Ratio Calculator
What is a ratio?
A ratio is a mathematical relationship between two numbers that expresses how many times the first number contains the second. For example, the ratio between cars in the street and people in them would be 1:5 (1 to 5) if each car can fit 5 people in it and they are loaded to the maximum. If there are just 2 persons in each car, the ratio would be 1:2. The 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.
There are more complex ratios where you might have more than two numbers in a single notation, like if you have 3 apples, 5 oranges and 2 bananas, the ratio between them is 3:5:2. Numbers in a ratio can be integers (whole numbers) or fractions and 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 can be visually represented using lengths or areas, and rarely - as volumes. An example is below:
ratio visualization
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, which would be misleading to the extent to which the ratio of areas does not correspond to the ratio of lengths.
Ratios are often used in business metrics, accounting and investment decision-making to calculate things like profit margin, markup, ROI, return on assets, and conversion rate among many others.
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.
Other than using our ratio calculator, 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 multiplying 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 numbers 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.
Reducing a ratio is done the same way as with fractions - you need to find the smallest integer that preserves the ratio, which is the greatest common divisor. If a ratio is given as 4:12, the GCD of 4 and 12 is 4, so we divide each side by 4 to get 1:4, which is the most convenient form of the ratio to use.
Odds as ratios
Odds, or probabilities of events occurring or not occurring, are expressed as ratios. The ratios must usually add up to 1 or 100%, for example 0.3:0.7 or 30:70, but most often they are presented in a from 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). They 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.
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.
