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This page sets out the method for writing & simplifying ratio in the order you would use it, then walks through the full lesson explanation. For three complete worked examples with the slips that cost marks, see the Writing & simplifying ratio topic page.
When you see 'simplest form', divide every part by the same factor until nothing else divides.
When the question asks for or , divide both parts by the one that must become 1.
2:3 is not the same as 3:2.
The first number belongs to the first thing named.
Divide both parts by the same number.
6:9 = 2:3 because both were divided by 3.
40 cm to 1 m is 40 cm to 100 cm, which is 2:5.
Do not write 40:1 when the units differ.
A ratio can have more than two numbers.
Divide all of them by the same factor.
2:3 = 4:6 = 20:30.
Multiplying both parts keeps the comparison.
The highest common factor gives the simplest form in one step.
You can divide by 2 and then by 3 if that is easier.
A ratio answers how many of one thing you have for how many of another. It is not the total, and it is not a fraction until the parts are added. Read left to right. If blue to red is 5:2, the 5 belongs to blue. Keep that order in every line of working.
Equivalent ratios are the same mix written with different numbers. Multiplying both parts by 4 turns 5:2 into 20:8. Each blue part is still matched with the same share of red. A larger batch can use the same ratio. The bar of parts gets longer, but the pattern does not change.
Method: 1. Name the things in the order the question uses. 2. Write one number for each thing, with colons between them. 3. Keep that order through every later step. 4. Multiply every part by the same number to build an equivalent ratio.
The highest common factor is the largest whole number that divides every part. For 18 and 24 the common factors include 2, 3 and 6, so 6 is the one to use. Dividing by a smaller common factor is allowed if you simplify again afterwards.
The same division has to hit every part. Dividing 18 by 6 and leaving 24 untouched gives 3:24, which is a different mix. When you have finished, the parts should have no common factor except 1. That is how you know the form is simplest.
Method: 1. List the factors of each part, or divide by small primes. 2. Choose the largest factor shared by every part. 3. Divide every part by that factor. 4. Stop when the only common factor left is 1.
A three-part ratio such as flour to sugar to butter still compares pieces of one mix. The order stays as the question gave it. The common factor must divide all three numbers, not just the first pair. A number that divides two parts and misses the third is not finished.
Divide, then write all three new parts. If they still share a factor, divide again. Adding the simplified parts tells you how many equal pieces are in the whole mix. That total becomes the denominator when one part is written as a fraction of the whole.
Method: 1. Write all three numbers in the given order. 2. Find a factor that divides every number. 3. Divide all three numbers by that factor. 4. Repeat until no common factor greater than 1 remains.
Mixed units make a ratio look simpler than it is. Fifteen centimetres to three metres is not 15:3. Change three metres into centimetres first, which is 300. The honest ratio is 15:300, and that simplifies to 1:20. Either convert the small unit up or the large unit down. Finish in one unit.
The form 1:n is a scale. Divide both parts by the first number. For 4:7 you get 1:1.75, because 7 ÷ 4 = 1.75. Map and model questions often need this after a unit change. Check by multiplying n by the first length. You should get the second length back.
Method: 1. Convert every measurement into the same unit. 2. Write the ratio in the order the question asks for. 3. Simplify by the highest common factor if the numbers are whole. 4. Divide each part by the left-hand number so the ratio starts with 1.
A ratio compares two quantities of the same kind, so both must be in the same unit before you write the numbers. A pole of 1.5 m and a pole of 60 cm cannot be written as 1.5 : 60. Change the metres to centimetres first: 1.5 m = 150 cm. The ratio is then 150 : 60. Choose the smaller unit, since it usually avoids decimals and gives whole numbers.
The final ratio has no units in it. Do not write "cm" after the numbers once the ratio is simplified. Common conversions are 1 m = 100 cm, 1 km = 1000 m, 1 kg = 1000 g, 1 litre = 1000 ml and 1 hour = 60 minutes. Always write down the conversion line before you simplify, so a slip in the conversion is not lost. A clear line such as 1.5 m = 150 cm also shows the examiner that you know what you are doing.
Method: 1. Spot the two different units in the question. 2. Convert to the smaller unit and write the conversion. 3. Write the ratio using the converted numbers, in the order given in the question. 4. Simplify, and leave the units out of the final ratio.
Tip: The order in the question is the order in the ratio. "Pole A to pole B" means A comes first.
A ratio in its simplest form uses whole numbers. If the ratio contains decimals, multiply both sides by 10, or 100 if needed, until both are whole. So 0.6 : 1.5 becomes 6 : 15, and dividing both by 3 gives 2 : 5. If there are fractions, multiply both sides by a common denominator. For 3/4 : 5/6 use 12 to get 9 : 10.
Take care with mixed numbers. Convert them to improper fractions first. A ratio such as 11/2 : 3 is 3/2 : 3, and multiplying by 2 gives 3 : 6, which simplifies to 1 : 2. In an exam the working should show the multiplier, for example "× 12", so the examiner can see the method even if the final numbers have a slip.
Method: 1. Convert any mixed numbers to improper fractions. 2. Multiply both sides by the same number to clear fractions or decimals. 3. Divide both sides by the highest common factor. 4. Check the answer by dividing the first by the second and comparing with the start.
To write a ratio in the form 1 : n, divide both parts by the first part. For 4 : 10, divide by 4 to get 1 : 2.5. To write it in the form n : 1, divide both parts by the second part, so 18 : 8 becomes 2.25 : 1. The number n does not have to be a whole number unless the question says "where n is an integer". Read this phrase carefully.
Map and model scales are always ratios of lengths in the same unit. A scale of 1 : 25 000 means 1 cm on the map is 25 000 cm in real life. Convert 25 000 cm to 250 m so the answer is easier to use. To find a map length from a real length, divide. To find a real length from a map length, multiply. Remember that areas use the scale squared.
Method: 1. For 1 : n, divide both parts by the first part. 2. For n : 1, divide both parts by the second part. 3. For scales, put both lengths in centimetres before writing the ratio. 4. Convert the real length to metres or kilometres at the end if that is sensible.
Tip: On a scale, map length times n gives real length. Real length divided by n gives map length.
Exam questions describe a ratio in words rather than giving it directly. If 3/8 of the pupils are boys, the rest, 5/8, are girls, so the ratio of boys to girls is 3 : 5. If two groups each have their own ratio, work out the number in each part first, then add the numbers of boys and the numbers of girls separately before writing a combined ratio.
For mixtures, add the amounts of each ingredient. If an alloy is made with copper and zinc in the ratio 7 : 3, then the fractions of the alloy are 7/10 and 3/10. Taking equal masses of two alloys lets you add the fractions of each metal, with a common mass such as 100 kg, then write the new ratio. Always simplify at the end, and check the order matches the question.
Method: 1. Turn the description into numbers of items or masses. 2. Add matching parts from different groups together. 3. Write the ratio in the order requested. 4. Simplify using the highest common factor.
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