How to Calculate Ratios and Simplify Them Step by Step

Suppose a box contains 12 red counters and 18 blue counters. The ratio of red to blue is 12:18. Both numbers share a common factor of 6, so divide both by 6 to get 2:3. That simplified ratio means there are 2 red counters for every 3 blue counters in the same proportion.

Order matters. The ratio of blue to red is 18:12, which simplifies to 3:2. It describes the same collection from the opposite direction, so the numbers swap places. Read the wording carefully before you write the ratio. This is a basic but essential part of calculating ratios.

Ratios can compare part to part or part to whole. With 12 red and 18 blue counters, there are 30 counters total. Red to total is 12:30, which simplifies to 2:5. That is different from the red-to-blue ratio of 2:3. Always identify exactly which quantities the question asks you to compare. Labels help here. Writing red:blue or red:total beside the numbers can stop you from simplifying the correct arithmetic for the wrong comparison.

A ratio only makes sense when the quantities are expressed in compatible units. If a ribbon is 2 metres long and another is 50 centimetres long, do not write 2:50. Convert first. Two metres is 200 centimetres, so the ratio is 200:50, which simplifies to 4:1.

Time questions often use the same idea. A ratio of 1 hour to 20 minutes must be converted to 60 minutes to 20 minutes before simplifying. The result is 3:1. The unit conversion changes the numbers but not the actual comparison.

This unit step is easy to miss when calculating ratios because the arithmetic itself may look simple. A quick check is to ask whether both terms describe the same kind of quantity in the same unit. If not, convert before you cancel anything.

Ratios are useful because equivalent ratios preserve the same relationship. If paint is mixed in a 2:3 ratio of blue to white, doubling both amounts gives 4:6 and tripling gives 6:9. These ratios all simplify to 2:3, so the mixture keeps the same proportion.

You can use a scale factor to find missing quantities. If 2 notebooks cost 6 units of currency at a constant rate, then 1 notebook costs 3 and 5 notebooks cost 15. You are keeping the cost-to-notebook ratio equivalent. A ratio table can make this pattern easy to see.

When checking how to calculate ratios, simplify both the original and your proposed answer. If they reduce to the same pair of numbers in the same order, they are equivalent. If not, one quantity was probably scaled differently from the other. This is useful in recipes, maps, scale drawings, mixtures, and rates where the same relationship must be preserved at a new size.

A very common ratio task is sharing a total in a given ratio. Suppose 45 sweets are shared between Amy and Ben in the ratio 4:5. First add the parts: 4 plus 5 makes 9 parts in total. Next find the value of one part by dividing the total by the number of parts, so 45 divided by 9 gives 5. Finally multiply: Amy receives 4 times 5, which is 20, and Ben receives 5 times 5, which is 25. Always check that the shares add back to the original total, here 20 plus 25 making 45.

Some questions give you the difference instead of the total. If Ben has 5 more sweets than Amy, the difference between their parts is one part, so one part is worth 5, and you can finish exactly as before. Ratios containing decimals or fractions need tidying first. For 1.5:2, multiply both terms by 2 to get 3:4. For 1/2:1/3, multiply both by 6, the lowest common multiple of the denominators, to get 3:2. You are scaling both terms by the same factor, so the relationship is unchanged but the numbers become whole and easier to use.

Ratios can compare more than two quantities. A simple concrete mix might use cement, sand and gravel in the ratio 1:2:3. To make 300 kilograms, add the parts to get 6, so one part is 50 kilograms, giving 50 of cement, 100 of sand and 150 of gravel. Sometimes you must combine two ratios. If a to b is 2:3 and b to c is 4:5, make the shared term equal. Multiply the first ratio by 4 and the second by 3 to get 8:12 and 12:15, so a to b to c is 8:12:15.

Map scales are ratios too. A scale of 1:25 000 means one unit on the map stands for 25 000 of the same unit on the ground. One centimetre on the map therefore represents 25 000 centimetres, which is 250 metres, and 4 centimetres represents a full kilometre. To find a real distance, measure on the map and multiply by 25 000, then convert units at the end. To go the other way, divide the real distance by the scale. Keeping both measurements in the same unit until the final step avoids most mistakes.

Some questions describe a ratio that changes when an amount is added or removed. Suppose a bag holds red and blue marbles in the ratio 3:5. Six red marbles are added, and the ratio becomes 3:4. How many blue marbles are there? The key observation is that the number of blue marbles did not change, so make the blue part the same number in both ratios. Five and four have a lowest common multiple of 20, so write 3:5 as 12:20 and 3:4 as 15:20.

Now the two ratios use parts of the same size. Red rose from 12 parts to 15 parts, an increase of 3 parts, and that increase was the 6 marbles that were added. So one part is worth 2 marbles. Blue is 20 parts, giving 40 blue marbles, and red started at 12 parts, which is 24 marbles. Check the answer: 24 red and 40 blue simplifies to 3:5, and after adding 6, 30 red and 40 blue simplifies to 3:4. The same method works whenever one quantity stays fixed while another changes.

To decide which of two ratios is larger, turn them into a form you can compare directly. Suppose one squash drink is mixed as concentrate to water 2:7 and another as 3:10. Writing each in the form 1:n gives 1:3.5 and about 1:3.33, so the second has less water for each part of concentrate and is stronger. Alternatively, find the fraction of concentrate in each drink: 2 out of 9 parts is about 22 percent, while 3 out of 13 is about 23 percent. Both methods agree, but only because each compares the same quantities in the same order.

Scale models bring a twist. If a model ship is built at a scale of 1:50, every length is divided by 50, but areas are divided by 50 squared and volumes by 50 cubed. The ratio of areas is therefore 1:2,500 and the ratio of volumes 1:125,000. So if the model's hull holds 2 litres of water, the real ship's hull holds 250,000 litres, or 250 cubic metres. Forgetting to square or cube the scale factor is one of the most common ratio mistakes, and the same reasoning explains why a giant built exactly like a scaled-up human would struggle to carry its own weight.

Calculating ratios is a matter of comparing quantities in the correct order, converting to compatible units, and dividing all terms by common factors. Distinguish part-to-part from part-to-whole ratios, and use equivalent ratios for scaling. Clear labels and units prevent most mistakes before the arithmetic even begins.