Ratios compare two quantities, which makes them useful in recipes, mixing paint, school maths and shopping. The confusing part is that bigger-looking numbers do not necessarily describe a bigger proportion. A mix of 2 parts cordial to 5 parts water may be stronger than one of 3 parts cordial to 10 parts water, even though 3 looks bigger than 2.
To compare ratios reliably, put the quantities in the same order, make the units consistent and convert them to an equivalent common form. Once you do that, you can compare proportions or rates without guesswork.
Understand what a ratio means
A ratio such as 2:3 says there are two parts of the first quantity for every three parts of the second. The order matters. A paint mix of two parts blue to three parts white is not the same as three parts blue to two parts white.
Ratios may describe parts within a whole, such as boys to girls in a group, or a relationship between two quantities, such as cups of water to rice.
Before calculating anything, label the terms. Write 'blue:white = 2:3' rather than just '2:3' if there is any chance you will reverse them.
Simplify ratios before comparing
If a drink uses 4 parts concentrate to 12 parts water, divide both numbers by 4 to get 1:3. The proportion is unchanged.
A second drink using 3:9 also simplifies to 1:3, so the two drinks have the same relative strength. Large absolute quantities do not affect the ratio.
Just as equivalent fractions are formed by multiplying or dividing numerator and denominator by the same nonzero factor, equivalent ratios use the same operation on both terms.
Compare the first quantity per second quantity
For 2:5 and 3:10, divide the first term by the second. The first ratio gives 2/5 = 0.4; the second gives 3/10 = 0.3.
If the ratio describes concentrate to water, the first mix uses more concentrate for each unit of water and is therefore stronger. State what 'stronger' means so the result is not ambiguous.
This method works well when the question asks which quantity is greater relative to another. Keep the order consistent across both ratios.
Use equivalent ratios with a common second term
You can avoid decimals by making the second terms equal. For 2:5 and 3:10, multiply the first ratio by 2 to get 4:10.
Now compare 4:10 against 3:10. The first contains four units of the first ingredient for the same ten units of the second, so it is greater in that sense.
This is the ratio counterpart of comparing fractions with a common denominator. It is often easier to explain in an exam or recipe than a string of decimal calculations.
Cross multiplication is another shortcut
For positive ratios a:b and c:d, compare the products a × d and c × b. In our example, 2 × 10 = 20 and 3 × 5 = 15, confirming that 2:5 represents the greater first-to-second quantity.
Make sure you are comparing the same kind of relationship on both sides. A ratio of sugar to flour cannot be directly compared to one of flour to sugar without reversing one of them.
Cross multiplication works because you are effectively comparing two fractions. It does not rescue an incorrectly interpreted word problem.
Distinguish part-to-part from part-to-whole
Suppose a bag holds two red marbles and three blue marbles. The red-to-blue ratio is 2:3. The fraction of all marbles that are red is 2/5, not 2/3.
This distinction matters in recipe mixtures. If concentrate:water is 1:4, the drink contains one part concentrate in five total parts, or 20% of the mixture.
Comparing the concentration of two drinks can be done by converting each to its first-part fraction of the total. Do not confuse that number with first-to-second ratio.
Compare unit rates when quantities differ
A pack of six batteries costing £4.80 has a price of £0.80 per battery. A pack of ten costing £7.50 has a price of £0.75 per battery.
To compare fairly, calculate the cost per one unit. The second pack is cheaper per battery even though its total price is higher.
This is a rate because money and batteries have different units, but the calculation follows the same principle as ratio comparison. OpenStax explains unit rates as rates with a denominator of one unit.
Convert different measurement units first
A recipe using 200ml of oil per kilogram of flour cannot be compared directly to one using 150ml per 500g until you match the flour units.
The first uses 200ml per 1000g, or 100ml per 500g. The second uses 150ml per 500g, so it has the higher oil-to-flour rate.
Unit conversion is part of the comparison, not an optional extra. The numbers alone tell the wrong story when their units differ.
Watch for reversed ratios
Suppose one classroom has a teacher-to-student ratio of 1:20 and another has 1:25. If comparing teachers available per student, the first classroom has more teachers relative to pupils.
If you reverse both to student-to-teacher ratios, they become 20:1 and 25:1. You may then talk about students per teacher instead, but the interpretation changes.
In any written answer, say what each term stands for and whether a larger or smaller ratio is better for the purpose.
Compare three-part ratios
A mixture with red:blue:white paint in the ratio 2:3:5 has ten total parts. Red is 2/10, blue is 3/10 and white is 5/10 of the whole.
To compare with another three-part mixture, first make sure the colours are listed in the same order. Then compare the colour proportions relevant to your question.
For example, a second mix of 1:1:2 has four parts total, so red is 1/4 or 25%, greater than the first mix's 20%. Yet that alone does not describe every difference between the mixtures.
Ratios and probability
If a bag contains three green and two yellow counters, green:yellow is 3:2. The probability of drawing green on a single random draw is 3/5.
If a second bag has green:yellow = 4:3, the green probability is 4/7, approximately 57.1%, compared with 60% in the first bag.
The first bag offers a slightly higher chance of drawing green. To reach that conclusion, convert part-to-part ratios to part-to-whole fractions.
Common mistakes that change the answer
Do not compare the first numbers without accounting for the second. Do not reverse the order in just one ratio. Do not compare millilitres with litres before conversion.
Another mistake is adding the terms when the question actually asks about first-to-second rates. Addition is useful for finding the total number of parts, but division is required to compare relative amounts.
When ratios include decimal terms, multiply both sides of each ratio by a suitable power of ten to obtain convenient whole numbers without changing the relationship.
A real-world method
Write the quantities and units, identify whether the question concerns part-to-part or part-to-whole, simplify each ratio and calculate a consistent fraction or unit rate.
Then explain the result in words. 'The second pack costs five pence less per battery' is more useful than simply writing 0.75 < 0.80, even though both are correct.