How to Split Anything Fairly Using a Ratio
Learn to simplify and compare ratios, divide a quantity in a given ratio, and test whether four numbers are in proportion using the extremes and means rule.
What is a ratio, and how do you simplify it?
A ratio compares two quantities of the same kind by division, written and read "a is to b". Because it is a comparison, a ratio has no unit of its own — and both quantities must be in the same unit before you compare them.
This page covers everything in the ICSE Class 6 Mathematics chapter on ratio and proportion: simplifying and comparing ratios, sharing a quantity in a given ratio, and testing four numbers for proportion.
This page covers everything in the ICSE Class 6 Mathematics chapter on ratio and proportion: simplifying and comparing ratios, sharing a quantity in a given ratio, and testing four numbers for proportion.
Why must both quantities be in the same unit first?
Because a ratio divides one quantity by the other, the units must cancel — and they only cancel when they match.
Find the ratio of 50 paise to 2 rupees. Comparing 50 to 2 would be meaningless. Convert first: 2 rupees is 200 paise, so the ratio is . Divide both parts by their HCF, 50, giving .
Another: the ratio of 40 cm to 2 m. Since 2 m is 200 cm, the ratio is .
A ratio is in its simplest form when the two parts have no common factor except 1 — in other words when they are co-prime.
For example, if a recipe uses 250 g of sugar to 1 kg of flour, the ratio is , so one part sugar to four parts flour whatever quantity you cook.
Note that a ratio only compares quantities of the same kind. Comparing 5 kg to 3 hours gives no ratio at all — those are different kinds of quantity.
Find the ratio of 50 paise to 2 rupees. Comparing 50 to 2 would be meaningless. Convert first: 2 rupees is 200 paise, so the ratio is . Divide both parts by their HCF, 50, giving .
Another: the ratio of 40 cm to 2 m. Since 2 m is 200 cm, the ratio is .
A ratio is in its simplest form when the two parts have no common factor except 1 — in other words when they are co-prime.
For example, if a recipe uses 250 g of sugar to 1 kg of flour, the ratio is , so one part sugar to four parts flour whatever quantity you cook.
Note that a ratio only compares quantities of the same kind. Comparing 5 kg to 3 hours gives no ratio at all — those are different kinds of quantity.
How do you compare two ratios?
Write each ratio as a fraction, convert them to like fractions using the LCM of the denominators, then compare the numerators.
Which is greater, or ? As fractions these are and . The LCM of 4 and 7 is 28, so
Since , we get .
Equivalent ratios are obtained by multiplying or dividing both parts by the same non-zero number, exactly like equivalent fractions. So — all describe the same comparison.
In real life this is how you scale a recipe: a mix of dal to water stays correct whether you cook 200 g and 300 ml or 400 g and 600 ml.
Which is greater, or ? As fractions these are and . The LCM of 4 and 7 is 28, so
Since , we get .
Equivalent ratios are obtained by multiplying or dividing both parts by the same non-zero number, exactly like equivalent fractions. So — all describe the same comparison.
In real life this is how you scale a recipe: a mix of dal to water stays correct whether you cook 200 g and 300 ml or 400 g and 600 ml.
How do you divide a quantity in a given ratio?
Add the parts of the ratio to find the total number of parts, work out the value of one part, then multiply.
Divide Rs 1,200 between two people in the ratio . The total is parts, so
The shares are and . Check: .
With three parts it works the same way. Divide 84 sweets among three children in the ratio . Total parts , so one part is , giving 12, 24 and 48 sweets. Again the check matters: .
Always add your answers back to the original total. If they do not match, you divided by the wrong number of parts — the commonest slip in this topic.
Divide Rs 1,200 between two people in the ratio . The total is parts, so
The shares are and . Check: .
With three parts it works the same way. Divide 84 sweets among three children in the ratio . Total parts , so one part is , giving 12, 24 and 48 sweets. Again the check matters: .
Always add your answers back to the original total. If they do not match, you divided by the wrong number of parts — the commonest slip in this topic.
Formula
How do you test whether four quantities are in proportion?
Four quantities are in proportion when the first ratio equals the second, written . Here and are the extremes and and are the means, and the test is:
Are 3, 4, 9, 12 in proportion? Extremes give and means give . They match, so yes.
The same rule finds a missing value. The fourth proportional to 2, 5, 8 satisfies , so
In a continued proportion , the repeated middle term is the mean proportional and is the third proportional. The mean proportional between 4 and 9 satisfies , so .
Units matter here too: convert before multiplying, or the products will not compare fairly.
Are 3, 4, 9, 12 in proportion? Extremes give and means give . They match, so yes.
The same rule finds a missing value. The fourth proportional to 2, 5, 8 satisfies , so
In a continued proportion , the repeated middle term is the mean proportional and is the third proportional. The mean proportional between 4 and 9 satisfies , so .
Units matter here too: convert before multiplying, or the products will not compare fairly.
Exam tip
Exam tip: writing the ratio the right way round
and are different ratios, and swapping them turns a correct method into a wrong answer.
The order follows the words of the question. "The ratio of boys to girls" puts boys first. If the question then asks for the ratio of girls to boys, you must invert it.
When dividing a quantity, keep the shares matched to the right people — write "A's share" and "B's share" beside your working rather than two bare numbers. In three-part questions especially, marks are lost by handing the largest share to the wrong person even when all the arithmetic is right.
The order follows the words of the question. "The ratio of boys to girls" puts boys first. If the question then asks for the ratio of girls to boys, you must invert it.
When dividing a quantity, keep the shares matched to the right people — write "A's share" and "B's share" beside your working rather than two bare numbers. In three-part questions especially, marks are lost by handing the largest share to the wrong person even when all the arithmetic is right.
Did you know
Why does a ratio have no unit?
A ratio divides one quantity by another of the same kind, so the units cancel out. Dividing 40 cm by 200 cm leaves the pure number — centimetres appear top and bottom and vanish.
That is what makes ratios so portable: a mix is the same instruction whether you are working in grams for a small batch or kilograms for a large one.
That is what makes ratios so portable: a mix is the same instruction whether you are working in grams for a small batch or kilograms for a large one.
Key takeaways
Ratio and proportion in 30 seconds
- A ratio compares two quantities of the same kind and has no unit; convert both to the same unit before simplifying.
- Simplify by dividing both parts by their HCF, so the parts end up co-prime.
- Compare ratios by writing them as fractions with a common denominator; multiplying or dividing both parts by the same number gives equivalent ratios.
- To divide a quantity, add the parts, find the value of one part, multiply for each share, then check the shares add back to the total.
- Four quantities are in proportion when the product of extremes equals the product of means; this finds the fourth, mean and third proportional.
You will remember all of this far better after answering five questions on it than after reading it twice.
- Simplify by dividing both parts by their HCF, so the parts end up co-prime.
- Compare ratios by writing them as fractions with a common denominator; multiplying or dividing both parts by the same number gives equivalent ratios.
- To divide a quantity, add the parts, find the value of one part, multiply for each share, then check the shares add back to the total.
- Four quantities are in proportion when the product of extremes equals the product of means; this finds the fourth, mean and third proportional.
You will remember all of this far better after answering five questions on it than after reading it twice.