumes. An example is below:

When visualizing, it is best to use thin lines or bars, as these are the least misleading. Wide bars may trigger subconscious comparison between their areas instead. This would be misleading to the extent to which the visual ratio of areas does not correspond to the visual ratio of lengths.