About Ratio and Proportion
A ratio compares two quantities of the same kind by showing how many parts of each there are. That is all it does, and that is why almost every question in this topic is solved the same way: find what one part is worth, then multiply. Almost every wrong answer comes from dividing by a single term of the ratio rather than by the total number of parts.
What you need to understand
- A ratio written a : b means there are a parts of the first quantity for every b parts of the second.
- The total number of parts is the sum of the terms. In the ratio 2 : 3 : 5 there are ten parts.
- A ratio does not tell you the quantities. The ratio 2 : 3 could describe 2 and 3, or 200 and 300.
- Multiplying or dividing every term by the same number leaves the ratio unchanged, which is what makes a ratio reducible to lowest terms.
- Two ratios are joined through their common term: to combine A : B with B : C, the two values of B must first be made equal.
- A difference between quantities corresponds to a difference between parts, never to a single part.
Formulas to remember
How to work through these questions
- Add the terms of the ratio to find the number of parts before doing anything else.
- Divide the total by that number of parts to get the value of one part.
- Multiply the value of one part by the term belonging to the quantity the question asks for.
- When two ratios share a middle term, equalise that term by multiplying both ratios, then read off the first and last terms.
- Check your answer by adding the shares back up, because they must return the original total.
Mistakes that cost marks
- Dividing the total by one term of the ratio instead of by the sum of all the terms.
- Giving the share of the wrong person, which happens whenever the parts are multiplied by the wrong term.
- Adding the corresponding terms of two ratios instead of multiplying them when the ratios are combined.
- Reporting the ratio itself when the question asked for an amount in rupees.
- Treating the parts as equal, which holds only when the ratio happens to be 1 : 1.
Worked example
Rs. 1200 is divided among A, B and C in the ratio 3 : 4 : 5. Find the share of B.
- Number of parts = 3 + 4 + 5 = 12.
- Value of one part = 1200 / 12 = Rs. 100.
- The share of B is 4 parts = 4 x 100 = Rs. 400.
- Check: A receives 300, B receives 400 and C receives 500. They add up to 1200, which is the original amount.
Answer: Rs. 400
Practice questions with answers
A few Ratio and Proportion questions with the full solution shown, so you can see
how the method is applied before you attempt the timed set.
Question 1
A sum of Rs. 144 is shared between A, B and C in the proportion 5 : 3 : 8. How much does A receive?
Answer: Option A — with explanation
The ratio gives the number of parts each person receives: 5 + 3 + 8 = 16 parts in all.
One part is worth 144 / 16 = Rs. 9.
A's share is 5 parts = 5 x 9 = Rs. 45.
Common mistakes
- gave C's share instead of A's: 72
- divided the whole amount by the number of parts in A's share: 28.8
- divided the amount equally among the three instead of in the given ratio: 9
Question 2
Two numbers are in the ratio 6 : 5. If their difference is 6, find the larger number.
Answer: Option A — with explanation
The difference of 6 - 5 = 1 parts is worth 6.
One part = 6 / 1 = 6.
The larger number is 6 parts = 6 x 6 = 36.
Common mistakes
- gave the smaller number: 30
- gave the difference instead of the number: 6
- added the two numbers together: 66
Question 3
A sum of Rs. 54 is shared between A, B and C in the proportion 8 : 3 : 7. How much does A receive?
Answer: Option B — with explanation
The ratio gives the number of parts each person receives: 8 + 3 + 7 = 18 parts in all.
One part is worth 54 / 18 = Rs. 3.
A's share is 8 parts = 8 x 3 = Rs. 24.
Common mistakes
- gave B's share instead of A's: 9
- gave C's share instead of A's: 21
- divided the amount equally among the three instead of in the given ratio: 3
Question 4
Two quantities are in the ratio 9 : 7, and the larger exceeds the smaller by 10. Find the larger quantity.
Answer: Option A — with explanation
The difference of 9 - 7 = 2 parts is worth 10.
One part = 10 / 2 = 5.
The larger number is 9 parts = 9 x 5 = 45.
Common mistakes
- gave the difference instead of the number: 10
- gave the smaller number: 35
- divided the difference by the wrong term: 12.86
Question 5
A sum of Rs. 228 is shared between A, B and C in the proportion 8 : 6 : 5. How much does A receive?
Answer: Option B — with explanation
The ratio gives the number of parts each person receives: 8 + 6 + 5 = 19 parts in all.
One part is worth 228 / 19 = Rs. 12.
A's share is 8 parts = 8 x 12 = Rs. 96.
Common mistakes
- divided the amount equally among the three instead of in the given ratio: 12
- gave C's share instead of A's: 60
- gave B's share instead of A's: 72
Frequently asked questions
What does the total number of parts mean?
It is the sum of the terms of the ratio. In 3 : 4 : 5 there are 12 parts, so the whole amount is split into 12 equal parts and each person takes their own number of them.
How do I combine two ratios that share a term?
Make the shared term equal in both. If A : B is 2 : 3 and B : C is 4 : 5, rewrite the first as 8 : 12 and the second as 12 : 15, which gives A : B : C = 8 : 12 : 15.
Can a ratio contain fractions or decimals?
Yes. A ratio is just a comparison, so 1/2 : 3/4 is valid. Multiplying both terms by 4 turns it into the whole number ratio 2 : 3.
What is the difference between a ratio and a proportion?
A ratio compares two quantities. A proportion states that two ratios are equal, as in 2 : 3 = 4 : 6. Every proportion can be solved by cross-multiplication.
